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 Title:
 Hydrodynamic lift in sediment transport.
 Series Title:
 Hydrodynamic lift in sediment transport.
 Creator:
 Benedict, Barry A.
 Place of Publication:
 Gainesville FL
 Publisher:
 University of Florida
 Publication Date:
 1968
Subjects
 Subjects / Keywords:
 Aerodynamic lift ( jstor )
Diameters ( jstor ) Gravel ( jstor ) Hemispheres ( jstor ) Hydrodynamics ( jstor ) Interpolation ( jstor ) Mathematical lattices ( jstor ) Subroutines ( jstor ) Velocity ( jstor ) Velocity distribution ( jstor )
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 University of Florida
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 University of Florida
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 Copyright Barry A. Benedict. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 000953375 ( alephbibnum )
16919111 ( oclc )

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Full Text 
HYDRODYNAMIC LIFT IN SEDIMENT
TRANSPORT
By
BARRY ARDEN BENEDICT
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1968
ACKNOWLEDGMENTS
The author wishes to express his sincere appreciation to
Dr. B. A. Christensen, Dr. J. H. Schmertmann, Dr. E. A. Farber,
and Dr. T. O. Moore for their service on his supervisory committee
and their interest in his work. The author is especially indebted
to the committee chairman, Dr. B. A. Christensen, whose guidance,
encouragement, and personal interest and enthusiasm were truly
valuable.
Appreciation is extended to the National Science Foundation,
under whose traineeship the author has been working.
The author also wishes to express his gratitude to the
University of Florida Computing Center for the computing facilities,
services, and aid extended by the Center.
Finally, the author wishes to extend thanks for the aid and
encouragement provided by his wife, who has persevered through many
trying times.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . . . . . .... ii
LIST OF TABLES . . . . . . . . . . . . vi
LIST OF FIGURES . .. ......... . . . . . .. vii
LIST OF SYMBOLS . . . . ... .. . . . . .. x
ABSTRACT ...... .. ........... .. ........ xiii
CHAPTER
I INTRODUCTION. .. . . . . . . . . ..1
II RELATED BACKGROUND ON LIFT FORCE STUDIES . . . . 4
2.1 Historical interest . . . . . . . 4
2.2 Neglect of lift force .. . . . . 5
2.3 Indications of significance of lift . . . 6
2.4 Use of potential theory in lift studies on
single bodies . . . . . . . . 7
2.4.1 General . . . . . . . 7
2.4.2 Applicability of potential flow . . 8
2.4.3 Work of Kutta and Joukowsky . . . 9
2.4.4 Fuhrmann's work and other studies . 11
2.4.5 Jeffreys' analysis . . . . . 12
2.4.6 Flow around a single sphere .. . . 14
2.5 Multiparticle studies .. . . . . . 19
2.5.1 Einstein and El Samni . . . . 19
2.5.2 Chepil . . . . . . . . 20
2.5.3 Chao and Sandborn . . . . . 22
2.6 Shapes of bodies studied . . . . . .. 22
2.7 Relation to work of dissertation . . . .. 23
III LOGARITHMIC VELOCITY DISTRIBUTION. . . . . . 25
3.1 Development of logarithmic velocity
distribution . . . . .. ... .. 25
3.2 Special problems of present expressions . 26
TABLE OF CONTENTS (Continued)
CHAPTER Page
3.2.1 Theoretical bed . . . . . ... 27
3.2.2 Conditions near the bed . . . ... 29
3.3 Use of proposed adjusted velocity distribution 31
3.3.1 Comparison of distributions at wall . 31
3.3.2 Comparison of distribution with
increasing y . . . . ..... 32
3.4 Determination of k and vf from experimental
data . . .. . . . . . . 33
3.5 Effect of sidewalls . . . . . ... 35
IV TWODIMENSIONAL WORK: EARLIER RESULTS . . . ... 41
4.1 Shapes studied . . . . . . .. 41
4.2 General methodsvelocity and pressure results 41
4.3 Lift integration . . . . . . ... 44
4.4 Calculation of predicted lift by use of
measured logarithmic velocity profile . . 47
4.5 Application to experimental results . . .. 49
V THREEDIMENSIONAL NUMERICAL SOLUTIONS . . . ... 52
5.1 General . . . . . . . .... . 52
5.2 Problem formulation . . . . . .... 53
5.2.1 Choice of solution method ...... 53
5.2.2, Depth of flow space ... . . . . 54
5.2.3 Boundary conditions . . . . ... 56
5.3 General finite differences approach . . .. 61
5.4 Finite differences equations: interior space 64
5.4.1 General lattice point ... .. . . . 65
5.4.2 Object point on planar noflow boundary 68
5.4.3 Object point on "foldedsymmetry"
boundary . . . . . . ... 70
5.4.4 Adjacent point on hemispherical surface 70
5.4.5 Graded lattice . . . . . ... 72
5.5 Finite difference equations: hemispherical
boundary . . . . . . . .. 78
5.5.1 General . . . . . . . . 78
5.5.2 Xdirection derivative . . . .. 79
TABLE OF CONTENTS (Continued)
CHAPTER Page
5.5.3 Ydirection derivative . . . .. 81
5.5.4 Zdirection derivative . . . ... 87
5.5.5 Adjacent zpoints and ypoints subjected
to normal derivative condition . 88
5.5.6 Final boundary formulation . . ... 91
5.5.7 Singular points . . . . . . .. 93
5.6 Velocity and lift calculations . . . .. 94
5.7 Implementation of solution . .. . . .. 99
VI RESULTS AND COMPARISONS . . . . . . .. .. 101
6.1 General . . . . . . . .... ... 101
6.2 Numerical results for Chepil arrangement . .. 102
6.3 Comparisons with Chepil's observations . . 110
6.3.1 Details of Chepil's work . . . .. 110
6.3.2 Comparison of lift forces . . .. . 116
6.4 Numerical results for closely packed
hemispheres . . . . . . . . 125
6.5 Comparison with EinsteinEl Samni observations 138
6.5.1 Physical details of experiments . . 138
6.5.2 Values of lift for hemisphere bed . .. 141
6.5.3 Lift on a gravel bed. . . . . 142
VII CONCLUSIONS AND FUTURE WORK . . . . . .. 146
APPENDIX . . . . . . . . . .. . . 149
NOTES ON FORTRAN IV COMPUTER PROGRAM . . . .. . 150
NOTES ON EQUIVALENT GRAIN SIZE . . . . . .. 200
REFERENCES . . . . . . . . . . . . . 201
BIOGRAPHICAL SKETCH . . . ... . . .. . . . 207
LIST OF TABLES
Table Page
1 Comparison of Proposed and Former Distributions ... .. 32
2 Chepil's Experimental Data . . . . . . ... 112
3 CL for Chepil's Work .................. 114
4 RoughnessGrain Size Ratios . . . . . . ... 115
5 Comparison of Theoretical with Chepil's Work ...... 123
LIST OF FIGURES
Figure Page
1 Flow around a Joukowsky profile . . . . ... 10
2 Calculated and measured pressure distribution around
aJoukowsky profile ...... .........10
a Joukowsky profile . . . . . . . . 10
3 Pressure distribution around a Fuhrmann body . . .. 11
4 Jeffreys' cylinder . . . . ..... . . 12
5 Pressures on a single sphere . . . . . ... 17
6 Arrangement of spheres and theoretical bed in
EinsteinEl Samni work .............. .28
7 Arrangement of spheres and theoretical bed in
Chepil's work . . . . . . . .... . 30
8 Sketches for sidewall effect . .. .. . .... 36
9 Evaluation of constant for sidewall analysis .... 38
10 Grains placed in rough bed configuration . . ... 42
11 Pressure distribution on twodimensional grain ..... 43
12 Piezometric head distribution on twodimensional grain 43
13 Distributions of surface hydrodynamic pressure decreases. 45
14 Lift coefficient CL . . . . 46
15 Theoretical bed used in relating logarithmic and
potential velocity profiles ......... .. 48
16 Definition sketch for experimental application .... 50
17 Comparison of theoretical and measured values . . .. 51
18 Solution space for closely packed hemispheres . . 58
19 Solution space for Chepil's arrangement . . . ... 59
vii
LIST OF FIGURES (Continued)
Figure
20 Foldedsymmetry boundary . . . . .
21 Sevenpoint finite difference scheme . .
22 General lattice point . . . . . .
23 Examples for object point on planar boundary
24 Arc interpolation for surface value . . .
Grading the lattice . . . . .
Xderivative condition at boundary .
Yderivatives . . . . . .
Special points for yderivative .
Zpoint for normal derivative condition
Pressures on area of hemisphere .
Chepil's case: velocities on y = 0
Chepil's case: velocities on y = 0.2a .
Chepil's case: velocities on y = 0.4a .
Chepil's case: velocities on y = 0.6a .
Chepil's case: velocities on y = 0.8a .
Chepil's case: velocities on x = 0 .
* *
37 Chepil's case: trace of some equipotential surfaces
in plane y = 0 .. ... .. . . . ... .
38 Measured and theoretical pressure distributions:
Vf = 68 cm/sec . .. . . . . . .
39 Measured and theoretical pressure distributions:
f = 91 cm/sec . . . . . . . . .
40 Measured and theoretical pressure distributions:
V = 128 cm/sec . . . . . . . .
viii
rr
r
109
117
118
119
Page
S 60
S 62
S 65
S 69
S 71
S 73
S 79
S 82
S 86
S 89
S 97
S 103
S 104
S 105
. 106
S 107
S 108
. .
* .
.*
r
. .
LIST OF FIGURES (Continued)
Figure Page
41 Measured and theoretical pressure distributions:
Vf = 159 cm/sec . . . . . . . .... 120
42 Chepil's hemisphere . . . . . . . .... 122
43 Closely packed hemispheres: velocities on y = 0 .... .127
44 Closely packed hemispheres: velocities on y = 0.2a .. 128
45 Closely packed hemispheres: velocities on y = 0.4a . 129
46 Closely packed hemispheres: velocities on y = 0.5a . 130
47 Closely packed hemispheres: velocities on y = 0.6a . .131
48 Closely packed hemispheres: velocities on y = 0.8a . 132
49 Closely packed hemispheres: velocities on x = 0 . . 133
50 Closely packed hemispheres: trace of some equipotential
surfaces in plane y = 0 . . . . . .... 134
51 Closely packed hemispheres: trace of some equipotential
surfaces in plane y = 0.5a . . . . . .. .135
52 Closely packed hemispheres: flow pattern on surface
viewed toward xyplane ... .. ..... .. . 136
53 Closely packed hemispheres: flow pattern on surface
viewed toward yzplane . . .. .. . . . .... 137
54 Velocity suppression near sidewall. ...... .. 140
LIST OF SYMBOLS
a sphere (or circular cylinder) radius
B coefficient in logarithmic velocity profile
CL lift coefficient based on total bed area
CLu lift coefficient based on projected area of grain
d grain diameter
d equivalent grain diameter
dA projected elemental area in xyplane
dA1 elemental surface area on hemisphere
g acceleration due to gravity
h full lattice increment
I point index, xdirection
J point index, ydirection
K point index, zdirection
k equivalent sand roughness
K diameter of hemispheres
s
k1 sidewall equivalent sand roughness
k2 bottom equivalent sand roughness
L hydrodynamic lift force
p pressure
p reference pressure, defined as used
pl reference pressure, defined as used
PI field point in xdirection
PI1 field point in xdirection
PJ field point in ydirection
PJI field point in ydirection
PK field point in zdirection
PK1 field point in zdirection
q total velocity at a point
qb velocity along base of the grain
r spherical coordinate
r roughnessgrain size ratio based on large hemisphere
diameter in Chepil's tests
R Reynolds number
Rew wall Reynolds number (vfk/v)
u velocity
U free stream velocity at a
u mean velocity
m
u velocity at uppermost point of hemisphere, sphere,
or elliptic cylinder
utop same as ut
u35 velocity at 0.35 Ks above theoretical bed
v velocity
v friction velocity (J7 /)
w velocity in zdirection
x Cartesian coordinate
y Cartesian coordinate; also elevation above datum
yb location of theoretical bed
yo distance at which velocity equals zero
fraction finer than (in soil gradation curve)
Cartesian coordinate; also elevation above datum
angle through "corners" of isovels
ratio of given lattice leg length to full increment, h
unit weight of fluid
spherical coordinate; also angle for arc interpolation
von Karman's constant
lift per unit area (total bed area)
lift per unit area (only projected area of grain)
kinematic viscosity of fluid
mass density of fluid
density of particle in Jeffreys' analysis
bed shear stress
potential function
value of potential function on boundary
overrelaxation factor
xii
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
HYDRODYNAMIC LIFT IN SEDIMENT TRANSPORT
by
Barry Arden Benedict
December, 1968
Chairman: Dr. B. A. Christensen
Major Department: Civil Engineering
Hydrodynamic lift is a force often neglected in studies of
sediment movement, despite being of the same order of magnitude as
the drag force. The goal of this dissertation is to demonstrate that
potential flow theory can be used to predict hydrodynamic lift in
sediment transport. It is known that concentration of streamlines,
rather than viscous forces, contribute most to lift. This is rein
forced by many early airfoil works using potential flow theory, which
yield good values for lift, even though exhibiting zero drag.
The theoretical methods used treat mean, steady flows with no
free surface or sidewall effects. Since potential flow involves an
inviscid fluid, the method considered is only applicable to studying
cases of real flow in the hydrodynamically rough range, where viscous
forces are negligible. The two sets of experimental work studied
involve flows in the rough range over beds of hemispheres arranged in
hexagonal patterns. One work uses hemispheres three diameters apart
center to center; the other uses touching hemispheres.
xiii
The analysis begins with a solution of the potential flow over
the two sets of hemispheres, solving here the threedimensional prob
lems by use of finite difference methods. The solutions enabled cal
culation of velocities near the hemispherical surface, relation to
pressures by Bernoulli's equation valid for rotational flow, and sub
sequent integration to find the lift force. The velocity distribution
near the surface (from potential flow) is then linked to the velocity
distribution in the actual flows considered, yielding values for lift
corresponding to the experimental works. This procedure can be used
for any rough bed and velocity field desired.
For the widely spaced hemispheres, theory produces (for three
cases most closely related to the theoretical model studied) values of
lift differing from measured values by 19 per cent, 13 per cent,.and
8 per cent. For the closely packed hemispheres, results from theory
are 16 per cent above measured values. This discrepancy is reduced to
perhaps 10 per cent or less if allowance is made for the sidewall
effects of the narrow flume used in the experimental work. Good
results are also obtained for a natural gravel bed replaced by a bed
of equal hemispheres.
The results show quite good agreement between theory and
experiment. Hence, the goal of this dissertation is accomplished, and
a new analytical tool is found effective in studying lift forces on
a rough bed. It is hoped that the tool will be useful in future work
in the field.
xiv
CHAPTER I
INTRODUCTION
One problem of great practical concern to the hydraulic engineer
is the development of a full knowledge of the transport of material in a
stream. Understanding sediment transport and developing advanced meth
ods in this area are physically and economically important in such prob
lems as scour evaluation and prevention, construction of stable channels,
planning and design of reservoirs, and the maintenance of harbor chan
nels. The problem is of true importance when some of the sediment magni
tudes involved are viewed. Brown [1]1 estimated in 1950 that the aver
age gross sediment production of the United States was on the order of
four billion tons per year. He also indicated that deposition of river
borne sediment in impounding reservoirs was estimated to be equivalent
to a loss each year of sufficient storage capacity to hold the annual
water supply for a city of 250,000 people. On a smaller scale,
Langbein and Leopold [2] note that watersheds composed of fine wind
blown soil, such as in western Iowa, yield as much as 2,000 tons of
sediment per square mile per year. The magnitude of the sediment prob
lem causes a continuing need for better understanding in sediment trans
port. One facet of this understanding must include the mechanism of
Numbers in brackets refer to the References.
a particle being moved from the stream bed. A need for a better defin
ition of the fundamental forces acting on such particles has prompted
the author's study of hydrodynamic lift in sediment transport, with
observations of flow characteristics in the vicinity of a rough bed.
Lift has been an often neglected force in fluvial hydraulics
despite the fact that it has the same order of magnitude as the bed
shear stress. This is reflected in the many stable channel design
procedures which neglect lift. These procedures may result in a design
which is too costly. Hence, studies of the lift force take on further
economic significance.
Most work on lift has been on single particles, with no con
sideration for the particle interaction existent in the actual stream.
Those works which have included interaction have generally measured
lift forces over a given area rather than in the form of pressure
distributions on the individual particles. The author intends to study
interaction effects, trying to provide a base for analytical studies
of lift forces.
First, potential flow theory will be used as a guide to provide
a means of studying the flow near the surface of a grain or bed shape
and then relating that flow to other flow characteristics. The poten
tial flow velocity profile will be related, for this work, to the
logarithmic velocity profile, furnishing a means for predicting the
pressure distribution and total lift force on single particles within
a series. It should be noted that forms of velocity profile other than
the logarithmic might be used if they are characteristic of the flow.
This dissertation will concern itself with cases involving a
steady, mean flow with free surface effects (wave resistance) being
insignificant. A report of some earlier results for twodimensional
bed shapes will be made. Then a numerical solution will be made for
two cases of flow over a bed of hemispherical particles. These solu
tions will be related to experimental measurements obtained elsewhere,
as will some of the twodimensional results.
It is the hope of the author to present herein an analytical
method wherein the hydrodynamic lift, often neglected, can be computed
with a reasonable degree of accuracy. It is also intended to form a
basis for future analytical and experimental work in sediment transport.
CHAPTER II
RELATED BACKGROUND ON LIFT FORCE STUDIES
2.1 Historical interest
For centuries engineers have been interested in the movement of
sediment due to flowing water, which Rouse and Ince call "A class of
flow phenomena inherently hydraulic in nature ." [3, p. 246].
Concerned with problems of scour, deposition of material, stable
channels, and the like, engineers have for years attempted to increase
their knowledge of sediment motion. Both empirical and theoretical
means have been employed in these attempts.
Domenico Guglielmini [3, p. 70] was perhaps representative of
the whole Italian school interested in flow resistance in open channels.
His work in the seventeenth century made some qualitative observations
which were very accurate, though his analytical work was more faulty.
Later, Pierre Louis Georges Du Buat [3, p. 1291 collected a
vast array of experimental results, including extensive data on the
beginning of sediment movement. His eighteenth century works over
shadowed that of other hydraulicians for about a century.
Work continued through various periods until the work of Grove
Karl Gilbert in the years around 1910. His tests on initial sediment
movement and various phases of transport covered a wide range. It has
been noted that ". . the results he presented in U.S.G.S. Professional
Paper No. 86 in 1914 still continue to be those most often quoted of
any in the field" [3, p. 225].
From the time of Gilbert's work, laboratory facilities increased,
enabling further, broader studies. As various governmental agencies
began to attack the problems in the United States, more and more quan
titative information became available. Additionally, interest in analyt
ical approaches to sediment problems grew with the advent of more
theories on movement. Consideration of the forces actually acting on
particles exposed to flow received increasing emphasis.
2.2 Neglect of lift force
As Leliavsky [4] indicated, the drag component of forces acting
on particles received the bulk of work earlier in this century. Indic
ative of this was the work by C. M. White [5] in 1940. White gave
great care to study of the drag, while the lift was treated only briefly,
with a guarded conclusion that lift did not exist. Leliavsky noted
in 1955 that there still exists ". . an almost unexplored aspect of
the problem, viz., the vertical component of the resultant of the
hydraulic forces applied to the grain, i.e., 'the lift'" [4, pp. 6465].
Similarly, Young stated, "Although some attention has been given
to the effect of the drag components on the behavior of suspensions,
little work has been done in connection with the determination of the
lift component" [6, p. 47].
This disregard for the lift force has resulted in a lack of
understanding of this force, with probably attendant shortcomings in
full understanding of drag. One area of very practical interest where
lift is often neglected is in stable channel design, where the widely
used method of E. W. Lane [7], as well as other methods, does not
include a lift force. It should be noted, however, that other methods
are beginning to include lift effects. This neglect of the lift force
is on the safe side but is economically costly.
2.3 Indications of significance of lift
As Leliavsky noted, contrary to White's results, Jeffreys [8]
and Fage [9] produced results which definitely indicate lift as an
important factor. Fage's experimental work provided evidence, and
Jeffreys' gave a theoretical approach. Studying a cylinder resting on
the flat bed of a deep stream and applying the principles of classical
hydrodynamics, Jeffreys found a relation for the lift and then for
a scour criterion. Use of realistic sizes indicated that lift alone
should be capable of dislodging particles.
Chang [10], considering the lift on particles as due to the
pressure of a velocity head, worked from the simple fact that the
particle would tend to lift when the vertical lifting force (or hydro
dynamic lift) plus the buoyant force equalled the particle's weight.
He also did work on drag and made the following comparison. "Theo
retically, the force required to lift a particle from the bottom of
a stream is about 40 per cent greater than that required to move it
along the bed" [10, p. 1282]. This certainly indicates the same order
of magnitude for drag and lift.
Further validity is given to the value of studying lift by
Yalin [11] who says "A consideration of the paths of saltating particles
by R. A. Bagnold reveals that saltation begins with a motion directed
'upward'; . However, if this is true, then, as has already been
maintained by various authors, the lift force . must be the cause
of the detachment" [11, p. 229]. Support for Yalin's statement is
offered by the work of Einstein [12], Bagnold [13], and Velikanow [14].
Young, doing work in 1960 on spheres in a cylindrical tube,
found the lift to be of the order of onehalf the drag for experiments
with a Reynolds number based on the pipe diameter and the mean veloc
ity in the range 3601115 (in the laminar range). He summarized,
"It is thus apparent that the lift force should not be overlooked in
studies related to the incipient motion of particles resting on stream
beds or pipe walls" [6, p. 57].
It seems apparent that sufficient evidence exists to prompt
efforts to increase understanding of the lift phenomena. The author
will now consider earlier works on lift.
2.4 Use of potential theory in lift studies on single bodies
2.4.1 General.Much of the analytical work done on lift has
dealt with single bodies or particles, especially spheres and circular
cylinders, isolated from any other bodies. An example of this approach
is the work done by Jeffreys [8] mentioned earlier.
The overwhelming amount of work on lift has been done in the
realm of aerospace engineering. While the principles thus developed
are applicable to hydraulics, the work concerns single bodies only,
often airfoils, struts, and the like. One of many works giving dis
cussions of several of these approaches is the famous work edited by
Goldstein [15]. Another, viewing historical development, is by
von Karman [16]. In a number of these, lift is studied by means of
potential flow.
2.4.2 Applicability of potential flow.The idea that lift
could be predicted by use of potential theory, even perhaps in those
areas where it might seem out of place, was offered by Prandtl. He
stated that any explanation of drag requires a consideration of
viscosity, ". . whereas the lift can be explained entirely without
the concept of viscosity so that the wellknown methods of the clas
sical hydrodynamics of the ideal fluid are applicable" [17, p. 159].
Part of the reason for some reluctance to use such methods here is the
fact that within a boundary layer adjacent to a surface, irrotational
flow does not exist. However, in the cases of interest, the sublayer,
where viscous and inertial forces are of the same order, over the surface
is very thin. Outside this layer, up to the point of separation, the
equations of inviscid fluid flow are valid. Hence, the pressures on
the outer edge of the sublayer can be found from such equations, and
since the pressure difference from the outer edge of the layer to the
surface is assumed negligible, the normal pressures on the surface, and
thus lift, can be predicted by inviscid flow principles. Also, since
the sublayer is thin, it is possible to discuss velocities "on the
surface," though the velocities actually considered are those a small
distance away at the edge of the boundary layer. Of course, the
actual velocities at the wall in a real fluid would be zero.
Reasoning such as Prandtl's has through the years prompted
many potential flow solutions to the lift problem, probably the most
famous of which are those by Kutta and Joukowsky.
2.4.3 Work of Kutta and Joukowsky.Generally, an understand
ing of the work of Kutta and Joukowsky begins with flow around a cir
cular cylinder. It is well known that no lift or drag forces are
predicted by any transformation of the symmetrical flow around a cylin
der. The addition of a vortex in the center of the cylinder produces
a streamline pattern, the effect of which is to yield a lift force.
This force is due to the circulation's tendency to increase the velocity
above the cylinder and decrease it below, thus causing a pressure dif
ference and a consequent lifting action.
Kutta [18] first applied the methods of conformal transforma
tion to transform the cylinder with circulation into a line inclined
to the flow. This produces a force acting perpendicular to the veloc
ity at infinity. This force is, of course, the one originally exerted
on the cylinder.
Kutta's work and use of transformations prompted further work.
Joukowsky [19] wanted to avoid difficulties at the sharp leading edge
of Kutta's plane, and he employed a mapping function by which a curvi
linear profile very similar to actual airfoil shapes was developed.
Flow around a Joukowsky profile is shown in Figure 1.
Numerous investigations of Joukowsky profiles have been carried
out. Joukowsky [20] himself performed experiments in 1912, and
Blumenthal [21] calculated the pressure distribution from theory
in 1913. Betz [22] provided experimental comparisons for the lift
and pressure distributions predicted by theory. Figure 2 shows this
comparison, and the agreement with theory is seen to be satisfactory.
The lower lift than that predicted can be accounted for by the fric
tion which causes flow separation from the profile near the end; the
resultant failure to attain the full pressure difference predicted
yields a smaller lift. The overall agreement is, however, good.
Figure 1.
Flow around a Joukowsky profile.
Profile
Calculated
Measured
Figure 2. Calculated and measured pressure distribution
around a Joukowsky profile. (From reference 17,
p. 181.)
The classic work of Kutta and Joukowsky, with subsequent
experimental verification of the validity of their approach, has led
to many other lift solutions by means of potential flow. These works
have generally produced similarly satisfactory results.
2.4.4 Fuhrmann's work and other studies.Prandtl and Tietjens
[23, p. 137] mention the work of Fuhrmann [24] who calculated, by poten
tial theory, and then measured the pressure distribution on some slender
bodies whose shapes were derived from sourcesink combinations. As
found in reference [15] and pictured in Figure 3, Fuhrmann's experi
ments revealed good agreement with theory except near the body's end, a
result common to other works and also expected. Rodgers [25] notes
this result in reference to several bodies of revolution analytically
treated by Lamb [26] and MilneThomson [27]. He attributes the devia
tions from theory primarily to generation of vorticity along the body.
i.
0.8  Measured
_ Potential Flow
0.6_
PPo i I
2
p v /2 1
o 0.2
0
0.4 x
0 0.2 0.4 0.6 0.8 1.0
x/L
Figure 3. Pressure distribution around a Fuhrmann body.
Other works where pressure distributions have been predicted
and checked experimentally could also be shown. The important factor
is not to go into details of numerous cases, but rather to point out
that there is strong historical backing for use of potential theory
to study lift forces. It should be recalled that these earlier uses
of potential theory have employed it to describe the entire flow
pattern. Only two further cases will be studied, the first being the
one case where potential flow was used to describe the entire flow
pattern in an effort to solve the problem of sediment movement.
2.4.5 Jeffreys' analysis.Jeffreys [8] dealt with a single
long circular cylinder resting on a bed in a twodimensional study,
as shown in Figure 4.
Uniform Flow
U
a Long Circular
Cylinder
Figure 4. Jeffreys' cylinder.
Jeffreys developed his work based on the complex potential for
this case,
na
W = raU coth () (21)
z
He found the amount by which the lift exceeded the weight of the
enclosed liquid, and thence wrote the following condition for move
ment of the cylinder:
( + 2 )U2 > ga (22)
3 9 p
where a = density of particle
g = acceleration due to gravity
a = cylinder radius
p = mass density of water
This can be written as
U2 > 1 ( ) ga (23)
1.43 p
Jeffreys noted that J. S. Owens [28] had earlier measured
the velocity required to move pebbles, finding
U2 = 1.65 ga (24)
The motion observed by Owens was not, however, a jumping or
lifting motion, but rather a rolling motion, yielding a difference
from values predicted by theory. However, as Jeffreys says, "The
proportionality of U to the square root of the linear dimensions is in
agreement with theory" [8, p. 276].
Use of a value for 7 of 2.7 times p in Jeffreys' equation yields
the following:
U2 > 1.19 ga
(25)
Differences from theory are primarily due to two causes.
First, the motion measured was rolling rather than totally lifting,
which (25) is based on. Second, the measured particles were three
dimensional, contacting the bed at only a small number of points,
while Jeffreys' cylinder made contact over the whole length of its
body. These effects are compounded by the fact that the Uvalues used
by the two men are not the same. Jeffreys employed a potential flow
field, but Owens made his measurements in a flow field exhibiting a
logarithmic velocity distribution. The differences involved here will
be discussed in Chapters III and IV. Despite the discrepancies, the
theory seems to provide a much better starting point than might first
be thought possible.
2.4.6 Flow around a single sphere.The results from measure
ments on flow around a single sphere will be presented because the shape
relates to this study and the results reveal some factors influencing
the actual flow pattern.
The flow involved is that around a single sphere suspended in
an otherwise uniform flow. For this case the potential can be
expressed [29] as
Ua
S= cos e + Ur cos e (26)
2r
2r
where a = radius of sphere
r = radial distance
0 = angle measured from horizontal and
through sphere center
U = free stream velocity at 0
Note that this is written for one meridian plane, since the
case is axisymmetric. The tangential surface velocity on r = a
can be found as
1 3
= U sin 8 = q (27)
Using Bernoulli's principle, with a pressure of po at the
forward stagnation point, yields the following expression for the
surface pressure, p.
p p = yz + q (28)
where z = elevation
y = unit weight of fluid
What is of interest here is to study only the vertical force
component over the upper half of the hemisphere. This will involve
integrating (28) over the upper surface. Integration of the first
term, z, would yield a hydrostatic lift (buoyancy) which would, in
fact, not even be measured, as the difference in piezometric head is
the measured value. Hence, integration of the last term in (28)
will yield the desired force component. It should be noted that this
would actually be the theoretically predicted value for the case
of a single hemisphere on a flat bed. However, the measurements herein
used are for a suspended sphere. The total vertical force can be
found as follows:
F = U2 sin2e dA (29)
v 2 4
o
where dA = 2a2 sin28 dG
yielding
2
S= a p 2 (210)
Note that if this was expressed per unit area (based on the area
na ), with U replaced by ut = (3/2)U, the velocity at the top of
the sphere, theory would show
2
3 u
3 t
F =  (211)
p 4 2
where F = vertical force per unit area
Measurements were made by Flachsbart [30], as shown in
Schlichting [31]. The results were very similar to those by Fage [32]
shown below. The measured values are shown below with the theoretical
curve. The values are plotted with reference to the stagnation pres
sure pl + ~ pU where pl is the pressure at where the velocity
equals U. This stagnation pressure is the p being used here. Replac
ing pl by po yields
S o= B 1 (212)
1 2
2pU
where B is the value shown in the graph. The expression thus found
for (p po) can be integrated numerically.
In the case of the higher Reynolds number, the force can be
expressed as (210) with a coefficient of 1.64, only about 3 per cent
below the theoretical. The lower Reynolds number produces a coefficient
of 1.32, about 20 per cent below the theoretical. It therefore seems
that above some critical Reynolds number the potential theory becomes
more and more capable of predicting pressures on a body outside the
separation zone. The effect of the Reynolds number involves the change
of laminar boundary layer at the body to turbulent as a critical range
of Reynolds number is reached. The boundary layer, becoming turbulent,
is capable of proceeding further downstream, therefore causing separa
tion to be delayed to points further and further back on the sphere.
Ppl
 0
1 2
pU
0 60 120
180
0 (degrees)
Figure 5. Pressures on a single sphere.
This allows the surface pressure distribution to more nearly approach
the theoretical. For even lower Reynolds numbers than those indicated,
the viscous forces would play an even greater role, thereby causing
further deviation in the flow from theoretical. These ideas are of
importance in relation to the limitations of work in this dissertation.
The forces evaluated in this section were only those in the
vertical direction on the upper surface of the sphere, with an eye to
noting agreement with theory.
For later comparison a value of CLu will be determined here
for the theoretical case of a single hemisphere on a flat bed. This
will entail integrating the pressure at the hemispherical base, found
from (28) and subtracting it from the force of (211) to give a
resultant vertical lift. Integration here occurs in a direction normal
to that in (29). Using 81 for this integration yields a relation
cos 0 = sin 0. Therefore, the ratio of the basal velocity (ub) to the
top velocity (u t) equals cos 81. The pressure decrease along the
bottom can be found as below.
2 2
t 22 2 1 t
Fv = P T (2a2) sin29 cos2 dO = 0 P T a2 (213)
The lift per unit area is therefore
F F u
S vvv vb t1 t
S= ,a = p  (2P.)
u 2 2
Tra
This lift.coefficient of 0.50 should form an upper bound for the
work to be done later.
At this point it seems appropriate to define certain lift
coefficients to be employed in this dissertation. The difference
lies in the area over which the force is considered. The subscript,
u, will denote those cases where the area considered is only that
directly beneath the body being considered, the projected area of the
grain. Thus, \ denotes lift per unit area based on the total bed
area, while X is based on the area of the grain projected onto the
U
bed. Similarly, the coefficients CL and CLu are used with the corre
sponding X's.
As indicated, most work done on lift has dealt with single
bodies. Attention will now be turned to systems with more than one
particle.
2.5 Multiparticle studies
2.5.1 Einstein and El Samni.It was natural that multi
particle studies should arise, as these begin to approach the sediment
conditions found in nature. Unfortunately, however, work in this area
has been limited. Some values for the lift force came from the work
by Einstein and El Samni [33,34]. Using the upper onehalf of
plastic spherical balls 0.225 feet in diameter placed in a hexagonal
pattern, they measured the lift force as a pressure difference. They
made the following statement.
The procedure in making such measurements was as follows:
if a lift force is exerted on the top layer of a stream bed,
the solid support of the sediment particles is relieved of part
of their load and this load is transmitted hydrostatically to
the fluid between the solid bed particles. Thus, it must be
possible to detect and measure this lift as a general pressure
increase of the pore fluid in the bed. [33, p. 521]
Their results enabled them to write
2
p = CL P (215)
where Ap = pressure difference
CL = lift coefficient
p = fluid density
u = velocity
They found CL a constant 0.178 if u was taken as the velocity 0.35
sphere diameters above the theoretical bed, determined by experiment
as 0.20 sphere diameters below the sphere tops. Further studies
which they made on natural gravel yielded the same expression for Ap
with some redefinition of u along lines consistent with Einstein's
earlier work [12]. This work forms the only example of a lift force
essentially integrated over a number of particles, though the distri
bution over individual particles was not ascertained. Support for the
rationale of measuring lift as a pressure difference is given by
Engelund and Hansen [35, p. 19] in discussing variations from hydro
static pressure due to streamline curvature.
2.5.2 Chepil.Chepil [36] performed experiments in a wind
tunnel on hemispherical elements placed on a plane bed in a hexagonal
pattern three diameters apart. He chose this spacing based on work
by Zingg [37], which indicated this is the average spacing between
particles erodible from a sand bed.' The processes of erosion of sand
by wind and by water involve essentially the same factors, as indicated
by, among others, Kadib [38].
Chepil measured pressures over the surface of one metal
hemisphere. This was done by placing pressure taps, starting at the
base of the hemisphere, 30 degrees apart along one line running parallel
with and another normal to the wind direction. The negative pressure
end of the manometer was connected to a tap on top of the hemisphere.
The remainder of the hemisphere pattern consisted of gravel hemispheres.
The hemispheres occupied 11 per cent of the total floor area. The lift
and drag forces on the hemisphere were determined by integrating the
measured pressure distributions and also, as a check, by means of two
torsion balances measuring the forces directly. Hemispheres of three
different sizes were used, and also some measurements were made on
relatively small sand and gravel mounds. Measurements were also made
at different points downstream in the tunnel to note the effects as the
air boundary layer developed to its full extent.
Chepil found that increasing the depth of the fluid boundary
layer, after a certain limiting depth is reached, has little effect on
the lift to drag ratio, though the depth of boundary layer had a
profound effect on the magnitude of both lift and drag. For the study,
it was found that
LIFT = 0.85 Drag (216)
For this study it was also found that the pressure difference between
the top and the bottom of the hemisphere is about 2.85 times the lift
per unit bed area directly under the hemispheres. Using the latter
finding, Chepil surmised that the CL from Einstein and El Samni [33]
should equal 0.178/2.85, or 0.0624. He then used equation (216)
with his drag and velocity measurements to attempt a correlation with
the case of closely packed hemispheres and found a CL of 0.0680.
Chepil concludes by noting that "This study shows that lift on
hemispherical surface projections, similar to soil grains resting on
a surface in a windstream, is substantial. Therefore, lift must be
recognized together with drag in determining an equilibrium or crit
ical condition between the soil grains and the moving fluid at the
threshold of movement of the grains" [36, p. 403].
2.5.3 Chao and Sandborn.Chao and Sandborn [39] also performed
experiments on spheres, but, by means of a transducer, they actually
measured the pressure distribution on the upper half of an element.
However, the type of flow they used bore no real relation to that in
nature's streams, and no attempts were made at analyzing velocities and
the like. Einstein used the flow of water in a channel with a measured
logarithmic velocity distribution. On the other hand, Chao and Sandborn
placed lead shot on a flat surface and blew a stream of air down onto
the particles, the air diverting horizontally at the flat surface.
Their conclusions included, "The present experimental results are of
primary interest in demonstrating that a problem exists. . More
extensive research is needed before there can exist a better understand
ing of the mechanism" [39, p. 203].
2.6 Shapes of bodies studied
With the exception of the vast amounts of work done on wings,
struts, and related areas, most effort has beenaimed at circular
cylinders and spheres, especially in work related to the sediment
problem. There have been, however, numerous indications of shape as
a parameter. Rouse defines sphericity [40, p. 777] and indicates it
as a probable factor. Studies have been made on shape influence,
including one by Krumbein [41] which specifically considers nonspherical
particles. Other works also exist for considering the effect of non
spherical particles.
Most design procedures today treat the bed material by conver
sion to a representative bed consisting of uniform spherical particles.
The conversion is generally based on factors such as particle density,
sizes, exposed areas, and volumes. Thus, there are means for relating
natural beds to the uniform spheres to be treated in this work.
2.7 Relation to work of dissertation
The preceding background material was presented in an effort to
point out those factors with special bearing on what the author is try
ing to accomplish in this dissertation. Generally, an attempt will be
made to advance the knowledge of lift through analytical work.
Analysis of the threedimensional cases corresponding to the
work of Einstein and El Samni and Chepil will be made. These multiple
particle systems allow consideration of particle interactions. Poten
tial flow theory will be used to study velocities on the surfaces, these
velocities then being linked to known logarithmic distributions.
Additionally, some twodimensional work and applications will be reported.
Support for the use of irrotational flow is gained from the lift predic
tions made on airfoils, as well as from a knowledge that viscosity is
not necessary to explain lift. Of course, in the airfoil cases as well
as here, the predicted drag is zero, obviously incorrect, but the
description of the magnitude of the lift is the primary goal here.
It should be very strongly noted, however, that whereas these earlier
works used the potential to describe the total flow field, a much
different use will be made herein. Essentially, potential flow theory
will be used as a guide in attempting to develop means of predicting
hydrodynamic lift. Success here might lead eventually to better
analytical approaches to some sediment problems, although it will
also be used in larger scale problems dealing with dunes and ripples.
Summarizing, it can be said that the author is relying on the
success of earlier approaches to related studies in an effort to find
a guide to a better understanding of hydrodynamic lift in sediment
transport.
CHAPTER III
LOGARITHMIC VELOCITY DISTRIBUTION
3.1 Development of logarithmic velocity distribution
Due to the use which will be made of the velocity distribution,
the author wishes to present briefly some background and to study some
specific points. Historically, three modern approaches to velocity
distributions in steady, uniform turbulent flow have arisen. The three
are the following: Prandtl [42), who introduced the concept of the
mixing length (related to the mean free path of particles) with momen
tum conserved; G. I. Taylor [43], who considered vorticity to be con
served along the mixing length; and von Karman [44], who developed a
similarity hypothesis for the problem.
Prandtl's derivation, beginning with the expression for shear
stress in fluid, is frequently cited in texts, such as [45]. It
neglects the viscous forces and considers only the socalled Reynolds
stresses, after 0. Reynolds [46]. The final expression can be written
as below.
V = B + ln (31)
v f k
where B = C + In k
k = equivalent sand roughness
The socalled von Karman constant, K, has been verified
experimentally, as noted by Bakhmeteff [47], to equal 0.40. The
values for B were obtained from experiments by Nikuradse [48,49],
who made tests on smooth pipes and then on pipes roughened by gluing
uniform sand grains of size k to the wall. Results in the transition
range between smooth and rough flow regimes by Colebrook [50] showed
different Bvalues for nonuniform roughness.
Hydrodynamically smooth and rough flow regimes are generally
defined in terms of the wall Reynolds number (vfk/v), Rew. The smooth
f ew
range, Rew less than 3.5 to 5, has a viscous sublayer of depth compar
able to bed roughness size and viscous forces play a dominant role.
In the rough range, Rew greater than 70, the roughness elements have
fully penetrated the sublayer, and viscous effects are negligible.
Most sediment problems in nature occur where flow is in the
rough range. This will be the range considered in the present work.
Substitution of the appropriate values for the rough range yields
S= 8.48 + 2.5 In (32)
vf k
Although this expression was derived from work in circular
pipes, it has been noted on many occasions that it is similarly
applicable in open channels.
3.2 Special problems of present expressions
Working with measurements near rough surfaces, Hwang and
Laursen noted that ". .. the adequacy of this logarithmic equation
can be debated, especially near the boundary, and the indeterminacy
of the zero datum presents difficulties. ." [51, p. 21]. These
essentially are the two problems with use of (32).
3.2.1 Theoretical bed.In a bed in which roughness elements
protrude, some reference datum must be decided upon. This reference
is usually referred to as the theoretical plane bed, and, if sufficient
information is available, its location can be computed. Generally,
satisfactory results are obtained by use of this datum, though problems
may arise for very irregular surfaces.
The basis for computation is to place the bed at such a level
that the water volume contained beneath it equals the volume of rough
ness elements protruding from the bed. Justification for use of this
definition can be gotten by reviewing the work of Einstein and El Samni
[33]. Using hemispheres of diameter k and placed in a hexagonal pat
tern, they found that for satisfactory results it was necessary to
assume the theoretical bed at a distance of 0.2 k below a tangent to
the tops of the spheres. Their arrangement appeared as shown in
Figure 6.
Realizing that the area considered (inside the dotted lines)
has dimensions 3k by v/3 k the following is obtained.
s s
Volume under theoretical bed = 3, k2 (33)
s b
1 3
Volume of particles =2 s k (34)
2 s
Equating these two and solving yields
(35)
yb = 0.302 ks
Plan
1 Flow
Theoretical Bed
y = 0 in equation (37)
0.2 ks by Einstein
and El Samni
Elevation
Figure 6. Arrangement of spheres and theoretical bed in
EinsteinEl Samni work.
Therefore, the distance down from the upper tangent is
0.198 k in excellent agreement with the 0.2 k found by experiment.
It might be noted that the value of 0.2 k seems to be satisfactory
even when the roughness pattern is more irregular. This was shown
experimentally by Einstein and El Samni [33]. Other instances could be
reviewed, but let it suffice to state that the above definition of the
/
theoretical bed is satisfactory.
At this point the conditions for the Chepil [36] case are shown
in Figure 7. The computations for the theoretical bed are also shown,
since this will be needed later. Considering the region inside the
dotted lines, with dimensions 3V1 Ks by 3Ks, the following is obtained:
Volume under theoretical bed = 9, K Yb (36)
s b (3)
1 3
Volume of particles = K (37)
6 s
Equating these and solving yields
Y = 0.0335 K (38)
b s
Therefore, for this pattern the distance down from the upper tangent to
the hemispheres is 0.467 K very near the plane bed itself.
3.2.2 Conditions near the bed.Experience indicates that
expression (32) is quite valid away from the bed. However, inspection
reveals that problems occur near the wall. As y approaches zero, the
value of v/vf, and hence of v, approaches minus infinity. This obvious
discontinuity presents a real problem in understanding what happens at
the bed, where sediment movement begins.
0
0
Flow
r
I '1
0
lan
Flow
__rw

'1
3K
I
Theoretical Bed
ri_...L\
Elevation
Figure 7.
Arrangement of spheres and theoretical bed
in Chepil's work.
P
3.3 Use of proposed adjusted velocity distribution
Christensen [52] is suggesting use of a slightly different
velocity distribution which he has developed. The new expression is
as follows:
v = 8.48 + 2.5 In (Y + 0.0338) (39)
vf k
Calculation will reveal one very desirable feature of (39),
that being prediction of a zero velocity for y equal to zero. The
added constant factor will continue to have a large effect very near
the wall, but as y/k increases, the effect of the additional term will
quickly become negligible, yielding essentially the same expression
as (32). This is correct, since (32) has proved adequate away from
the wall.
3.3.1 Comparison of distributions at wall.In addition to
the effect of zero velocity at the wall as compared with the infinite
value predicted by (32), other features can be noted by looking at
the change of velocity with distance from the wall, or dv/dy shown
below.
dv f
Former: = 2.5
dy y (310)
For y 0 dv/dy 
y O dv/dy 0
dv 1
Proposed: = 2.5 v ] (311)
dy f y + 0.0338k
For y = 0, dv/dy = 74.0 v /k Finite
y dv/dy 0
In both cases, the derivative approaches zero for large
yvalues, as it should. However, at the wall, a definite discontin
uity exists by the former method. The new proposal predicts a finite
change at the bed, a far more reasonable development. The author
feels that the proposed distribution, with its continuous curve, will
enable better studies of action near a rough bed.
3.3.2 Comparison of distribution with increasing y.The
increasing value of (y/k) will eventually negate the effect of the
added term in (39). Table 1 shows the difference in velocity indi
cated'by (32) and (39) for some (y/k) values. The difference in
hydrodynamic force, proportional to the velocity squared, is also
shown. It can be seen that the effect of the added term in (39) is
dissipated very rapidly.
TABLE 1
COMPARISON OF PROPOSED AND FORMER DISTRIBUTIONS
Variation between Force
(32) and (39) y/k Variation
1% 0.99 2%
3% 0.61 6%
5% 0.29 10%
3.4 Determination of k and vf from experimental data
Essentially, in either equation (32) or (39), it is neces
sary to find k and vf to fully describe the profile in a given
situation. If the distribution is first measured to be logarithmic,
k and vf may then be determined by considering the velocity at two
different depths [45].
Call the total flow depth d. Consider then a depth pd (p,
a fraction), where the velocity is v Consider also a second depth
p
qd with velocity v .
k, the equivalent sand roughness
vf, the friction velocity
From these measurements the following can be written, using first
expression (32).
Depth 1:
v
2 = 8.48 + 2.5 In Md (312)
vf k
Depth 2:
v
= 8.48 + 2.5 In q (313)
vf k
Combination of these equations and elimination of vf yield
29.7 pd (314)
(v )/(v v )
( p '
Elimination of k yields
V. v
vf = (315)
2.5 In (P)
q
Frequently the depths chosen are for p = 0.90 and q = 0.15.
Use of the proposed expression in (39) for equations such as
(312) and (313) above, does not lend itself as readily to direct
elimination of variables and solution for k and vf. Therefore, some
consideration must be given to the relative values of the terms.
As noted earlier, for terms where y/k is greater than one, the effect
of the added 0.0338 is less than 1 per cent and can presumably be
neglected. If indeed p = 0.90 and q = 0.15 were used, for most
practical cases y/k would be far greater than one for both depths,
and the expressions for k and vf would be precisely those found in
(314) and (315).
In any case, the mechanism is available for computation of k
and vf through observations. These parameters can then be used in
computing forces, velocities, and the like according to equations
to be subsequently derived.
3.5 Effect of sidewalls
As will be seen later, the experimental flume to be studied is
rather narrow. For this reason, possible effects of the sidewalls on
the velocity will be discussed here.
Sayre and Albertson mentioned this problem, when, in reference
to their analysis, they indicated that one assumption which they made
was . that the channel is of sufficient width, or that the bed
roughness is so great relative to the sidewall effect" [53, p. 124].
Rouse [54, pp. 276277] points out the effect that sidewalls have,
through secondary flows, in varying the isovels and in depressing the
region of maximum velocity below the surface of the flow.
In order to provide some quantitative means of evaluating the
sidewall effect, aside from the qualitative evaluation of experimen
tally obtained velocity profiles and isovels, the following approx
imate analysis is presented. Figure 8 indicates the assumptions util
ized listed below.
Assumptions:
1. Idealized isovel picture, enabling passing of
line through corners.
2. Idealized pressure distribution (varying linearly,
as indicated in Figure 8).
3. T1 proportional to yl in some way; T2 proportional
to Y2 in the same way.
4. Logarithmic velocity distribution in center line
profile.
Roughness:
Idealized
Isovels
Roughness:
Probable True
Tdistribution
Idealized
Shear
Distribution
Sketches for sidewall effect.
Figure 8.
The following expressions can be written relating the shear
stresses.
1.0 1
=  = tan f (316)
2.0 Y2
T1.0 h + T2.0(W h tan a) = yh WSb (317)
This can be written as
T1.0 + tan ') = WSb (318)
Equation (316) results from assuming 71 and 72 to both be
proportional to.their respective yvalues in the same way. The other
expressions merely equate the weight of fluid acting on the wetted
perimeter and the shear forces resisting it.
Obviously the value of tan Q0 is of primary importance'in this
analysis of sidewall effect. The equation enabling evaluation of the
angle comes from equating expressions for the velocity at a point such
as A in Figure 8, computed first with reference to the channel bottom
and then with reference to the side. For this analysis the form (32)
will be used for the velocity.
[8.48 + 2.5 In [ ] A = [8.48 + 2.5 In []} T (319)
k1 2 p
Rewriting this and using the similar proportionality of
T1 and T2 to yl and y2 yields
[In 29.7 ] = [In 29.7 .] L 2 (320)
1 2
100
F(M)
'I = y/k
Evaluation of constant for sidewall analysis.
100
Figure 9.
Introducing y/k = T enables the writing of
,/I F(I,) = /F(T2 ) (321)
where
F(C) = [In (29.7 1)] 7 /T
The curve shown in Figure 9 indicates that the Ffunction can
be approximated by a function of the form
F(J) = alb (322)
This would enable the writing of
1/b
1 27 (y_1/k) y_ k2
S_ 2 =(323)
Sy2/k2) 2 1
Using the fact that = tan C, it can be written that
Y2
l(1/2b)
tan a = () (324)
2
Evaluation from Figure 9 reveals that bcequals about 0.664. Thus,
0.248
k2
tan = 1) (325)
The last equation provides an opportunity for an
approximate evaluation of the sidewall's effect on the isovels and
hence on the center line profile. The likelihood of simulating two
dimensional flow in the center region can then be estimated.
40
Essentially, the work above is an attempt to check beforehand
the probable validity of all the previous discussion on logarithmic
distributions in this chapter. The chapter, in general, was intended
to point out phases of the velocity distribution applicable to the
present work.
CHAPTER IV
TWODIMENSIONAL WORK: EARLIER RESULTS
4.1 Shapes studied
In earlier work the author has studied some twodimensional bed
shapes. In one case a convenient potential function (that formed from
an infinite row of vortices) was chosen, and the grain shape produced
by this was taken. Here the solution was in a closed mathematical form.
In other cases, a series of elliptic cylinders was chosen, and the
solution for the irrotational flow obtained, in terms of the stream
function, by finitedifference methods. Some of the results obtained
and methods used will be noted briefly here as a prelude to the three
dimensional work of this paper.
4.2 General methodsvelocity and pressure results
The potential flow solution for flow around the given shape was
first obtained, using Laplace's equation involving the stream function.
The solution was used to compute velocities along the upper surface of
the shape. Then, applying Bernoulli's equation for a streamline along
the upper surface streamline, the pressure distribution could be found.
The grain shape obtained from the potential of an infinite series of
vortices is shown in Figure 10, and Figures 11 and 12 show the computed
pressure distribution on these repeating grains. It can be seen that
Figure 11 indicates, in the difference between the upper curve and its
Main Stream Flow
cosh Y = 2 + cos X
PO
vT
/ Stagnant
/ b/d = 0.561
\
NS.
Figure 10. Grains placed in rough bed configuration.
c,,
Figure 11.
()bottom
Y bottom
PO
(t)
y top
Pressure distribution on twodimensional grain.
z+
Y
)
(z + =bo m
y bottom y
+7T
(z + )
y top
Figure 12. Piezometric head distribution on twodimensional grain.
P
Y
,,,
dotted counterpart, the buoyant lift, while the remaining pressure
decrease on the upper surface contributes to hydrodynamic lift.
In the experimental work, the actual quantity to be measured
is the difference in piezometric head between upper and lower surfaces.
Since the situation below the grain is hydrostatic, the difference can
be taken between a point on the surface and any point in the region
below.
Results for velocitysquared distributions for the elliptic
cylinders can be found in Figure 13.
4.3 Lift integration
Integrating the vertical pressures over the grain surface will
yield the lift. The lift per unit area, X, can then be expressed by
S=c 2 u2 (41)
L2 t
where CL = lift coefficient
ut = velocity at top of grain = utop
This lift coefficient was found to have a value of 0.500 for the hyper
bolic cosine grain shapes, and the results for the elliptic cylinders
are found in Figure 14. Note the parameter b/a represents the thickness
tolength ratio of the ellipse. The numerical solution of the Laplace
equation requires a completely bounded solution space. Studies of flow
solutions for single elliptic cylinders and for the hyperbolic cosine
grains indicated that the streamlines became essentially straight lines
45
1.0
0.3
22
utop
0.8
5.0 \3.0 \ 2.0 \
0
0 1.0
x/a
Dashed Line: cosh bed shape
Figure 13. Distributions of surface hydrodynamic pressure
decreases.
0.5 1.0
Figure 14. Lift coefficient CL.
Lj
5.0 b/a
at distances of two to three times the particle height. Hence,
a solution space was chosen which was four times the height of the
particle in question.
4.4 Calculation of predicted lift by use of
measured logarithmic velocity profile
Figure 15 illustrates the basis for relating the potential
predictions to the actual turbulent flow. The general procedure can
be outlined in a few steps. These steps are cited under the assump
tion that it is established that the turbulent velocity profile is
indeed logarithmic in nature and that measurements have been made.
1. For the relationship (39)
yYb
S= 8.48 + 2.5 In ( + 0.0338)
f k
evaluate vf and k as outlined in Section 3.4. Notice the use of
(y yb) to indicate that this refers to distance above the theo
retical bed.
2. Then, using (39) at the top of the grain, evaluate ut
from the measured data.
3. Calculate the lift by equation (42).
2
u
Unit area: X = CL p (42)
Thus, by relating the velocity profiles, it is possible to
make a prediction of the lift.
Potential
Velocity
Profile
Logarithmic
Velocity
Profile
Theoretical Bed
Figure 15. Theoretical bed used in relating logarithmic and potential velocity profiles.
The discussion above and Figure 15 indicate the case for the
hyperbolic cosine grains and a logarithmic velocity profile, but any
other.bed shape or velocity profile could have been involved.
4.5 Application to experimental results
At the time of the earlier work, the author referred to some
measurements made by Vanoni and Hwang [55] over a part of an alluvial
bed which had been allowed to form twodimensional ripples under flow.
The bed was artificially stabilized and then pressure measurements were
taken. Measurements were used corresponding to Vanoni's C series, with
a flow depth of 0.350 feet above the mean bed.
The same methods as indicated earlier were used, solving first
for potential flow in the solution space indicated in Figure 16. It was
assumed, for ease of solution, that the streamlines had horizontal
tangents at the two ends of the space. The solution also was developed
using the dividing streamline in both cases as a portion of the bound
ary, thus assuming a region of no flow beneath the streamline. Fig
ure 17 indicates the results of the computations, which seem to approx
imate the measurements well.
This chapter was presented to provide a link between the
earlier work and the work of this dissertation,
Flow
I I
Si SI
I DLviding Streamline ,
......... I d
Actual Bed
Figure 16. Definition sketch for experimental application.
0.8
Theory  
Measured
pp
1 2
pu
P um \\
0.4 
S!/
0.0
0 0.2 0.4 0.6 0.8 1.0
Pr: reference pressure Bed Location
u : mean velocity
m
Figure 17. Comparison of theoretical and measured values.
CHAPTER V
THREEDIMENSIONAL NUMERICAL SOLUTIONS
5.1 General
Previously it was indicated that irrotational flow theory would
be utilized to study velocities on a threedimensional bed surface
toward the end of making an analytical evaluation of hydrodynamic lift
on such a surface. The results of some previous similar twodimensional
studies were presented in Chapter IV. In this dissertation emphasis is
placed on a comparison of the analytical results obtained by the writer
with those obtained by Einstein and El Samni [33,34] and Chepil [36].
Evaluation of lift in these cases by the proposed method will require
solution of the potential flow equation for the two arrangements of
hemispheres as indicated in Figures 6 and 7.
A potential flow solution implies the solution of Laplace's
equation (51) in the given flow space.
2 2 2
2+ 2 + 2 = 0 (51)
2 2 6 2
bx by z
Laplace's equation is of elliptic type and hence its solution
is fully determined by conditions on the boundary enclosing the solu
tion space. On the boundary must be specified either values of the
potential function, cp, or its derivative normal to the boundary.
5.2 Problem formulation
5.2.1 Choice of solution method.There are numerous ways of
attempting a solution for Laplace's equation in a given case.
Robertson [56] gives a listing of several such methods, some of which
were considered for the solutions to be made herein.
The method of separation of variables was eliminated because
no convenient coordinate system is available to represent the boundary
involved in these cases. The method of integral equations,in which the
differential equation (51) is expressed instead in integral form and
a solution made,was eliminated because some past works indicated that
threedimensional studies by this means had resulted in exorbitant
computation time on the computer. Another method which has been
employed is the method of integral transforms where the number of var
iables in the equation is reduced to two, and the resulting two
dimensional problem solved by numerical methods. These twodimensional
results are then inverted back to the threedimensional case to obtain
the desired final result. Tranter [57,58] shows some examples of this
approach. This method, however, has not been frequently used, for it
is dependent upon finding the proper transform kernel, which is not
usually possible. The method frequently used to obtain desired body
shapes in a flow is that of the summation of singularities and their
effects, such as sources and sinks distributed throughout the flow
volume. The distributions and relative strength of these singular
ities would be varied until their accumulative effect produced a surface
across which no flow occurred that was sufficiently similar to the
54
hemispherical surface being treated here. While the latter method
showed promise, there were numerous problems involved, such as a choice
of distribution and types of singularities and possible stability
problems in the numerical process required to vary these factors and
approach a solution. Having considered these other alternatives, the
author finally chose the method of finite differences for the solution
since this method enables obtaining the desired accuracy, while, at
the same time, providing a mathematically stable numerical analysis.
The potential flow case being considered is one in which the
depth of flow is essentially infinite. However, the choice of finite
differences method of solution makes treatment of such an infinite flow
space impossible. Therefore, the geometry of flow patterns of other
cases was reviewed with an eye to choosing a depth of flow which would
enable solution of the problem without adversely affecting the desired
results.
5.2.2 Depth of flow space.Work in two dimensions reported in
Chapter IV indicated that a depth of four times the height of the bed
element was adequate to approach the straight streamlines associated
with free stream flow. To further check this, consider the flow around
a single sphere, where, from [29]
Ua3
p =  cos 0 + Ur cos 0 (52)
2r
This is the same case described in Section 2.4.6. The expression for
the tangential velocity component is shown below, along with its value
when = rT/2, or points immediately above the center of the sphere.
3
1 F =p (+U sin 0)(1 + a ) (53)
r 2r
For 9 = n/2,
3
7 = U(l + a) (54)
2r
Equation (54) expresses the horizontal velocity across the
8 = T/2 axis above the center of the sphere. In a free stream such
velocity would simply be U. Hence, the velocity at any distance r can
be compared with U as an indicator of how closely flow at that depth
approaches the free stream. Using (54), it can be seen that the
velocity shown there is only 1 per cent greater than U when r = 3.7a.
Thus, 4a would provide less than 1 per cent deviation.
If (52), representing one plane in an axisymmetric problem,
is converted to Cartesian coordinates x and z for y = 0, (55) is
obtained.
Ua x
a x3 + Ux (55)
Ss3/2
where S =(x2 + z2)(2)(3)
From this, expressions for the x and zvelocity components can be
found, along with the ratio of the latter to the former. If this is
done, and z is chosen as 4a, with x as +a, it is found that the
vertical velocity component there is less than 1 per cent of the
horizontal component. Thus, at this point also, y = 4a gives a good
approximation of free stream flow, with streamlines horizontal.
For these reasons as well as earlier studies, a flow space of depth
four times the hemisphere height was chosen for the solution space
for (51).
Choosing a depth for the flow space assumes streamlines at
that elevation to be horizontal. Thus, at any greater depth than this
the indicated flow lines above that level will also be horizontal, and
the solution along a lower boundary will remain the same. It might
also be noted that the choice of such a solution space eliminates any
consideration of free surface effects, such as wave resistance. Flow
of greater depth relative to particle size may then be superimposed
on the flow space solved for with the knowledge that the solution of
distribution of velocity and pressure along the lower boundary will
remain essentially the same. The error here is determined by how
closely the stream surface at the chosen elevation approaches a flat
plane. It should also be noted that the depth chosen would form an
even better approximation in the cases of more closely packed hemi
spheres being treated here, since the stream surfaces in these cases
more rapidly approach free stream conditions.
5.2.3 Boundary conditions.Due to the periodic nature of the
hemispheres in the solutions, numerous conditions of symmetry are avail
able which enable reduction of the solution to handling a typical
repeated portion of the total flow space. Figures 18 and 19 represent
the portions chosen for the solutions of this dissertation. It can
be seen that the two drawings are different in that Figure 18 could be
divided by "symmetry" one more time. For purposes of the finite
differences solution, however, this was not done. Two alternative
approaches were available, but neither seemed as practical as simply
including two hemispherical portions in the solution space. One of
these alternatives was assuming an equipotential surface between the
two hemispherical portions and making finite differences solutions in
which this assumed potential surface was to be checked and itself
adjusted and treated as a variable in a solution. However, this would
have involved a complete solution of the problem for each assumed sur
face and would have required, therefore, exorbitant computation time.
The other alternative was to choose a plane between the two hemispheres
and use it along with conditions of "symmetry" to reduce by onehalf
the number of points involved in the differences solution. In the case
of Figure 18, the proximity of the hemispheres introduces complexities
which offset the benefits gained in computation time. For the Chepil
case, however, the latter approach could be used easily 'as indicated
in Figure 19.
The constant xplanes passing through the hemispheres are
equipotential surfaces. All other boundaries in the solution spaces
represent surfaces across which there is no flow, with the exception
of the added symmetry boundary shown in Figure 19. This means that the
derivative of the potential function normal to the given surface is
equal to zero. Thus, the following conditions hold.
On constant yplanes through hemispheres:
a = 0 (56)
y
On upper zsurface:
= 0 (57)
az
xconstant plane:
p = p2 = constant
Figure 18. Solution space for closely packed hemispheres.
xconstant plane:
"foldedsymmetry" axis
\
z =4a 
xconstant plane:
cp= p = constant
y f i^^'
Solution space for Chepil's arrangement.
Figure 19.
On lower zsurface:
zp
7z 0
(58)
On hemispherical surfaces:
= 0 (59)
6h
The selection of the symmetry plane for the Chepil case is
based on a type of "folded" symmetry. This is indicated in Figure 20
and equations (510) and (511).
x +xo
1 0
y=0
Constant 2plane
Figure 20. Foldedsymmetry boundary.
K'P'
y = yl
y0
Yo
'PB
YO
x Xo
1 0
PA 'P = ( B P2) (510)
PB = (2 + 1) A (511)
The problem at hand is therefore to solve Laplace's equation
in the regions indicated,subject to the foregoing boundary conditions.
5.3 General finite differences approach
The method of finite differences consists of replacing a function
continuous within a region by its values at certain points in that
region. The equation to be solved is then expressed in difference form
at each of these points, and the problem becomes one of solving a set
of simultaneous equations corresponding to the points chosen. As noted
earlier, one of the reasons for choosing this method for the present
work is that, due to the nature of the elliptic equations, ". . there
is no problem with stability of difference equations approximating
elliptic partial differential equations" [59].
Some consideration was given to the type of coordinate system
to be used. The spherical coordinate system had some advantages in
expressing the noflow condition at one hemispherical boundary, but was
less efficient at other boundaries and at the second hemisphere. Hence,
the ordinary rectangular Cartesian coordinatesx, y, and zwere chosen.
A general sevenpoint scheme was chosen for the difference solution, as
shown in Figure 21, with equal increments of length h in all three
coordinate directions. For this scheme, the difference equation becomes
that of (512), as found in works by Allen [603, Collatz [61), and
Allen and Dennis [62]. Section 5.4 will give details of how this
is obtained.
E (i 6 cp = 0
i=l
6
(p = Cp
S6 i=l i
L=1
(512)
(513)
6
where E pi represents the 6 points around p .
i=l
Figure 21.
Sevenpoint finite difference scheme.
Similarly, if a net, or lattice, were placed over the entire
solution space, an equation could be written for each point of the net
of the same type as (512). Thus, Laplace's equation could be written
in the form of a matrix equation as
AX = B (514)
where A is the matrix of coefficients of cp, X is the matrix containing
p elements, and B is a matrix determined by the boundary conditions.
Therefore, inversion of the matrix A should enable a solution for X,
the potential field. However, A is usually a very large matrix which
has many zero entries. Fox [63, p. 185] states that, "There is no point
in ever evaluating an inverse Ai for the purpose of solving equations
of the form AX = B. Elimination and back substitution or its compact
equivalents are always faster. . On the subject of sparse matrices
(many zero entries) such as occur here, Fox [63, p. 189] says that,
"Iterative methods are used . for matrices of large order but with
many zero coefficients. .. "
The foregoing comments bring to mind the methods often termed
relaxation methods. These procedures were popularized before the rise
of computers by such people as Southwell [64], Shaw [65], and Allen (60].
As digital computers have become more available, variations in the
relaxation processes have led to expressions more applicable to the
computer. That method wherein the iterative methods are applied to
the points of the net singly and in an orderly repetitive fashion,
through numerous iterations, has become known as the Liebmann method
[66], which employs equation (513) when a rectangular Cartesian system
is used.
In order to accelerate convergence of the iterative method, the
process known as overrelaxation [67] is used, employing an overrelax
ation factor, w, and utilizing an equation of the form of (515).
6
cp0 = {i=1 + (1 u) c (515)
where cp' indicates the value from the preceding iteration. Obviously,
0
as in equation (513), the value of cp from (515) will show little
change as convergence is neared.
If the value of w is equal to one, the method is called the
Liebmann method, while the name extrapolated Liebmann is applied for
cases where w is not equal to unity. The latter case will be employed
here. Much work has been done, at least in twodimensional cases, in
finding an optimum w to give most rapid convergence [59]. Approximate
indications are that the optimum wvalue for the cases herein, lies
below about 1.80. However, no great amount of work will be done to
refine this value, as such work could easily involve very extensive
time.
The solutions of this dissertation will therefore be carried
out using a sevenpoint finite difference scheme in rectangular Cartesian
coordinates, applying the extrapolated Liebmann iterative method. The
ensuing sections will develop the needed relationships for use in
special situations.
5.4 Finite differences equations: interior space
This section will present the equations needed to handle all
the different situations which arise in the solutions. While many
applications of finite differences have been made in two dimensions,
few cases other than problems involving simple cubes, and the like,
have been treated in three dimensions. This condition has caused some
new procedures to be employed in the following work.
For clarity, certain conventions will be followed in presenting
the equations and their descriptions. First, each point will be given
three subscripts representing the three coordinate directions, with I
denoting the xdirection, J denoting the ydirection, and z represented
by K. Hence, the subscript I+1 implies the next point in the positive
xdirection beyond a point at I. Additionally, where one or more of
the variables is a constant for the investigations of a special relation
ship, the figure will be drawn in two dimensions, eliminating the third,
constant dimension.
5.4.1 General lattice point.Equation (513) was presented
as representing those cases where six adjacent points are available,
all at a distance h. However, frequently one or more points lie at
some distance other than h. For this reason, the general equation will
be developed for the case of Figure 22, where all six lengths are dif
ferent from h.
z
I,J,K+
y
chsh
I1,J,K 2h / 4h
S3h 1h PI+1,J,K
S6h
J1, ,JKK
I,J Kl
Figure.22. General lattice point.
Difference expressions can be formed for this case.
Forward difference:
S I+1+JK IJ,K
1h
Backward difference:
= II,J,K I1J,K
'2h
(516)
(517)
From these the following equation for the second derivative can be
obtained.
I+1J K I,J,K
1h
YI,J,K Y1,J2K
C2h
(1 +)h
(0l +0'2)h
SI+I,J,K IJK
I (1+ 2 )
SI,J,K I1,J,K
2 (l +2)
Similar expressions can be found for the other directions.
I ,J+1,K I,,K
S(PIJK+1 IJ),K
SQ'( +Q' )
5 5 6
SIJ,K ,J1,K
 I,J,K IJ,K1
6 (5 6)
(520)
(521)
Adding these three expressions yields an approximation for
the Laplacian.
(C) =
^ A &
u^ P
1 2 2
h
2 x 2
(518)
(519)
1h2 2
hr
2 6 2
12 2
2 2
1 2 2 I+1,J,K I,J,K IJK IlJK
h V p= ai(cI+a ) a (a +2)
2 1 +2 2 12
+ IJ+1K I,J,K I,JK I.,J1,K
%3(a%'') 3 (a%+a4)
+ I,J,K+l YIJ,K %IJ,K CI,J,K1 0 (522)
5 (5 6 6 5+6
Since the desire at a given point is to solve for the potential, or cp
value there, the equation above must be solved for cq(I,J,K).
( I+1,J, K +I1,J,K I,J+1,K
+ + +
CIJ = 1 1 1 (523)
al(al+ 2 2()l+(y 2a ) + 3(a'Y3+)
1 1 1
4(043+40 ) +5( 5+6) + 6(Ca5+016)
Note that if all legs are of length h, which implies that all crvalues
are 1.0, equation (523)reduces to (513), as it should.
Equation (523) is applicable for all interior points of the
solution space. The six points may contain among them points that
lie on a regular lattice point, on a hemispherical surface point where
p is being calculated, on or beyond a noflow boundary, on or beyond
the "foldedsymmetry" boundary of Figure 19, or on one of 'the two
equipotential planes at the xextremities of the solution space.
The next few sections will cover these cases as well as the grading of
the lattice. For clarity, the point cPIJ,,K will be termed the object
point, while the six surrounding points will be called adjacent points.
The case for an adjacent point, lying on a regular lattice point or on
a hemispherical surface point where c is being calculated, will be
omitted, as these simply involve the substitution of the present value
of cp for that adjacent point. The same is true when an adjacent point
lies on one of the equipotential planes.
5.4.2 Object point on planar noflow boundary.The term
planar is intended to exclude the hemispherical boundaries, which will
be treated in Section 5.5. Therefore, the planar surfaces indicated
are the constant ybounding planes and the constant zbounding planes,
as illustrated in Figures 18 and 19. If the object point lies on one
or two of such planes, one or more of the adjacent points will lie
outside the solution space, and it must be replaced by an equivalent
expression involving points on the interior of the space. For this
purpose, use is made of the fact that the derivative of cp normal to the
planes is zero. Then the value at such an exterior point can be
expressed as equaling the corresponding point inside the space. Fig
ure 23 illustrates what is meant. Here, as later, CB represents the
cpvalue at a surface point.
Constant zplane Constant zplane
I ,J+1,K
y=o 0'I,J,K y=0 I,J,K
I I
CP= I,J+1,K = B
(a) (b)
Figure 23. Examples for object point on planar boundary.
The preceding requirements can be summarized as follows, where
the equations imply replacing the left member by the right member for
use in equation (523).
For z = 0:
I,J,K = ,J,K+ (524)
For z on upper bound:
I,J,K+1 I=I,J,K1 (525)
For y = 0:
I,J1,K I,J+1,K (526)
For y = a (or y = 3a):
I,J+1,K = YI,J1,K (527)
Note that in the last two equations the terms may represent regular
lattice points, as in Figure 23a, or points on the hemispheres, as
in Figure 23b.
5.4.3 Object point on "foldedsymmetry" boundary.This
problem arises only in the case with hemispheres spaced wider apart,
as in Figure 7. One of the points, in the sevenpoint difference
scheme lies beyond the boundary, but the symmetry of the flow pattern
makes it possible to express this value at this point by a value inside
the given space. Equations (510) and (511) show the expressions
needed. The value thus obtained for this external point is then
inserted into its proper place in the sevenpoint formula.
5.4.4 Adjacent point on hemispherical surface.In some
instances, the lattice point being treated may have as adjacent points
one, two, or as many as three points which lie on a hemispherical bound
ary. At first, the author hoped to expand some of these points in
terms of other values in the field, while still applying the normal
derivative boundary condition (59) to most surface points. It was
found, however, that this approach created stability problems in some
portions of the iterative solutions. Therefore, the application of
the difference equations for (59) was extended to obtain a value for
all adjacent points falling on a hemisphere, with a small number of
exceptions. Discussion of the noflow condition and its use will occur
in ensuing sections.
The single exception occurs when both adjacent ypoints lie on
hemispherical surfaces. In order to obtain values for these two points,
a process which will be called arc'interpolation will be used. This
entails interpolating a value for the point from two surface points
where a value exists. The two points selected lie in the same zplane,
and the interpolation is made by the arc length between the three
points. These lengths along the arc are proportional to the angles
shown in Figure 24, and the ratio of lengths can be replaced by the
ratio of the angles, written in (528).
Constant zplane
ch
Il,J,K 0 I"
S3h
Point 0: cpI,
iPB
PI1,JI,K.
F2
Figure 24. Arc interpolation for surface value.
PIlJK,J1,K I1,J,K
1 2
(528)
A similar expression can be written for the point on the other
hemisphere, and the two values are then inserted into the general
sevenpoint equation.
It would have been possible to use the difference expressions
at these ypoints. It was felt, however, that in the regions where
both points lie on a hemisphere, the accuracy of the arc interpolation
method is of the same order as the expressions which would have to be
used for (59) in such cases. Therefore, arc interpolation was chosen
as a more convenient manner of obtaining values for the points which
were still consistent with the remainder of the potential field.
The evaluation of cp at the six adjacent points has been discussed,
with the exception of surface points to which the boundary condition
equation is applied, which will be covered in Section 5.5 and following.
5.4.5 Graded lattice.The greatest accuracy is desired near
the boundaries to enable velocity calculations on those surfaces. Those
points which are farther away from the hemispheres can be handled,
therefore, with a larger mesh spacing, h. This is more true the
nearer the approach to a free stream condition. An advantage is gained
in computatio, as using a larger lattice cube at some distance can
greatly reduce the number of points at which a solution is sought.
In the present cases, it was decided to use a larger spacing in the
upper onehalf of the solution space. Allen and Dennis [68] used a
scheme similar to that which is shown in Figure 25 and which will be
used here.
First, consideration will be given to those points where the
distances involved are all h or 2h (ac = 2 = 1). The problem can be
considered on three zplanes. First is the plane CDTH. Here, letters
will be used for points rather than the usual convention, as it may
allow greater clarity. For points such as C, D,. H, or J, six adjacent
points exist at a distance 2h; for points like Q, R, S, T, and U, the
six needed points lie at a distance h. Similarly, for all points in
plane EFKL, there are six points 2h away.
Q
S
U
J^
C
.4
R
 .
p
I 1
E
M
1* I
K
Figure 25. Grading the lattice.
D
h
y
1
'
I
I

There are two types of points in the zplane intermediate to
the above two; points such as V and those such as G. For points like V,
the following expression can be quickly written:
2e = 'A + 29V (529)
ox2 h2
The other four needed points lie in plane CDEF and can be utilized,
since the Laplacian is invariant with respect to a rotation of axes,
and thus
.y . + 2D E *F V (530)
by2 az2 (/2 h)2
Summing the two preceding equations leads to the final difference
expression.
2(cpA+pG) + C + D + E+ PF 8yPV = 0 (53i)
For points such as G, a triple McLaurin series expansion [68]
enables a result as below, with neglect of terms 0(h ).
C + D + E + cF + H + TJ + (PK + L. 8(G = 0 (532)
Due to the irrational numbers (square root of three) arising
from the hexagonal patterns of the hemispherical elements,.the succeed
ing planes of constant x treated in the grading process are not always
equidistant. This causes littleproblem for those points where
6 adjacent points are used, as these are treated precisely as indicated
by equation (523), the general equation. However, in the two cases
in the intermediate plane, equations (531) and (532) must be altered.
In the case of points such as V, rewrite (529) for the case where G
is at a distance h and A at a point a2h away.
2h22 c2(l)A +G l  4 (533)
6K2 L2(1+0 ) 2 2
Adding this to (530) and equating to zero produces the desired result.
C D + } 4 1 + p = 0 (534)
Points such as G require more attention. The triple McLaurin
expansion used to obtain (532) is such that, due to symmetry, all
oddorder terms (first derivative, third derivative, and so on) cancel
between the terms. However, an unequal xspacing causes those terms
involving a 6x to remain in the equation, though those oddorder terms
in the other directions still disappear. Here, the 0(h3) terms will
be neglected so that of the following form can be written for cp(x,y,z)
at some point 6x, 6y, and 6z from co.
cp(x,y,z) = Cp + 6x L + 6y . + 6z 6C
2 2 2
(1x)12 J + (6y)2 + 0(z)2 }
x by z
2 2
+ 6x 6y 2! + 8x 6z 62 + 6y 6z 2 (
b bay axaz ayaz
(535)
where all derivatives are taken at point 0. Expansions similar to this
are discussed by Olmsted [69].
The crossderivative terms cancel due to symmetry in directions
other than x, as do the first derivatives except for the xderivative,
which can be approximated by
V1 P (536)
(1+02 )h x G
and the second xderivative, which can be approximated by
1 2 2PV 0p q0G
h 2 =+ ( + (537)
x2 G 0(1+2) 1+2 2
Then (535) yields, when applied successively to all eight of the
adjacent points used by object point G, the following:
~ 2
i 8pG 4h2 (V22 G 2(2 l)h2 62cp
bx
4(a21)
I+ 12 V cP = 0 (538)
where cpi indicates the sum of the eight adjacent points. Setting
the Laplacian equal to zero yields an expression the same as (532),
except for including the last terms in the equation above.
One further point, such as N, is slightly different from V,
though the ideas are similar.
h2 G N1 2N (539)
hy
where (Nl) is the point lying a distance h from N and not shown
in Figure 25.
In plane DEKJ:
2 2 YD + + y + 4N
S, D E K J (540)
ax2 z 2 (,r2 h)2
The resulting difference equation is
2(CG+PN1) + D + CE + CPK + J 8N = 0 (541)
Treatment of points such as N becomes somewhat different when the
xintervals are not equal. A McLaurin expansion will also be applied
in this instance, and,as for point G, the first and second derivatives
will respect to x remain in the difference equation, as shown below.
.2( PG +N) + + (PE + K + J 8N
2
2h(O1)  h2("21) 2 = 0 (542)
2 6x 2 2
x
Evaluation of these two derivatives requires values at a point where
no value is being computed, as seen in Figure 25. Thus, such a
value is obtained by a linear interpolation between two known points,
such as D and E.
The foregoing completes the discussion on graded nets and some
special problems they create. Application of the grading process will
take place away from the hemispherical elements, as a finer lattice
is desired there.
5.5 Finite difference equations: hemispherical boundary
5.5.1 General.The only boundary condition which has not yet
been discussed is the normal derivative condition on the hemispherical
surfaces and expressed in equation (59). The surfaces of the two
hemispherical portions being treated can be expressed by
2 2 2
x + y + z = 1 (543)
(xv3)2 + (y1)2 + z = 1 (544)
Both of these are for spheres of unit radius. Since the gradient of
a scalar function indicates the normal to a surface represented by
that function, the desired derivative is found by taking the gradient
of the above surfaces, yielding
x + y + z + = 0 (545)
ax 6y az
(x) J + (y1) a + z 0 (546)
ax 6y dz
In order to assure conformation to the conditions of the
problem, (545) and (546) must be incorporated into the differences
solution. The surface points encountered will be thus treated, the
only exceptions being those few points mentioned in 5.4.4. The aim
will be to express the first derivatives at the surfaces in difference
form and use.them in equations (545) and (546). The ensuing sections
will describe the formation of these terms.
5.5.2 Xdirection derivative.In this section the general
means for expressing the boundary derivatives will be indicated.
Consider the situation of Figure 26.
PBI CP
II ,,J,K ^I+I,J,K
Constant zplane 1
\I+2,J,K
y
Figure 26. Xderivative condition at boundary.
Indicated in Figure 26 is afictitious cp value located outside the
solution volume. This point will be utilized in the derivative and
eliminated by expressing it in terms of points in the field. The
difference expression for the derivative at the boundary can be written
as (547).
pI Spher
Sphere 1: =4
Sphere 2:
2 h
ax 2a 2h
(547)
A Taylor's expansion can be written for the fictitious point, expand
ing it about the point (I+1). Neglecting terms of order h3 yields
the following.
Sphere 1:
SI+2 B
S= 9I 2 {rh
I+C1 1 (1+al1)h J
1
2h2 1
l 1 I+2 B/c +11
2: 2 L 1 (548)
(1 + 1
Then, from (547) and (548), the derivative is approximated by
12 I
23 1
h B + 2y,+1 + 2i I1+2 (549)
ax 1+011I+1 1+P+ 1+2
1 1
Similarly, on Sphere 2, the fictitious point can be found by expand
ing about the point (11), yielding an equation like (548). Subse
quent substitution into (547) and rearrangement yield
Sphere 2:
3 2 21
=h 7O 0 +B 3 2 21c (550)
+2 B +Q I2 I
Not in all cases, however, do two lattice points exist in a
direction away from the boundary, thus causing a need for some other
means of finding the derivative. Two cases exist: first, the next
field point might lie on the next hemisphere, or it might lie on a
lattice point with the second field point lying on a sphere. In both
instances, the choice here is to use a simple linear expression involv
ing a oneway forward or backward difference rather than a centered
difference. These difference expressions simply involve taking the
difference in value between the two points and dividing it by the
distance between the two.
T+1 B
Sphere 1: h = +
alh
(551)
Sphere 2: = B Yh
5.5.3 Ydirection derivative.For determining the derivative
in the ydirection, the same ideas exist as for the xderivatives.
However, since the points needed for an expansion in the ydirection
generally do not lie at a lattice intersection, special means must be
used to develop these points. For an indication of the problem, see
Figure 27. For simplicity, call the needed points PJ and PJ1. Their
value will be discussed. First, a series expansion can be written for
the fictitious points shown.
Sphere 1:
SPJ1 cp 2 cp + PJ1 2PJ
p = PJ 2h ii 2h + 2 2 (552)
h
Rearranging yields the derivative
Sphere 1:
h 2h =C B  + 2PJ  (553)
Similar expansion of the fictitious point at the second sphere results
in (554).
zconstant
(a)
zconstant
(b)
Figure 27. Yderivatives.
A+1
A1
Sphere 2:
_'_PJ 3PB PJ1
h h 2 =  2PJ + T (554)
The values indicated by PJ and PJ1 must be established by
interpolation, since they do not, in general, lie at a lattice point.
The interpolation will utilize points in the lattice with expansion
to be made about the nearer of two surrounding points in the xdirection,
shown in Figure.27a as PI.,J+1,K and cPI+1,J+1,K( C and PA).
First, consider the case where Ph is less than 1 h. Here, PJ
o
can be expressed as follows, for Sphere 1.
Sphere 1:
B PA+1 'C
PJ = CPA B( +1. C
A A '+{ +
0
CA+1 PC 21 1
+ B2 A (555)
aI + X + B
o
where A: (I+1,J+1,K)
A+l: (I+2,J+1,K)
C: (I,J+1,K)
A similar expression can be written by the other point needed, PJ1,
involving points one increment in the ydirection. In addition, the
same process yields an expression for points located from the second
hemisphere, indicated in Figure 27b.
Sphere 2:
+ + A+
$ +0+1
0
+ B 0 (556)
a + + a
o
where A: (I1,J1,K)
Ai: (I2,J1,K)
C: (I,J1,K)
Again a very similar expansion provides a value for PJ1.
To gain accuracy the expansions used are to be developed about
the point nearest the desired point. In the case where a is less
o
than B, a slightly different approach becomes more convenient due to
the proximity of the spherical surface. For that reason, the point at
a distance a will be expressed by expanding about PJ (or PJ1), and
subsequently an equation obtained for the latter point. This is
illustrated below.
Sphere 1:
9A PJ PJ PC
P = Pa o + 7 (_
0 + +V
o
o % C
PJ"l 0 Cpc + { .CPo (558)
0 a +
Again, in the same manner, an equation can be obtained for the second
sphere.
(+1 =1 1 + (559)
o o o
Both equations (558) and (559) are the same expressions to
be used for PJ1 at either sphere, with cPC and cpA replaced by points
one increment further away from the spheres.
All the immediately preceding equations depend for their
expansions upon the existence of certain lattice points. In some
instances, points such as A+l (or Al) lie on a spherical boundary.
The above equations may be used, however, with a proper value of .
In the case where the points such as A fall on a boundary surface,
a linear interpolation between points A and C will be used. This is
a firstorder approximation, assuming a oneway derivative between
points A and C equal to the difference in their values divided by the
distance between them. The same approach will be used for either PJ
or PJ1 from either sphere where the conditions warrant.
The foregoing equations for cp/6y, (553) and (554), are
based on those cases where two points such as PJ and PJ1 actually exist.
In some cases, they will not both be available, and these cases must be
treated in a different manner. Figure 28 demonstrates the two cases
to be met, shown with reference to obtaining the derivative on the
first sphere.
Constant zplanes
y
\I ,B 2
I CDB 2=PJ
J+1
CPB1
(a) (b)
Figure 28.
Special points for yderivative.
The case in Figure 28a will be treated by first evaluating PJ
by a linear interpolation between the two points (I,J + 1,K) and
(I+1,J+1,K). Then, the value of PJ is used in a forward difference
expression to form
h a PJ OB1
h = 
(560)
In the situation of Figure 28b, the same expression is used,
with PJ being determined by the process of arc interpolation, described
in an earlier section.
One further instance where a special case arises, involves
points near, but not at, y = 0 or y = 1, the two planar noflow bound
aries. Here, the value of PJ1 may correspond to a point on another
hemisphere outside the solution space, such as shown in Figure 23b.
As in that case, PJ1 will equal CB. The equations involving these
SJ+2
+
L

Full Text 
CHAPTER IV
TWODIMENSIONAL WORK: EARLIER RESULTS
4.1 Shapes studied
In earlier work the author has studied some twodimensional bed
shapes. In one case a convenient potential function (that formed from
an infinite row of vortices) was chosen, and the grain shape produced
by this was taken. Here the solution was in a closed mathematical form
In other cases, a series of elliptic cylinders was chosen, and the
solution for the irrotational flow obtained, in terms of the stream
function, by finitedifference methods. Some of the results obtained
and methods used will be noted briefly here as a prelude to the three
dimensional work of this paper.
4.2 General methodsvelocity and pressure results
The potential flow solution for flow around the given shape was
first obtained, using Laplace's equation involving the stream function.
The solution was used to compute velocities along the upper surface of
the shape. Then, applying Bernoulli's' equation for a streamline along
the upper surface streamline, the pressure distribution could be found.
The grain shape obtained from the potential of an infinite series of
vortices is shown in Figure 10, and Figures L1 and 12 show the computed
pressure distributionon these repeating grains. It can be seen that
Figure 11 indicates, in the difference between the upper curve and its
41
125
Along this line, it is interesting to look at the 1.27 cm
/
hemisphere. Use of y^values from (66) yields X^values in fair
agreement with Chepil's X This indicates that a reasonable predic
tion of lift could be made with a proper designation of theoretical
bed. Knowledge of flow in the vicinity of a rough bed will have to be
increased through experimental work to enable a real study of this
i
problem. The agreement of the theory of this dissertation with the
measurements is quite reasonable for the 5.08 cm element. Evidence
indicates that the future work may make it possible to develop this
accuracy for the other situations (1.27 cm and 2.54 cm) also. This '
is desirable, as these two cases are quite representative of many
natural beds subjected to flow.
As a final note in this section, consider the conversion of
the coefficient, to a coefficient representing the lift over the
total bed area, C rather than just under the hemispheres. This will
li
yield a coefficient for equation (577) equal to 0.0408.
6.4 Numerical results for closely packed hemispheres
The hemispherical arrangement considered in this section is
the hexagonal pattern with hemispheres touching. The results of the
numerical differences solution are shown graphically for certain sec
tions through the hemisphere. Pressures and velocities are plotted.
Also included are some plots of equipotential surfaces as they cut
through certain planes.
Integration of pressures over the surface to yield a lift
force gave the following coefficient for equation (577).
141
velocity. Yet he also reports adherence of the flow to a logarithmic
velocity profile, such as equation (32). From such a profile, note
that the ratio of the velocities at two different depths is equal to
a constant, independent of roughness or friction velocity. Hence,
a constant 0.178 for C based on u, should be some different constant
for another reference velocity. However, El Samni indicates otherwise
For example, he shows values for C based on u varying from 0.29 to
lJ t
0.36; equation (32) would indicate 0.308, while (39), the revised
profile, indicates 0.281. Both values are based on El Samni's theo
retical bed located 0.20 k below the tops of the hemispheres. This
inconsistency caused some concern, and some source of the discrepancy
was sought by this author.
Using (32), it can be shown that u^,. = 5.85 v^ and
u. = 4.45 v_. If the 5.85 is held constant, it can be seen that CT
t t 1*
values (based on u^) of 0.29 and 0.36 would indicate u. = 4.58 v_
t t f
and u^ = 4.12 v^. These differ from the 4.45 value by 3 and 8 per
cent. This is merely an indication that the discrepancies may well
have had their basis in the experimental values.
6.5.2 Values of lift for hemisphere bed.The value of
determined by the differences solution is shown in Section 6.4 to be
0.359 based on u^. El Samni has a value of 0.178, based on u^,..
Refer to equation (39) for the velocity profile. Here a careful
examination of the theoretical bed location (point where y = 0) is
in order. El Samni found a point 0.20 k beneath the hemisphere tops
as the distance required to give a straight line plot (with slope v^)
44
dotted counterpart, the buoyant lift, while the remaining pressure
decrease on the upper surface contributes to hydrodynamic lift.
In the experimental work, the actual quantity to be measured
is the difference in piezometric head between upper and lower surfaces.
Since the situation below the grain is hydrostatic, the difference can
be taken between a point on the surface and any point in the region
below.
Results for velocitysquared distributions for the elliptic
cylinders can be found in Figure 13.
4.3 Lift integration
Integrating the vertical pressures over the grain surface will
yield the lift. The lift per unit area, can then be expressed by
x = cl! \ ,1).
where C = lift coefficient
L
u^ = velocity at top of grain = u^
This lift coefficient was found to have a value of 0.500 for the hyper
bolic cosine grain shapes, and the results for the elliptic cylinders .
are found in Figure 14. Note the parameter b/a represents the thickness
tolength ratio of the ellipse. The numerical solution of the Laplace
equation requires a completely bounded solution space. Studies of flow
solutions for single elliptic cylinders and for the hyperbolic cosine
grains indicated that the streamlines became essentially straight lines
Flow
cp =
100 90 80 70 60 50 40 30 20 10 0
Figure 50. Closely packed hemispheres: trace
of some equipotential surfaces in
plane y = 0.
o o o
178
1 PH I ( I, J,K+1) ) /ALSUM +
2 (BETA**2)MPHI( 12, J,K4D 4 PHI(I,J,K+1)/
3
4 1.0/(ALFA72(J,K,D+BETA))/ALSUM
4972 ALSUM = 1.0 + ALFA72(J,K,2 ) + BETA
IF (ALFA72IJ,K,2) .LT. BETA) GO TO 3854
PK1 = PHI(Il,J,K+2) BETAMPHIII2J,K+2)
1 PHI(I,J,K + 2))/ALSUM
2 + (BETA**2)*(PHI(I2,J,K+2) PHI(I,J,K+2)/
3 (ALFA72(J,K,2)+BETA) PHI(11,J,K+2)*(1.0 +
4 1.0/(ALFA72(J,K,2)+BETA)))/ALSUM
GO TC 975
3853 Cl = ALFA72IJ,K,1) + BETA
PK =(PHI(I,J,K+1) ALFA72IJ,K,1 )*
1 (PHKIyJfK+l) PHI(I1,J,K + 1))
2 /C1 (ALFA72(J,K,l)**2)*(PHI(I,J K+D/ALFA72 ( J ,K, 1) +
3 PHI(Il,J,K+l)/BETA)/Cl )/C10 ALFA72(JK,1)/BETA)
IF (K .EG. KL0W1) GO TO 4972
GO TO 972
3854 Cl = ALFA72(J,K,2) + BETA
PK1 =(PH ICI,J,K+2) ALFA72IJ,K,2)*
1 tPHK I, J,K+2) PHH Il,J,K+2) i
2 /Cl (ALFA721J,K,2)**2)(PH I(I,J,K+2)/ALFA72IJ,K,2) +
3 PHI(Il,J,K+2)/BETA)/Cl J/C1.0 ALFA72(J,K,2)/BETA)
GO TO 975
981 S3 = 0.0
S3A = 0.0
GO TO 990
971 BETA = {XXIN IT(J,K+1))/H
PK = PHIt11,J,K+1) + (PH I(I,J,K +1) PHI(I1*JK+1))BETA/
1 (BETA4ALFA72(J,K,1) )
GO TO 972
973 BETA = (XXIN IT(J,K42))/H
PK1 = PHI(11,JK42 ) 4 (PHKI,J,K42) 
1 PHIlI1,J,K42))*BETA/(BETA+ALFA72(J,K,2)i
975 S3 = Z*(PK1/2.0^2.CPK)
S3A = (3 .0 )*Z/2.0
990 SUMI = SI 4 S2 4 S3
SUM2 = S1A 4 S2A 4 S3A
TAM = SUM1/SUM2
IF (ABSITAMPHItI,J,K)).GT.EPS) EPS=ABSITAMPHHI,J,K))
PHI(IJ,K) = OMEGATAM + TAGPHI(I,J,K)
869 RETURN
ENO
SUBROUTINE ZBOUND
C
C SUBROUTINE ZBOUND IS FOR HANDLING THOSE POINTS ON THE HEMI
C SPHERICAL SURFACE ENCOUNTERED AS ONE OF THE ADJACENT ZPOINTS
99
From the preceding it is seen that it will be necessary to
compute the surface velocities and then carry out the integration
cited in equations (575) and (576) to obtain the lift. This step
forms the concluding step, then, to the solution of the flow problem
based on finite differences. The integration will be carried.out
numerically on the computer and will not be discussed here.
*
5.7 Implementation of solution
This section will discuss briefly actual solution of the
problem on the computer, but only insomuch as it relates to the mathe
matical formulation. The actual computer programs with some brief
notes on them can be found in the Appendix.
A value of 1.70 was used for the overrelaxation factor, io,
described earlier. The iterations were begun by using initially poten
tial values as they would exist for a free stream. The value of the
lattice increment, h, used to obtain the solutions Was equal to one
tenth of the hemispherical radius. This lattice increment was used for
the lower onehalf of the solution space, up to twice the hemispherical
radius. The increment was then doubled in size for the remaining Upper
half of the space, using the grading procedure outlined in Section 5.4.5
Convergence of the iterative solution canbe judged in many ways
Frequently the value of the maximum change within the field during each
iteration is taken as a gauge. This value Will decrease as the iter
ations continue, and some minimum value may be chosen beyond which
convergence is assumed sufficient. However, in the present work, the
item of most interest is the distribution of velocities (and hence
24
as here, the predicted drag is zero, obviously incorrect, but the
description of the magnitude of the lift is the primary goal here.
It should be very strongly noted, however, that whereas these earlier
works used the potential to describe the total flow field, a much
different use will be made herein. Essentially, potential flow theory
will be used as a guide in attempting to develop means of predicting
hydrodynamic lift. Success here might lead eventually to better
analytical approaches to some sediment problems, although it will
also be used in larger scale problems dealing with dunes and ripples.
Summarizing, it can be said that the author is relying on the
success of earlier approaches to related studies in an effort to find
a guide to a better understanding of hydrodynamic lift in sediment
transport.
33
3.4 Determination of k and v^ from experimental data
Essentially, in either equation (32) or (39), it is neces
sary to find k and v^ to fully describe the profile in a given
situation. If the distribution is first measured to be logarithmic,
k and may then be determined by considering the velocity at two
different depths [45].
Call the total flow depth d. Consider then a depth pd (p,
a fraction), where the velocity is v Consider also a second depth
qd with velocity v .
k, the equivalent sand roughness
v^, the friction velocity
From these measurements the following can be written, using first
expression (32).
Depth 1:
v ,
Â£=8.48*2.5 In Â£Â£ (312)
vf k
Depth 2:
2. = 8.48 + 2.5 In (313)
v k
o o o
194
IF (AES(TAMPHI(I ,J,K)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PH I (I, J, K ) = OMEGA*TAM TAGPHI(I,J,K)
795 CONTINUE
C THE FOLLOWING STEPS CARRY THE ITERATIVE PROCESS FROM THE
C FIRST ROW WITH ALL SPACINGS OF 2H TO THE UPPER LIMITS OF THE
C SOLUTION SPACE.
K3 = KGRACE +3
DO 799 K=K3KMAX
DO 799 J=ItJMAX2
DO 799 1 = 3,IMIT2 2
KMIT = KMAX 1
C STEPS FOR CONDITION DPHI/DZ = 0 ON UPPER LIMIT OF SOLUTION
C SPACE.
IF (K .LE. KMIT) BOG = PHI(I,J,K4l)
IF (K .EC. KMAX) BOG = PHI(I,J,K1)
IF (J .EC. I .OR. J .EQ. JMAX) GO TO 783
IF ( I .EQ. IMIT2) GC TO 784
C ORDINARY DIFFERENCE FORMULA.
TAM = (PHI(I + 2,J,K) 4 PHI ( 12 J*K) 4 PHI (I J42 K)
1 PH I (I, J2 ,K ) + BOG + PHK I, J,K1) )/6.0
IF (ABS(TAMPHI(I,JK))GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(I,J,K) = OMEGA*TAM 4 TAGPHI (I, J,K)
GO TO 799
783 IF{J .EC. 1) L=J+2
IF (J .EQ. JMAX) L=J2
IF (I .EQ. IMIT2) GO TO 785
C FOR TREATMENT OF DPHI/DY = 0 CONDITION.
TAM = (PH 1(142,J,K) 4 PH I (12 J, K ) 4 BOG 4 PHI(I,J,K1) 4
1 2.0*PHI(I,L,K))/6.0
IF (ABS(TAMPHKI,J,K)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PH I(I,J,K) = CMEGA*T AM 4 TAG*PHI(I,J,K)
GO TO 799
C UNEQUAL XSPACING
784 TAM = (PHIlI2,J,K)/(2.0*BIP) 4 PHI(142,J,K)/(BIP*BI Pi) 4
1 PHI(I,JK1) 4 BOG 4 PH I(I,J42,K) 4 PHI(I,J2,K))/BING
IF (ABS(TAMPHI (I,J,K)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI{I,J,K) = OMEGA*TAM 4 TAG*PHI(IJ,K)
GO TC 799
C COMBINATION OF EFFECTS IN 783 AND 784.
785 TAM = (PHI(I2,J,K)/(2.0*BIP) 4 PHI(142*JK)/(BIP*BIP1) 4
1PHKI,J,K1)4 BOG 4 2.0*PHI(I,L,K)J/BING
IF (ABS(TAMPHI(I,J,K)).GT.EPS) EPS=ABS(TAMPHI{I,J,K))
PHI(I,J,K) = OMEGA*TAM 4 TAG*PHI(I,J,K)
799 CONTINUE
RETURN
END
SUBROUTINE VELOC
C
61
9b = (^ + 9^ 9A (511)
The problem at hand is therefore to solve Laplace's equation
in the regions indicated, subject to the foregoing boundary conditions. t
5.3 General finite differences approach
The method of finite differences consists of replacing a function
continuous within a region by its values at certain points in that
region. The equation to be solved is then expressed in difference form
at each of these points, and the problem becomes one of solving a set
of simultaneous equations corresponding to the points chosen. As noted
earlier, one of the reasons for choosing this method for the present
work is that, due to the nature of the elliptic equations, "... there
is no problem with stability of difference equations approximating
elliptic partial differential equations" [59].
Some consideration was given to the type of coordinate system
to be used. The spherical coordinate system had some advantages in
expressing the noflow condition at one hemispherical boundary, but was
less efficient at other boundaries and at the second hemisphere. Hence,
the ordinary rectangular Cartesian coordinatesx, y, and zwere chosen.
A general sevenpoint scheme was chosen for the difference solution, as
shown in Figure 21, with equal increments of length h in all three
coordinate directions. For this scheme, the difference equation becomes
that of (512), as found in works by Allen [60], Collatz [61], and
o o o
161
DQ 5 J=1,JMAX
5PZ(I,J) = 100.0 ( 1CO.O/SQRT(3.00))*X
SET VALUES FOR KEYED POINT TO BE USED FOR ANY POINT WHERE ALL
ALFA VALUES EQUAL
ALFA(MARK 11) =
ALFA(MARK,2) =
ALFA(MARK* 3) =
ALFAt MARK4) =
ALFA(MARK 5) =
169 RETURN
END
1.000, AS
l.COOCOO
1.000000
l.OOOCCO
1.000000
1.000000
DENOTED BY NO(IJK) = 0
SUBROUTINE ALFBDR
C
C SUBROUTINE ALFBDR IS USED TO CALCULATE INTERPOLATION FACTORS
C TO BE USED TO FINDFIELD VALUES IN TERMS OF NEARBY POINTS IN
C THE LATTICE THE VALUES THUS OBTAINED WILL BE USED IN SERIES
C EXPANSION EXPRESSIONS FOR DERIVATIVES AT THE HEMISPHERICAL
C SURFACES. THIS SUBROUTINE ALSO KEYS THE SURFACE POINTS,
C THROUGH AN ARRAY MIG1(J,K) FOR HEMISPHERE 1 ANO MIG2(J,K) FOR
C THE SECOND. IN THIS ARRAY THE THREE POSSIBLE VALUES HAVE
C THE FOLLOWING MEANINGS. MIGl = 10 TWO FIELD POINTS ARE
C AVAILABLE FOR THE SERIES EXPANSION FOR THE DERIVATIVE IN THAT
C DIRECTION MIG = 20 TWO POINTS ARE AVAILABLE, BUT THE
C SECOND LIES ON THE OTHER HEMISPHERE, AND A LINEAR INTERPCLA
C TION WILL BE EMPLOYED FCR THE VALUE OF THE FIRST POINT
C MIG = 30 ONLY ONE POINT IS AVAILABLE, AND IT LIES ON THE
C OTHER HEMISPHERE, WHERE ARC INTERPOLATION WILL BE USED TO
C OBTAIN A VALUE
C
COMMON ALFAOI21,21,2 ) ,ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETAl(21,21),
3 THETA2(21,21),IBDRYl(21,2l),IBDRY2(21,21),PZ(36,21),
4 TIP(100),TOP{11,11),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEYi,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,B0G,IG1*IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,ZTAG,OMEGAXPBETAKIPKEY,NO(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
CCMMCN IMIG(21),PIG,KIN,KIN1,FL0
C EINSTEINEL SAMNI ARRANGEMENT
INTEGER S
KL0W1 = KLOW 1
KL0W2 = KLOW 2
JMIT = JMAX 1
JMIT2 = JMAX 2s
C INITIALIZE ALL ARRAYS TO BE SET HEREIN
DO 310 J=1,JMAX
DO 310 K=1,KLCW
85
Again, in the same manner, an equation can be obtained for the second
sphere.
Both equations (558) and (559) are the same expressions to
be used for PJ1 at either sphere, with cp^ and cp^ replaced by points
one increment further away from the spheres.
All the immediately preceding equations depend for their
expansions upon the existence of certain lattice points. In some
instances, points such as A+l (or Al) lie on a spherical boundary.
The above equations may be used, however, with a proper value of Q.
In the case where the points such as A fall on a boundary surface,
a linear interpolation between points A and C will be used. This is
a firstorder approximation, assuming a oneway derivative between
points A and C equal to the difference in their values divided by the
distance between them. The same approach will be used for either PJ
or PJ1 from either sphere where the conditions warrant.
The foregoing equations for 3cp/Sy, (553) and (554), are
based on those cases where two points such as PJ and PJ1 actually exist
In some cases, they will not both be available, and these cases must be
treated in a different manner. Figure 28 demonstrates the two cases
to be met, shown with reference to obtaining the derivative on the
first sphere.
Figure 31. Chepil's case: velocities on y = 0.
The analysis begins with a solution of the potential flow over
the two sets of hemispheres, solving here the threedimensional prob
lems by use of finite difference methods. The solutions enabled cal
culation of velocities near the hemispherical surface, relation to
pressures by Bernoulli's equation valid for rotational flow, and sub
sequent integration to find the lift force. The velocity distribution
near the surface (from potential flow) is then linked to the velocity
distribution in the actual flows considered, yielding values for lift
corresponding to the experimental works. This procedure can be used
for any rough bed and velocity field desired.
For the widely spaced hemispheres, theory produces (for three
cases most closely related to the theoretical model studied) values of
lift differing from measured values by 19 per cent, 13 per cent,, and
8 per cent. For the closely packed hemispheres, results from theory
are 16 per cent above measured values. This discrepancy is reduced to
perhaps 10 per cent or less if allowance is made for the sidewall
effects of the narrow flume used in the experimental work. Good
results are also obtained for a natural gravel bed replaced by a bed
of equal hemispheres.
The results show quite good agreement between theory and
experiment. Hence, the goal of this dissertation is accomplished, and
a new analytical tool is found effective in studying lift forces on
a rough bed. It is hoped that the tool will be useful in future work
in the field.
xiv
o o o o o non
165
C ARC INTERPOLATION FACTORS
121 FI Y H
ALFA02(J K,1) = 100
ALFA02(JK2) = 1.00
A = Y ALFA(NK,3)*H
THET A2(J,K) = (ATAN2(Y,XINIT(J,KMATAN2(A,X))/
1(ATAN2(Y,XINIT(J,K))ATAN2(F1,XINITCJl,KJ))
MIG2(J *K) = 2
GO TO 201
C LINEAR INTERPOLATION FACTORS
122 AIB = IBCRY2(J K) 1
ALFA02( J K, 1) = 1.00
ALFA02(J,K,2) = 1.00
IF ( IBDRY2(J,K).LT.IEDRY2(J1,K)) Cl = AIB*H XEND(J,KI
IF ( IBDRY2(J,K).EQ.IBDRY2(J1,K)) Cl = XEND(J1,K>
1 XEND(JK)
IF ((AIBH XINIT(J1K)).GT.H) C2 XEND(J,K) 
1 (A IBl .G)H
IF ((AIB*H XINIT(J~1,K)J.LT.H) C2 = XEN0(J,K) 
1 XINIT(J1K)
CCCN2(J K) = C1/(C1 + C2)
MIG2(J,K) = 3
130 S = 2
201 CONTINUE
200 CONTINUE
RETURN
END
SUBROUTINE OIVIDE
SUBROUTINE DIVIDE CALCULATES THE DIVISORS TO BE USED AT EACH
POINT IN THE LATTICE FOR THE SIXADJACENTPOINT
REPRESENTATION OF THE LAPLACIAN.
COMMON ALFA0(21,21,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72I21t21,2)tXENDl21,21),XINIT(21,21),MIG1(2121),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA1(2121),
3 THETA2(21,21),IBORY1(21,21),IBDRY2(21,21),PZ(36,21),
4 TIP(100)T0P(1111),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2JMI0MARKNCAS,KL0W1IMIDTAB1,TAB2*
6 TAE3,S1,S1A,S2S2A,S3S3A,BING,BIPB0G,IG1,1G2IJK,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,NO(36,21,21)
COMMON DIV(2500),ALFAl2500,5),PHI(36,21,32)
COMMON IMIG(21)PIG,KIN,KIN1,FL0
C EINSTEINEL SAMNI ARRANGEMENT
DIV(MARK) = 3.000000
DO 400 N=1,KIP
DIV(N) = 1.0/(ALFA(N,1)*(ALFA(N,1) f ALFA(N,2)J)
1 + 1.0/(ALFA N,2)*(ALFA(N,1) + ALFA(N,2)))
55
 = ( + U sin 6)(1 +
For 0 = tt/2,
J;
r 50
U(1 + ir)
2r
(53)
(54)
Equation (54) expresses the horizontal velocity across the
0 = tt/2 axis above the center of the sphere. In a free stream such
velocity would simply be U. Hence, the velocity at any distance r can
be compared with U as an indicator of how closely flow at that depth
approaches the free stream. Using (54), it can be seen that the
velocity shown there is only 1 per cent greater than U when r = 3.7a.
Thus, 4a would provide less than 1 per cent deviation.
If (52), representing one plane in an axisymmetric problem,
is converted to Cartesian coordinates x and z for y = 0, (55) is
obtained.
'P = Â¡+Ux <55)
where S = (x2 + z2)(2)(2/3)
From this, expressions for the x and zvelocity components can be
found, along with the ratio of the latter to the former. If this is
done, and z is chosen as 4a, with x as +a, it is found that the
vertical velocity component there is less than 1 per cent of the
horizontal component. Thus, at this point also, y = 4a gives a good
approximation of free stream flow, with streamlines horizontal.
27
of the zero datum presents difficulties. ." [51, p. 2l]. These
essentially are the two problems with use of (32).
3.2.1 Theoretical bed.In a bed in which roughness elements
protrude, some reference datum must be decided upon. This reference
is usually referred to as the theoretical plane bed, and, if sufficient
information is available, its location can be computed. Generally,
satisfactory results are obtained by use of this datum, though problems
may arise for very irregular surfaces.
The basis for computation is to place the bed at such a level
that the water volume contained beneath it equals the volume of rough
ness elements protruding from the bed. Justification for use of this
definition can be gotten by reviewing the work of Einstein and El Samni
[33], Using hemispheres of diameter k and placed in a hexagonal pat
s
tern, they found that for satisfactory results it was necessary to
assume the theoretical bed at a distance of 0.2 kg below a tangent to
the tops of the spheres. Their arrangement appeared as shown in
Figure 6.
Realizing that the area considered (inside the dotted lines)
has dimensions 3ko by \/5" ko, the following is obtained.
(33)
Volume under theoretical bed = 3/73* k y,
s 'b
Volume of particles
Equating these two and solving yields
1 3
=2 n ks
(34)
y, = 0.302 k
Jb s
(35)
80
A Taylors expansion can be written for the fictitious point, expand
3
ing it about the point (1+1). Neglecting terms of order h yields
the following.
Sphere 1:
 vx V fenM
lh j_l_ ^1+2 + C^B/o'1 ^I+l^1*
(548)
2(1+l>
Then, from (547) and (548), the derivative is approximated by
h
&p _
dx
3
1+a^
2cp
1+1
120?
+ cp
1+a^ yI+2
(549)
Similarly, on Sphere 2, the fictitious point can be found by expand
ing about the point (11), yielding an equation like (548). Subse
quent substitution into (547) and rearrangement yield
Sphere 2:
h
3
i*2
b
+
2c*21
l+a2 ^12
29
11
(550)
Not in all cases, however, do two lattice points exist in a
direction away from the boundary, thus causing a need for some other
means of finding the derivative. Two cases exist: first, the next
field point might lie on the next hemisphere, or it might lie on a
lattice point with the second field point lying on a sphere. In both
instances, the choice here is to use a simple linear expression involv
ing a oneway forward or backward difference rather than a centered
23
problem. There have been, however, numerous indications of shape as
a parameter. Rouse defines sphericity [40, p. 777] and indicates it
as a probable factor. Studies have been made on shape influence,
including one by Krumbein [41] which specifically considers nonspherical
particles. Other works also exist for considering the effect of non
spherical particles.
Most design procedures today treat the bed material by conver
sion to a representative bed consisting of uniform spherical particles.
The conversion is generally based on factors such as particle density,
sizes, exposed areas, and volumes. Thus, there are means for relating
natural beds to the uniform spheres to be treated in this work.
2.7 Relation to work of dissertation
The preceding background material was presented in an effort to
point out those factors with special bearing on what the author is try
ing to accomplish in this dissertation. Generally, an attempt will be
made to advance the knowledge of lift through analytical work.
Analysis of the threedimensional cases corresponding to the
work of Einstein and El Samni and Chepil will be made. These multiple
particle systems allow consideration of particle interactions. Poten
tial flow theory will be used to study velocities on the surfaces, these
velocities then being linked to known logarithmic distributions.
Additionally, some twodimensional work and applications will be reported.
Support for the use of irrotational flow is gained from the lift predic
tions made on airfoils, as well as from a knowledge that viscosity is
not necessary to explain lift. Of course, in the airfoil cases as well
53
5.2 Problem formulation
5.2.1 Choice of solution method.There are numerous ways of
attempting a solution for Laplace's equation in a given case.
Robertson [56] gives a listing of several such methods, some of which
were considered for the solutions to be made herein.
The method of separation of variables was eliminated because
no convenient coordinate system is available to represent the boundary
involved in these cases. The method of integral equations, in. which the
differential equation (51) is expressed instead in integral form and
a solution made,was eliminated because some past works indicated that
threedimensional studies by this means had resulted in exorbitant
computation time on the computer. Another method which has been
employed is the method of integral transforms where the number of var
iables in the equation is reduced to two, and the resulting two
dimensional problem solved by numerical methods. These twodimensional
results ar then inverted back to the threedimensional case to obtain
the desired final result. Tranter [57,58] shows some examples of this
approach. This method, however, has not been frequently used, for it
is dependent upon finding the proper transform kernel, which is not
usually possible. The method frequently used to obtain desired body
shapes in a flow is that of the summation of singularities and their
effects, such as sources and sinks distributed throughout the flow
volume. The distributions and relative strength of these singular
ities would be varied until their accumulative effect produced a surface
across which no flow occurred that was sufficiently similar to the
118
Flow
/
Average of Chepil's 2.54 cm radius
front and back values hemisphere
Pressures plotted radially as difference between given point
and uppermost point of hemisphere. Above circle, positive;
below circle, negative.
Figure 39. Measured and theoretical pressure distributions:
Vf = 91 cm/sec.
164
IF (IBDRYKJ,K).GT.IECRY1(J+1K)) Cl = XINIT(J,K) AIB*H
IF (IBDRYKJ,K).EQ.IBDRY1
1 XINIT(J+lfK)
IF ((XENCtJ+1,K) AIB*H).GT.H) C2 = (AIB+1.0)*HXINIT(J,K)
IF ((XENC(J+1,K) AIBH).LT.H) C2 = XENDJ+i*K)XINIT(J,K)
CCDN1(JK) = Cl/lCl + C2)
HIGl(JfK) =3
GO TO 130
C INTERPOLATION FACTORS ON SECOND SPHERE
C COMMENTS SIMILAR TO THOSE FOR FIRST SPHERE
210 ALFA721J,K,1) = 1.00
AH = (
IF (K .GT. KL0W1) ALFA72(J,K,1) = AH
IF (K .GT. KLUW1) GO TO 208
AG = (XEND(J K + l) X)/H
ALFA72 ( J K 1) = AMINK AG AH)
IF (ALFA72(JKtl).GT.1.00) ALFA72(JK1) = 1.00
208 ALFA721JK,2) = 1.00
IF (K .GT. KL0W2) ALFA72(J,K,2) = AH
IF (K .GT. KL0W2) Go TO 209
AG = { XEND(J K+2) X)/H
ALFA72(J K,2) = AMINKAG.AH)
IF (ALFA721J,K,2).GT.l.CO) ALFA72(JK,2) = 1.00
209 BET = 1.0 Z**2.0
BET2 = XENC(J,K)2.C
IF IJ .EG. 2) GO TO 1221
IF ((BET2 + Z**2) .GE. 1.00) Y2 = 0.00
IF ((BET2 + Z**2) .GE. 1.00) GO TO 221
Y2 = SORT(BET BET2)
221 IF (ABSUY2YJ/H) .LT.1.0) GO TO 121
IF (ABSl(Y2YJ/H).GT.1.0 .AND. ABS((Y2YJ/H).LT.2.0)
1 GO TO 122
1221 ALFA02(J K,1) = 1.00
IF (J .LT. 2) ALFA02(JK* 1) = AH
IF U .LT. 2) GO TO 215
AG = (XENO(JlfK) X)/H
ALFAO2(J tK11) = AMINl(AGtAH)
IF (ALFA021J,K,1).GT.1.00) ALFA02(JK1) = 1.00
215 ALFA02(J K,2) = 1.00
IF (J .LT. 3) ALFA021J,K,2) = AH
IF (J .LT. 3) GO TO 115
AG = (XEND(J2 *Ki X)/H
ALFA0 2 ( J K, 2 ) = AMINK AG AH)
IF (ALFA02(J,K,2).GT.1.00) ALFA02(J.K2) 1.00
115 MIG2(JK ) = 1
GO TO 201
231 ALFA72(J K,1) = 1.00
ALFA72(J,K,2) = 1.00
232 ALFA02(JKy1) = 1.00
ALFA02(J,K,2) = 1.00
GO TO 115
45
Dashed Line: cosh bed shape
Figure 13. Distributions of surface hydrodynamic pressure
decreases.
TABLE OF CONTENTS (Continued)
CHAPTER Page
5.5.3 Ydirection derivative 81
5.5.4 Zdirection derivative 87
5.5.5 Adjacent zpoints and ypoints subjected
to normal derivative condition .... 88
5.5.6 Final boundary formulation 91
5.5.7 Singular points 93
5.6 Velocity and lift calculations 94
5.7 Implementation of solution . 99
VI RESULTS AND COMPARISONS 101
6.1 General 101
6.2 Numerical results for Chepil arrangement .... 102
6.3 Comparisons with Chepil's observations . . 110
6.3.1 Details of Chepil's work 110
6.3.2 Comparison of lift forces 116
6.4 Numerical results for closely packed
hemispheres 125
6.5 Comparison with EinsteinEl Samni observations 138
6.5.1 Physical details of experiments ..... 138
6.5.2 Values of lift for hemisphere bed .... 141
6.5.3 Lift on a gravel bed 142
VII CONCLUSIONS AND FUTURE WORK 146
APPENDIX 149
NOTES ON FORTRAN IV COMPUTER PROGRAM 150
NOTES ON EQUIVALENT GRAIN SIZE 200
REFERENCES 201
BIOGRAPHICAL SKETCH 207
v
74
There are two types of points in the zplane intermediate to
the above two; points such as V and those such as G. For points like V,
the following expression can be quickly written:
d2cp ^A^G^V
Sx2 h2
(529)
The other four needed points lie in plane GDEF and can be utilized,
since the Laplacian is invariant with respect to a rotation of axes,
and thus
d2cp [ d2cp
Sy2 5z2 Cv/2 h)2
(530)
Summing the two preceding equations leads to the final difference
expression.
2(CPA+CPG} + + + ^E + ^F 8cf)V = (5_31)
For points such as G, a triple McLaurin series expansion [68]
4
enables a result as below, with neglect of terms 0(h ).
+ 9D + ^E + VF + + + \ + \ 8
Due to the irrational numbers (square root of three) arising
from the hexagonal patterns of the hemispherical elements,.the succeed
ing planes of constant x treated in the grading process are not always
equidistant. This causes little problem for those points where
6 adjacent points are used, as these are treated precisely as indicated
by equation (523), the general equation. However, in the two cases
in the intermediate plane, equations (531) and (532) must be altered.
117
Flow
Chepil's measured values
Values from theory
Average of Chepils
front and back values
Pressure scale:
1" = 100 dynes/cm^
2.54 cm radius
hemisphere
Pressures plotted radially as difference between given point
and uppermost point of hemisphere. Above circle, positive;
below circle, negative.
Figure 38. Measured and theoretical pressure distributions:
v = 68 cm/s ec.
204
42. Prandtl, L. Zeitschrift fur Angewandte Mathematik und Mechanik,
vol. 5, 1925.
43. Taylor, G. I. "The Transport of Vorticity and Heat Through Fluids
in Turbulent Motion," Proceedings of the Royal Society of London,
Series A, vol. 135, 1932.
44. von Karman, T. Gottinger Nachrichten, MathematikPhysical Klasse,
1930, pp. 5876.
45. Christensen, B. A. "Fundamental Hydraulics with an Introduction to
Fluid Mechanics," Unpublished' Preliminary Text* University of
Florida, 1963.
46. Reynolds, 0. "On The Dynamical Theory of Incompressible Viscous
Fluids and the Determination of the Criterion," Philosophical
Transactions, Series A, 1895.
47. Bakhmeteff, B. The Mechanics of Turbulent Flow. Princeton, New
Jersey: Princeton University Press, 1936.
48. Nikuradse, J. "Laws of Turbulent Flow in Smooth Pipes,"
Forschungsheft No. 356, Berlin, 1932.
49. Nikuradse, J. "Laws of Flow in Rough Pipes," National Advisory
Committee for Aeronautics, Technical Memorandum 1292,
Washington, 1950.
50. Colebrook, C. F. "Turbulent Flow in Pipes with Particular Refer
ence to the Transition Region between the Smooth and Rough
Pipe Laws," Journal of the Institution of Civil Engineers,
London, 1939.
51. Hwang, LiSan, and Laursen, Emmett M. "Shear Measurement Technique
for Rough Surfaces," Journal of the Hydraulics Division, Proceed
ings, ASCE, vol. 89, No. HY2, March 1963, pp. 1937.
52. Christensen, B. A. "Turbulent Velocity Distribution Near a Rough
Bed," Under preparation. To be submitted to Journal of the
Hydraulics Division, ASCE, 1968.
53. Sayre, W. W., and Albertson, M. L. "Roughness Spacing in Rigid
Open Channels," Journal of the Hydraulics Division, Proceedings,
ASCE, vol. 87, No. HY3, 1963, pp. 121150.
54. Rouse, Hunter. Fluid Mechanics for Hydraulic Engineers. New York:
Dover Publications, Inc., 1961, 422 pp.
CHAPTER I
INTRODUCTION
One problem'of great practical concern to the hydraulic engineer
is the development of a full knowledge of the transport of material in a
stream. Understanding sediment transport and developing advanced meth
ods in this area are physically and economically important in such prob
lems as scour evaluation and prevention, construction of stable channels
planning and design of reservoirs, and the maintenance of harbor chan
nels. The problem is of true importance when some of the sediment magni
tudes involved are viewed. Brown [lj estimated in 1950 that the aver
age gross sediment production of the United States was on the order of
four billion tons per year. He also indicated that deposition of river
borne sediment in impounding reservoirs was estimated to be equivalent
to a loss each year of sufficient storage capacity to hold the annual
water supply for a city of 250,000 people. On a smaller scale,
Langbein and Leopold [2] note that watersheds composed of fine wind
blown soil, such as in western Iowa, yield as much as 2,000 tons of
sediment per square mile per year. The magnitude of the sediment prob
lem causes a continuing need for better understanding in sediment trans
port. One facet of this understanding must include the mechanism of
Numbers in brackets refer to the References.
1
160
IF (I .GT. IBDRY1(J,1)) GO TO 153
ZINIT = X IN IT(J I )
ALFA(NP 5) = (ZZINIT )/H
IF ( ALFA(NP5) .GT. 1.00) ALFA(NP,5) = 1.00
GO TO 1011
153 ALFA(NP,5) = 1.00
IF (I .EG. IMAX .AND. J .EQ. 1) GO TO 101
135 IF (1.0 (XSQRTO.OO) )**2 (Y1.0)**2) 1011 1011 142
142 ALFAlNP 5) = (Z SQRTI1.0(XSQRTI3.00)) **2
1 (Yl.0)*2 ))/H
IF (ALFA(NP 5) .GT. 1.00) ALFA(NP*5) = 1.00
GO TG 1011
C POINT ON FIRST HEMISPHERE
150 DO 151 L= 1,5
151 ALFA(NP L ) = 1.00
IF (((XEND(J K) X)/H) .LT. 1.00) ALFA(NP,2)=
1 (XEND(J,K)X)/H
GO TO 108
C POINT ON SECOND HEMISPHERE
160 DO 161 L= 15
161 ALFA(NP,L) = 1.00
IF U(X X IN I T( J K ) )/H) .LT. 1.00) ALFA(NP,1)=
1 (XXIN IT(J K))/H
GO TO 107
C CHECK TO SEE IF ANY OF THE ALFA VALUES FOR THE CURRENT POINT
C POINT ARE LESS THAN ONE. IF SO, PROCEED. IF NOT, THEN SET
C THE VALUE OF NO(I,J,K) = 0, THUS KEYING IT AS A POINT WHERE
C ALL ALFA VALUES ARE EQUAL TO 1.000
1011 KIR x 0
DO 1305 L= 15
IF (ALFA(NPL) .LT. 1.0) KIR = KIR + 1
1305 CONTINUE
IF U .EQ. ID1 .OR. I .EQ. ID2) GO TO 101
IF (KIR .EQ. 0) NP = NP 1
IF (KIR .EQ. 0) N0(I,J,K) = 0
101 CONTINUE
GO TO 102
C INITIALIZE PHI(I,J,K) AS UNIFORM STREAM POTENTIAL FOR THOSE
C POINTS ABOVE THE TOPS OF THE HEMISPHERES
110 DO 111 J1=1,JMAX
GO 111 1=1,IMAX
AI = I
X = (AI1.0)*H
IF (I .EQ. IMAX) X = SQRT(3.CO)
111 PHI(I J1,K) = 100.0 (1C0.0/SQRT3.0))*X
102 CONTINUE
C INITIALIZE, AS A UNIFORM STREAM POTENTIAL, THE ARRAY PZ(I,J),
C WHICH IS TO BE USED IN SUBROUTINE ZBOUND.
1306 DO 5 1=1,IMAX
AI = I
X = (AI1.0)*H
17
of 1.32, about 20 per cent below the theoretical. It therefore seems
that above some critical Reynolds number the potential theory becomes
more and more capable of predicting pressures on a body outside the
separation zone. The effect of the Reynolds number involves the change
of laminar boundary layer at the body to turbulent as a critical range
of Reynolds number is reached. The boundary layer, becoming turbulent,
is capable of proceeding further downstream, therefore causing separa
tion to be delayed to points further and further back on the sphere.
58
xconstant plane:
cp = cp^ = constant
Figure 18. Solution space for closely packed hemispheres.
3
This dissertation will concern itself with cases involving a
steady, mean flow with fr.ee surface effects (wave resistance) being
insignificant. A report of some earlier results for twodimensional
bed shapes will be made. Then a numerical solution will be made for
two cases of flow over a bed of hemispherical particles. These solu
tions will be related to experimental measurements obtained elsewhere,
as will some of the twodimensional results.
It is the hope of the author to present herein an analytical
method wherein the hydrodynamic lift, often neglected, can be computed
with a reasonable degree of accuracy. It is also intended to form a
basis for future analytical and experimental work in sediment transport.
173
C CHECK FCR OBTAINING PK BY LINEAR INTERPOLATION WHEN PROXIMITY
C OF TWO HEMISPHERES MAKES SERIES EXPANSION DIFFICULT
IF (XP .GE. XEND(J,K+1)) GO TO 871
BETA = (XPXJ/H
ALSUM = ALFA(NT2,2) 4 ALFA7(J,K,1) + BETA
IF (K .EQ. 1) GO TO 881
IF (J .EQ. 1 .AND. K .EQ. JIPI) GO TO 883
IF (J .EQ. 1 .AND. K .EQ. JIP2) GO TO 883
C CHOOSE NEARER POINT FOR EXPANSION TO FIND PK.
IF
C SERIES EXPANSION TO EVALUATE PK
PK = PH 11I4lfJ,K+1) BETA*! PHK 1+2, J,K+l)
1 PHI(I, J,K+1))/(ALSUM)
2 4 (BETA*2)*(PHI(I+2,J,K+1)/ALFA(NT2,2) 4 PHI(I,J,K4l)/
3 (BETA+ALFA7(J,K,1)) PH I (I 41,J,K+l)*(1.O/ALFACNT2,2)
4 1.0/(ALFA7(J,K,1)+BETA)))/ALSUM
C CHECK FCR LINEAR INTERPOLATION FOR PK1.
872 IF (XP .GE. XEND(J,K+2)) GO TO 873
BETA = (XPXJ/H
ALSUM = ALFA(NT32) 4 ALFA7(J,K,2) + BETA
C CHOOSE NEARER POINT (TO PK1) FOR EXPANSION TO EVALUATE PK1.
IF (ALFA7J,K,2) .LT. BETA) GO TO 2857
PK1 = PHKI + l, J,K + 2) BETAMPHKI + 2, J,K+2) PHI(I,J,K+2))/
1 (ALSUM) 4 (BETA**2)*(PHI(I+2,J,K+2)/ ALFA(NT32) 4
2 PHI(I,J,K+2)/(BETA+ALFA7(J,K,2)) PHI(1 + 1 JK+2)*
3 (1.0/ALFA(NT32) 4 1.0/(ALFA7J,K,2)+BETA)))/ALSUM
GO TO 875
C SERIES 4070 AND 4872 ARE EXACTLY ANALOGOUS TO 870 AND 872
C ABOVE, BUT WERE INCLUDEC AS NO VALUE IS STORED FOR ALFA FOR
C THOSE POINTS ABOVE K = KLOW. THIS VALUE IS THE 1.0 IN THE
C ALSUM EXPRESSIONS.
4870 BETA = (XPXJ/H
ALSUM = 1.0 4 BETA 4 ALFA7(J,K1)
IF (ALFA7(J,K,1) .LT. BETA) GO TO 2855
PK = PHK 141, J,K4l) BETA*(PHI(I42,J,K41) PHI (I J,K+l))
1 /ALSUM 4 (BETA*2)*(PHI(I42,J,K+l) +PHI(I,J,K+1)/
2 (BETA+ALFA7(J,K,1)) PHI (1 + 1,J,K+1)*(1.0 +
3 1.0/(ALFA7(J,K,1)+BETA)))/ALSUM
4872 ALSUM = 1.0 + BETA + ALFA7(J,K,2)
IF (ALFA7(J ,K,2) .LT. BETA) GO TO 2857
PK1 = PH 1(1 + 1,JK + 2) BETA*(PHI(I+2*JK+2) PHI(I,J,K+2))/
1ALSUM + (BETA*2)*(PHI(I+2JK+2) + PHI(I,J,K+2)/
2 (BETA+ALFA7(JK2)) PHI(I+1,J,K+2*(1.0 +
3 1.0/(ALFA7(J,K,2) +BETA)))/ALSUM
GO TO 875
C FOR DPHI/DZ = 0 AT Z = 0.
881 S3 = 0.0
S3A = 0.0
GO TO 890
883 PK = PHK I,J,K + l)
PK 1 = PHK I,J,K+2)
171
851 IF (J .EC. JMAX) GO TO 1851
NT = NO(I+1,J+1,K)
IF (NT .EQ. 0) NT = MARK
XP = AIH
IF (J .EC. 1) GO TO 680
IF (K .EC. 1 .AND. J .EQ. JIPI) GO TO 882
IF (K .EQ. 1 .AND. J .EQ. JIP2) GO TO 882
C CHECK TO SEE IF LINEAR INTERPOLATION, RATHER THAN SERIES
C EVALUATION, SHALL BE MACE FOR PJ.
IF (XP .GE. XEND(J+1K)) GO TO 854
BETA = (XP X )/H
C EXPAND ABOUT NEARER POINT FOR EVALUATING PJ.
IF (ALFAO(J,K,1) .LT. BETA) GO TO 2851
ALSUM = ALFA(NT,2) + BETA + ALFA0(J,K,1)
C PJ IS THE VALUE TO BE USED FOR THE FIRST POINT IN THE SERIES
C EXPANSION USED IN EVALUATION OF DPHI/DY AT THE SURFACE. PJ
C IS ITSELF FOUND FROM OTHER FIELD VALUES BY SERIES EXPANSION.
PJ = PHI ( I+1,J+1, K) BETAMPHK 1+2, J + 1,K) PHI (I,J+1,K) )/
1 ALSUM + (BETA**2)*(PHI(I+2,J+l,K)/ALFA(NT,2) +
2 PHI(I,J+1K)/(BETA+ALFA0(JK,1)) PHI(1+1,J+l,K)*
3 (1.0/ALFA(NT,2) + 1.0/(ALFAO(J,K,1)+BETA)))/ALSUM
IF (J .EC. JMIT) GO TO 1850
856 NT 1 = NO( 1 + 1,J+2,K)
IF (NT1 .EQ. 0) NT1 = MARK
C CHECK.ON USE OF LINEAR INTERPOLATION FOR PJi DUE TO PROXIMITY
C OF THE HEMISPHERES, RATHER THAN USING SERIES EXPANSION.
IF (XP .GE. XEND(J+2,K)) GO TO 857
C EXPAND ABOUT NEARER POINT FOR EVALUATING PJi.
IF (ALFAO(J,K,2) .LT. BETA) GO TO 2853
ALSUM = ALFA(NTl,2) + BETA + ALFA0(J,K,2)
PJI = PHI(1+1J+2,K) BETA*(PHI(1+2,J+2,K) 
1 PH I(I,J + 2K))/ALSUM + (BETA**2)*(PHI(1 + 2,J + 2,K)/
2 ALFA(NT 1,2) + PHI(I,J + 2,K)/(BETA+ALFAO(J,K,2)) 
3 PHI(I+1,J+2,K)*(1.0/ALFA(NTl,2) +
4 1.0/(ALFA0(J,K,2)+BETA)))/ALSUM
8701 S2 = (1.0)*Y*(2.0*PJ PJ1/2.0)
S2A = (3.0)*Y/2.0
GO TO 870
C CASE FOR DPHI/DY = 0 ON Y = 0.
880 S2 = 0.0
S2A = 0.0
GO TO 870
882 PJ = PH I(I,J+l,K)
PJI = PH 1(1,J + 2,K)
GO TO 87C1
C LINEAR INTERPOLATION FOR PJ WHEN PJI EXISTS.
854 BETA = (XEND(J+1,K) X)/H
NTE = NO( IJ+1,K)
IF (NTE .EC. 0) NTE = MARK
PJ = PHI(IJ+1K) + (PHI(I+1,J+1,K) PHI(I,J+1K))*
1 (ALFA(NTE,2)BETA)/ALFA(NTE,2)
o o o o o
174
GO TO 875
C 2855 AND 2857 ARE FOR CASES WHERE PK AND PK1 ARE NEARER TO
C PHI(I) THAN TO PHIU + l).
2855 Cl = ALFA7lJ,K,1) 4 BETA
PK = (P H IC IyJtK+1) 4 ALFA7(J*K1)*(PHI{I+ 11JtK+1)~
1 PHIII,J,K41))/Cl IALFA71J,K,1)**2I*
2 ( PHIU + 1, J,K4l J/BETA 4 PHIUtJK+l)/
3 ALFA7(JK1))/Cl )/<1.0 ALFA7IJ,K,1)/BETAJ
IF (K .EG. KLOW1) GO TO 4872
GO TO 872
2857 Cl = ALFA7(J K 2) 4 BETA
PK1 = (PHIU.J.K42) 4 ALFA7IJ,K,2)*IPHI(I41,JfK42)
1 PHI(I,J,K42 ) )/C1 IALFA7IJ,K2)**2)*
2 IPHII141,J,K42)/BETA 4 PHI(I,J,K+2)/
3 ALFA7IJ,K,2))/C1 )/11.0 ALFA7{J,K,2)/BETA)
GO TO 875
C LINEAR INTERPOLATION FOR PK.
871 BETA = (XEND(J,K41) X)/H
PK = PH I(11J K41) 4 {PHI(l4ltJ,K4l) PHI(I,J,K4l))*
1 ALFA7(J,K,1)/(ALFA7(J,K,1J4BETA)
GO TO 872
C LINEAR INTERPOLATION FOR PK1.
873 BETA = {XEND(J,K42) X)/H
PK 1 = PHIU,J,K42) 4 (PHH I41,J,K42) PHIII t JK42)) *
1 ALFA7IJ,K,2)/IALFA7IJ,K,2)4BETA)
875 S3 = (2.0*PK PK1/2.0)*(1.C)*Z
S3A = (3 .0 )*Z/2.0
C THE S TERMS REPRESENT THE CONTRIBUTION TO DPHI/DN = 0 FROM THE
C THREE DIRECTIONAL PHIDERIVATIVES THE TERMS OF THE FORM SN
C ( N AN INTEGER ) REPRESENT ALL TERMS BUT THOSE CONTAINING
PHIB, THE BOUNDARY PHI VALUE. THE TERMS OF FORM SNA ARE THE
C COEFFICIENTS FOR PHIB. N=10 X, N=20 Y* N=30 Z.
890 SUMI = SI 4 S2 4 S3
IF INCAS .EQ. 3) GO TO 869
SUM2 = S1A 4 S2A 4 S3A
TAM = SUM1/SUM2
IF (ABSITAMPHIII,J,KH.GT.EPS) EPS=ABSITAMPHIIIJK))
PHIII,J,K) = OMEGA*TAM 4 TAG*PHI11,J,K)
GO TO 869
C COMPUTATIONS FOR BOUNDARY SURFACE POINTS ON SECOND HEMISPHERE.
9950 CALL BDDER2
869 RETURN
END
SUBROUTINE BDDER2 _
SUBROUTINE BDDER2 HAS THE FUNCTION OF APPLYING THE NORMAL DERI
VATIVE
C BOUNDARY CONDITION FOR THOSE POINTS LYING ON THE SECOND HEMISP
32
In both cases, the derivative approaches zero for large
yvalues, as it should. However, at the wall, a definite discontin
uity exists by the former method. The new proposal predicts a finite
change at the bed, a far more reasonable development. The author
feels that the proposed distribution, with its continuous curve, will
enable better studies of action near a rough bed.
3.3.2 Comparison of distribution with increasing y.The
increasing value of (yA) will eventually negate the effect of the
added term in (39). Table 1 shows the difference in velocity indi
cated' by (32) and (39) for some (yA) values. The difference in
hydrodynamic force, proportional to the velocity squared, is also
shown. It can be seen that the effect of the added term in (39) is
dissipated very rapidly.
TABLE 1
COMPARISON OF PROPOSED AND FORMER DISTRIBUTIONS
Variation between
(32) and (39)
yA
Force
Variation
1%
0.99
2%
3%
0.61
6%
5%
0.29
10%
142
of u versus log (y/k). However, this is based on a velocity profile
such as (32). Use of (39) would require a straight line plot of u
versus log (y/k + 0.0338). Hence, the measured values would imply a
different theoretical bed when extrapolated according to (39). This
value would be 0.20 k 0.0338 k = 0.1662 k, still only 16 per cent
below the value 0.198 calculated in Chapter III. This new, theoretical
bed will be used to remain consistent with the actual velocity data
El Samni obtained.
2
Use of (39), then, yields (u,c/u.) as 1.73. Note that this
step relates the velocity at the top of the grain, and hence the veloc
ities and pressures over its surface, to the flow given by the velocity
profile. This is quite similar to Figure 15. Applying the ratio 1.73
to (69) yields C = 0.207 based on u._, only 16 per cent above
El Samni's 0.178. If the 0.189 value based on the sidewall analysis of
Section 6.5.1 is used as a norm, the theoretical value is less than
10 per cent greater. In either case, the agreement between theory and
measurement is quite good. This result in itself shows great promise
for the hypothesis of this dissertationthat potential flow theory can
be used as a guide to predict hydrodynamic lift.
6.5.3 Lift on a gravel bed.El Samni also made some lift
measurements for a bed composed of gravel. He then attempted to relate
his findings to those for the bed of plastic hemispheres. Here the
author will try to relate his theoretical results to the gravel bed.
The theoretical bed was found by extrapolation to zero velocity
to be located 0.040 feet below the toppoints of the gravel. Also,
140
Figure 54. Velocity suppression near sidewall.
This effect is in addition to the possible total suppression
of the velocity distribution by the sidewalls, causing u^ at the center
line to be less than that for a wider flume. El Samni gives some pos
sible indication of this by his finding a coefficient for (32) of
8.40 rather than 8.48. Even if this were assumed as totally due to
sidewalls, the influence is only about 2 per cent.
One other feature of_El Samni's work is his conclusion that
the value of CT was constant for all flows only when the velocity at
li
0.35 diameters above the theoretical bed (u^,.) was used as a reference
16
where
yielding
dA = 2a^ sin^6 d6
27 2 U
F = a TT o TT
v 16 H 2
(210)
Note that if this was expressed per unit area (based on the area
2
na ), with U replaced by u^_ = (3/2)U, the velocity at the top of
the sphere, theory would show
2
(211)
3 Ut
F = i. *
4 2
where F^ = vertical force per unit area
Measurements were made by Flachsbart [30], as shown in
Schlichting [3l], The results were very similar to those by Fage [32]
shown below. The measured values are shown below with the theoretical
curve. The values are plotted with reference to the stagnation pres
12
sure Pj + pU where p^ is the pressure at where the velocity
equals U. This stagnation pressure is the pQ being used here. Replac
ing Pj, by pQ yields
(212)
where j3 is the value shown in the graph. The expression thus found
for (p pQ) can be integrated numerically.
In the case of the higher Reynolds number, the force can be
expressed as (210) with a coefficient of 1.64, only about 3 per cent
below the theoretical. The lower Reynolds number produces a coefficient
105
0.0 0.2 0.4 0.6 0.8 1.0
x/a
Figure 33. Chepil's case: velocities on y 0.4a.
86
Constant zplanes
(b)
Figure 28. Special points for yderivative
The case in Figure 28a will be treated by first evaluating PJ
by a linear interpolation between the two points (I ,J + 1,K) and
(I+1,J+1,K). Then, the value of PJ is used in a forward difference
expression to form
h
dcp
dy ~
pj 9,
B1
0i_
(560)
In the situation of Figure 28b, the same expression is used,
with PJ being determined by the process of arc interpolation, described
in an earlier section.
One further instance where a special case arises, involves
points near, but not at, y = 0 or y = 1, the two planar noflow bound
aries. Here, the value of PJ1 may correspond to a point on another
hemisphere outside the solution space, such as shown in Figure 23b.
As in that case, PJ1 will equal 9 The equations involving these
D
26
The socalled von Karman constant, h, has been verified
experimentally, as noted by Bakhmeteff [47], to equal 0.40. The
values for B were obtained from experiments by Nikuradse [48,49],
who made tests on smooth pipes and then on pipes roughened by gluing
uniform sand grains of size k to the wall. Results in the transition
range between smooth and rough flow regimes by Colebrook [50] showed
different Bvalues for nonuniform roughness.
Hydrodynamically smooth and rough flow regimes are generally
defined in terms of the wall Reynolds number (v_k/v), R The smooth
f ew
range, R&w less than 3.5 to 5, has a viscous sublayer of depth compar
able to bed roughness size and viscous forces play a dominant role.
In the rough range, greater than 70, the roughness elements have
fully penetrated the sublayer, and viscous effects are negligible.
Most sediment problems in nature occur where flow is in the
rough range. This will be the range considered in the present work.
Substitution of the appropriate values for the rough range yields
= 8.48 2.5 In f (32)
vÂ£ k
Although this expression was derived from work in circular
pipes, it has been noted on many occasions that it is similarly .
applicable in open channels.
3,2 Special problems of present expressions
Working with measurements near rough surfaces, Hwang and
Laursen noted that ". . the adequacy of this logarithmic equation
can be debated, especially near the boundary, and the indeterminacy
69
Constant
zplane
^.J+ljK
o
II
>>
*^1, J, K
/
,9
/
(a)
Constant zplane
Figure 23. Examples for object point on planar boundary.
The preceding requirements can be summarized as follows, where
the equations imply replacing the left member by the right member for
use in equation (523).
For z = 0:
^1, J,K1 ~ ^IjJ^+l
(524)
For z on upper bound:
,J,K+1 = ^IjJjKl
For y = 0:
^I.Jl.K = ^1,J+1,K
For y = a (or y = 3a):
CpI,J+l,K = ^IjJljK
(525)
(526)
(527)
Note that in the last two equations the terms may represent regular
lattice points, as in Figure 23a, or points on the hemispheres, as
in Figure 23b.
LIST OF SYMBOLS
sphere (or circular cylinder) radius
coefficient in logarithmic velocity profile
lift coefficient based on total bed area
lift coefficient based on projected area of grain
grain diameter
equivalent grain diameter
projected elemental area in xyplane
elemental surface area on hemisphere
acceleration due to gravity
full lattice increment
point index, xdirection
point index, ydirection
point index, zdirection
equivalent sand roughness
diameter of hemispheres
sidewall equivalent sand roughness
bottom equivalent sand roughness
hydrodynamic lift force
pressure
reference pressure, defined as used
reference pressure, defined as used
field point in xdirection
113
then a recomputation of y including also the volume of the upper
gravel layer in the calculations. Assume the gravel layer to be
spread out as a layer of tightly packed hemispheres of diameter 0.5 cm.
This was chosen as representative of the range of gravel sizes, 0.2 to
0.64 cm. It is impossible to select the most representative size,
since no gravel analysis is available. Consider an area containing
onequarter of a larger hemisphere and measuring 3a by 3*/5 a/2.
n = Gf) = 1Sn/3 a2 (63)
n = number of gravel grains of diameter 0.5 cm
Vg = volume of gravel = (){j Tr(.25)^}*n (64)
14 3
= volume hemisphere = OgHy tt a ) (65)
Combining the volumes and dividing by the area provides a relation
ship for y the distance to the theoretical bed.
yQ = 0.131 + 0.0670a (66)
a, y in cm
Valid only for gravel of 0.5 cm diameter
Evaluation of yQ by this means yields values of 0.216 cm,
0.301 cm, and 0.471 cm for the respective cases. The latter two
values agree very well with Chepil's values, indicating a sound basis
for this approach. The first value is some 40 per cent to 75 per cent
higher depending on which yQ is used. This difference could be due
120
F low
Chepil's measured values
Values from theory
Pressure scale:
2
1 = 400 dynes/cm
Average of Chepil's
front and back values
2.54 cm radius
hemisphere
Pressures plotted radially as difference between given point
and uppermost point of hemisphere. Above circle, positive;
below circle, negative.
Figure 41. Measured and theoretical pressure distributions:
v^ = 159 cm/sec.
78
5.5 Finite difference equations: hemispherical boundary
5.5.1 General.The only boundary condition which has not yet
been discussed is the normal derivative condition on the hemispherical
surfaces and expressed in equation (59). The surfaces of the two
hemispherical portions being treated can be expressed by
x + y + z =1 (543)
(xV3)2 + (y1)2 + z2 = 1 (544)
Both of these are for spheres of unit radius. Since the gradient of
a scalar function indicates the normal to a surface represented by
that function, the desired derivative is found by taking the gradient
of the above surfaces, yielding
xg+yÂ£.z^=0 (545)
ox ay oz
(xa/3) + (y1) Â£ + z Â£ = 0 (546)
In order to assure conformation to the conditions of the
problem, (545) and (546) must be incorporated into the differences
solution. The surface points encountered will be thus treated, the
only exceptions being those few points mentioned in 5.4.4. The aim
will be to express the first derivatives at the surfaces in difference
form and use them in equations (545) and (546). The ensuing sections
will describe the formation of these terms.
49
The discussion above and Figure 15 indicate the case for the
hyperbolic cosine grains and a logarithmic velocity profile, but any
other, bed shape or velocity profile could have been involved.
4.5 Application to experimental results
At the time of the earlier work, the author referred to some
measurements made by Vanoni and Hwang [55] over a part of an alluvial
bed which had been allowed to form twodimensional ripples under flow.
The bed was artificially stabilized and then pressure measurements were
taken. Measurements were used corresponding to Vanonis C series, with
a flow depth of 0.350 feet above the mean bed.
The same methods as indicated earlier were used, solving first
for potential flow in the solution space indicated in Figure 16. It was
assumed, for ease of solution, that the streamlines had horizontal
tangents at the two ends of the space. The solution also was developed
using the dividing streamline in both cases as a portion of the bound
ary, thus assuming a region of no flow beneath the streamline. Fig
ure 17 indicates the results of the computations, which seem to approx
imate the measurements well.
This chapter was presented to provide a link between the
earlier work and the work of this dissertation*
59
xconstant plane:
"foldedsymmetry" axis
Figure 19. Solution space for Chepil's arrangement.
37
The following expressions can be written relating the shear
stresses.
tan Qi
(316)
0 h + t2 q(W h tan O') = yh WS^
This can be written as
(317)
T1.0 T2.0(h tan 1
ws.
(318)
Equation (316) results from assuming and t^ to both be
proportional to. their respective yvalues in the same way. The other
expressions merely equate the weight of fluid acting on the wetted
perimeter and the shear forces resisting it.
Obviously the value of tan Qi is of primary importance in this
analysis of sidewall effect. The equation enabling evaluation of the
angle comes from equating expressions for the velocity at a point such
as A in Figure 8, computed first with reference to the channel bottom
and then with reference to the side. For this analysis the form (32)
will be used for the velocity.
yn /Tn j
{8.48 + 2.5 In [ri]) V = {8.48 + 2.5 In [ .. .
1 kj J P k2" p
Rewriting this and using the similar proportionality of
and T2 to y^ and yields
(319)
y y
[In 29.7 ] = [In 29.7 ^
(320)
LIST OF TABLES
Table Page
1 Comparison of Proposed and Former Distributions ..... 32
2 Chepil's Experimental Data 112
3 for Chepil's Work 114
4 RoughnessGrain Size Ratios 115
5 Comparison of Theoretical with Chepil's Work 123
vi
104
Chepil's case: velocities on y = 0.2a.
Figure 32.
205
55. Vanoni, V., and Hwang, L. S. "Relation Between Bed Forms and
Friction in Streams," Journal oÂ£ the Hydraulics Division,
Proceedings, ASCE, vol. 93, No. HY3, May 1967, pp. 121144.
56. Robertson, J. M. Hydrodynamics in Theory and Application.
Englewood Cliffs, New Jersey: PrenticeHall, Inc., 1965, 652 pp.
57. Tranter, C. J. "The Combined Use of Relaxation Methods and FoUfier
Transforms in the Solution of Some ThreeDimensional Boundary
Value Problems," Quarterly Journal of Mechanics and Applied
Mathematics, vol. 1, 1948, pp. 281286.
58^ Tranter, C. J. Integral Transforms in Mathematical Physics.
New York: John Wiley and Sons, Inc., 1966, 139 pp.
59. Dingle, A. N., and Young, Charles. Computer Applications in the
Atmospheric Sciences. College of Engineering* University of
Michigan, 1965, 256 pp.
60. Allen, D. N. deG. Relaxation Methods in Engineering and Science.
New York: McGrawHill, Inc., 1954, 257 pp.
61. Collatz, Luther. The Numerical Treatment of Differential Equations
New York: SpringerVerlag New York, Inc., 1966, 568 pp.
62. Allen, D. N. deG., and Dennis, S. C. R. "The Application of
Relaxation Methods to the Solution of Differential Equations
in Three Dimensions. I. Boundary Value Problems," Quarterly
Journal of Mechanics and Applied Mathematics, vol. 4, 1951,
pp. 199208.
63. Fox, L. An Introduction to Numerical Linear Algebra. New York:
Oxford University Press, 1965, 327 pp.
64. Southwell, R. V. Relaxation Methods in Theoretical Physics.
Oxford University Press, 1946.
65. Shaw, F. S. An Introduction to Relaxation Methods. New York:
Dover Publications, Inc., 1953, 396 pp.
66. Fox, L. Numerical Solution or Ordinary and Partial Differential
Equations. Reading, Massachusetts: AddisonWesley Publishing
Company, Inc., 1967, 509 pp.
67. Forsythe, G. E., and Wasow, W. R. FiniteDifference Methods for
Partial Differential Equations. New York: John Wiley and Sons,
Inc., 1960, 444 pp.
187
NT = NO(12 J K)
IF (NTEG 0) NT = NARK
IF (ALFA{NT4) .LT. 1.0) PB1 = PYl(I2tK)
PIl = PHI(I2JK) (ALFA(Nf4)/ALFA(NT4))*
l (P81PHK I2J*K) )
362 PB = PHH ItJ+lfK + 1)
NB = N0(ItJ,K+1)
IF (NB .EC. 0) NB = PARK
IF (ALFA(NB*4) .LT. 1.0) PB = PY1(I,K+1)
PK = PHI( 11Jt K + l) +(ALFA(N,4)/ALFA(NB,4))(PBPHI(IfJfK+1))
PB1 = PH I(If J +1,K + 2)
IF (K .EC. KL0W1) NB = MARK
IF (K .EG. KL0W1) GO TO 365
NB = N0(IfJfK+2)
IF (NB .EQ. 0) NB = MARK
IF (ALFA(NB f4) .LT. 1.0) PB1= PYl(ltK+2)
365 PK1 = PHI(I,J,K+2)+(ALFA(N,4)/ALFA(NB,4))*
1 (PB1PH I (It JtK + 2))
IF (MULE .EQ. 1) GO TO 1362
SI = (l.C)*(XSQRT(3.0))*(PIl/2.0 2.0*PI)
SiA = (XSQRT(3.0))*(3.0/2.0)
1362 S3 = Z*(PK1/2.0 PK2.0)
S3A = (3.0 )*Z/2.0
GO TO 378
1379 S2 = 2.0*(Y+ALFA(N,4)*H1.0)*PHI(I,J,K)/ALFA(Nf4)
S2A = 2.0*(Y+ALFA(Nf4)*H1.0)/ALFA(N,4)
GO TO 1378
C CNE POINT AVAILABLE IN XDIRECTION.
363 SI = (XSQRT13.00))*PI
SIA = X SQRT(3.00)
MULE =1
GO TO 362
C ARC INTERPOLATION FOR XPOINT.
364 XT = SQRT(1.0 Z**2 (Y+ALFA(N,4)*H)**2)
17 = IBDRY1(J + 1 K)
A = Y + H
B =Y + ALFA(N,4)*H
THM = (ATAN2(B,XT) ATAN2(Y,XINIT(JtK)))/
1 (ATAN2(A,XINIT(J+l.K)) ATAN2(YfXlNIT(J,K)))
PI = PHI(IlfJ,K) + THM*(PHI(I7J+1,K) PHI(I1,J,K))
XDEL = (X XTJ/H
SI = (XSQRT(3.00))*PI/X0EL
SIA = (XSQRT(3.00))/XDEL
MULE = 1
GO TO 362
378 SUMI = SI + S2 + S3
SUM2 = SIA + S2A + S3A
C FINAL CALCULATION OF A NEW VALUE FOR PHI AT THE BOUNDARY
C SURFACE POINT.
IF {IK ID .EQ. 2) PYl(IfK) = (SUM1/SUM2)*0MEGA +TAG*PY1(I,K)
IF (IK ID .EQ. 2) BING = PY1(I,K)
Flow
O o
Figure 7. Arrangement of spheres and theoretical bed
in Chepil's work.
47
at distances of two to three times the particle height; Hence,
a solution space was chosen which was four times the height of the
particle in question.
4.4 Calculation of predicted lift by use of
measured logarithmic velocity profile
Figure 15 illustrates the basis for relating the potential
predictions to the actual turbulent flow. The general procedure can
be outlined in a few steps. These steps are cited under the assump
tion that it is established that the turbulent velocity profile is
indeed logarithmic in nature and that measurements have been made.
1. For the relationship (39)
yy
= 8.48 + 2.5 In ( 0.0338)
vÂ£ k
evaluate v^ and k as outlined in Section 3.4. Notice the use of
(y y^) to indicate that this refers to distance above the theo
retical bed.
2. Then, using (39) at the top of the grain, evaluate u^
from the measured data.
3. Calculate the lift by equation (42).
2
ut
Unit area: h = C p r (42)
Li Z
Thus, by relating the velocity profiles, it is possible to
make a prediction of the lift.
43
Figure 11. Pressure distribution on twodimensional grain.
Figure 12. Piezomettfic head distribution on twodimensional grain.
63
AX = B (514)
where A is the matrix of coefficients of cp, X is the matrix containing
9 elements, and B is a matrix determined by the boundary conditions.
Therefore, inversion of the matrix A should enable a solution for X,
the potential field. However, A is usually a very large matrix which
i
has many zero entries. Fox [63, p. 185] states that, "There is no point
in ever evaluating an inverse A1 for the purpose of solving equations
of the form AX = B. Elimination and back substitution or its compact
equivalents are always faster. . ." On the subject of sparse matrices
(many zero entries) such as occur here, Fox [63, p. 189] says that,
"Iterative methods are used . for matrices of large order but with
many zero coefficients. ..."
The foregoing comments bring to mind the methods often termed
relaxation methods. These procedures were popularized before the rise
of computers by such people as Southwell [64], Shaw [65], and Allen [60]
As digital computers have become more available, variations in the
relaxation processes have led to expressions more applicable to the
computer. That method wherein the iterative methods are applied to
the points of the net singly and in an orderly repetitive fashion,
through numerous iterations, has become known as the Liebmann method
[66], which employs equation (513) when a rectangular Cartesian system
is used.
In order to accelerate convergence of the iterative method, the
process known as overrelaxation [67] is used, employing an overrelax
ation factor, u), and utilizing an equation of the form of (515).
143
a lift coefficient of 0.178 was found to apply, based on a velocity at
a distance 0.058 feet above this theoretical bed. El Samni considered
the first as 1/5 the representative grain diameter and the second as
0.35 of the diameter. This gave values of 0.200 feet and 0.166 feet.
The gravel analysis showed these to equal dg^ and d^, where the
subscript indicates per cent finer by weight. El Samni also justified
the value of 0.20 feet by noting that this value gave him B = 8.40 for
a profile similar to (32). This value of B was the average he found
for runs with the plastic hemispheres. However, the value used as k
in (32) is the roughness, which does not necessarily equal the grain
size except in a case such as the closely packed hemispheres or
Nikuradse's sand roughness. Schlichting [31] was one of the early
experimenters in this area, conducting extensive tests showing roughness
dependent on size, shape, spacing, and arrangement of the roughness
elements. Koloseus [72] reports additional work, while Rouse [73] also
reviews work in the area. All these results verify that the value of
the representative grain diameter might differ from the roughness, or
0.20 feet here.
This author would like to replace the upper layer of gravel
by an equivalent hemispherical bed, the theoretical bed of which is
at the same plane as that of the gravel. Then, the predicted lift on
this equivalent bed can be compared with the measured lift. The equiv
alent diameter, de> for this purpose, would ordinarily be obtained
from the grainsize distribution of the bed material. Here only the
d^g and dg^ ^ sizes are available. An approximation will be made by
assuming the grainsize curve to be a straight line passing through
196
C TRAVERSE IN XDIRECTION ALONG THE GIVEN YSECTION.
DO
650
11 = 1,
IT
All
=
11
XI
= (
All1.0)
H
JIP
1 =
15
IF
(JMAX .EQ.
11)
JIPI
= 7
IF
UMAX .EQ.
21)
JIPI
= 13
JIP2 =
15
IF
UMAX .EQ.
11)
JIP2
= 9
IF
UMAX .EQ.
21)
JIP2
= 17
IF
( 11
.EG. 1
.AND
. J
.EQ.
1
)
I
= 11
IF
(11
.EG. 1
.AND
. J
.EG.
1
)
GO
TO
1650
IF
(11
.EG. 1
.AND
. J
.EQ.
J
I
PI)
I =
11
IF
(11
.EC. 1
AND
. J
.EQ.
J
I
PI)
GO
TO 1650
IF
(11
.EQ. 1
.AND
. J
.EQ.
J
I
P2)
I =
11
IF
(11
.EC. 1
.AND
. J
.EQ.
J
IP2)
GO
TO 1650
IF
(II
.EQ. 1)
GO
TO 650
645 IF
(Z
.LT. 0.6
) GO
TO
1645
IF
(XI
.LT. XINITUtK
D) GO
TO
641
C CALCULATION GF VELOCITIES FOR POINTS HANOLED BY SUBROUTINE
C BDDER.
1645 KEY = 1
I = IBDRYK J Kl)
AI = I
K = Kl
AK = K
Z = (AK1.0)*H
X = XINIT(JyK)
IF (K .EG. 1) GO TO 639
N = NG(IJK)
CALL BDDER
L = L + l
PIG = PHI { I yJ K)
GO TO 642
C CALCULATION OF VELOCITIES FOR SURFACE POINTS HANDLEO BY
C SUBROUTINE ZBOUND.
641 I = II
AI = I '
X = X1
IF (I .EG. JIPI .AND. J .EQ. 1) GO TO 650
IF (I .EQ. JIP2 .AND. J .EQ. 1) GO TO 1645
AK = K
Z = I AK1.0)*H
N = NC(IyJyK)
KEY = 0
CALL ZBOUND
L = L + l
PIG = PZ(I,J)
GO TO 642
C CALCULATION OF VELOCITIES FOR X 0* WHERE DPHI/DZ AND DPHI/DY
C BOTH EQUAL ZERO.
109
Figure 37.
Chepils case: trace of some equipotential
surfaces in plane y. = *
162
NIG1(J,K) =0
MIG2 (J,K) = 0
THETA 1(J #K) = 0.00
THETA2(J K) = 0.00
CCON1(J K) = 0.00
310 CC0N2(J K ) = 0.00
C SET Z VALUE
DO 200 K= 1KLOW
AK = K
Z = (AK1.0)*H
C SET Y VALUE
CO 200 J= 1 JMAX
AJ = J
Y = (AJ1.0)*H
IB1 = IBDRYl(J K)
IF (NCAS. EQ. 1) IB2 = IBDRY2(J,K)
IF INCAS .EQ. 2) IB2 = IB1
S= 1
C TRAVERSE IN X DIRECTION
CO 201 1= IB1 IB2
AI = I
AH = (XINlT(JfK) (AI1.0)*H)/H
NK = N0(I,J,K)
IF ( I.EG.IB1) X = XINITIJ,K)
IF (NCAS .EQ. 2) GO TO 1201
IF (I .EC. IB 1 .AND. I .EQ. 1) GO TO 211
IF (I .EQ. IB2) X = XEND(J tK)
IF (I .EG. IB2 .AND. I .EQ. IMAX) GO TO 231
IF (I.GT.IB1 .AND. I.LT.IB2) GO TO 201
IF (S.EQ.2) GO TO 21C
C CALCULATE INTERPOLATION FACTORS ON FIRST SPHERE
C Z DIRECTION FACTORS
C FACTOR FCR POINT PK
1201 ALFA7(J,K,1) = 1.00
IF (K .GT. KL0W1) ALFA7(J,K,1) = AH
IF (K .GT. KLOWl) GO TO 205
AG = (X X IN IT ( J Kf 1) ) /H
ALFA7{J K,1) = AN IN 1(AG# AH)
IF (ALFA7(J,K,1).GT.1.00) ALFA7(J,K,1) = 1.00
C FACTOR FCR POINT PK1
205 ALFA7(J,K,2) = 1.00
IF (K .GT. KL0W2) ALFA7(J,K#2) = AH
IF (K .GT. KL0W2) GO TO 206
AG = (X XINIT(J.K+2))/H
ALFA7 ( JK 2 ) = AMINKAG.AH)
IF (ALFA7(J,K,2).GT.1.00) ALFA7(JK2)=1.00
C CHECK TO SEE WHICH OF THREE SITUATIONS Y DIRECTION FITS FOR
C THE CURRENT POINT MIG1 =1#2, OR 3.
206 BET 1.0 Z**2.0
IF (NCAS .EQ. 2) GO TO 1220
IF (J .EQ. JMIT) GO TO 1220
10
in 1913. Betz [22] provided experimental comparisons for the lift
and pressure distributions predicted by theory. Figure 2 shows this
comparison, and the agreement with theory is seen to be satisfactory.
The lower lift than that predicted can be accounted for by the fric
tion which causes flow separation from the profile near the end; the
resultant failure to attain the full pressure difference predicted
yields a smaller lift. The overall agreement is, however, good.
Figure 1. Flow around a Joukowsky profile
10
Figure 2. Calculated and measured pressure distribution
around a Joukowsky profile. (From reference 17,
p. 181.)
88
xdirection about the nearer of two surrounding points. In fact, the
equations developed therein can be used directly, replacing PJ with
PK and points A, C, and A+l (or Al) with their appropriate points for
the zderivative. Hence, equations (555) and (558) will apply to
points above Sphere 1, and equations (556) and (559) will hold for
cases above Sphere 2.
Again, as in the case of y, certain points near the hemisphere
bases will require a linear interpolation between known points to form
an expression for PK or PK1. Notice, however, that problems such as
those indicated in Figure 28 do not occur in the zdirection, since the
hemispheres do not overhang one another.
This section closes discussion of surface derivatives for the
cases of those points on the surface encountered by a line along which
both y and z are constant. It has been noted, however, that other points
encountered as adjacent points in the zdirection and ydirection would
be subjected also to the normal derivative difference expression.
Discussion of these points follows, with a subsequent section to combine
all the spatial derivatives into a normal derivative.
5.5.5 Adjacent zpoints and ypoints subjected to normal deriva
tive condition.The preceding subsections of 5.5 have covered only
those points on the hemispheres encountered as adjacent points in the
xdirection. Those points have counterparts in the y and zdirections
which will be discussed here. Much of the detail will be omitted, since
most of the methods employed parallel those of the immediately preceding
sections.
193
2 PHI(I+l,Jl,K+l))/8.0
IF (ABS(TAMPHI(1+1,J,K)J.GT.EPS) EPS*ABS(TAMPHIU + l,J,K))
PHI(I +1, J K ) = T AM*OMEGA + TAG*PHI(1+1,J,K)
GO TO 790
C SERIES 7760 FOR UNEQUAL XSPACING.
776 BUG7 = 0.5MTAB71.0 )/(1.0+TAB7)
TAM = (PHIt I1,J1,K1) + PH I {I1 J + l,K1)
1 + PHI(I1,J1,K+1) +
2 PH I(I1 J + l,K+1) + PHI(1+1,J1,K1) + PH1(1+1,J+l,Kl) +
3 PHI ( 1 + 1 J1K+1} + PHIU + 1, J+l,K+l) )/8.0 BUG7
4 *(PHI(1 + 1J K PHI(I1,J,K)) + BUG*(PHHI+1,JK)/BUG1
5 PHKI1, J KJ/BIP1J/2.0
TAM = TAM/(l.O+BUG/t 2.0*TAB7))
IF ( ABSTAMPHK I, J,K) ) .GT. EPS) TAM=TAB7*
1 PHI (11,J,K J/BIPl
IF (ABS(TAMPHIU,J,K)J.GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PH I(IJ K) a QMEGA*T AM + TAG*PHI(I,J,K)
790 CONTINUE
C FIRST ROW AT NEW LEVEL WITH MESH LENGTH 2H.
K = KGRADE +2
00 795 J=1, JMAX 2
DO 795 1=3IMIT22
IF (J .EG. 1 .OR. J .EQ. JMAX) GO TO 777
IF ( I .EQ. IMIT2) GO TO 779
C DIFFERENCE EQUATION BASED ON SIX ADJACENT POINTS
C A DISTANCE 2H AWAY FROM OBJECT POINT.
TAM = (PHI(I2,J,K) + PHI( 1 + 2,J K) + PHI(IJ,K2) +
1 PH I (I J K+l) + PH I (I, J21K ) + PHK IJ + 2,K ) ) /6.0
IF (ABS(TAMPHI(IJK)).GT.EPS) EPS=ABS(TAMPHI(IJ,K))
PH I(I,J,K) = OMEGA*T AM + TAG*PHI(I,JK)
GO TO 795
C SERIES 7770 FOR DPHI/DY = 0 CONDITION (Y=0 OR Y=A).
777 IF (J .EG. 1) L =J+2
IF (J .EG. JMAX) L=J2
IF (I .EG. IMIT2) GO TO 781
TAM = (PHI(I2J,K) + PHI(I+2JK) + PHI(I,J,K2) +
1 PH I(I J K + l) + 2.0*PHI(I,L,K))/6.0
IF (ABS(TAMPHI(IJK)).GT.EPS) EPS=ABS(TAMPHI(I,JK))
PHIt I J K) = OMEGA*T AM + TAG*PHI(I,J,K)
GC TO 795
C POINTS WITH UNEQUAL XSPACING NEAR X = SQRT(3.00).
779 TAM = (PHI(I,J,K2) + PHI(IJ,K+1) + PHI(I,J2,K) +
1 PHI(I,J + 2,K) + PH 1(1 + 2,J,K)/(BIP*(1.0+TAB7))
2 + PHI(I2,J,K)/(2.C*BIP))/BING
IF (ABS(TAMPHI(IJ,K)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PH I ( I,J,K) = OMEGA*TAM + TAG*PHI(I,J,K)
GO TO 795
C POINTS WITH COMBINED EFFECTS OF 777 AND 779
781 TAM = (PHI(I2,J,K)/(2.0*BIP) +
1 PHI(I+2,J,K)/(BIP*(1.0+TAB7)) 4
2 PH I(I J K2) + PHI(I,J K + 1) 4 2.0*PHI(ILK)I/BING
o o o o o no o
184
THM = (ATAN2(Z,C) ATAN2{AE ))/(ATAN2(Z.C) ATAN2
TAM = PHI(I6,J,K) THM*(PHI(I7,J,K1) PHIII6,J,K))
PZ1(IJ) = TAM*OMEGA PZ1(I,J)*TAG
GG TO 1668
1611 PI = TUM
NUT = 2
GO TC 1605
1612 PI1 = TUM
NUT = 3
GO TO 1605
1613 PJ = TUN
NUT = A
GO TO 1605
1614 PJi = TUN
1620 S2 = ( 1.0)CY1.0)*(PJl/2.0 2.0*PJ)
S2A = 3.0*(Y1.0)/2.0
1622 IF (NULE .EQ. 1) GO TO 1621
SI = (l.C)(XSQRT(3.00))*C PI 1/2.0 2.0*PI)
SI A = (XSCRT13.00))3.0/2.0
1621 SUN1 = SI S2 S3
SUM2 = S1A S2A + S3A
1623 PZl(ItJ) = (SUMI/SUM 2 ) *OMEGA + TAG*PZUI,J)
1668 BING = PZ1(I J)
1669 RETURN
1009 FGRNAT (1IF 10.5/10F10.5)
ENO
SUBROUTINE YBOUND
SUBROUTINE YBOUND SERVES THE FUNCTION OF SOLVING THE NOFLOW
BOUNDARY CONDITION ON A HEMISPHERICAL SURFACE WHEN ONE OF THE
TWO ADJACENT POINTS IN THE YDIRECTION FALLS ON THE HEMISPHERE
CONMGN ALFA0(21,21,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,212),XEND{21,21),XINIT(21,21),MIG1( 21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA 1(21,21),
3 THETA2(21,21),IBDRY1(21,21),IBDRY2(21,21),PZ(36,21),
4 TIP(100),TOP(11,11 ),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JNID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,NOC36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FLO
C EINSTEINEL SAMNI ARRANGEMENT
DIMENSION PY(37,21),PY1(37,21)
C THE ARRAY PY CONTAINS THE CURRENT VALUES FOR THOSE POINTS
C LYING ON THE FIRST HEMISPHERE, WHILE PY1 IS FOR POINTS ON THE
C SECOND HEMISPHERE.
KL0W1 KLOW l
28
Figure 6. Arrangement of spheres and theoretical bed in
EinsteinEl Samni work.
185
KAB = O
MULE = O
C INITIALIZE PY AND PY1 THE FIRST TIME ENTERING YBOUND.
IF (KEY1 .EG. 1) GO TO 376
DO 377 Kl=l,KLOW
DO 377 11=1,IMAX '
All = II
XI = l A111.0)H
IF (II.EC. IMAX) XI = SQRT(3.00)
PYKIl.Kl) = 100.0 (100.0/SQRT(3.00))*X1
377 PY(11K1) = 100.0 (100.0/SQRT(3.0))XI
KEY1 = 1
C IF IMID EQUALS 2, THIS IS A KEY TO GO TO THE PORTION OF
C YBOUND DEALING WITH THE SECOND HEMISPHERE.
376 IF (IMID .EQ. 2) GO TO 379
C POINTS ON FIRST HEMISPHERE.
IF ( J .EC. JMAX) GO TO 1376
C VALUES FOR DPHI/DY
S2 = (l.O)MYALFA(N,3)*H)*(2.0*PHI(I,J,K) + PHI(I,J+1,K)
1 *(1.02.0*ALFA(N3))/(1.0+ALFA(N3) ))
S2A = (1.0)*(YALFA(N3)*H)*(3.0/(1. O+ALFA (N,3 ) ) )
1377 IF (((XEND(J K) X)/H) .LT. 1.0) GO TO 374
C EVALUATE PI.PI1 (FOR DPHI/DX) AND PK,PK1 (FOR DPHI/OZ) ALL BY
C LINEAR INTERPOLATION BETWEEN OTHER POINTS IN THE FIELD.
C THESE FOUR VALUES ARE THEN USED LATER IN SERIES EXPANSIONS TO
C EVALUATE THE INDICATED DERIVATIVES.
PB = PH I(1 + 1,J1,K)
NT = NO(1+1,J,K)
IF (NT .EC. 0) NT = MARK
IF (ALFA(NT,3) .LT. 1.0) PB = PY(I+1,K)
PI = PHI(I+1,JK) (ALFA(N,3)/ALFA(NT ,3))*
1 (PH I(1 +1,J,K)PB)
IF (((XEND(J,K) X)/H) .LT. 2.0) GO TO 373
PB1 = PHI(1+2,J1,K)
NT = N0(1+2,JK)
IF (NT.EC. 0) NT = MARK
IF (ALFA(NT,3) .LT. 1.0) PB1 = PY(I + 2,K)
PI1 = PH 1(1+2,J,K) (ALFA(N,3)/ALFA(NT,3 ) )*
1 (PH I(1 + 2,J,K )PBl)
372 PB = PHK I, J1,K + 1)
NB = NO(I,J,K+1)
IF (NB .EQ. 0) NB = MARK
IF (ALFA(NB,3 ) .LT. 1.0) PB = PY(I,K+1)
PK = PHI(I,JK+1) (ALFA(N,3)/ALFA(NB,3))*
1 (PHK I, J.K+DPB)
PB1 = PHI(I,JlK + 2 )
IF (K .EC. KL0W1) NB = MARK
IF (K .EQ. KL0W1) GO TO 375
NB NO(I,J,K+2)
IF (NB .EQ. 0) NB = MARK
IF (ALFA(NB,3) .LT. 1.0) PB1= PYU.K+2)
172
GC TG 856
C LINEAR INTERPOLATION FOR PJ1.
857 BETA = (XEND(J42,K) X)/H
NTE = NCI IJ+2K)
IF INTE .EQ. 0) NTE = HARK
PJ1 = PHIII,J42,K) + (PHI(I+1J+2K)PHI(IJ+2K))*
1IALFA(NTE,2)BETA)/ALFA(NTE*2)
S2 = I1.0)*Y*(2.0*PJ PJ1/2.0)
S2A = (3.0)*Y/2.0
GO TO 870
1851 PH II I, J, K ) = 100.0
GO TO 869
1850 S2 = (1.0)Y*PJ*2.0
S2A = {1.0 )*Y*20
GO TO 870
C 2851 AND 2853 ARE EXPRESSIONS FOR PJ AND PJ1 WHEN THEY LIE
C NEARER PHIII) THAN PHIII+l).
2851 Cl = ALFAOIJ,K,1) + BETA
PJ = (PHI!I,J41,K) + ALFAOIJ,K,l)*IPHI(I+lfJ+1.K
1 PHIII,J+1K))/C1 (ALFAOIJ,K,1)**2)*
2 I PH 11 1 +1 yJ+1K)/BETA + PHI I IJ+1,K)/
3 ALFAOIJ,K,1))/Cl )/11.0 ALFAOIJ,K,1)/BETA)
IF (J .EG. JMIT) GO TO 1850
GO TO 856
2853 Cl = ALFAO(J,K 2 ) + BETA
PJ1 = (P HI (IJ+2 K) 4 ALFAOIJ,K2)*(PHI(l4lfJ+2,K)
1 PHIII,J42,KJ/C1 (ALFAOIJ,K,2)**2)*
2 I PH I ( I + 1,J+2,K)/BETA + PHI I I,J42,K)/
3 ALFAOIJ,K,2))/Cl )/tl.O ALFAO(J,K,2)/BETA I
GO TO 8701
C Y POINTS OBTAINED BY ARC INTERPOLATION.
852 II = IBDRY2IJ41,K)
12 = IBDRY2IJ,K)
PJ = PHI IIl,j4lfK) THETA1IJK)*
1 IPHI(I1,J41,K) PHI(I2,J,K))
S2 = I1.0)*YPJ/ALFA(N,4)
S2A = I1.0)*Y/ALFAINA)
GO TO 870
C Y POINTS BY LINEAR INTERPOLATION.
853 PJ = PHIII,J41,K) CC0N1(J,K)*(PHI(IJ+1K) 
1 PHI I 141,J41,K))
S2 = (10)*Y*PJ
S2A = I1.0)*Y
IF (J .EG. JMIT) S2 = 2.0*S2
IF (J .EG. JMIT) S2A = 2.0*S2A
C FACTORS FOR Z DERIVATIVE.
870 IF IK .EC. KL0W1) GO TO 4870
NT2 = NO!141,J,K41)
IF INT2 .EQ. 0) NT2 = MARK
NT3 = NO!141,J,K42)
IF INT3 .EQ. 0) NT3 = MARK
190
C EINSTEINEL SAMNI ARRANGEMENT
C COMPUTE CONSTANTS NEEDED FOR CALCULATIONS NEAR X SQRT(3.00)
C DUE TO UNEQUAL XSPACING THERE.
IMIT = IMAX 1
IMIT2 = IMAX 2
JMIT = JMAX 1
AI MAX = IMAX
TAB7 = (SQRTO.OO) tAIMAX2.0)*H)/H
BUG = 1.0 TAB72
BUG1 = TAB7*(1.0+TAB7)
BIP = (3.0+TAB7)/8.0
BIP1 = 1.0 + TAB7
BING = 4.0 + (0.5 + 1.0/(1.0+TAB7))/<0.125*<3.0+TAB7))
C MAKE COMPUTATIONS FOR LAST FULL ZVALUE WHERE POINTS ARE
C LOCATED A DISTANCE H APART IN HE XYPLANE.
C SERIES THROUGH 760 HANDLES THOSE POINTS WHICH HAVE SIX
C ADJACENT POINTS ALL LYING H AWAY.
DO 760 J = 2 ,JMIT,2
DO 760 I = 2 IMIT
IF (I .EC. IMIT) GO TO 751
TAM = (PH I(I 11J K) + PHI(I+1,J,K) 4 PHI(I,J1,K) 4
l PH KI J+lK) + PHI(IJ,K1) PHI(I,J,K + 1))/6.0
752 IF (ABS{TAMPHI(IJ K )).GT.EPS) EPS=ABS( TAMPHK IJ,K))
PHI(ItJfK) = OMEGA*TAM 4 TAG*PHI(I,J,K)
GO TO 760
C STEP 751 IS FOR THE UNEQUAL XSPACINGS AT X=SQRT(3.00).
751 TAM = (PHKI,J+1,K) + PHI(I,J1,K) 4
1 PHIUfJtK + L) + PHI(IJ,Ki) 4
2 PHI (I + 1JK)*TAB2 4 PH I(I1.J,K)*TAB3)/TAB 1
GO TO 752
760 CONTINUE
C THE SERIES OF STEPS THROUGH 770 ARE FOR THOSE POINTS (STILL
C IN THE SAME ZPLANE) WHICH HAVE (ALTERNATING FROM POINT TO
C POINT) ALL SIX ACJACENT POINTS EITHER AT A DISTANCE H OR 2H.
DO 770 J= 1 JMAX 2
00 770 1 = 2,IMIT,2
IF ( J .EC. 1) GO TO 762
IF (J .EC. JMAX) GO TO 763
IF (I .EQ. IMIT) GO TO 761
C POINTS WITH SIX ACJACENT POINTS H AWAY.
TAM = (PHKI1,J,K) 4 PHI(I41,J,K) 4 PHI(I,J1,K) 4
1 PHI(I,J41,K) 4 PHK I, J ,K1) 4 PHK I,J,K41) )/6.0
IF (ABS(TAMPHI(I,J,K)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(I,J,K) = OMEGATAM 4 TAGPHKI,J,K)
C POINTS WITH SIX ADJACENT POINTS AT A DISTANCE 2H.
TAM = (PHK I lJK) + PHI(I + 3JK) 4 PHI (1.41, J42.K) 4
1 PHI(I41,J2,K) 4 PHI(141,J,K+2) 4 PHI(I + l,JK2))/6.0'
IF (ABS(TAMPHI(141,J,K)).GT.EPS) EPS=ABS(TAMPHI(141,J,K))
PHKI+1J*K) = TAM*OMEGA 4 TAG*PHI(141,J,K)
GO TO 770
C SERIES 7610 FOR UNEQUAL XSPACING NEAR X = SQRTO.O)
PI1
PJ
PJ1
PK
PK1
q
%
r
r
c
R
e
R
ew
u
U
u
m
u.
top
u
35
w
X
y
yb
yo
field point in xdirection
field point in ydirection
field point in ydirection
field point in zdirection
field point in zdirection
total velocity at a point
velocity along base of the grain
spherical coordinate
roughnessgrain size ratio based on large hemisphere
diameter in Chepil's tests
Reynolds number
wall Reynolds number (v^k/v)
velocity
free stream velocity at 00
mean velocity
velocity at uppermost point of hemisphere, sphere,
or elliptic cylinder
same as u^_
velocity at 0.35 Kg above theoretical bed
velocity
friction velocity O/w'p")
velocity in zdirection
Cartesian coordinate
Cartesian coordinate; also elevation above datum
location of theoretical bed
distance at which velocity equals zero
xi
11
The classic work of Kutta and Joukowsky, with subsequent
experimental verification of the validity of their approach, has led
to many other lift solutions by means of potential flow. These works
have generally produced similarly satisfactory results.
2.4.4 Fuhrmann's work and other studies.Prandtl and Tietjens
[23, p. 137J mention the work of Fuhrmann [24] who calculated, by poten
tial theory, and then measured the pressure distribution on some slender
bodies whose shapes were derived from sourcesink combinations. As
found in reference [15] and pictured in Figure 3, Fuhrmanns experi
ments revealed good agreement with theory except near the bodys end, a
/
result common to other works and also expected. Rodgers [25] notes
this result in reference to several bodies of revolution analytically
treated by Lamb [26] and MilneThomson [27]. He attributes the devia
tions from theory primarily to generation of vorticity along the body.
1.0
0.8
1 1
Measured
0.6
 Potential
Flow
1
1
PPG
0.4
1
1
P Vq/2
0.2
1
1
0
J
_
0.2
0.4
i\
\\
\\
0
0
2
0
.4
0
.6
0.8
1.
x/L
Figure 3. Pressure distribution around a Fuhrmann body.
HYDRODYNAMIC LIFT IN SEDIMENT
TRANSPORT
By
BARRY ARDEN BENEDICT
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1968
ACKNOWLEDGMENTS
The author wishes to express his sincere appreciation to
Dr. B. A. Christensen, Dr. J. H. Schraertmann, Dr. E. A. Farber,
and Dr. T. 0. Moore for their service on his supervisory committee
and their interest in his work. The author is especially indebted
to the committee chairman, Dr. B. A. Christensen, whose guidance,
encouragement, and personal interest and enthusiasm were truly
valuable.
Appreciation is extended to the National Science Foundation,
under whose traineeship the author.has been working.
The author also wishes to express his gratitude to the
University of Florida Computing Center for the computing facilities,
services, and aid extended by the Center.
Finally, the author wishes to extend thanks for the aid and
encouragement provided by his wife, who has persevered through many
trying, times.
11
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF TABLES vi
LIST OF FIGURES vii
LIST OF SYMBOLS . x
ABSTRACT . xiii
CHAPTER
I INTRODUCTION ... 1
II RELATED BACKGROUND ON LIFT FORCE STUDIES 4
2.1 Historical interest 4
2.2 Neglect of lift force 5
2.3 Indications of significance of lift 6
2.4 Use of potential theory in lift studies on
single bodies 7
2.4.1 Genera 1 7
2.4.2 Applicability of potential flow 8
2.4.3 Work of Kutta and Joukowsky 9
2.4.4 Fuhrmann's work and other studies .... 11
2.4.5 Jeffreys' analysis 12
2.4.6 Flow around a single sphere ....... 14
2.5 Multiparticle studies 19
2.5.1 Einstein and El Samni 19
2.5.2 Chepil 20
2.5.3 Chao and Sandborn 22
2.6 Shapes of bodies studied 22
2.7 Relation to work of dissertation 23
III LOGARITHMIC VELOCITY DISTRIBUTION 25
3.1 Development of logarithmic velocity
distribution ..... 25
3.2 Special problems of present expressions .... 26
iii
TABLE OF CONTENTS (Continued)
CHAPTER Page
3.2.1 Theoretical bed 27
3.2.2 Conditions near the bed 29
3.3 Use of proposed adjusted velocity distribution 31
3.3.1 Comparison of distributions at wall ... 31
3.3.2 Comparison of distribution with
increasing y 32
3.4 Determination of k and v_ from experimental
data 33
3.5 Effect of sidewalls 35
IV TWODIMENSIONAL WORK: EARLIER RESULTS 41
4.1 Shapes studied 41
4.2 General methodsvelocity and pressure results 41
4.3 Lift integration 44
4.4 Calculation of predicted lift by use of
measured logarithmic velocity profile .... 47
4.5 Application to experimental results 49
V THREEDIMENSIONAL NUMERICAL SOLUTIONS 52
5.1 General 52
5.2 Problem formulation 53
5.2.1 Choice of solution method 53
5.2.2 Depth of flow space 54
5.2.3 Boundary conditions 56
5.3 General finite differences approach 61
5.4 Finite differences equations: interior space 64
5.4.1 General lattice point . 65
5.4.2 Object point on planar noflow boundary 68
5.4.3 Object point on "foldedsymmetry"
boundary 70
5.4.4 Adjacent point on hemispherical surface 70
5.4.5 Graded lattice 72
5.5 Finite difference equations: hemispherical
boundary 78
5.5.1 General 78
5.5.2 Xdirection derivative 79
iv
TABLE OF CONTENTS (Continued)
CHAPTER Page
5.5.3 Ydirection derivative 81
5.5.4 Zdirection derivative 87
5.5.5 Adjacent zpoints and ypoints subjected
to normal derivative condition .... 88
5.5.6 Final boundary formulation 91
5.5.7 Singular points 93
5.6 Velocity and lift calculations 94
5.7 Implementation of solution . 99
VI RESULTS AND COMPARISONS 101
6.1 General 101
6.2 Numerical results for Chepil arrangement .... 102
6.3 Comparisons with Chepil's observations . . 110
6.3.1 Details of Chepil's work 110
6.3.2 Comparison of lift forces 116
6.4 Numerical results for closely packed
hemispheres 125
6.5 Comparison with EinsteinEl Samni observations 138
6.5.1 Physical details of experiments ..... 138
6.5.2 Values of lift for hemisphere bed .... 141
6.5.3 Lift on a gravel bed 142
VII CONCLUSIONS AND FUTURE WORK 146
APPENDIX 149
NOTES ON FORTRAN IV COMPUTER PROGRAM 150
NOTES ON EQUIVALENT GRAIN SIZE 200
REFERENCES 201
BIOGRAPHICAL SKETCH 207
v
LIST OF TABLES
Table Page
1 Comparison of Proposed and Former Distributions ..... 32
2 Chepil's Experimental Data 112
3 for Chepil's Work 114
4 RoughnessGrain Size Ratios 115
5 Comparison of Theoretical with Chepil's Work 123
vi
LIST OF FIGURES
Figure Page
1 Flow around a Joukowsky profile 10
2 Calculated and measured pressure distribution around
a Joukowsky profile . . 10
3 Pressure distribution around a Fuhrmann body 11
4 Jeffreys' cylinder 12
5 Pressures on a single sphere 17
6 Arrangement of spheres and theoretical bed in
EinsteinEl Samni work 28
7 Arrangement of spheres and theoretical bed in
Chepil's work 30
8 Sketches for sidewall effect ........ 36
9 Evaluation of constant for sidewall analysis 38
10 Grains placed in rough bed configuration 42
11 Pressure distribution on twodimensional grain 43
12 Piezometric head distribution on twodimensional grain 43
13 Distributions of surface hydrodynamic pressure decreases. 45
14 Lift coefficient C . 46
Li
15 Theoretical bed used in relating logarithmic and
potential velocity profiles 48
16 Definition sketch for experimental application ..... 50
17 Comparison of theoretical and measured values 51
18 Solution space for closely packed hemispheres 58
19 Solution space for Chepil's arrangement .... 59
vii
LIST OF FIGURES (Continued)
Figure Page
20 Foldedsymmetry boundary 60
21 Sevenpoint finite difference scheme .... 62
22 General lattice point 65
23 Examples for object point on planar boundary 69
24 Arc interpolation for surface value 71
25 Grading the lattice 73
26 Xderivative condition at boundary 79
27 Yderivatives 82
28 Special points for yderivative 86
29 Zpoint for normal derivative condition 89
30 Pressures on area of hemisphere 97
31 Chepil's case: velocities on y = 0 103
32 Chepil's case: velocities on y = 0.2a 104
33 Chepils case: velocities on y = 0.4a 105
34 Chepil's case: velocities on y = 0.6a 106
35 Chepil's case: velocities on y = 0.8a 107
36 Chepil's case: velocities on x = 0 108
37 Chepil's case: trace of some equipotential surfaces
in plane y = 0 109
38 Measured and theoretical pressure distributions:
v^ = 68 cm/sec 117
39 Measured and theoretical pressure distributions:
v^ = 91 cra/sec 118
40 Measured and theoretical pressure distributions:
v^ = 128 cm/sec 119
viii
LIST OF FIGURES (Continued)
Figure Page
41 Measured and theoretical pressure distributions:
= 159 cm/sec 120
42 Chepil's hemisphere 122
43 Closely packed hemispheres: velocities on y = 0 127
44 Closely packed hemispheres: velocities on y = 0.2a . 128
45 Closely packed hemispheres: velocities on y = 0.4a . 129
46 Closely packed hemispheres: velocities on y = 0.5a ... 130
47 Closely packed hemispheres: velocities on y = 0.6a ... 131
48 Closely packed hemispheres: velocities on y = 0.8a . 132
49 Closely packed hemispheres: velocities on x = 0 133
50 Closely packed hemispheres: trace of some equipotential
surfaces in plane y = 0 134
51 Closely packed hemispheres: trace of some equipotential
surfaces in plane y = 0.5a 135
52 Closely packed hemispheres: flow pattern on surface
viewed toward xyplane . 136
53 Closely packed hemispheres: flow pattern on surface
viewed toward yzplane 137
54 Velocity suppression near sidewall 140
*
ix
LIST OF SYMBOLS
sphere (or circular cylinder) radius
coefficient in logarithmic velocity profile
lift coefficient based on total bed area
lift coefficient based on projected area of grain
grain diameter
equivalent grain diameter
projected elemental area in xyplane
elemental surface area on hemisphere
acceleration due to gravity
full lattice increment
point index, xdirection
point index, ydirection
point index, zdirection
equivalent sand roughness
diameter of hemispheres
sidewall equivalent sand roughness
bottom equivalent sand roughness
hydrodynamic lift force
pressure
reference pressure, defined as used
reference pressure, defined as used
field point in xdirection
PI1
PJ
PJ1
PK
PK1
q
%
r
r
c
R
e
R
ew
u
U
u
m
u.
top
u
35
w
X
y
yb
yo
field point in xdirection
field point in ydirection
field point in ydirection
field point in zdirection
field point in zdirection
total velocity at a point
velocity along base of the grain
spherical coordinate
roughnessgrain size ratio based on large hemisphere
diameter in Chepil's tests
Reynolds number
wall Reynolds number (v^k/v)
velocity
free stream velocity at 00
mean velocity
velocity at uppermost point of hemisphere, sphere,
or elliptic cylinder
same as u^_
velocity at 0.35 Kg above theoretical bed
velocity
friction velocity O/w'p")
velocity in zdirection
Cartesian coordinate
Cartesian coordinate; also elevation above datum
location of theoretical bed
distance at which velocity equals zero
xi
z
a
a.
V
e
H
k
k
U
V
P
CT
T
O
'Pb
U)
fraction finer than (in soil gradation curve)
Cartesian coordinate; also elevation above datum
angle through "corners" of isovels
ratio of given lattice leg length to full increment, h
unit weight of fluid
spherical coordinate; also angle for arc interpolation
von Karman's constant
lift per unit area (total bed area)
lift per unit area (only projected area of grain)
kinematic viscosity of fluid
mass density of fluid
density of particle in Jeffreys analysis
bed shear stress
potential function
value of potential function on boundary
overrelaxation factor
xii
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
HYDRODYNAMIC LIFT IN SEDIMENT TRANSPORT
by
Barry Arden Benedict
December, 1968
Chairman: Dr. B. A. Christensen
Major Department: Civil Engineering
Hydrodynamic lift is a force often neglected in studies of
sediment movement, despite being of the same order of magnitude as
the drag force. The goal of this dissertation is to demonstrate that
potential flow theory can be used to predict hydrodynamic lift in
sediment transport. It is known that concentration of streamlines,
rather than viscous forces, contribute most to lift. This is rein
forced by many early airfoil works using potential flow theory, which
yield good values for lift, even though exhibiting zero drag.
The theoretical methods used treat mean, steady flows with no
free surface or sidewall effects. Since potential flow involves an
inviscid fluid, the method considered is only applicable to studying
cases of real flow in the hydrodynamically rough range, where viscous
forces are negligible. The two sets of experimental work studied
involve flows in the rough range over beds of hemispheres arranged in
hexagonal patterns. One work uses hemispheres three diameters apart
center to center; the other uses touching hemispheres.
xiii
The analysis begins with a solution of the potential flow over
the two sets of hemispheres, solving here the threedimensional prob
lems by use of finite difference methods. The solutions enabled cal
culation of velocities near the hemispherical surface, relation to
pressures by Bernoulli's equation valid for rotational flow, and sub
sequent integration to find the lift force. The velocity distribution
near the surface (from potential flow) is then linked to the velocity
distribution in the actual flows considered, yielding values for lift
corresponding to the experimental works. This procedure can be used
for any rough bed and velocity field desired.
For the widely spaced hemispheres, theory produces (for three
cases most closely related to the theoretical model studied) values of
lift differing from measured values by 19 per cent, 13 per cent,, and
8 per cent. For the closely packed hemispheres, results from theory
are 16 per cent above measured values. This discrepancy is reduced to
perhaps 10 per cent or less if allowance is made for the sidewall
effects of the narrow flume used in the experimental work. Good
results are also obtained for a natural gravel bed replaced by a bed
of equal hemispheres.
The results show quite good agreement between theory and
experiment. Hence, the goal of this dissertation is accomplished, and
a new analytical tool is found effective in studying lift forces on
a rough bed. It is hoped that the tool will be useful in future work
in the field.
xiv
CHAPTER I
INTRODUCTION
One problem'of great practical concern to the hydraulic engineer
is the development of a full knowledge of the transport of material in a
stream. Understanding sediment transport and developing advanced meth
ods in this area are physically and economically important in such prob
lems as scour evaluation and prevention, construction of stable channels
planning and design of reservoirs, and the maintenance of harbor chan
nels. The problem is of true importance when some of the sediment magni
tudes involved are viewed. Brown [lj estimated in 1950 that the aver
age gross sediment production of the United States was on the order of
four billion tons per year. He also indicated that deposition of river
borne sediment in impounding reservoirs was estimated to be equivalent
to a loss each year of sufficient storage capacity to hold the annual
water supply for a city of 250,000 people. On a smaller scale,
Langbein and Leopold [2] note that watersheds composed of fine wind
blown soil, such as in western Iowa, yield as much as 2,000 tons of
sediment per square mile per year. The magnitude of the sediment prob
lem causes a continuing need for better understanding in sediment trans
port. One facet of this understanding must include the mechanism of
Numbers in brackets refer to the References.
1
2
a particle being moved from the stream bed. A need for a better defin
ition of the fundamental forces acting on such particles has prompted
the author's study of hydrodynamic lift in sediment transport, with
observations of flow characteristics in the vicinity of a rough bed.
Lift has been an often neglected force in fluvial hydraulics
despite the fact that it has the same order of magnitude as the bed
shear stress. This is reflected in the many stable channel design
procedures which neglect lift. These procedures may result in a design
which is too costly. Hence, studies of the lift force take on further
economic significance.
Most work on lift has been on single particles, with no con
sideration for the particle interaction existent in the actual stream.
Those works which have included interaction have generally measured
lift forces over a given area rather than in the form of pressure
distributions on the individual particles. The author intends to study
interaction effects, trying to provide a base for analytical studies
of lift forces.
First, potential flow theory will be used as a guide to provide
a means of studying the flow near the surface of a grain or bed shape
and then relating that flow to other flow characteristics. The poten
tial flow velocity profile will be related, for this work, to the
logarithmic velocity profile, furnishing a means for predicting the
pressure distribution and total lift force on single particles within
a series. It should be noted that forms of velocity profile other than
the logarithmic might be used if they are characteristic of the flow.
3
This dissertation will concern itself with cases involving a
steady, mean flow with fr.ee surface effects (wave resistance) being
insignificant. A report of some earlier results for twodimensional
bed shapes will be made. Then a numerical solution will be made for
two cases of flow over a bed of hemispherical particles. These solu
tions will be related to experimental measurements obtained elsewhere,
as will some of the twodimensional results.
It is the hope of the author to present herein an analytical
method wherein the hydrodynamic lift, often neglected, can be computed
with a reasonable degree of accuracy. It is also intended to form a
basis for future analytical and experimental work in sediment transport.
CHAPTER II
REIATED BACKGROUND ON LIFT FORCE STUDIES
2.1 Historical interest
For centuries engineers have been interested in the movement of
sediment due to flowing water, which Rouse and Ince call "A class of
flow phenomena inherently hydraulic in nature . [3, p. 246].
Concerned with problems of scour, deposition of material, stable
channels, and the like, engineers have for years attempted to increase
their knowledge of sediment motion. Both empirical and theoretical
means have been employed in these attempts.
Domenico Guglielmini [3, p. 70] was perhaps representative of
the whole Italian school interested in flow resistance in open channels.
(
His work in the seventeenth century made some qualitative observations
which were very accurate, though his analytical work was more faulty.
Later, Pierre Louis Georges Du Buat [3, p. 129] collected a
vast array of experimental results, including extensive data on the
beginning of sediment movement. His eighteenth century works over
shadowed that of other hydraulicians for about a century.
Work continued through various periods until the work of Grove
Karl Gilbert in the years around 1910. His tests on initial sediment
movement and various phases of transport covered a wide range. It has
been noted that "... the results he presented in U.S.G.S. Professional
4
5
Paper No. 86 in 1914 still continue to be those most often quoted of
any in the field" [3, p. 225],
From the time of Gilbert's work, laboratory facilities increased,
enabling further, broader studies. As various governmental agencies
began to attack the problems in the United States, more and more quan
titative information became available. Additionally, interest in analyt
ical approaches to sediment problems grew with the advent of more
theories on movement. Consideration of the forces actually acting on
particles exposed to flow received increasing emphasis.
2.2 Neglect of lift force
As Leliavsky [4] indicated, the drag component of forces acting
on particles received the bulk of work earlier in this century. Indic
ative of this was the work by C. M. White [5] in 1940. White gave
great care to study of the drag, while the lift was treated only briefly,
with a guarded conclusion that lift did not exist. Leliavsky noted
in 1955 that there still exists ". .an almost unexplored aspect of
the problem, viz., the vertical component of the resultant of the
hydraulic forces applied to the grain, i.e., 'the lift'" [4, pp. 6465].
Similarly, Young stated, "Although some attention has been given
to the effect of the drag components on the behavior of suspensions,
little work has been done in connection with the determination of the
lift component" [6, p. 47].
This disregard for the lift force has resulted in a lack of
understanding of this force, with probably attendant shortcomings in
full understanding of drag. One area of very practical interest where
6
lift is often neglected is in stable channel design, where the widely
used method of E. W. Lane [7], as well as other methods, does not
include a lift force. It should be noted, however, that other methods
are beginning to include lift effects. This neglect of the lift force
is on the safe. side but is economically costly.
2.3 Indications of significance of lift
As Leliavsky noted, contrary to White's results, Jeffreys [8]
and Fage [9] produced results which definitely indicate lift as an
important factor. Fage's experimental work provided evidence, and
Jeffreys' gave a theoretical approach. Studying a cylinder resting on
the flat bed of a deep stream and applying the principles of classical
hydrodynamics, Jeffreys found a relation for the lift and then for
a scour criterion. Use of realistic sizes indicated that lift alone
should be capable of dislodging particles.
Chang [10], considering the lift on particles as due to the
pressure of a velocity head, worked from the simple fact that the
particle would tend to lift when the vertical lifting force (or hydro
dynamic lift) plus the buoyant force equalled the particle's weight.
He also did work on drag and made the following comparison. "Theo
retically, the force required to lift a particle from the bottom of
a stream is about 40 per cent greater than that required to move it
along the bed" [10, p. 1282]. This certainly indicates the same order
of magnitude for drag and lift.
Further validity is given to the value of studying lift by
Yalin [11] who says "A consideration of the paths of saltating particles
7
by R. A. Bagnold reveals that saltation begins with a motion directed
'upward*; . However, if this is true, then, as has already been
maintained by various authors, the lift force must be the cause
of the detachment" [11, p. 229]. Support for Yalin's statement is
offered by the work of Einstein [123> Bagnold [13], and Velikanow [14].
Young, doing work in 1960 on spheres in a cylindrical tube,
found the lift to be of the order of onehalf the drag for experiments
with a Reynolds number based on the pipe diameter and the mean veloc
ity in the range 3601115 (in the laminar range). He summarized,
"It is thus apparent that the lift force should not be overlooked in
studies related to the incipient motion of particles resting on stream
beds or pipe walls" [6, p. 57].
It seems apparent that sufficient evidence exists to prompt
efforts to increase understanding of the lift phenomena. The author
will now consider earlier works on lift.
2.4 Use of potential theory in lift studies on single bodies
2.4.1 General.Much of the analytical work done on lift has
dealt with single bodies or particles, especially spheres and circular
cylinders, isolated from any other bodies. An example of this approach
is the work done by Jeffreys [8] mentioned earlier.
The overwhelming amount of work on lift has been done in the
realm of aerospace engineering. While the principles thus developed
are applicable to hydraulics, the work concerns single bodies only,
often airfoils, struts, and the like. One of many works giving dis
cussions of several of these approaches is the famous work edited by
8
Goldstein [15]. Another, viewing historical development, is by
von Karman [16]. In a number of these, lift is studied by means of
potential flow.
2.4.2 Applicability of potential flow.The idea that lift
could be predicted by use of potential theory, even perhaps in those
areas where it might seem out of place, was offered by Prandtl. He
stated that any explanation of drag requires a consideration of
viscosity, "... whereas the lift can be explained entirely without
the concept of viscosity so that the wellknown methods of the clas
sical hydrodynamics of the ideal fluid are applicable" [17, p. 159].
Part of the reason for some reluctance to use such methods here is the
fact that within a boundary layer adjacent to a surface, irrotational
flow does not exist. However, in the cases of interest, the sublayer,
where viscous and inertial forces are of the same order, over the surface
is very thin. Outside this layer, up to the point of separation, the
equations of inviscid fluid flow are valid. Hence, the pressures on
the outer edge of the sublayer can be found from such equations, and
since the pressure difference from the outer edge of the layer to the
surface is assumed negligible, the normal pressures on the surface, and
thus lift, can be predicted by inviscid flow principles. Also, since
the sublayer is thin, it is possible to discuss velocities "on the
surface," though the velocities actually considered are those a small
distance away at the edge of the boundary layer. Of course, the
actual velocities at the wall in a real fluid would be zero.
9
Reasoning such as Prandtl's has through the years prompted
many potential flow solutions to the lift problem, probably the most
famous of which are those by Kutta and Joukowsky.
2.4.3 Work of Kutta and Joukowsky.Generally, an understand
ing of the work of Kutta and Joukowsky begins with flow around a cir
cular cylinder. It is well known that no lift or drag forces are
predicted by any transformation of the symmetrical flow around a cylin
der. The addition of a vortex in the center of the cylinder produces
a streamline pattern, the effect of which is to yield a lift force.
This force is due to the circulation's tendency to incrase the velocity
above the cylinder and decrease it below, thus causing a pressure dif
ference and a consequent lifting action.
Kutta [18] first applied the methods of conformal transforma
tion to transform the cylinder with circulation into a line inclined
to the flow. This produces a force acting perpendicular to the veloc
ity at infinity. This force is, of course, the one originally exerted
on the cylinder.
Kutta's work and use of transformations prompted further work.
Joukowsky [19] wanted to avoid difficulties at the sharp leading edge
of Kutta's plane, and he employed a mapping function by which a curvi
linear profile very similar to actual airfoil shapes was developed.
Flow around a Joukowsky profile is shown in Figure 1.
Numerous investigations of Joukowsky profiles have been carried
out. Joukowsky [20] himself performed experiments in 1912, and
Blumenthal [2l] calculated the pressure distribution from theory
10
in 1913. Betz [22] provided experimental comparisons for the lift
and pressure distributions predicted by theory. Figure 2 shows this
comparison, and the agreement with theory is seen to be satisfactory.
The lower lift than that predicted can be accounted for by the fric
tion which causes flow separation from the profile near the end; the
resultant failure to attain the full pressure difference predicted
yields a smaller lift. The overall agreement is, however, good.
Figure 1. Flow around a Joukowsky profile
10
Figure 2. Calculated and measured pressure distribution
around a Joukowsky profile. (From reference 17,
p. 181.)
11
The classic work of Kutta and Joukowsky, with subsequent
experimental verification of the validity of their approach, has led
to many other lift solutions by means of potential flow. These works
have generally produced similarly satisfactory results.
2.4.4 Fuhrmann's work and other studies.Prandtl and Tietjens
[23, p. 137J mention the work of Fuhrmann [24] who calculated, by poten
tial theory, and then measured the pressure distribution on some slender
bodies whose shapes were derived from sourcesink combinations. As
found in reference [15] and pictured in Figure 3, Fuhrmanns experi
ments revealed good agreement with theory except near the bodys end, a
/
result common to other works and also expected. Rodgers [25] notes
this result in reference to several bodies of revolution analytically
treated by Lamb [26] and MilneThomson [27]. He attributes the devia
tions from theory primarily to generation of vorticity along the body.
1.0
0.8
1 1
Measured
0.6
 Potential
Flow
1
1
PPG
0.4
1
1
P Vq/2
0.2
1
1
0
J
_
0.2
0.4
i\
\\
\\
0
0
2
0
.4
0
.6
0.8
1.
x/L
Figure 3. Pressure distribution around a Fuhrmann body.
12
Other works where pressure distributions have been predicted
and checked experimentally could also be shown. The important factor
is not to go into details of numerous cases, but rather to point out
that there is strong historical backing for use of potential theory
to study lift forces. It should be recalled that these earlier uses
of potential theory have employed it to describe the entire flow
pattern. Only two further cases will be studied, the first being the
one case where potential flow was used to describe the entire flow
pattern in an effort to solve the problem of sediment movement.
2.4.5 Jeffreys' analysis.Jeffreys [8] dealt with a single
long circular cylinder resting on a bed in a twodimensional study,
as shown in Figure 4.
Uniform Flow
Figure 4. Jeffreys' cylinder.
Jeffreys developed his work based on the complex potential for
this case,
TTfl
W = rraU coth ()
z
(21)
He found the amount by which the lift exceeded the weight of the
enclosed liquid, and thence wrote the following condition for move
ment of the cylinder:
13
(22)
where
o = density of particle
g = acceleration due to gravity
a = cylinder radius
p = mass density of water
This can be written as
(23)
Jeffreys noted that J. S. Owens [28] had earlier measured
the velocity required to move pebbles, finding
U2 = 1.65 ga
(24)
The motion observed by Owens was not, however, a jumping or
lifting motion, but rather a rolling motion, yielding a difference
from values predicted by theory. However, as Jeffreys says, "The
proportionality of U to the square root of the linear dimensions is in
agreement with theory" [8, p. 276].
Use of a value for <7 of 2.7 times p in Jeffreys* equation yields
the following:
U2 > 1.19 ga
(25)
14
Differences from theory are primarily due to two causes.
First, the motion measured was rolling rather than totally lifting,
which (25) is based on. Second, the measured particles were three
dimensional, contacting the bed at only a small number of points,
while Jeffreys' cylinder made contact over the whole length of its
body. These effects are compounded by the fact that the Uvalues used
by the two men are not the same. Jeffreys employed a potential flow
field, but Owens made his measurements in a flow field exhibiting a
logarithmic velocity distribution. The differences involved here will
be discussed in Chapters III and IV. Despite the discrepancies, the
theory seems to provide a much better starting point than might first
be thought possible.
2.4.6 Flow around a single sphere.The results from measure
ments on flow around a single sphere will be presented because the shape
relates to this study and the results reveal some factors influencing
the actual flow pattern.
The flow involved is that around a single sphere suspended in
an otherwise uniform flow. For this case the potential can be
expressed [29] as
Ua3
cp = cos 0 + Ur cos 0 (26)
2r
where a = radius of sphere
r = radial distance
0 = angle measured from horizontal and
through sphere center
U = free stream velocity at 00
15
Note that this is written for one meridian plane, since the
case is axisymmetric. The tangential surface velocity on r = a
can be found as
1 ocp 3 TT a
"7^=2Usin0=q C2"7)
Using Bernoulli's principle,'with a pressure of p^ at the
forward stagnation point, yields the following expression for the
surface pressure, p.
PQ P = Yz + 2 q2 (28)
where z = elevation
Y = unit weight of fluid
What is of interest here is to study only the vertical force
component over the upper half of the hemisphere. This will involve
integrating (28) over the upper surface. Integration of the first
term, z, would yield a hydrostatic lift (buoyancy) which would, in
fact, not even be measured, as the difference in piezometric head is
the measured value. Hence, integration of the last term in (28)
will yield the desired force component. It should be noted that this
would actually be the theoretically predicted valtle for the case
of a single hemisphere on a flat bed. However, the measurements herein
used are for a suspended sphere. The total vertical force can be
found as follows:
F = f f* T U2 sin26 dA
v 2 4
o
(29)
16
where
yielding
dA = 2a^ sin^6 d6
27 2 U
F = a TT o TT
v 16 H 2
(210)
Note that if this was expressed per unit area (based on the area
2
na ), with U replaced by u^_ = (3/2)U, the velocity at the top of
the sphere, theory would show
2
(211)
3 Ut
F = i. *
4 2
where F^ = vertical force per unit area
Measurements were made by Flachsbart [30], as shown in
Schlichting [3l], The results were very similar to those by Fage [32]
shown below. The measured values are shown below with the theoretical
curve. The values are plotted with reference to the stagnation pres
12
sure Pj + pU where p^ is the pressure at where the velocity
equals U. This stagnation pressure is the pQ being used here. Replac
ing Pj, by pQ yields
(212)
where j3 is the value shown in the graph. The expression thus found
for (p pQ) can be integrated numerically.
In the case of the higher Reynolds number, the force can be
expressed as (210) with a coefficient of 1.64, only about 3 per cent
below the theoretical. The lower Reynolds number produces a coefficient
17
of 1.32, about 20 per cent below the theoretical. It therefore seems
that above some critical Reynolds number the potential theory becomes
more and more capable of predicting pressures on a body outside the
separation zone. The effect of the Reynolds number involves the change
of laminar boundary layer at the body to turbulent as a critical range
of Reynolds number is reached. The boundary layer, becoming turbulent,
is capable of proceeding further downstream, therefore causing separa
tion to be delayed to points further and further back on the sphere.
18
This allows the surface pressure distribution to more nearly approach
the theoretical. For even lower Reynolds numbers than those indicated,
the viscous forces would play an even greater role, thereby causing
further deviation in the flow from theoretical. These ideas are of
importance in relation to the limitations of work in this dissertation.
The forces evaluated in this section were only those in the
vertical direction on the upper surface of the sphere, with an eye to
noting agreement with theory.
For later comparison a value of C^u will be determined here
for the theoretical case of a single hemisphere on a flat bed. This
will entail integrating the pressure at the hemispherical base, found
from (28) and subtracting it from the force of (211) to give a
resultant vertical lift. Integration here occurs in a direction normal
to that in (29). Using 0^ for this integration yields a relation
cos 0j = sin 0. Therefore, the ratio of the basal velocity (u^) to the
top velocity (u^_) equals cos 9^. The pressure decrease along the
bottom can be found as below.
o J7 9 9
F = p t (2a ) f sin 9 cos 0, d
vb 2 Jo 1 1 ]
u.
(213)
The lift per unit area is therefore
\
u
F F ...
v vb
TTa
(214)
This lift coefficient of 0.50 should form an upper bound for the
work to be done later.
19
At this point it seems appropriate to define certain lift
coefficients to be employed in this dissertation. The difference
lies in the area over which the force is considered. The subscript,
u, will denote those cases where the area considered is only that
directly beneath the body being considered, the projected area of the
grain. Thus, \ denotes lift per unit area based on the total bed
area, while is based on the area of the grain projected onto the
bed. Similarly, the coefficients and C are used with the corre
sponding ^*s.
As indicated, most work done on lift has dealt with single
bodies. Attention will now be turned to systems with more than one
particle.
2.5 Multiparticle studies
2.5.1 Einstein and El Samni.It was natural that multi
particle studies should arise, as these begin to approach the sediment
conditions found in nature. Unfortunately, however, work in this area
has been limited. Some values for the lift force came from the work
by Einstein and El Samni [33,34]. Using the upper onehalf of
plastic spherical balls 0.225 feet in diameter placed in a hexagonal
pattern, they measured the lift force as a pressure difference. They
made the following statement.
The procedure in making such measurements was as follows:
if a lift force is exerted on the top layer of a stream bed,
the solid support of the sediment particles is relieved of part
of their load and this load is transmitted hydrostatically to
the fluid between the solid bed particles. Thus, it must be
possible to detect and measure this lift as a general pressure
increase of the pore fluid in the bed. [33, p. 52l]
20
Their results enabled them to write
2
Ap = CL p Â¡y (215)
where Ap = pressure difference
CL = lift coefficient
p = fluid density
u = velocity
They found a constant 0.178 if u was taken as the velocity 0.35
sphere diameters above the theoretical bed, determined by experiment
as 0.20 sphere diameters below the sphere tops. Further studies
which they made on natural gravel yielded the same expression for Ap
with some redefinition of u along lines consistent with Einstein's
earlier work [12 ]. This work forms the only example of a lift force
essentially integrated over a number of particles, though the distri
bution over individual particles was not ascertained. Support for the
rationale of measuring lift as a pressure difference is given by
Engelund and Hansen [35, p. 19] in discussing variations from hydro
static pressure due to streamline curvature.
2.5.2 Chepil.Chepil [36] performed experiments in a wind
tunnel on hemispherical elements placed on a plane bed in a hexagonal
pattern three diameters apart. He chose this spacing based on work
by Zingg [37], which indicated this is the average spacing between
particles erodible from a sand bed.' The processes of erosion of sand
by wind and by water involve essentially the same factors, as indicated
by, among others, Kadib [38].
21
Chepil measured pressures over the surface of one metal
hemisphere. This was done by placing pressure taps, starting at the
base of the hemisphere, 30 degrees apart along one line running parallel
with and another normal to the wind direction. The negative pressure
end of the manometer was connected to a tap on top of the hemisphere.
The remainder of the hemisphere pattern consisted of gravel hemispheres.
The hemispheres occupied 11 per cent of the total floor area. The lift
and drag forces on the hemisphere were determined by integrating the
measured pressure distributions and also, as a check, by means of two
torsion balances measuring the forces directly. Hemispheres of three
different sizes were used, and also some measurements were made on
relatively small sand and gravel mounds. Measurements were also made
at different points downstream in the tunnel to note the effects as the
air boundary layer developed to its full extent.
Chepil found that increasing the depth of the fluid boundary
layer, after a certain limiting depth is reached, has little effect on
the lift to drag ratio, though the depth of boundary layer had a
profound effect on the magnitude of both lift and drag. For the study,
it was found that
LIFT = 0.85 Drag (216)
For this study it was also found that the pressure difference between
the top and the bottom of the hemisphere is about 2.85 times the lift
per unit bed area directly under the hemispheres. Using the latter
finding, Chepil surmised that the C from Einstein and El Samni [33]
should equal 0.178/2.85, or 0.0624. He then used equation (216)
22
with his drag and velocity measurements to attempt a correlation with
the case of closely packed hemispheres and found a C of 0.0680.
Li
Chepil concludes by noting that "This study shows that lift on
hemispherical surface projections, similar to soil grains resting on
a surface in a windstream, is substantial. Therefore, lift must be
recognized together with drag in determining an equilibrium or crit
ical condition between the soil grains and the moving fluid at the
threshold of movement of the grains" [36, p. 403].
2.5.3 Chao and Sandborn.Chao and Sandborn [39] also performed
experiments on spheres, but, by means of a transducer, they actually
measured the pressure distribution on the upper half of an element.
However, the type of flow they used bore no real relation to that in
nature's streams, and no attempts were made at analyzing velocities and
the like. Einstein used the flow of water in a channel with a measured
logarithmic velocity distribution. On the other hand, Chao and Sandborn
placed lead shot on a flat surface and blew a stream of air down onto
the particles, the air diverting horizontally at the flat surface.
Their conclusions included, "The present experimental results are of
primary interest in demonstrating that a problem exists. . More
extensive research is needed before there can exist a better understand
ing of the mechanism" [39, p. 203].
2.6 Shapes of bodies studied
With the exception of the vast amounts of work done on wings,
struts, and related areas, most effort has beenaimed at circular
cylinders and spheres, especially in work related to the sediment
t
23
problem. There have been, however, numerous indications of shape as
a parameter. Rouse defines sphericity [40, p. 777] and indicates it
as a probable factor. Studies have been made on shape influence,
including one by Krumbein [41] which specifically considers nonspherical
particles. Other works also exist for considering the effect of non
spherical particles.
Most design procedures today treat the bed material by conver
sion to a representative bed consisting of uniform spherical particles.
The conversion is generally based on factors such as particle density,
sizes, exposed areas, and volumes. Thus, there are means for relating
natural beds to the uniform spheres to be treated in this work.
2.7 Relation to work of dissertation
The preceding background material was presented in an effort to
point out those factors with special bearing on what the author is try
ing to accomplish in this dissertation. Generally, an attempt will be
made to advance the knowledge of lift through analytical work.
Analysis of the threedimensional cases corresponding to the
work of Einstein and El Samni and Chepil will be made. These multiple
particle systems allow consideration of particle interactions. Poten
tial flow theory will be used to study velocities on the surfaces, these
velocities then being linked to known logarithmic distributions.
Additionally, some twodimensional work and applications will be reported.
Support for the use of irrotational flow is gained from the lift predic
tions made on airfoils, as well as from a knowledge that viscosity is
not necessary to explain lift. Of course, in the airfoil cases as well
24
as here, the predicted drag is zero, obviously incorrect, but the
description of the magnitude of the lift is the primary goal here.
It should be very strongly noted, however, that whereas these earlier
works used the potential to describe the total flow field, a much
different use will be made herein. Essentially, potential flow theory
will be used as a guide in attempting to develop means of predicting
hydrodynamic lift. Success here might lead eventually to better
analytical approaches to some sediment problems, although it will
also be used in larger scale problems dealing with dunes and ripples.
Summarizing, it can be said that the author is relying on the
success of earlier approaches to related studies in an effort to find
a guide to a better understanding of hydrodynamic lift in sediment
transport.
CHAPTER III
LOGARITHMIC VELOCITY DISTRIBUTION
3.1 Development of logarithmic velocity distribution
Due to the use which will be made of the velocity distribution,
the author wishes to present briefly some background and to study some
specific points. Historically, three modern approaches to velocity
distributions in steady, uniform turbulent flow have arisen. The three
are the following: Prandtl [42], who introduced the concept of the
mixing length (related to the mean free path of particles) with momen
tum conserved; G. I. Taylor [43], who considered vorticity to be con
served along the mixing length; and von Karman [44], who developed a
similarity hypothesis for the problem.
Prandtl's derivation, beginning with the expression for shear
stress in fluid, is frequently cited in texts, such as [45]. It
neglects the viscous forces and considers only the socalled Reynolds
stresses, after 0. Reynolds [46]. The final expression can be written
as below.
(31)
where
B = C +  In k
k = equivalent sand roughness
25
26
The socalled von Karman constant, h, has been verified
experimentally, as noted by Bakhmeteff [47], to equal 0.40. The
values for B were obtained from experiments by Nikuradse [48,49],
who made tests on smooth pipes and then on pipes roughened by gluing
uniform sand grains of size k to the wall. Results in the transition
range between smooth and rough flow regimes by Colebrook [50] showed
different Bvalues for nonuniform roughness.
Hydrodynamically smooth and rough flow regimes are generally
defined in terms of the wall Reynolds number (v_k/v), R The smooth
f ew
range, R&w less than 3.5 to 5, has a viscous sublayer of depth compar
able to bed roughness size and viscous forces play a dominant role.
In the rough range, greater than 70, the roughness elements have
fully penetrated the sublayer, and viscous effects are negligible.
Most sediment problems in nature occur where flow is in the
rough range. This will be the range considered in the present work.
Substitution of the appropriate values for the rough range yields
= 8.48 2.5 In f (32)
vÂ£ k
Although this expression was derived from work in circular
pipes, it has been noted on many occasions that it is similarly .
applicable in open channels.
3,2 Special problems of present expressions
Working with measurements near rough surfaces, Hwang and
Laursen noted that ". . the adequacy of this logarithmic equation
can be debated, especially near the boundary, and the indeterminacy
27
of the zero datum presents difficulties. ." [51, p. 2l]. These
essentially are the two problems with use of (32).
3.2.1 Theoretical bed.In a bed in which roughness elements
protrude, some reference datum must be decided upon. This reference
is usually referred to as the theoretical plane bed, and, if sufficient
information is available, its location can be computed. Generally,
satisfactory results are obtained by use of this datum, though problems
may arise for very irregular surfaces.
The basis for computation is to place the bed at such a level
that the water volume contained beneath it equals the volume of rough
ness elements protruding from the bed. Justification for use of this
definition can be gotten by reviewing the work of Einstein and El Samni
[33], Using hemispheres of diameter k and placed in a hexagonal pat
s
tern, they found that for satisfactory results it was necessary to
assume the theoretical bed at a distance of 0.2 kg below a tangent to
the tops of the spheres. Their arrangement appeared as shown in
Figure 6.
Realizing that the area considered (inside the dotted lines)
has dimensions 3ko by \/5" ko, the following is obtained.
(33)
Volume under theoretical bed = 3/73* k y,
s 'b
Volume of particles
Equating these two and solving yields
1 3
=2 n ks
(34)
y, = 0.302 k
Jb s
(35)
28
Figure 6. Arrangement of spheres and theoretical bed in
EinsteinEl Samni work.
29
Therefore, the distance down from the upper tangent is
0.198 k in excellent agreement with the 0.2 k found by experiment,
s s
It might be noted that the value of 0.2 kg seems to be satisfactory
even when the roughness pattern is more irregular. This was shown
experimentally by Einstein and El Samni [33]. Other instances could be
reviewed, but let it suffice to state that the above definition of the
/
theoretical bed is satisfactory.
At this point the conditions for the Chepil [36] case are shown
in Figure 7. The computations for the theoretical bed are also shown,
since this will be needed later. Considering the region inside the
dotted lines, with dimensions 3K by 3K the following is obtained:
s s
Volume under theoretical bed = 9^/3* K Y, (36)
s b
13
Volume of particles = TT K (37)
6 s
Equating these and solving yields
Y = 0.0335 K (38)
b s
Therefore, for this pattern the distance down from the upper tangent to
the hemispheres is 0.467 very near the plane bed itself.
3.2.2 Conditions near the bed.Experience indicates that
expression (32) is quite valid away from the bed. However, inspection
reveals that problems occur near the wall. As y approaches zero, the
value of v/v and hence of v, approaches minus infinity. This obvious
discontinuity presents a real problem in understanding what happens at
the bed, where sediment movement begins.
Flow
O o
Figure 7. Arrangement of spheres and theoretical bed
in Chepil's work.
31
3.3 Use of proposed adjusted velocity distribution
Christensen [52] is suggesting use of a slightly different
velocity distribution which he has developed. The new expression is
as follows:
= 8.48 + 2.5 In (f + 0.0338)
v_ k
(39)
Calculation will reveal one very desirable feature of (39),
that being prediction of a zero velocity for y equal to zero. The
added constant factor will continue to have a large effect very near
the wall, but as y/k increases, the effect of the additional term will
quickly become negligible, yielding essentially the same expression
as (32). This is correct, since (32) has proved adequate away from
the wall.
3.3.1 Comparison of distributions at wall.In addition to
the effect of zero velocity at the wall as compared with the infinite
value predicted by (32), other features can be noted by looking at
the change of velocity with distance from the wall, or dv/dy shown
below.
Former: ~ =2.5
dy y
dy
(310)
For y * 0 dv/dy * 00
y dv/dy * 0
QV
Proposed: = 2.5 v
(311)
For y = 0, dv/dy = 74.0 v^/k Finite
y  , dv/dy 0
32
In both cases, the derivative approaches zero for large
yvalues, as it should. However, at the wall, a definite discontin
uity exists by the former method. The new proposal predicts a finite
change at the bed, a far more reasonable development. The author
feels that the proposed distribution, with its continuous curve, will
enable better studies of action near a rough bed.
3.3.2 Comparison of distribution with increasing y.The
increasing value of (yA) will eventually negate the effect of the
added term in (39). Table 1 shows the difference in velocity indi
cated' by (32) and (39) for some (yA) values. The difference in
hydrodynamic force, proportional to the velocity squared, is also
shown. It can be seen that the effect of the added term in (39) is
dissipated very rapidly.
TABLE 1
COMPARISON OF PROPOSED AND FORMER DISTRIBUTIONS
Variation between
(32) and (39)
yA
Force
Variation
1%
0.99
2%
3%
0.61
6%
5%
0.29
10%
33
3.4 Determination of k and v^ from experimental data
Essentially, in either equation (32) or (39), it is neces
sary to find k and v^ to fully describe the profile in a given
situation. If the distribution is first measured to be logarithmic,
k and may then be determined by considering the velocity at two
different depths [45].
Call the total flow depth d. Consider then a depth pd (p,
a fraction), where the velocity is v Consider also a second depth
qd with velocity v .
k, the equivalent sand roughness
v^, the friction velocity
From these measurements the following can be written, using first
expression (32).
Depth 1:
v ,
Â£=8.48*2.5 In Â£Â£ (312)
vf k
Depth 2:
2. = 8.48 + 2.5 In (313)
v k
34
Combination of these equations and elimination of v^ yield
k =
29.7 pd
<Â£>
(v )/(v V )
p p q
(314)
Elimination of k yields
v
f
v v
p q
2.5 In (Â£)
q
(315)
Frequently the depths chosen are for p = 0.90 and q = 0.15.
Use of the proposed expression in (39) for equations such as
(312) and (313) above, does not lend itself as readily to direct
elimination of variables and solution for k and v^. Therefore, some
consideration must be given to the relative values of the terms.
As noted earlier, for terms where y/k is greater than one, the effect
of the added 0.0338 is less than 1 per cent and can presumably be
neglected. If indeed p = 0.90 and q = 0.15 were used, for most
practical cases y/k would be far greater than one for both depths,
and the expressions for k and v^ would be precisely those found in
(314) and (315).
In any case, the mechanism is available for computation of k
and Vg through observations. These parameters can then be used in
computing forces, velocities, and the like according to equations
to be subsequently derived.
35
3.5 Effect of sidewalls
As will be seen later, the experimental flume to be studied is
rather narrow. For this reason, possible effects of the sidewalls on
the velocity will be discussed here.
Sayre and Albertson mentioned this problem, when, in reference
to their analysis, they indicated that one assumption which they made
was . that the channel is of sufficient width, or that the bed
roughness is so great relative to the sidewall effect" [53, p. 124].
Rouse [54, pp. 276277] points out the effect that sidewalls have,
through secondary flows, in varying the isovels and in depressing the
region of maximum velocity below the surface of the flow.
In order to provide some quantitative means of evaluating the
sidewall effect, aside from the qualitative evaluation of experimen
tally obtained velocity profiles and isovels, the following approx
imate analysis is presented. Figure 8 indicates the assumptions util
ized listed below.
Assumptions:
1. Idealized isovel picture, enabling passing of
line through corners.
2. Idealized pressure distribution (varying linearly,
as indicated in Figure 8).
3. proportional to y^ in some way; proportional
to y in the same way.
4. Logarithmic velocity distribution in center line
profile.
Roughness
Idealized
Isovels
/
/
/
\
\
\
I
\
h
/
!
I
' Roughness:
\
J:
k
2
Figure 8. Sketches for sidewall effect.
37
The following expressions can be written relating the shear
stresses.
tan Qi
(316)
0 h + t2 q(W h tan O') = yh WS^
This can be written as
(317)
T1.0 T2.0(h tan 1
ws.
(318)
Equation (316) results from assuming and t^ to both be
proportional to. their respective yvalues in the same way. The other
expressions merely equate the weight of fluid acting on the wetted
perimeter and the shear forces resisting it.
Obviously the value of tan Qi is of primary importance in this
analysis of sidewall effect. The equation enabling evaluation of the
angle comes from equating expressions for the velocity at a point such
as A in Figure 8, computed first with reference to the channel bottom
and then with reference to the side. For this analysis the form (32)
will be used for the velocity.
yn /Tn j
{8.48 + 2.5 In [ri]) V = {8.48 + 2.5 In [ .. .
1 kj J P k2" p
Rewriting this and using the similar proportionality of
and T2 to y^ and yields
(319)
y y
[In 29.7 ] = [In 29.7 ^
(320)
38
TI = y A
Figure 9. Evaluation of constant for sidewall analysis.
39
Introducing y/k = Tj enables the writing of
FC^) = ^ F(T]2)
(321)
where
F(T) = [In (29.7 Tj) ] v'T
The curve shown in Figure 9 indicates that the Ffunction can
be approximated by a function of the form
F(T) = atf
(322)
This would enable the writing of
T
'2
\^Â¡i
l/b
(ylAl) yl k2
(y2A2) y2 kl
(323)
Using the fact that = tan it can be written that
l(l/2b)
tan or = ()
k2
<324)
Evaluation from Figure 9 reveals that bcequals about 0.664. Thus,
, 0.248
tan o, = ()
k2
(325)
The last equation provides an opportunity for an
approximate evaluation of the sidewall's effect on the isovels and
hence on the center line profile. The likelihood of simulating two
dimensional flow in the center region can then be estimated.
40
Essentially, the work above is an attempt to check beforehand
the probable validity of all the previous discussion on logarithmic
distributions in this chapter. The chapter, in general, was intended
to point out phases of the velocity distribution applicable to the
present work.
CHAPTER IV
TWODIMENSIONAL WORK: EARLIER RESULTS
4.1 Shapes studied
In earlier work the author has studied some twodimensional bed
shapes. In one case a convenient potential function (that formed from
an infinite row of vortices) was chosen, and the grain shape produced
by this was taken. Here the solution was in a closed mathematical form
In other cases, a series of elliptic cylinders was chosen, and the
solution for the irrotational flow obtained, in terms of the stream
function, by finitedifference methods. Some of the results obtained
and methods used will be noted briefly here as a prelude to the three
dimensional work of this paper.
4.2 General methodsvelocity and pressure results
The potential flow solution for flow around the given shape was
first obtained, using Laplace's equation involving the stream function.
The solution was used to compute velocities along the upper surface of
the shape. Then, applying Bernoulli's' equation for a streamline along
the upper surface streamline, the pressure distribution could be found.
The grain shape obtained from the potential of an infinite series of
vortices is shown in Figure 10, and Figures L1 and 12 show the computed
pressure distributionon these repeating grains. It can be seen that
Figure 11 indicates, in the difference between the upper curve and its
41
Main Stream Flow
JSJ
Figure 10.
Grains placed in rough bed configuration.
p
fO
43
Figure 11. Pressure distribution on twodimensional grain.
Figure 12. Piezomettfic head distribution on twodimensional grain.
44
dotted counterpart, the buoyant lift, while the remaining pressure
decrease on the upper surface contributes to hydrodynamic lift.
In the experimental work, the actual quantity to be measured
is the difference in piezometric head between upper and lower surfaces.
Since the situation below the grain is hydrostatic, the difference can
be taken between a point on the surface and any point in the region
below.
Results for velocitysquared distributions for the elliptic
cylinders can be found in Figure 13.
4.3 Lift integration
Integrating the vertical pressures over the grain surface will
yield the lift. The lift per unit area, can then be expressed by
x = cl! \ ,1).
where C = lift coefficient
L
u^ = velocity at top of grain = u^
This lift coefficient was found to have a value of 0.500 for the hyper
bolic cosine grain shapes, and the results for the elliptic cylinders .
are found in Figure 14. Note the parameter b/a represents the thickness
tolength ratio of the ellipse. The numerical solution of the Laplace
equation requires a completely bounded solution space. Studies of flow
solutions for single elliptic cylinders and for the hyperbolic cosine
grains indicated that the streamlines became essentially straight lines
45
Dashed Line: cosh bed shape
Figure 13. Distributions of surface hydrodynamic pressure
decreases.
0.1 0.2 0.5 1.0 5.0 b/a
Figure 14. Lift coefficient C .
F
o\
47
at distances of two to three times the particle height; Hence,
a solution space was chosen which was four times the height of the
particle in question.
4.4 Calculation of predicted lift by use of
measured logarithmic velocity profile
Figure 15 illustrates the basis for relating the potential
predictions to the actual turbulent flow. The general procedure can
be outlined in a few steps. These steps are cited under the assump
tion that it is established that the turbulent velocity profile is
indeed logarithmic in nature and that measurements have been made.
1. For the relationship (39)
yy
= 8.48 + 2.5 In ( 0.0338)
vÂ£ k
evaluate v^ and k as outlined in Section 3.4. Notice the use of
(y y^) to indicate that this refers to distance above the theo
retical bed.
2. Then, using (39) at the top of the grain, evaluate u^
from the measured data.
3. Calculate the lift by equation (42).
2
ut
Unit area: h = C p r (42)
Li Z
Thus, by relating the velocity profiles, it is possible to
make a prediction of the lift.
Figure 15. Theoretical bed used in relating logarithmic and potential velocity profiles.
p
oo
49
The discussion above and Figure 15 indicate the case for the
hyperbolic cosine grains and a logarithmic velocity profile, but any
other, bed shape or velocity profile could have been involved.
4.5 Application to experimental results
At the time of the earlier work, the author referred to some
measurements made by Vanoni and Hwang [55] over a part of an alluvial
bed which had been allowed to form twodimensional ripples under flow.
The bed was artificially stabilized and then pressure measurements were
taken. Measurements were used corresponding to Vanonis C series, with
a flow depth of 0.350 feet above the mean bed.
The same methods as indicated earlier were used, solving first
for potential flow in the solution space indicated in Figure 16. It was
assumed, for ease of solution, that the streamlines had horizontal
tangents at the two ends of the space. The solution also was developed
using the dividing streamline in both cases as a portion of the bound
ary, thus assuming a region of no flow beneath the streamline. Fig
ure 17 indicates the results of the computations, which seem to approx
imate the measurements well.
This chapter was presented to provide a link between the
earlier work and the work of this dissertation*
Figure 16. Definition sketch for experimental application.
0.2
reference pressure
0.4
0.6 0.8 1.0
Bed Location
u : mean velocity
m
Figure 17. Comparison of theoretical and measured values.
CHAPTER V
THREEDIMENSIONAL NUMERICAL SOLUTIONS
5.1 General
i
Previously it was indicated that irrotational flow theory would
be utilized to study velocities on a threedimensional bed surface
toward the end of making an analytical evaluation of hydrodynamic lift
on such a surface. The results of some previous similar twodimensional
studies were presented in Chapter IV. In this dissertation emphasis is
placed on a comparison of the analytical results obtained by the writer
with those obtained by Einstein and El Samni [33,34] and Chepil [36].
Evaluation of lift in these cases by the proposed method will require
solution of the potential flow equation for the two arrangements of
hemispheres as indicated in Figures 6 and 7.
A potentil flow solution implies the solution of Laplace's
equation (51) in the given flow space.
I = 0 (51)
cbc dy dz
Laplace's equation is of elliptic type and hence its solution
is fully determined by conditions on the boundary enclosing the solu
tion space. On the boundary must be specified either values of the
potential function, 9 or its derivative normal to the boundary.
52
53
5.2 Problem formulation
5.2.1 Choice of solution method.There are numerous ways of
attempting a solution for Laplace's equation in a given case.
Robertson [56] gives a listing of several such methods, some of which
were considered for the solutions to be made herein.
The method of separation of variables was eliminated because
no convenient coordinate system is available to represent the boundary
involved in these cases. The method of integral equations, in. which the
differential equation (51) is expressed instead in integral form and
a solution made,was eliminated because some past works indicated that
threedimensional studies by this means had resulted in exorbitant
computation time on the computer. Another method which has been
employed is the method of integral transforms where the number of var
iables in the equation is reduced to two, and the resulting two
dimensional problem solved by numerical methods. These twodimensional
results ar then inverted back to the threedimensional case to obtain
the desired final result. Tranter [57,58] shows some examples of this
approach. This method, however, has not been frequently used, for it
is dependent upon finding the proper transform kernel, which is not
usually possible. The method frequently used to obtain desired body
shapes in a flow is that of the summation of singularities and their
effects, such as sources and sinks distributed throughout the flow
volume. The distributions and relative strength of these singular
ities would be varied until their accumulative effect produced a surface
across which no flow occurred that was sufficiently similar to the
54
hemispherical surface being treated here. While the latter method
showed promise, there were numerous problems involved, such as a choice
of distribution and types of singularities and possible stability
problems in the numerical process required to vary these factors and
approach a solution. Having considered these other alternatives, the
author finally chose the method of finite differences for the solution
since this method enables obtaining the desired accuracy, while, at
the same time, providing a mathematically stable numerical analysis.
The potential flow case being considered is one in which the
depth of flow is essentially infinite. However, the choice of finite
differences method of solution makes treatment of such an infinite flow
space impossible. Therefore, the geometry of flow patterns of other
cases was reviewed with an eye to choosing a depth of flow which would
enable solution of the problem without adversely affecting the desired
results.
5.2.2 Depth of flow space.Work in two dimensions reported in
Chapter IV indicated that a depth of four times the height of the bed
element was adequate to approach the straight streamlines associated
with free stream flow. To further check this, consider the flow around
a single sphere, where, from [29]
Ua3
cp =  cos 0 + Ur cos 0 (52)
2r
This is the same case described in Section 2.4.6. The expression for
the tangential velocity component is shown below, along with its value
when 0 = tt/2, or points immediately above the center of the sphere.
55
 = ( + U sin 6)(1 +
For 0 = tt/2,
J;
r 50
U(1 + ir)
2r
(53)
(54)
Equation (54) expresses the horizontal velocity across the
0 = tt/2 axis above the center of the sphere. In a free stream such
velocity would simply be U. Hence, the velocity at any distance r can
be compared with U as an indicator of how closely flow at that depth
approaches the free stream. Using (54), it can be seen that the
velocity shown there is only 1 per cent greater than U when r = 3.7a.
Thus, 4a would provide less than 1 per cent deviation.
If (52), representing one plane in an axisymmetric problem,
is converted to Cartesian coordinates x and z for y = 0, (55) is
obtained.
'P = Â¡+Ux <55)
where S = (x2 + z2)(2)(2/3)
From this, expressions for the x and zvelocity components can be
found, along with the ratio of the latter to the former. If this is
done, and z is chosen as 4a, with x as +a, it is found that the
vertical velocity component there is less than 1 per cent of the
horizontal component. Thus, at this point also, y = 4a gives a good
approximation of free stream flow, with streamlines horizontal.
56
For these reasons as well as earlier studies, a flow space of depth
four times the hemisphere height was chosen for the solution space
for (51).
Choosing a depth for the flow space assumes streamlines at
that elevation to be horizontal. Thus, at any greater depth than this
the indicated flow lines above that level will also be horizontal, and
the solution along a lower boundary will remain the same. It might
also be noted that the choice of such a solution space eliminates any
consideration of free surface effects, such as wave resistance. Flow
of greater depth relative to particle size may then be superimposed
on the flow space solved for with the knowledge that the solution of
distribution of velocity and pressure along the lower boundary will
remain essentially the same. The error here is determined by how
closely the stream surface at the chosen elevation approaches a flat
plane. It should also be noted that the depth chosen would form an
even better approximation in the cases of more closely packed hemi
spheres being treated here, since the stream surfaces in these cases
more rapidly approach free stream conditions.
5.2.3 Boundary conditions.Due to the periodic nature of the
hemispheres in the solutions, numerous conditions of symmetry are avail
able which enable reduction of the solution to handling a typical
repeated portion of the total flow space. Figures 18 and 19 represent
the portions chosen for the solutions of this dissertation. It can
be seen that the two drawings are different in that Figure 18 could be
divided by "symmetry" one more time. For purposes of the finite
57
differences solution, however, this was not done. Two alternative
approaches were available, but neither seemed as practical as simply
including two hemispherical portions in the solution space. One of
these alternatives was assuming an equipotential surface between the
two hemispherical portions and making finite differences solutions in
which this assumed potential surface was to be checked and itself
adjusted and treated as a variable in a solution. However, this would
have involved a complete solution of the problem for each assumed sur
face and would have required, therefore, exorbitant computation time.
The other alternative was to choose a plane between the two hemispheres
and use it along with conditions of "symmetry" to reduce by onehalf
the number of points involved in the differences solution. In the case
of Figure 18, the proximity of the hemispheres introduces complexities
which offset the benefits gained in computation time. For the Chepil
case, however, the latter approach could be used easily as indicated
in Figure 19.
The constant xplanes passing through the hemispheres are
equipotential surfaces. All other boundaries in the solution spaces
represent surfaces across which there is no flow, with the exception
of the added symmetry boundary shown in Figure 19. This means that the
derivative of the potential function normal to the given surface is
equal to zero. Thus, the following conditions hold.
On constant yplanes through hemispheres:
^=0 (56)
By
On upper zsurface:
(57)
58
xconstant plane:
cp = cp^ = constant
Figure 18. Solution space for closely packed hemispheres.
59
xconstant plane:
"foldedsymmetry" axis
Figure 19. Solution space for Chepil's arrangement.
60
On lower zsurface:
j*p =
dz
0
(58)
On hemispherical surfaces:
*Â£ =
dh
0
(59)
The selection of the symmetry plane for the Chepil case is
based on a type of "folded" symmetry. This is indicated in Figure 20
and equations (510) and (511).
Figure 20. Foldedsymmetry boundary.
61
9b = (^ + 9^ 9A (511)
The problem at hand is therefore to solve Laplace's equation
in the regions indicated, subject to the foregoing boundary conditions. t
5.3 General finite differences approach
The method of finite differences consists of replacing a function
continuous within a region by its values at certain points in that
region. The equation to be solved is then expressed in difference form
at each of these points, and the problem becomes one of solving a set
of simultaneous equations corresponding to the points chosen. As noted
earlier, one of the reasons for choosing this method for the present
work is that, due to the nature of the elliptic equations, "... there
is no problem with stability of difference equations approximating
elliptic partial differential equations" [59].
Some consideration was given to the type of coordinate system
to be used. The spherical coordinate system had some advantages in
expressing the noflow condition at one hemispherical boundary, but was
less efficient at other boundaries and at the second hemisphere. Hence,
the ordinary rectangular Cartesian coordinatesx, y, and zwere chosen.
A general sevenpoint scheme was chosen for the difference solution, as
shown in Figure 21, with equal increments of length h in all three
coordinate directions. For this scheme, the difference equation becomes
that of (512), as found in works by Allen [60], Collatz [61], and
62
Allen and Dennis [62]. Section 5.4 will give details of how this
is obtained.
6
2 cp. 6 cp =0 (512)
i=l 1
or
1
^o = g S cpL (513)
1=1
6
where 2 cp^ represents the 6 points around cpQ.
Figure 21. Sevenpoint finite difference scheme.
Similarly, if a net, or lattice, were placed over the entire
solution space, an equation could be written for each point of the net
of the same type as (512). Thus, Laplace's equation could be written
in the form of a matrix equation as
63
AX = B (514)
where A is the matrix of coefficients of cp, X is the matrix containing
9 elements, and B is a matrix determined by the boundary conditions.
Therefore, inversion of the matrix A should enable a solution for X,
the potential field. However, A is usually a very large matrix which
i
has many zero entries. Fox [63, p. 185] states that, "There is no point
in ever evaluating an inverse A1 for the purpose of solving equations
of the form AX = B. Elimination and back substitution or its compact
equivalents are always faster. . ." On the subject of sparse matrices
(many zero entries) such as occur here, Fox [63, p. 189] says that,
"Iterative methods are used . for matrices of large order but with
many zero coefficients. ..."
The foregoing comments bring to mind the methods often termed
relaxation methods. These procedures were popularized before the rise
of computers by such people as Southwell [64], Shaw [65], and Allen [60]
As digital computers have become more available, variations in the
relaxation processes have led to expressions more applicable to the
computer. That method wherein the iterative methods are applied to
the points of the net singly and in an orderly repetitive fashion,
through numerous iterations, has become known as the Liebmann method
[66], which employs equation (513) when a rectangular Cartesian system
is used.
In order to accelerate convergence of the iterative method, the
process known as overrelaxation [67] is used, employing an overrelax
ation factor, u), and utilizing an equation of the form of (515).
64
9
o
0)
6
(1 cu) cp'
O
(515)
where C{/ indicates the value from the preceding iteration. Obviously,
as in equation (513), the value of cpQ from (515) will show little
change as convergence is neared.
If the value of cu is equal to one, the method is called the
Liebmann method, while the name extrapolated Liebmann is applied for
cases where U) is not equal to unity. The latter case will be employed
here. Much work has been done, at least in twodimensional cases, in
finding an optimum cu to give most rapid convergence [59]. Approximate
indications are that the optimum covalue for the cases herein, lies
below about 1.80. However, no great amount of work will be done to
refine this value, as such work could easily involve very extensive
t ime.
The solutions of this dissertation will therefore be carried
out using a sevenpoint finite difference scheme in rectangular Cartesian
coordinates, applying the extrapolated Liebmann iterative method. The
ensuing sections will develop the needed relationships for use in
I
special situations.
5.4 Finite differences equations: interior space
This section will present the equations needed to handle all
the different situations which arise in the solutions. While many
applications of finite differences have been made in two 'dimensions,
few cases other than problems involving simple cubes, and the like,
65
have been treated in three dimensions. This condition has caused some
new procedures to be employed in the following work.
For clarity, certain conventions will be followed in presenting
the equations and their descriptions. First, each point will be given
three subscripts representing the three coordinate directions, with I
denoting the xdirection, J denoting the ydirection, and z represented
by K. Hence, the subscript 1+1 implies the next point in the positive
xdirection beyond a point at I. Additionally, where one or more of
the variables is a constant for the investigations of a special relation
ship, the figure will be drawn in two dimensions, eliminating the third,
constant dimension.
5.4.1 General lattice point.Equation (513) was presented
as representing those cases where six adjacent points are available,
all at a distance h. However, frequently one or more points lie at
some distance other than h. For this reason, the general equation will
be developed for the case of Figure 22, where all six lengths are dif
ferent from h.
Figure.22. General lattice point*
66
Difference expressions can be formed for this case.
Forward difference:
6I+1,J,K ~ ^
Q^h
I,J,K
Backward difference:
&Â£ = yi.J.K ~ ^Il.J.K
Ax o^h
(516)
(517)
From these the following equation for the second derivative can be
obtained.
9.
1+1,J,K
" ,J,K ~ ^Il.J.K
Tsra5 
V
ff2h
(Oii+Q'2)h
(518)
or
JL2. ~ ,J,K ^I.J.R ~ ^Il.J.K
ax'
a (q? +a )
2 1 2J
(519)
Similar expressions can be found for the other directions,
1,2 a2cp CPI.J+1,K ^X.J.K CpItJ,K ^I.Jl.K
2 *ay2 WV WV
(520)
L2 = ^I^K+l ~ ^1 ,J,K ~ ^I.J.Kl
2 az2 WV WV
(521)
Adding these three expressions yields an approximation for
the Laplacian.
67
2 V2 CpI+I>J;K ~ Cf'1>J>K ~ ^11,3,K
2h ^ a (o +a ) a (o? +a )
lv 1 2J 21 2
^J+^K ^I^jK ^I,J1,K
+ c (a +a ) ~ a (o +a, )
3v3r 4V34
^I^K+l
a_(Qf +of )
3 5 6
~ ^I^Kl
a*(0lTi + a&')
OJO
= O
(522)
Since the desire at a given point is to solve for the potential, or cp
value there, the equation above must be solved for cp(IjJ>K).
/ *I+
1+1,J,K ^11,3,K + yI,J+1,K 1
0il(ai+a2') + a2('ai+0i2^ Q'3(Q3+Qi4)
^1, Jl, K ^I^K+l ^I^Kl
+ a (a+a ) + or (a +a ) + a* (Qv+Q,<) )
v 4 3 4 556 656'
9.
I,J,K
(523)
, a/,ia5+a/,) (a +al (or+of,) 7
' 434 556 656
Note that if all legs are of length h, which implies that all arvalues
are 1.0, equation (523) reduces to (513), as it should.
Equation (523) is applicable for all interior points of the
solution space. The six points may contain among them points that
lie on a regular lattice point, on a hemispherical surface point where
cp is being calculated, on or beyond a noflow boundary, on or beyond
the "foldedsymmetry" boundary of Figure 19, or on one of 'the two
equipotential planes at the xextremities of the solution space.
68
The next few sections will cover these cases as well as the grading of
the lattice. For clarity, the point cp ^ will be termed the object
point, while the six surrounding points will be called adjacent points.
The case for an adjacent point, lying on a regular lattice point or on
a hemispherical surface point where 9 is being calculated, will be
omitted, as these simply involve the substitution of the present value
of cp for that adjacent point. The same is true when an adjacent point
lies on one of the equipotential planes.
5.4.2 Object point on planar noflow boundary.The tern
planar is intended to exclude the hemispherical boundaries, which will
be treated in Section 5.5. Therefore, the planar surfaces indicated
are the constant ybounding planes and the constant zbounding planes,
as illustrated in Figures 18 and 19. If the object point lies on one
or two of such planes, one or more of the adjacent points will lie
outside the solution space, and it must be replaced by an equivalent
expression involving points on the interior of the space. For this
purpose, use is made of the fact that the derivative of cp normal to the
planes is zero. Then the value at such an exterior point can be
expressed as equaling the corresponding point inside the space. Fig
ure 23 illustrates what is meant. Here, as later, cp represents the
cpvalue at a surface point.
69
Constant
zplane
^.J+ljK
o
II
>>
*^1, J, K
/
,9
/
(a)
Constant zplane
Figure 23. Examples for object point on planar boundary.
The preceding requirements can be summarized as follows, where
the equations imply replacing the left member by the right member for
use in equation (523).
For z = 0:
^1, J,K1 ~ ^IjJ^+l
(524)
For z on upper bound:
,J,K+1 = ^IjJjKl
For y = 0:
^I.Jl.K = ^1,J+1,K
For y = a (or y = 3a):
CpI,J+l,K = ^IjJljK
(525)
(526)
(527)
Note that in the last two equations the terms may represent regular
lattice points, as in Figure 23a, or points on the hemispheres, as
in Figure 23b.
70
5.4.3 Object point on "foldedsymmetry" boundary.This
problem arises only in the case with hemispheres spaced wider apart,
as in Figure 7. One of the points, in the sevenpoint difference
scheme lies beyond the boundary, but the symmetry of the flow pattern
makes it possible to express this value at this point by a value inside
the given space. Equations (510) and (511) show the expressions
needed. The value thus obtained for this external point is then
inserted into its proper place in the sevenpoint formula.
5.4.4 Adjacent point on hemispherical surface.In some
instances, the lattice point being treated may have as adjacent points
one, two, or as many as three points which lie on a hemispherical bound
ary. At first, the author hoped to expand some of these points in
terms of other values in the field, while still applying the normal
derivative boundary condition (59) to most surface points. It was
found, however, that this approach created stability problems in some
portions of the iterative solutions. Therefore, the application of
the difference equations for (59) was extended to obtain a value for
all adjacent points falling on a hemisphere, with a small number of
exceptions. Discussion of the noflow condition and its use will occur
in ensuing sections.
The single exception occurs when both adjacent ypoints lie on
hemispherical surfaces. In order to obtain values for these two points
a process which will be called arc interpolation will be used. This
entails interpolating a value for the point from two surface points
where a value exists. The two points selected lie in the same zplane,
71
and the interpolation is made by the arc length between the three
points. These lengths along the arc are proportional to the angles
shown in Figure 24, and the ratio of lengths can be replaced by the
ratio, of the angles, written in (528).
Constant zplane
Figure 24. Arc interpolation for surface value.
1
= ^1 1,J,K + i+2 fal.Jl.K ^IljJ,!^
(528)
A similar expression can be written for the point on the other
hemisphere, and the two values are then inserted into the general
sevenpoint equation.
It would have been possible to use the difference expressions
at these ypoints. It was felt, however, that in the regions where
both points lie on a hemisphere, the accuracy of the arc interpolation
72
method is of the same order as the expressions which would have to be
used for (59) in such cases. Therefore, arc interpolation was chosen
as a more convenient manner of obtaining values for the points which
were still consistent with the remainder of the potential field.
The evaluation of cp at the six adjacent points has been discussed,
with the exception of surface points to which the boundary condition
equation is applied, which will be covered in Section 5.5 and following.
5.4.5 Graded lattice.The greatest accuracy is desired near
the boundaries to enable velocity calculations on those surfaces. Those
points which are farther away from the hemispheres can be handled,
therefore, with a larger mesh spacing, h. This is more true the
nearer the approach to a free stream condition. An advantage is gained
in computation, as using a larger lattice cube at some distance can
greatly reduce the number of points at which a solution is sought.
In the present cases, it was decided to use a larger spacing in the
upper onehalf of the solution space. Allen and Dennis [68] used a
scheme similar to that which is shown in Figure 25 and which will be
used here.
First, consideration will be given to those points where the
distances involved are all h or 2h = or = 1). The problem can be
considered on three zplanes. First is the plane CDJH Here, letters
will be used for points rather than the usual convention, as it may
allow greater clarity. For points such as G, D,. H, or J, six adjacent
points exist at a distance 2h; for points like Q, R, S, T, and U, the
six needed points lie at a distance h. Similarly, for all points in
plane EFKL, there are six points 2h away.
73
Figure 25. Grading the lattice.
74
There are two types of points in the zplane intermediate to
the above two; points such as V and those such as G. For points like V,
the following expression can be quickly written:
d2cp ^A^G^V
Sx2 h2
(529)
The other four needed points lie in plane GDEF and can be utilized,
since the Laplacian is invariant with respect to a rotation of axes,
and thus
d2cp [ d2cp
Sy2 5z2 Cv/2 h)2
(530)
Summing the two preceding equations leads to the final difference
expression.
2(CPA+CPG} + + + ^E + ^F 8cf)V = (5_31)
For points such as G, a triple McLaurin series expansion [68]
4
enables a result as below, with neglect of terms 0(h ).
+ 9D + ^E + VF + + + \ + \ 8
Due to the irrational numbers (square root of three) arising
from the hexagonal patterns of the hemispherical elements,.the succeed
ing planes of constant x treated in the grading process are not always
equidistant. This causes little problem for those points where
6 adjacent points are used, as these are treated precisely as indicated
by equation (523), the general equation. However, in the two cases
in the intermediate plane, equations (531) and (532) must be altered.
75
In the case of points such as V, rewrite (529) for the case where G
is at a distance h and A at a point a^h away.
2h
2 SjÂ£
r t9 v 4
^2 U,ar2) i2 V
(533)
Adding this to (530) and equating to zero produces the desired result.
 4 {' + 9v = 0 (534>
fc ^ p M
2 2
Points such as G require more attention. The triple McLaurin
expansion used to obtain (532) is such that, due to symmetry, all
oddorder terms (first derivative, third derivative, and so on) cancel
between the terms. However, an unequal xspacing causes those terms
involving a &x to remain in the equation, though those oddorder terms
3
in the other directions still disappear. Here, the 0(h ) terms will
be neglected so that of the following form can be written for cp(x,y,z)
at some point 6x, 6y, and 6z from cp^.
cp(x,y,z)
= 9,
dcÂ£
dz
+
(6x)
+ 6x
2 + (6y)2 I + (dz)
dx dy
Sy
dcp
dxdy
+
6x
6z
d2cp
dxdz
~2
S_Â£
dz2
6y 6z
>+ o(h3)
d2cp
dydz,
(535)
where all derivatives are taken at point 0. Expansions similar to this
are discussed by Olmsted [69],
76
The crossderivative terms cancel due to symmetry in directions
other than x, as do the first derivatives except for the xderivative,
which can be approximated by
&p
(l+C^h Sx
(536)
and the second xderivative, which can be approximated by
ih2JÂ£
Sx
9,
a2(l+a2)
1+n
(537)
Then (535) yields, when applied successively to all eight of the
adjacent points used by object point G, the following:
2 cp. 8cp 4h2(V2cp) 2(Qi2l)h2
L G G 2 5x2
4(a1)
" Tk? {cpv V = 0 <5"38>
where Â£ cp^ indicates the sum of the eight adjacent points. Setting
the Laplacian equal to zero yields an expression the same as (532),
except for including the last terms in the equation above.
One further point, such as N, is slightly different from V,
though the ideas are similar.
By
9.
9
Nl
 29
N
(539)
where (Nl) is the point lying a distance h from N and not shown
in Figure 25.
77
In plane DEKJ:
32cq  a2cp Id Pe + fj %
dx2 dz2 (jjl h)2
The resulting difference equation is
(540)
^Wl5 + 9D + + + 'Pj 8CPN = C5"41)
Treatment of points such as N becomes somewhat different when the
xintervals are not equal. A McLaurin expansion will also be applied
in this instance, and,as for point G, the first and second derivatives
will respect to x remain in the difference equation, as shown below.
2
2
 2h(o1) Â§ h2(cl) = 0 (542)
2 ox 2^2
dx
Evaluation of these two derivatives requires values at a point where
no value is being computed, as seen in Figure 25. Thus, such a
value is obtained by a linear interpolation between two known points,
such as D and E.
The foregoing completes the discussion on graded nets and some
special problems they create. Application of the grading process will
take place away from the hemispherical elements, as a finer lattice
is desired there.
78
5.5 Finite difference equations: hemispherical boundary
5.5.1 General.The only boundary condition which has not yet
been discussed is the normal derivative condition on the hemispherical
surfaces and expressed in equation (59). The surfaces of the two
hemispherical portions being treated can be expressed by
x + y + z =1 (543)
(xV3)2 + (y1)2 + z2 = 1 (544)
Both of these are for spheres of unit radius. Since the gradient of
a scalar function indicates the normal to a surface represented by
that function, the desired derivative is found by taking the gradient
of the above surfaces, yielding
xg+yÂ£.z^=0 (545)
ox ay oz
(xa/3) + (y1) Â£ + z Â£ = 0 (546)
In order to assure conformation to the conditions of the
problem, (545) and (546) must be incorporated into the differences
solution. The surface points encountered will be thus treated, the
only exceptions being those few points mentioned in 5.4.4. The aim
will be to express the first derivatives at the surfaces in difference
form and use them in equations (545) and (546). The ensuing sections
will describe the formation of these terms.
79
5.5.2 Xdirection derivative.In this section the general
means for expressing the boundary derivatives will be indicated.
Consider the situation of Figure 26.
Figure 26. Xderivative condition at boundary.
Indicated in Figure 26 is a fictitious cp7 value located outside the
solution volume. This point will be utilized in the derivative and
eliminated by expressing it in terms of points in the field. The
difference expression for the derivative at the boundary can be written
as (547).
Sphere 1:
Vl ~ *
20? h
*Â£. =
dx
cp cp.
11
2Q2h
(547)
Sphere 2:
80
A Taylors expansion can be written for the fictitious point, expand
3
ing it about the point (1+1). Neglecting terms of order h yields
the following.
Sphere 1:
 vx V fenM
lh j_l_ ^1+2 + C^B/o'1 ^I+l^1*
(548)
2(1+l>
Then, from (547) and (548), the derivative is approximated by
h
&p _
dx
3
1+a^
2cp
1+1
120?
+ cp
1+a^ yI+2
(549)
Similarly, on Sphere 2, the fictitious point can be found by expand
ing about the point (11), yielding an equation like (548). Subse
quent substitution into (547) and rearrangement yield
Sphere 2:
h
3
i*2
b
+
2c*21
l+a2 ^12
29
11
(550)
Not in all cases, however, do two lattice points exist in a
direction away from the boundary, thus causing a need for some other
means of finding the derivative. Two cases exist: first, the next
field point might lie on the next hemisphere, or it might lie on a
lattice point with the second field point lying on a sphere. In both
instances, the choice here is to use a simple linear expression involv
ing a oneway forward or backward difference rather than a centered
81
difference. These difference expressions simply involve taking the
difference in value between the two points and dividing it by the
distance between the two.
Sphere 1:
Sphere 2:
^1+1 ~
SCÂ£ ~ ^Il
5x Q^h
(551)
5.5.3 Ydirection derivative.For determining the derivative
in the ydirection, the same ideas exist as for the xderivatives.
However, since the points needed for an expansion in the ydirection
generally do not lie at a lattice intersection, special means must be
used to develop these points. For an indication of the problem, see
Figure 27. For simplicity, call the needed points PJ and PJl. Their
value will be discussed. First, a series expansion can be written for
the fictitious points shown.
Sphere 1:
cp' = PJ 2h {
iPJ1
4h2
fcpB + PJl 2PJ^
l 2h J
' 21
l v2 J
(552)
Rearranging yields the derivative
Sphere 1:
(553)
Similar expansion of the fictitious point at the second sphere results
in (554).
Figure 27. Yderivatives.
83
Sphere 2:
h
2PJ +
PJ1
2
(554)
The values indicated by PJ and PJ1 must be established by
interpolation, since they do not, in general, lie at a lattice point.
The interpolation will utilize points in the lattice with expansion
to be made about the nearer of two surrounding points in the xdirection,
shown in Figure. 27a as J.tl>Kc and
First, consider the case where f3h is less than afQh. Here, PJ
can be expressed as follows, for Sphere 1.
Sphere 1:
PJ
4
,cp
A+l
a' + $+a
a + a + B
o
where
A: (I+1,J+1,K)
A+l: (I+2,J+1,K)
C: (I,J+1,K)
(555)
A similar expression can be written by the other point needed, PJ1,
involving points one increment in the ydirection. In addition, the
same process yields an expression for points located from the second
hemisphere, indicated in Figure 27b.
84
Sphere 2:
pj =
A ^ a +0+0 J
o
+ &
p+a
3A1 f 1 1 *1
a' "Hv' P+CV
at' + p + a
(556)
where A: (I1,J1,K)
Al: (12 ,J1 ,K)
C: (I,J1,K)
Again a very similar expansion provides a value for PJ1.
To gain accuracy the expansions used are to be developed about
the point nearest the desired point. In the case where atq is less
than P, a slightly different approach becomes more convenient due to
the proximity of the spherical surface. For that reason, the point at
a distance at will be expressed by expanding about PJ (or PJ1), and
subsequently an equation obtained for the latter point. This is
illustrated below.
Sphere 1:
rn PT PT r
= PJ a e
o la + P J o
cpA PJ PJ cpr
Ot
(557)
p + a
r a i a r s TA/p
Pjjl = cp + ^ cp _cp } _
l p J Yc a +p 17a o of +
{V. *
(558)
85
Again, in the same manner, an equation can be obtained for the second
sphere.
Both equations (558) and (559) are the same expressions to
be used for PJ1 at either sphere, with cp^ and cp^ replaced by points
one increment further away from the spheres.
All the immediately preceding equations depend for their
expansions upon the existence of certain lattice points. In some
instances, points such as A+l (or Al) lie on a spherical boundary.
The above equations may be used, however, with a proper value of Q.
In the case where the points such as A fall on a boundary surface,
a linear interpolation between points A and C will be used. This is
a firstorder approximation, assuming a oneway derivative between
points A and C equal to the difference in their values divided by the
distance between them. The same approach will be used for either PJ
or PJ1 from either sphere where the conditions warrant.
The foregoing equations for 3cp/Sy, (553) and (554), are
based on those cases where two points such as PJ and PJ1 actually exist
In some cases, they will not both be available, and these cases must be
treated in a different manner. Figure 28 demonstrates the two cases
to be met, shown with reference to obtaining the derivative on the
first sphere.
86
Constant zplanes
(b)
Figure 28. Special points for yderivative
The case in Figure 28a will be treated by first evaluating PJ
by a linear interpolation between the two points (I ,J + 1,K) and
(I+1,J+1,K). Then, the value of PJ is used in a forward difference
expression to form
h
dcp
dy ~
pj 9,
B1
0i_
(560)
In the situation of Figure 28b, the same expression is used,
with PJ being determined by the process of arc interpolation, described
in an earlier section.
One further instance where a special case arises, involves
points near, but not at, y = 0 or y = 1, the two planar noflow bound
aries. Here, the value of PJ1 may correspond to a point on another
hemisphere outside the solution space, such as shown in Figure 23b.
As in that case, PJ1 will equal 9 The equations involving these
D
87
cases are derived in the same manner as (553) and (554), the only
difference arising in the distance between points PJ and PJ1. These
equations are included below.
Sphere 1 (y = a):
frp PJ cp' 2CpB 2PJ
3y 2cv3h a3 cv3
(561)
Sphere 2 (y = 0):
, &Â£_ cp7 PJ 2PJ
5y 2a4h " C*4
(562)
The preceding subsection has described the means for obtain
ing a value for dcp/dy at the hemispherical boundaries. It remains
now to compute dcp/dz..
5.5.4 Zdirection derivative.The computation of the value of
dcp/dz is in many instances virtually the same as the case for dcp/dy.
In fact, Figure 27a serves as a representation of the situation, except
that the plane is now one of constant y, rather than constant z, and
the points PJ and PJ1 will be replaced by PK and PK1. Expansion as
in (552) enables removal of the fictitious cp7value and expressions
for the derivatives.
Sphere 1 and Sphere 2
h Sr~ hiÂ£Â£irE} i%*2PK (563)
The values of the points PK and PK1 can be expressed just as
indicated in Section 5.5.3 for the yderivative, with expansion in the
88
xdirection about the nearer of two surrounding points. In fact, the
equations developed therein can be used directly, replacing PJ with
PK and points A, C, and A+l (or Al) with their appropriate points for
the zderivative. Hence, equations (555) and (558) will apply to
points above Sphere 1, and equations (556) and (559) will hold for
cases above Sphere 2.
Again, as in the case of y, certain points near the hemisphere
bases will require a linear interpolation between known points to form
an expression for PK or PK1. Notice, however, that problems such as
those indicated in Figure 28 do not occur in the zdirection, since the
hemispheres do not overhang one another.
This section closes discussion of surface derivatives for the
cases of those points on the surface encountered by a line along which
both y and z are constant. It has been noted, however, that other points
encountered as adjacent points in the zdirection and ydirection would
be subjected also to the normal derivative difference expression.
Discussion of these points follows, with a subsequent section to combine
all the spatial derivatives into a normal derivative.
5.5.5 Adjacent zpoints and ypoints subjected to normal deriva
tive condition.The preceding subsections of 5.5 have covered only
those points on the hemispheres encountered as adjacent points in the
xdirection. Those points have counterparts in the y and zdirections
which will be discussed here. Much of the detail will be omitted, since
most of the methods employed parallel those of the immediately preceding
sections.
89
First, points met in the zdirection will be discussed, begin
ning with those points at a sufficient distance from the other hemisphere
that all needed surrounding values are available. Figure 29 portrays
the situation here, where the plane indicated could be either one of
constant x or constant y.
Constant yplane
Figure 29. Zpoint for normal derivative condition.
In this case, z becomes analogous to x in the previous discus
sions, as it lies on a lattice line. Hence, it can be expanded in
terms of lattice points, and, with appropriate replacements, equa
tion (549) can be used for the derivative, &p/dz. Then, points PI
and PI1 (PJ and PJ1) must be found to evaluate Scp/dx and cXp/dy.
Because the situations are similar, one expansion can serve all four
needed points. Emphasis will again be placed on the point nearer the
desired value. In this case, however, an expansion like that of (557)
will be used for both cases,.not just the point nearer the boundary
(C in this case). Note also that C might itself lie on the boundary.
Consider the case for PI, using G = oc^.
90
Sphere 1
PI nearer A:
PI nearer C:
(564)
(565)
(566)
(567)
The equations above can be used to express each of the desired
four interpolated values and hence allow computation of dcp/dx and
Scp/dy from an equation which is the same as (553). The derivatives
thus obtained can therefore be utilized in the equations of Section
5.5.1 just as indicated in the following section.
91
The equations derived for Sphere 1 can also be used for points
on Sphere 2, since these equations are for interpolations in the same
direction. Here C still represents the point at the lower zelevation,
either on the surface or a lattice point, and A depicts the point at
the higher zelevation, similar to Figure 29.
Just as was the case earlier, there will be areas where not all
of. the x or ypoints needed for equation (565) or equation (567)
will be available. Then it will be necessary to use a oriepoint, one
way derivative approximation as shown in the xdirection discussions.
As in that case, the single point is obtained for use in an equation
such as (551) by one of two means. Points on the other hemisphere
can be evaluated by the arc interpolation procedure, while other points
are expressed by interpolation between lattice points.
For the points met in the ydirection, the problems and methods
are the same as for those in the x and zdirections. Hence, no dis
cussion will be given here except to state that equations similar to
those for the other cases will be written and used. The difference
expressions will be utilized to form the normal derivative condition
just as indicated in the next section.
5.5.6 Final boundary formulation.In Section 5.5.1, expres
sions were shown for dcp/dn in terms of the three derivatives with
respect to coordinate axes. With these three derivatives expressed in
difference form a difference equation for the normal derivative is
available. Each of the three derivatives has been shown to contain
9, the function value at the surface point being treated, and some
B
92
other values from the field. Therefore, replacement of 3cp/dx, dcp/dy,
and &p/Sz by their difference forms enables a subsequent equation for
cd The latter equation becomes the one to be employed at that point
to impose the noflow condition to the solution. There are numerous
expressions for the three derivatives, developed in Sections 5.5.1
through 5.5.4. It is not intended here to list all possible combina
tions, but rather to illustrate by defining the equation for the cases
where all points such as PJ are available. Using these points with
equations (549), (553), and (563) and cancelling a common h, enables
writing from (551), after rearranging, the following:
r ^i i
X12CPI + 1,J,K 1+0^ CPI+2,J,KJ
+ y{2PJ + z{2PK (568)
Similarly, on the second sphere, use of equations (550),
(554), and (563) allows an expression for co on the second
D
hemisphere.
Sphere 2:
2or2l
1,J,K T^T
+
12 ,J
(569)
93
Other equations could be formed and would hold in the special
situations discussed earlier for which other means were used to obtain
a derivative. These will not be cataloged here, however.
These equations above thus complete the considerations of the
finite difference equations for the solutions herein. The next step
will involve the use of the final solution in calculating surface
velocities.
5.5.7 Singular points.Many of the preceding equations are
necessary only for the case of closely packed hemispheres, as certain
situations do not arise in the Chepil arrangement. Each case has a
singularity at the stagnation point, where the velocity equals zero.
This point presents no real problem for the differences solution, as
the value of the velocity is still finite.
In the closely packed hemispheres, however, three other singular
points exist within the solution space. These are the points where the
hemispheres touch. Mathematically, the velocities would approach infin
ite values at these points because of the convergence of streamlines
at those points. This presents no problem as to the potential value
at these points, for this is known already. There are, however, two
considerations which must be made. First, for evaluating the lift on
the hemisphere, a more meaningful velocity at the base can be found by
extrapolating to a finite value from the distribution over the rest of
the surface. This avoids the obviously incorrect negative infinity
pressure predicted mathematically at the singular points.
94
A second concern is the possibility of error in the differences
solution for regions very near the singular points. To detect any
propagation of such errors and also to aid in the extrapolation to a
finite velocity, solutions will be made for varying values of h, the
lattice spacing. The solutions thus obtained can be compared to eval
uate any problems near the various singular points.
In order to avoid certain problems about the tangent point
midway in the solution space, advantage was taken of certain symmetries.
The value of cp at this midpoint for any zvalue is the same. It equals
the average of the values at the ends of the space. Hence, in some of
the lower regions where no lattice point exists between hemispheres,
the midpoint cpvalue can be used as a point in the difference equa
tions. The equations are quite similar to others developed. Inclu
sion of such an approach in the differences solution aided convergence
in the region of this particular singularity.
5.6 Velocity and lift calculations
Once the potential (9) field has been sufficiently determined
through the differences solution described, it is then possible to
calculate the velocities created by that potential field. Of interest
here are the velocities on the surfaces of the hemispherical elements,
which will enable computation of pressures and lift forces. The
velocities can be calculated from the potential as shown in (570),
where u, v, and w represent velocity components in the x, y, and z
directions, respectively.
95
(570)
The means for calculating these three directional derivatives at the
surface have already been discussed in Section 5.5. Therefore, let it
simply be said that the surface velocities will be calculated using
the surface derivative expressions already available. The directional
components of velocity can be combined to yield the magnitude of the
velocity, q, at any point through (571)
(571)
Hence, the distribution of velocities over the surface can
be found. From these, it is desired to obtain the pressures acting
on the surface in accordance with these velocities. The general means
for all the relations needed herein were presented in a discussion of
earlier twodimensional works in Chapter IV. First, the system of
particles or elements treated will be considered as spheres whose lower
half is immersed in fluid, which is stagnant, being therefore subject
to be calculated flow only on the upper surfaces of the elements.
This situation is analogous to the case which is assumed as an approx
imation for the natural condition, with only the upper portion of the
grains exposed to flow.
To relate velocities to pressures, Bernoulli's principle will
be employed. This is developed for flow along a streamline by, among
many, Prandtl [70]. Irrotational flow is not assumed for such a deri
vation, and thus the surface velocities related to it need not be from
96
an irrotational flow. In this case they are, but the purpose is to
discover if such predicted velocities form a good representation of the
actual. If they do, then since rotational flow is allowed in this form
of Bernoulli's theorem, the pressures thus predicted would be similarly
representative. The equation follows.
2
z +Â£ + Â§= constant (572)
V 2g
where p = pressure
z = elevation
Notice that this constant is a constant only for the given
streamline and would differ from one streamline to another. However,
in this case, each streamline on the hemispherical surface will pass
through the stagnation point on the hemisphere. This is located at
coordinates (1, 0, 0) on Sphere 1.
It can be seen that at a stagnation point, with z = 0 and
q = 0, the constant of (572) becomes the static pressure at that
point, pQ. This, therefore, is the constant to be used in employing
(572) to determine surface pressure. This simply means that all sur
face pressures will be found relative to pQ. This static pressure
acting alone would give rise to a lift known as the buoyant, or hydro
static lift. The addition of a flow will produce the added lift
called hydrodynamic lift, which is of interest here. Figure 30 illus
trates an elemental surface area on the hemisphere.
97
dA = area in xyplane
Figure 30. Pressures on area of hemisphere.
Evaluation of the lift on the particles must take into account
the finite flow occurring along the basal diameter of the hemisphere
in the xyplane. This velocity also reduces the pressure at the base
below pQ. Application of (572) shows that the pressure, p, at any
point on the surface is found as below.
(573)
98
The lift force will require integrating the pressure differ
ences existing between pressures on the hemispheres and corresponding
pressures at the base. Call the velocity at a given point at the base
and then integrate the difference. Integration of the terms involv
ing z will yield a term y times the volume of the hemisphere, which is
the buoyant lift. The remaining terras indicate the hydrodynamic lift,
L, as shown below.
L = / / f(q2
surface &
where dA is the elemental surface area projected onto, the horizontal.
To express this in a better form, use the velocity, u^at the upper
most part of the hemisphere, as a reference.
L
_ p 2
2 Ut
where
2 2
X.
surface
dA
(575)
(576)
A more appropriate expression uses lift per unit area, with L divided
by either the total bed area or the projected area of the grain. The
'
former yields X and the latter X .
u
CL 2
u
2
t
X
u
= C
Â£
Lu 2
u
2
t
(577)
99
From the preceding it is seen that it will be necessary to
compute the surface velocities and then carry out the integration
cited in equations (575) and (576) to obtain the lift. This step
forms the concluding step, then, to the solution of the flow problem
based on finite differences. The integration will be carried.out
numerically on the computer and will not be discussed here.
*
5.7 Implementation of solution
This section will discuss briefly actual solution of the
problem on the computer, but only insomuch as it relates to the mathe
matical formulation. The actual computer programs with some brief
notes on them can be found in the Appendix.
A value of 1.70 was used for the overrelaxation factor, io,
described earlier. The iterations were begun by using initially poten
tial values as they would exist for a free stream. The value of the
lattice increment, h, used to obtain the solutions Was equal to one
tenth of the hemispherical radius. This lattice increment was used for
the lower onehalf of the solution space, up to twice the hemispherical
radius. The increment was then doubled in size for the remaining Upper
half of the space, using the grading procedure outlined in Section 5.4.5
Convergence of the iterative solution canbe judged in many ways
Frequently the value of the maximum change within the field during each
iteration is taken as a gauge. This value Will decrease as the iter
ations continue, and some minimum value may be chosen beyond which
convergence is assumed sufficient. However, in the present work, the
item of most interest is the distribution of velocities (and hence
100
pressures) over the hemisphere. For this reason, convergence here
would best be judged in terms of the velocities and their distribution.
The point where velocities have stabilized within reasonable limits
will then be used as a criterion for convergence here.
The program for the Chepil arrangement was run through 70 iter
ations, at which point the maximum change in the field was 0.014 for
a field of values ranging from 0.00 to 100.00. This run took slightly
less than 14 minutes on the IBM 360 computer. The value for C^u, the
lift coefficient, was within 0.2 per cent of its final value after
30 iterations. The added iterations primarily refined the velocities
and pressures.
The solution for the closely packed hemispheres was made using
three different values for h: a/6, a/10, and a/20, where a is the
radius of the hemisphere. Each run was made through 50 iterations,
with computer times of 4.5, 6.5, and 17.8 minutes, respectively. The
corresponding maximum field changes were 0.027, 0.039, and 0.050.
One note should be made concerning the h = a/20 solution. Due to
the extensive computer storage and time required, savings in these two
areas were made by limiting the height of the solution space. Values
of cp for elevation z = 1.10a were found from the h = a/10 solution
and then used as given, fixed data. The a/20 solution then proceeded
within this space.
The results obtained from the differences solutions will be
presented in Chapter VI.
CHAPTER VI
RESULTS AND COMPARISONS
6.1 General
The foregoing numerical work has been conducted to enable
testing a hypothesis, namely, that potential flow theory can be used
to describe the hydrodynamic lift forces experienced in sediment
transport. Several assumptions are involved in the analytical
approach and with regard to possible applications and limitations.
First, the flow studied is a mean, steady flow. There are no forces
involved from either free surfaces or sidewalls. Probably the most
telling restriction for application is that the flow should exist in
the hydrodynamicslly rough range. This is usually considered to hold
for wall Reynolds numbers (v^k/v) of 70 or greater. Essentially, in
this range the surface roughness elements completely penetrate the
laminar sublayer and this sublayer is of negligible effect.. For flows
in the ranges below the rough range, the viscous forces play a more
dominant role. In these ranges the results from ideal flow theory
would be less and less valid. A number of examples of this effect
were offered in Chapter II, and the same should hold in the present
work.
With the results of the numerical solutions in hand, attempts
will now be made to study the validity of the abovementioned hypoth
esis as a possible tool for sediment transport studies.
101
102
6.2 Numerical results for Chepil arrangement
The arrangement considered in this section is that of hemi
spheres placed three diameters apart (center to center) in a hexag
onal pattern on a horizontal bed. The values of interest from the
finite differences solution are the pressures and velocities as dis
tributed over the surface. It is possible to present the results in
numerous ways, but the threedimensional character of the solution
makes clarity sometimes difficult. Here* the results will be shown on
traces of the hemisphere found in planes parallel to the zxplane or
the yzplane. Some of the plots may look unusual as the velocity at '
the end of the trace doesn't equal zero. This is because the veloc
ity shown is the total velocity at each point. There is a velocity
along the basal circumference of the sphere. These distributions are
. \
shown on the following pages.
Integration of the pressures over the surface as discussed
in Chapter V yields a lift coefficient, based on the lift force per
unit area directly under the hemisphere as follows.
Chepil's arrangement:
C = 0.405 (61)
Lu
This coefficient is used with equation (577) to evaluate the lift.
The value of C = 0.405 compares with that of 0.50 for
a single hemisphere shown in Section 2.4.6. This is not unexpected,
as the threediameter separation approaches the singleelement case.
Work done by Michaels .[7l], who analytically treated a single row
Figure 31. Chepil's case: velocities on y = 0.
104
Chepil's case: velocities on y = 0.2a.
Figure 32.
105
0.0 0.2 0.4 0.6 0.8 1.0
x/a
Figure 33. Chepil's case: velocities on y 0.4a.
106
0.0 0.2 0.4 0.6 0.8 1.0
x/a
Figure 34. Chepil's case: velocities on y 0.6a.
1.0
Figure 35. Chepils case: Velocities on y = 0.8a.
108
Figure 36* Chepil's case: velocities on x = 0
109
Figure 37.
Chepils case: trace of some equipotential
surfaces in plane y. = *
110
of spheres, indicates that for his case a spacing of three diameters
would very closely approach the case of infinite spacing. Hence, the
coefficient of (61) is within the upper bound set for it.
6.3 Comparisons with Chepil's observations
6.3.L Details of Chepil's work.The pattern Chepil studied
has already been mentioned. Tests were made in a wind tunnel. One of
the hemispheres was a metal element with holes every thirty degrees on
a plane parallel to the flow direction and a plane normal to the flow.
The remaining hemispheres were formed of gravel (ranging from 2 to
6.4 mm in diameter) as was the floor surrounding all the hemispheres.
Three radii were used for these hemispheres1.27 cm, 2.54 cm, and 5.08
cm. He then used a means of averaging the values found on those two
planes over the surface of the hemisphere and hence integrating that
supposed average pressure to determine the lift. The drag was deter
mined by similar means, using the actual differences of pressure between
front and back of the hemisphere.
The velocity profiles found for the observations were expressed
in a form as (62).
(62)
where
Vj, = friction velocity
y^ = distance at which velocity equals zero
In the only set of velocity data given for the 2.54 cm hemisphere,
Chepil shows his lowest measurement at about 3.5 cm, or about 1 cm
Ill
above the top of the hemispheres. Chepil defines the limits of the
expanding boundary layer in the wind tunnel as the greatest distance
y to which the equation (62) is still valid.
The work done by Chepil on the three sizes mentioned falls in
the hydrodynamicslly rough range, with Reynolds numbers of form
(v^a/v) from 517 to 6840, where a is the hemispherical radius. This
is similar to the wall Reynolds number, differing by the ratio (a/k).
Hence it is to be expected that the lift is independent of the
Reynolds number. If this is true, some consistency should be expected
in relating lift to a form of some lift coefficient times a reference
velocity head.
Some investigation was necessary to provide a relatively con
sistent picture. Generally, the velocity picture here will be related
with (62) to remain consistent with Chepil. Chepil determined a
value for yQ in (62) by extrapolating his velocity measurements to the
point of zero velocity. Table 2 shows values which Chepil found,
including yQ for the three sizes. Using these values of yQ for the
equation (62) and choosing as a velocity that at the elevation of
the hemisphere tops resulted in lift coefficients varying from
0.169 to 0.374, or over a twofold difference. Such a variance is not
reasonable. Some of the factors involved in the experimental work
needed to be studied. First, recall that in Chapter III, a value of
0.0670a was found for the value of y^ for the given pattern. This was
derived on a volume basis, considering a smooth surface between hemi
spheres. However, Chepil*s work had a rough gravel surface. Consider
TABLE 2
CHEPIL'S EXPERIMENTAL DATA
Radius,
cm
yo
cm
R =
e
v a/v
V
dynes
2
cm
\ ,
u
dynes
2
cm
1.27
0.12
517
3.9
13
1.27
0.12
715
7.5
22
1.27
0.15
1007
14.8
42
1.27
0.15
1254
23.0
83
2.54
0.30
1232
5.5
19
2.54
0.30
1651
10.5
37
2.54
0.30
2317
19.6
79
2.54
0.30
2885
30.4
130
5.08
0.46
2772
7.0
33
5.08
0.46
3962
14.4
88
5.08
0.46
5290
25.6
165
5.08
0.46
6840
42.7
287
113
then a recomputation of y including also the volume of the upper
gravel layer in the calculations. Assume the gravel layer to be
spread out as a layer of tightly packed hemispheres of diameter 0.5 cm.
This was chosen as representative of the range of gravel sizes, 0.2 to
0.64 cm. It is impossible to select the most representative size,
since no gravel analysis is available. Consider an area containing
onequarter of a larger hemisphere and measuring 3a by 3*/5 a/2.
n = Gf) = 1Sn/3 a2 (63)
n = number of gravel grains of diameter 0.5 cm
Vg = volume of gravel = (){j Tr(.25)^}*n (64)
14 3
= volume hemisphere = OgHy tt a ) (65)
Combining the volumes and dividing by the area provides a relation
ship for y the distance to the theoretical bed.
yQ = 0.131 + 0.0670a (66)
a, y in cm
Valid only for gravel of 0.5 cm diameter
Evaluation of yQ by this means yields values of 0.216 cm,
0.301 cm, and 0.471 cm for the respective cases. The latter two
values agree very well with Chepil's values, indicating a sound basis
for this approach. The first value is some 40 per cent to 75 per cent
higher depending on which yQ is used. This difference could be due
114
to the increased possibility of experimental discrepancies in the
smaller hemisphere. Chepil himself directly indicates a discrepancy
by showing two distinct values of yQ for the same boundary surface.
The term yQ should not be dependent upon the flow conditions for a
given boundary arrangement. As a means of evaluating the lift coeffi
cient, the values obtained by (66) will be used in (62). Here the
2
lift equation can be rewritten, using T0 = P vf an<* u = v^.
This yields (67).
C
Lu
(67)
Using (67) and Chepil's data from Table 2, values. for. C^./are Calcu
lated as shown below in Table 3, based on u = it ^ .
t top
TABLE 3
CT FOR CHEPIL'S WORK
Lu
1.27 cm,
y from
Jo
(66)
2.54 cm,
y from
Jo
(66)
5.08 cm,
yQ from
(66)
5.08 cm,
yo =
0.46 cm
0.340
0.244
0.266
0.262
0.300
0.248
0.346
0.340
0.290
0.284
0.365
0.358
0.368
0.302
0.380
0.373
115
Also included in the table are lift coefficients based on Chepil's
yQ for the largest hemisphere. It is at this size where the value of
yQ from experiments is most likely to relate to a similar yQ for the
theoretical case studied. This is because the gravel particles on
the bed have a much smaller effect than for the smallest hemispheres.
Even the values presented in Table 3 do not provide complete consist
ency, but some trends exist. Considering the values based on Chepil's
y where the closest relation to theory is expected, the value would
be around 0.36. This value of C^ is only about. 11 per cent less than
the theoretically found value, which is very satisfactory agreement.
Some benefit may be gained from translating the velocity
distribution (62) into a rough bed form such as found in Chapter III.
If this is done, it is possible to find that k, the equivalent sand
roughness, equals 29.7 y^. Using this relation, Table 4 can be formed,
relating y k, and the ratio.of roughness to grain size, herein
denoted rc> which is k divided by the diameter of the metal hemisphere.
TABLE 4
ROUGHNESSGRAIN SIZE RATIOS
Chepil's Data
From (66)
yo
k
r
c
V
k
r
c
0.12
3.57
1.40
1 0.216
6.42
2.53
0.15
4.46
1.75


0.30
8.91
1.75
0.301
8.95
.1.76
0.46
13.67
1.34
! ,
0.471
13.99
1.37
116
The decrease in r^, especially for those values from (66),
indicates a very definite contribution from the substantial roughness
of the gravel used for the bed. This same gravel was used to form the
other bodies in the hexagonal pattern. As the size of hemisphere
grows larger, the gravel plays less and less of a role in determining
the flow.
6.3,2 Comparison of lift forces.Values of lift will be com
puted from equation (576), using Chepil's velocity distribution and
the lift coefficient from Section 6.3.1.
These values will be reported later. First, comparison will be
made with the pressure distributions that Chepil gives which are for
only the 2.54 cm hemisphere. The four figures on ensuing pages will
show his measured distributions, this author's predicted distributions,
and distributions corresponding to an average of front and back pres
sures for Chepil. The predicted distributions follow the numerical
results shown earlier in this chapter. The maximum pressure differ
1 2
ence is taken for this prediction as p u^, where u^_ is evaluated
from (62).
The values show some discrepancies, especially for the lower
pressure runs.. However, the agreement seems good for the case of
highest v^. Therefore, it was decided to compute lift on the hemi
spheres by the same means used by Chepil and compare those values with
those computed by Chepil from his measurements.
It is interesting to note that in all four instances shown,
Chepil's measurements yield a higher pressure at thirty degrees up
117
Flow
Chepil's measured values
Values from theory
Average of Chepils
front and back values
Pressure scale:
1" = 100 dynes/cm^
2.54 cm radius
hemisphere
Pressures plotted radially as difference between given point
and uppermost point of hemisphere. Above circle, positive;
below circle, negative.
Figure 38. Measured and theoretical pressure distributions:
v = 68 cm/s ec.
118
Flow
/
Average of Chepil's 2.54 cm radius
front and back values hemisphere
Pressures plotted radially as difference between given point
and uppermost point of hemisphere. Above circle, positive;
below circle, negative.
Figure 39. Measured and theoretical pressure distributions:
Vf = 91 cm/sec.
119
Flow
Chepils measured values
Values from theory
Pressure scale:
2
1" = 400 dynes/cm
Avpragp of Ghepil's
front and back values
2.54 cm radius
hemisphere
Pressures plotted radially as difference between given point
and uppermost point of hemisphere. Above circle, positive;
below circle, negative.
Figure 40. Measured and theoretical pressure distributions:
v = 128 cm/sec.
r *.
120
F low
Chepil's measured values
Values from theory
Pressure scale:
2
1 = 400 dynes/cm
Average of Chepil's
front and back values
2.54 cm radius
hemisphere
Pressures plotted radially as difference between given point
and uppermost point of hemisphere. Above circle, positive;
below circle, negative.
Figure 41. Measured and theoretical pressure distributions:
v^ = 159 cm/sec.
121
the hemisphere than at the base. This increase is generally about
10 per cent. Ordinarily a type of stagnation zone would be expected
to exist in this region, with only slight pressure decrease. The
slight discrepancy here may well be due to the effects of the gravel
bed surrounding the hemispheres. These particles near the base of the
hemisphere likely create disturbances and shed vortices which could
7
affect conditions at the base, while possibly not having influence up
at thirty degrees. Such a situation could result in a pressure decrease
at the basal point due to motion while not decreasing the higher point.
Chepil considered three surface zones, relative to positions of
his pressure measurements. Averaging the pressure on two planes over
this zone, he then projected this pressure vertically downward to get
the lift. Figure 42 shows a plan view of the zones, A, B, and C.
The pressure PP^ represents the pressure difference between the
bottom and zone A, with PP and PP being similarly defined. Equa
tion (68) is then used for the lift per unit area occupied by the
hemisphere.
\ = 0.062 (PPA) + 0.394 (PP ) + 0.544 (PP,,) (68)
u A B C
Application of (68) was made, using pressures as shown in earlier
figures, with the results shown in Table 5. Also shown in the table
are values of lift based on equation (577) with C 0.405. In
liU
122
Plan
Flow Elevation
^"
Figure 42. Chepil's hemisphere.
123
TABLE 5
COMPARISON OF THEORETICAL WITH CHEPIL'S WORK
a,
cm
X
u
Chepil's
work
dynes
2
cm
y from Table 2
o
y from
'o
(66)
CLu
0.348
\T
CLu
0.405
CLu
0.348
CLu
0.405
1.27
13
23.6
27.4
13.3
15.5
1.27
22
45.4
52.9
25.6
29.8
1.27
42
73.4
85.5
50.5
59.0
1.27
83
114.0
133.0
78.5
91.3
2.54
19
27.3
31.8
27.2
31.6
2.54
37
52.1
60.6
52.0
60.5
2.54
79
97.3
113.0
96.8
113.0
2.54
130
151.0
175.0
150.0
175.0
5.08
33
43.8
51.0
43.0
50.0
5.08
88 .
90.3
105.0
88.5
103.0
5.08
165
160.0
186.0
158.0
183.0
5.08
287
268.0
311.0
262.0
306.0
124
Replacement of the pressure differences in (68) by their
values from the numerical results will result in CT = 0.348. This
value is somewhat lower than the 0.405 found by integrating over the
entire surface. Computation of X^ using each C^u is included in
Table 5. The value of u^ is also chosen for this table in two ways.
First, Chepils values for yQ are chosen, and then, for comparison,
values based on (66) are used. Therefore, four X values based on
u
theoretical results are included.
A number of statements may be made concerning the results in
Table 5. First, there is not too good agreement for the smaller sizes,
with u^ based on Chepil's yQ. However, for the last three values in
the table, the agreement is much better, being high by 19, 13, and
8 per cent. It is for this largest hemisphere that the theoretical
model is most closely simulated. The differences between the two cases
lie in the gravel floor and gravel hemispheres in Chepil's work versus
a smooth floor and identical hemispheres in the theory.
One possible source of error exists in v^, the friction veloc
ity. Chepil obtained this by extrapolating his velocity curves and
taking the slope of the uln y plot. This is related to possible
error in y found by extrapolating velocity measurements to zero
velocity. Such extrapolations from elevations, where Chepil apparently
ended his velocity measurements, could have been faulty without affect
ing his work. Any change of y however, can be seen to cause quite
a change in the value of X^ through indicating a different ut
125
Along this line, it is interesting to look at the 1.27 cm
/
hemisphere. Use of y^values from (66) yields X^values in fair
agreement with Chepil's X This indicates that a reasonable predic
tion of lift could be made with a proper designation of theoretical
bed. Knowledge of flow in the vicinity of a rough bed will have to be
increased through experimental work to enable a real study of this
i
problem. The agreement of the theory of this dissertation with the
measurements is quite reasonable for the 5.08 cm element. Evidence
indicates that the future work may make it possible to develop this
accuracy for the other situations (1.27 cm and 2.54 cm) also. This '
is desirable, as these two cases are quite representative of many
natural beds subjected to flow.
As a final note in this section, consider the conversion of
the coefficient, to a coefficient representing the lift over the
total bed area, C rather than just under the hemispheres. This will
li
yield a coefficient for equation (577) equal to 0.0408.
6.4 Numerical results for closely packed hemispheres
The hemispherical arrangement considered in this section is
the hexagonal pattern with hemispheres touching. The results of the
numerical differences solution are shown graphically for certain sec
tions through the hemisphere. Pressures and velocities are plotted.
Also included are some plots of equipotential surfaces as they cut
through certain planes.
Integration of pressures over the surface to yield a lift
force gave the following coefficient for equation (577).
126
CL = 0.359 (69)
Comparison shows this value still lower than the coefficient for the
hemispheres at three diameters, which is expected. Note that this
C is based on the lift per unit area, and the velocity u The
L t
area used here was the total bed area, rather than merely that under,
the hemisphere. The difference in this case is slight, but the concept
of lift per unit area of total bed has more meaning in sediment trans
port studies.
Some items should be noted concerning the results shown.
First, the velocities for y = 0.40a and y = 0.50a are shown as com
puted using two values of h and three values of h, respectively.
Indicated on the velocity plots are the extrapolated values actually
used in the lift integrations. These values were obtained based on
consideration of the general trend of the curve and the variation due
to changing h. Extrapolations were necessary in both x and ydirec
tions near singular points as discussed in Section 5.5.7.
Values of C were obtained, based on u for all three h values
L t
a/6, a/10, and a/20. The C values were 0.301, 0.349, and 0.359,
il
respectively. The last value is used here without any attempt to
extrapolate it to h = 0, since it differs by only about 3 per cent
from the a/10 result.
Also included in the figures are two plots of flow patterns
over the hemispherical surface. One is viewed parallel to the xyplane
the other parallel to the yzplane. Note the flow near the middle
singular point, up over the point of contact.
1.0
I
u
Ut
or
()2
Ut
4
i

\ \
v u
ut
(JL)2 \
ut
i Flow
r
\j
0.0 0.2 0.4
0.6 0*8 1.0
x/a
Figure 43. Closely packed hemispheres: velocities on y 0.
128
/
Figure 44. Closely packed hemispheres: velocities on y = 0.2a
129
0.0 0.2 0.4 0.6 0.8 1.0
x/a '
Figure 45. Closely packed hemispheres: velocities on y = 0.4a.
130
x/a
Figure 46. Closely packed hemispheresi velocities on y 0.5a.
131
0.0 0.2 0.4 0.6 0.8 1.0
x/a
Figure 47. Closely packed hemispheres: velocities on y = 0.6a.
132
x/a
Figure 48. Closely packed hemispheres: velocities on y = 0.8a.
133
Figure 494 Closely packed hemispheres: velocities on x = 0.
Flow
cp =
100 90 80 70 60 50 40 30 20 10 0
Figure 50. Closely packed hemispheres: trace
of some equipotential surfaces in
plane y = 0.
Figure 51. Closely packed hemispheres: trace of some
equipotential surfaces in plane y = 0.5a.
136
y
Figure 52. Closely packed hemispheres: flow pattern on surface
viewed toward xyplane.
137
1
u/ut
1.0
Vectors plotted are resultants of
dcp/dy and dcp/dz at the points
of the tails of the vectors
Figure 53. Closly packed hemispheres: flow pattern on surface
viewed toward yzplane.
138
6.5 Comparison with EinsteinEl Samni observations
6.5.1 Physical details of experiments.The experiments of
interest now [33,34] used plastic hemispheres placed on the flume bed
in a hexagonal pattern. This is essentially the same as the theoretical
case studied. The flows studied were typified by wall Reynolds numbers
Cv k/v) in the range 3330 to 5580. These are very definitely in the
^ v
1
hydrodynamically rough range, where the laminar effects are negligible.
Hence, it is to be expected that the lift would not be a function of
the Reynolds number.
The theoretical model for which a solution has been made treats
a bed of infinite extent. This would be approximated Well over an
entire flume length in the experimental flow. However, some differences
might be expected to occur due to sidewall influences in a narrow flume.
Also, the bed had a finite width of perhaps five elements, as the flume
used in [33,34] was 12 inches wide. Section 3.5 was presented with an
eye to evaluating this effect approximately.
The flume had a depth of 36 inches. Section 3.5, equation (325)
shows that the ratio between sidewall and bottom roughness is needed in
order to evaluate o, the angle through the "corners" of the idealized
isovels. The bottom equivalent sand roughness, k^, equals the diameter
of the plastic hemispheres, or 0.225 feet. It is much more difficult
to estimate the sidewall roughness, although a picture in [34] indicates
a sideboard which is rougher than most This observation prompted a
first choice of k^ = 0.001 feet, or about 0.3 mm. Use of this value in
equation (325) enables evaluation of Qi to be 14.6 degrees. The line
at angle Qf will meet the center line at about 23 inches above the bed,
139
so that any portion of the flow above that depth is calculable
directly from the sidewall. It should be noted that there is some
possibility that El Samni's wotfk had even greater sidewall influence.
He noted that velocities taken in horizontal traverses through the
flow plotted as straight .lines versus distance from the sidewall on
semilogarithmic paper. No allowance was needed for a theoretical bed
because of the small roughness of the sidewall.
The concern here is how much effect will be created down at
the level of flow near the top of the hemispheres or grains, for the
lift is largely determined by these velocities. The amount of any
effect will give an idea of how much the finite bed might differ from
the infinite bed studied theoretically. The following figure indicates
how the velocity decreases at u^_ near the wall. From the values indi
cated, an average velocity across the flume can be computed at the
level of the tops of the hemispheres. This velocity indicates an aver
2 2
age u^ about 94.2 per cent of the u^ at the center of the flume. This
then is a very approximate measure of the decrease of lift below that
for a very wide bed. If the value above is applied to C = 0.178 for
the flume bed average, a value of = 0.189 is indicated for a very
wide bed. A check reveals that 10 per cent variation corresponds to
a = 24.5 degrees and a sidewall roughness k^ of about 0.009 feet, or
nine times that used and probably too large. Hence, the original value
chosen will be used as a gauge. In any event, 5 per cent is not a very
great influence and indicates reasonable similarity between the theo
retical and actual situations.
140
Figure 54. Velocity suppression near sidewall.
This effect is in addition to the possible total suppression
of the velocity distribution by the sidewalls, causing u^ at the center
line to be less than that for a wider flume. El Samni gives some pos
sible indication of this by his finding a coefficient for (32) of
8.40 rather than 8.48. Even if this were assumed as totally due to
sidewalls, the influence is only about 2 per cent.
One other feature of_El Samni's work is his conclusion that
the value of CT was constant for all flows only when the velocity at
li
0.35 diameters above the theoretical bed (u^,.) was used as a reference
141
velocity. Yet he also reports adherence of the flow to a logarithmic
velocity profile, such as equation (32). From such a profile, note
that the ratio of the velocities at two different depths is equal to
a constant, independent of roughness or friction velocity. Hence,
a constant 0.178 for C based on u, should be some different constant
for another reference velocity. However, El Samni indicates otherwise
For example, he shows values for C based on u varying from 0.29 to
lJ t
0.36; equation (32) would indicate 0.308, while (39), the revised
profile, indicates 0.281. Both values are based on El Samni's theo
retical bed located 0.20 k below the tops of the hemispheres. This
inconsistency caused some concern, and some source of the discrepancy
was sought by this author.
Using (32), it can be shown that u^,. = 5.85 v^ and
u. = 4.45 v_. If the 5.85 is held constant, it can be seen that CT
t t 1*
values (based on u^) of 0.29 and 0.36 would indicate u. = 4.58 v_
t t f
and u^ = 4.12 v^. These differ from the 4.45 value by 3 and 8 per
cent. This is merely an indication that the discrepancies may well
have had their basis in the experimental values.
6.5.2 Values of lift for hemisphere bed.The value of
determined by the differences solution is shown in Section 6.4 to be
0.359 based on u^. El Samni has a value of 0.178, based on u^,..
Refer to equation (39) for the velocity profile. Here a careful
examination of the theoretical bed location (point where y = 0) is
in order. El Samni found a point 0.20 k beneath the hemisphere tops
as the distance required to give a straight line plot (with slope v^)
142
of u versus log (y/k). However, this is based on a velocity profile
such as (32). Use of (39) would require a straight line plot of u
versus log (y/k + 0.0338). Hence, the measured values would imply a
different theoretical bed when extrapolated according to (39). This
value would be 0.20 k 0.0338 k = 0.1662 k, still only 16 per cent
below the value 0.198 calculated in Chapter III. This new, theoretical
bed will be used to remain consistent with the actual velocity data
El Samni obtained.
2
Use of (39), then, yields (u,c/u.) as 1.73. Note that this
step relates the velocity at the top of the grain, and hence the veloc
ities and pressures over its surface, to the flow given by the velocity
profile. This is quite similar to Figure 15. Applying the ratio 1.73
to (69) yields C = 0.207 based on u._, only 16 per cent above
El Samni's 0.178. If the 0.189 value based on the sidewall analysis of
Section 6.5.1 is used as a norm, the theoretical value is less than
10 per cent greater. In either case, the agreement between theory and
measurement is quite good. This result in itself shows great promise
for the hypothesis of this dissertationthat potential flow theory can
be used as a guide to predict hydrodynamic lift.
6.5.3 Lift on a gravel bed.El Samni also made some lift
measurements for a bed composed of gravel. He then attempted to relate
his findings to those for the bed of plastic hemispheres. Here the
author will try to relate his theoretical results to the gravel bed.
The theoretical bed was found by extrapolation to zero velocity
to be located 0.040 feet below the toppoints of the gravel. Also,
143
a lift coefficient of 0.178 was found to apply, based on a velocity at
a distance 0.058 feet above this theoretical bed. El Samni considered
the first as 1/5 the representative grain diameter and the second as
0.35 of the diameter. This gave values of 0.200 feet and 0.166 feet.
The gravel analysis showed these to equal dg^ and d^, where the
subscript indicates per cent finer by weight. El Samni also justified
the value of 0.20 feet by noting that this value gave him B = 8.40 for
a profile similar to (32). This value of B was the average he found
for runs with the plastic hemispheres. However, the value used as k
in (32) is the roughness, which does not necessarily equal the grain
size except in a case such as the closely packed hemispheres or
Nikuradse's sand roughness. Schlichting [31] was one of the early
experimenters in this area, conducting extensive tests showing roughness
dependent on size, shape, spacing, and arrangement of the roughness
elements. Koloseus [72] reports additional work, while Rouse [73] also
reviews work in the area. All these results verify that the value of
the representative grain diameter might differ from the roughness, or
0.20 feet here.
This author would like to replace the upper layer of gravel
by an equivalent hemispherical bed, the theoretical bed of which is
at the same plane as that of the gravel. Then, the predicted lift on
this equivalent bed can be compared with the measured lift. The equiv
alent diameter, de> for this purpose, would ordinarily be obtained
from the grainsize distribution of the bed material. Here only the
d^g and dg^ ^ sizes are available. An approximation will be made by
assuming the grainsize curve to be a straight line passing through
144
these two known points. The diameter dg can be found, aB shown by
Christensen [74] and as outlined in the Appendix, by
n 1 dy
_1 [ Js
d J0 d
e u
(610)
where d = grain diameter ,
y = fraction finer than
3 '
Evaluation of d by this means gives a diameter of 0.178 feet. This
e
equivalent diameter, d^, is based on converting the top layer of
natural material to an equivalent layer. This new layer will have the
same weight of material and the same surface area exposed to flow over
a given bed area sufficient to contain all grain sizes in the natural
sediment.
The theoretical bed, as used earlier, is 0.1662 d below the
e
grain top, or 0.0296 feet in this instance. The lift coefficient based
on u^ is still 0.359 from theory. Finding the ratio of u^ (at 0.0296
feet above the theoretical bed, y ) to the velocity at 0.058 feet above
y enables computation of C based on the latter velocity. Doing so
yields a value of 0.202 for C which compares with the 0.178 from El
Li
Samni's measurements. This value is based on the use of k = d = 0.178
e
feet. However, a better duplication of the natural conditions would be
obtained by using the roughness determined to be valid by El Samni,
0.20 feet. Using this value yields C = 0.199 for the velocity at
0.058 feet. The latter CT is only about 12 per cent above the measured
L
0.178.
145
It therefore seems likely that where sufficient information
is available about a natural bed and its material, values of lift may
be predicted from analysis based on some idealized model. Relation
ships between natural beds and idealized counterparts will have to be
developed more fully as future work occurs.
CHAPTER VII
CONCLUSIONS AND FUTURE WORK
The goal of this dissertation was to demonstrate that potential
flow theory could be used to predict hydrodynamic lift in sediment
transport. Such predictions are made by relating the pressure distri
bution from potential flow theory to known flow characteristics.
Two bed configurations were studied analytically and related to avail
able experimental observations. The work of Chepil [36] was treated
first. He used metal hemispheres, three diameters apart center to
center. Due to a gravel floor, the test situation differed from the
one treated analytically. For three cases with Chepil*s largest hemi
sphere, where the theoretical and natural conditions were most similar,
the predicted lift was 19 per cent, 13 per cent, and 8 per cent above
the corresponding measured values.
Next, a hexagonal pattern of closely packed hemispheres was
treated. Theory yielded a lift 16 per cent higher than that measured
by Einstein and El Samni [33,34]. If allowances are made for possible
sidewall effects of the narrow flume used in their work, the discrep
ancy decreases to perhaps less than 10 per cent.
Einstein and El Samni also made measurements for a gravel bed.
This author made use of an Approximate gravel gradation curve to
establish an equivalent bed of equal hemispheres. Use of the lift
146
147
value from theory for the equivalent bed yielded a lift less than 12
per cent higher than that measured.
The results above indicate satisfactory agreement and strongly
support the validity of using potential flow theory to study hydro
dynamic lift. All the measurements were from flows in the hydrodynam
ics lly rough range.' It is in this range that the theoretical approach
is expected to hold valid, for viscous effects are negligible. Use
of such methods in flows not in the rough range may be quite risky.
Extensive experimental work is needed for studying lift.
Related to this is a need for establishing knowledge of the velocity
distribution as it varies from point to point over the bed. Emphasis
should also be placed on defining an appropriate theoretical bed.
Studies should consider the use of idealized beds to replace natural
ones along with what characteristics this idealized bed should possess
to most adequately portray the natural conditions.
The approach of this dissertation provides now an analytical
means of studying many bed forms and patterns. These forms may arise
on a macro scale (dunes and ripples) or on a micro scale (individual
sediment particles). It is to be expected that any arrangement studied
will have relation to some particular aspect of sediment movement.
The analytical results would provide a basis for comparison with
experimental results and reasons for those results, Also, in some
instances, analysis might aid in evaluation of which cases to treat
experimentally and what to look for.
148
All the analytical and experimental work mentioned has a
common goalthe development of improved design methods in sediment
transport. It is hoped that the use of potential theory to study
hydrodynamic lift, advocated and shown valid in this dissertation,
will aid in reaching these ultimate design goals. .
i
\
\
APPENDIX
NOTES ON FORTRAN IV COMPUTER PROGRAM
The program listed on the following pages was written in
Fortran IV and used for the solution of the flow over the closely
packed hemispheres. The program used for the other case had a few
differences for the author's convenience, but it differed essentially
only in that many features of the listed program were unnecessary.
As a result, only a listing of the more general program is included
here.
Numerous comment cards are included within the various sub
routines to define usage and describe steps taken. However, to provide
a summary, brief descriptions of the functions of each of the fifteen
subroutines are offered below.
The MAIN program is primarily a control section, calling into
play the routines used in the iterative process and calling for input
and output.
Subroutine INOUT reads in initial data, and, in the a/20 run,
interpolated cpvalues are calculated. Output for iteration number
and maximum field change, as well as the potential field itself comes
from here. INOUT also calls subroutine VEUOC to calculate a lift
coefficient.
Subroutine XLIMIT calculates xvalues on hemispherical surfaces
s
corresponding to lattice lines and also assigns lattice line numbers
to those points.
150
151
Subroutine ALFINT calculates the leg lengths af^ shown in
Figure 22. Also, the cpfield is set to initial values corresponding
to flow in a uniform stream.
Subroutine ALFBDR calculates the lengths shown in Figure 27,
needed for interpolation of the PJ and PJlvalues shown there, which
are used for series expansions to evaluate derivatives at the surface.
Corresponding values for the zdirection are also calculated.
Subroutine DIVIDE computes divisors for use in the sevenpoint
scheme when not all Of.values are 1.0.
i
Subroutine BELOW treats all points in the iterative process
from z = 0 to the tops of the hemispheres, applying the general differ
ence equation subject to all boundary conditions. When one of the six
adjacent points falls on a hemisphere, BELOW calls the appropriate, sub
routine: BDDER, BDDER2, ZBOUND, ZB02, or YBOUND.
Subroutine BDDER applies the normal derivative boundary condi
tion for adjacent points in the xdirection lying on the first hemi
sphere in the region.
Subroutine BDDER2, which is similar to BDDER, treats points on
the second hemisphere in the xdirection.
Subroutines ZBOUND and ZB02 treat adjacent points in the z
direction which fall on a hemisphere. ZBOUND handles those points on
hemisphere one, while ZB02 handles those on hemisphere two.
Subroutine YBOUND makes the calculations for those adjacent
points in the ydirection falling on either hemisphere.
152
Subroutine BETWN controls those points in the solution space
between the tops of the hemispheres and the portion where the lattice
is graded.
Subroutine GRADE treats the transition from lattice size h to
size 2h and the remaining points beyond the transition.
Subroutine VELOC computes, from the present potential field,
velocities on the surface of the hemisphere and integrates velocities
squared to yield a lift coefficient. VELOC calls BDDER and ZBOUND to
evaluate surface derivatives of cp. Desired values are also written
out from this subroutine.
o o o o
153
//HYDRQ2 JOB (1432,41,020,152000),BENEDICT,B.A.
// EXEC F4GDX FORTRAN G, COMPILE, PUNCH OBJ.
//FORTSYS IN DD
MSGLE
VEL= 1
DECK, EXE
CUTE
MAIN PROGRAM FOR ITERATION SOLUTION OF LAPLACES EQUATION
OVER A BED COMPOSED OF HEMISPHERICAL ELEMENTS.
COMMON ALFAO 21,21,2 ),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2121,21),CC0Nl(212i)CC0N2l212i),THETA1(21,21),
3 THETA2( 21,21), IBDRY1{ 21,21) IBDRY2(21,21) PZ ( 36,21) ,
A TIP(100),TOP(11,11 ),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCASKLQW1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,B0G,IG1,IG2,I,J,K,
7 AIAJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,N0(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,2132)
COMMON IMIG(21),PIG,KIN,KIN1,FL0
C OMEGA IS AN OVERRELAXATION FACTOR FOR THE ITERATION PROCESS.
KIN1 = 0
KIN = 0
C REAO INITIAL DATA
CALL INOUT
C ROUTINE TO LOCATE THE SURFACES OF THE HEMISPHERES, BOTH IN
C TERMS OF I, THE ARRAY XVARIABLE AND THE ACTUAL VALUES
C OF X ON THE SURFACE FOR Z,Y VALUES CORRESPONDING
C TO LATTICE LINES.
CALL XLIMIT
C CALCULATE REMAINING NEEDED INITIAL VALUES AND DATA.
CALL ALFINT
CALL ALFBOR
CALL DIVIDE
IF (KEYTAG .EQ. 0 .OR. KEYTAG EQ. 1) GO TO 3
C SECTION FOR A/20 RUN, BEGINNING WITH GIVEN PHI VALUES.
KIN =1
CALL INOUT
3 KIP = 0
C STATEMENT TO ALLOW STOPPING AFTER CHECKING INITIAL DATA.
IF (KEYTAG .EQ. 3) GO TO 69
C PORTION OF RELAXATION ITERATION PROGRAM FOR REGION BELOW UPPER
C POINTS OF THE HEMISPHERES.
1 NCAS =1
EPS = 0.00
WRITE (6,1011) EPS
CALL BELOW
C PORTION OF SOLUTION SPACE ABOVE HEMISPHERES BUT BEFORE
C GRADED NET BEGINS.
IF (KEYTAG .EQ. 2) GO TO 10
5 CALL BETWN
WRITE (6,1011) EPS
non
154
IF (KEYTAG .EQ. 1) GO TO 10
K = KGRADE
C CALL ROUTINE FOR GRADING NET, OR CHANGING SIZE OF H.
750 CALL GRADE
10 KIP = KIP +1
KIN = 2
C CALL INOUT TO WRITE ITERATION NUMBER, MAXIMUM FIELD
C CHANGE, AND TO CALL FOR VELOCITY CALCULATIONS.
CALL INCUT
IF (KINI .EQ. 1) GO TO 69
IF (KIP .LT. 50) GO TO 1
1006 FORMAT (1110)
1009 FORMAT (11F10.5/10F10.5)
1011 FORMAT (9H EPSILON F10.3)
69 STOP
END
SUBROUTINE INOUT
C
C SUBROUTINE INOUT IS TO READ IN ANY DATA REQUIRED FOR THE
C SOLUTION AND TO PRINT OUT ANY RESULTS AS THEY ARE DESIRED
C THIS SUBROUTINE ALSO CALLS SUBROUTINE VELOC TO COMPUTE VELO
C CITIES AND CALCULATE A LIFT COEFFICIENT*
C
COMMCN ALFAO(21,21,2),ALFA7(21,21,Z),ALFA02(21,21,2),
1 ALFA72(21,21,2),XENC(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA 1(21,21),
3 THETA2(21,21),IBDRY1(21,21),IBDRY2(21,21),PZ(36,21),
4 TIPI 100),TOP(11,11),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1iSlA,S2,S2A,S3,S3A,BING,BIP,BOGtlGl,IG2,I,J,K,
7 AI ,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,N0{36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FLO
KL0W1 = KLOW 1
IF (KIN .EQ. 0) GO TO 20
IF (KIN .EQ. 1) GO TO 21
C WRITE ITERATION NUMBER AND MAXIMUM FIELD CHANGE.
WRITE (6,1010) KIP
WRITE (6,1011) EPS
C CHECK TO SEE WHETHER TO CALCULATE VELOCITIES.
IF
(KIP .EG.
30)
GO
TO
2
IF
(KIP .EQ.
40)
GO
TO
2
IF
(KIP .EQ.
50)
GO
TO
2
C EPSMIN = ESTABLISHED MINIMUM FIELD CHANGE WHICH WILL
C SIGNIFY SUFFICIENT CONVERGENCE.
IF (EPS .LE. EPSMIN) GO TO 2
C CHECKS TO STOP IF SOLUTION NOT CONVERGING PROPERLY.
IF (EPS .GT. 15.0) GO TO 2
155
IF (KIP .GE. 30 .ANO. EPS .GT. 0.5) GO TO 2
IF (KIP .LE. 50) GO TO 2069
C STEPS TO WRITE OUT POTENTIAL FIELD AND CALL SUBROUTINE
C VELOC TO CALCULATE VELOCITIES.
2 WRITE (6,1012)
IF UMAX .ECU 11) GO TO 201
IF UMAX .EQ. 21) GO TO 202
WRITE (6,1007) tl(PHKI,JfK),J=l,JMAX),I=lIMAX),K=
22 CALL VELOC
IF (EPS .LE. EPSMIN) KIN1 = 1
IF (EPS .GT. 15.0) KIN1 = 1
IF (KIP .GE. 30 .AND. EPS .GT. 0.5) KIN1 = 1
GO TO 2069
201 WRITE (6,1000) (((PH I(I,J,K),J=1,JMAX),1 = 1,IMAX),K =
GO TO 22
202 WRITE (6,1009) (((PH I(I,J,K),J = 1,JMAX),I = 1,I MAX),K=
K2 = KLOW +1
WRITE (6,1000) (((PHI(I,J,K)*J=ltJMAX,2),I=l,IMIT2
GO TO 22
C READ IN INITIAL DATA
20 READ (5,1002) H
READ (5,1002) EPSMIN
READ (5,1002) OMEGA
READ (5,1005) NCAS,JMAX,IMAX
READ (5,1005) KLOW,KGRADE,KEYTAG
READ (5,1005) KMAX,KEY1,KEY2
READ (5,1005) JMID,MARK,KIP
READ (5,1001) (IMIG(L),L=1,JMAX)
TAG = 1.0 OMEGA
GO TO 2069
C SPECIAL SECTION FOR A/20 RUN. READ IN DATA FOR Z = 1.
C AND Z = 1.10, WHICH WILL BE USED AS FIXED DATA FOR
C THE SOLUTION INSIDE THE REDUCED SOLUTION SPACE.
21 JMIT = JMAX 1
IMIT = IMAX 1
READ (5,1008) (((PH I(I,J,K),J = 1,JMAX,2),1 = 1,IMIT,2)
1 K=22,23 )
C INTERPOLATE BETWEEN POINTS FROM DATA GIVEN FROM A/IO
C SOLUTION
211 DO 4 K=22,KMAX
DO 4 J = 2,JMIT,2
DO 14 1=1,IMIT,2
14 PHI(I,J,K) = 0.5*(PHI(I,J1,K) + PHI(I,J+1,K))
PHK IMAX, J,K) = 0.0000
4 CONTINUE
DO 6 K=22,KMAX
DO 6 J=1,JMAX
DO 6 1=2,IMAX,2
IF (I .EC. IMAX) PHI(IJK) = 0.0000
IF (I .EQ. IMAX) GO TO 6
1,KMAX)
1,KMAX)
l,KLOW)
),K=K2,K
MAX,2)
05
o o o
156
PHI{IJ,K) = 0.5*(PHI(I1,J,K) PHI(I+1,J,K))
6 CONTINUE
1000
FORMAT
( IX, 11F10.5 )
1001
FORMAT
(1115)
1002
FORMAT
(IF 10.3 )
1003
FORMAT
( 15,2F10.6,15)
1004
FORMAT
(6F10.6)
1005
FORMAT
(315)
1006
FORMAT
(1110)
1007
FORMAT
(7F10.5)
1008
FORMAT
(6F10.4/5F10.4)
1009
FORMAT
(11F10.5/10F10.5)
1010
FORMAT
(6H LOOP 13)
1011
FORMAT
(9H EPSILON F8.3)
1012
FORMAT(
16H POTENTIAL FIELO)
1013
FORMAT
(115)
1014
FORMAT
(5F10.6)
1015
2069
FORMAT(8F10.5)
RETURN
END
SUBROUTINE XLIMIT
C
C SUBROUTINE XLIMIT HAS AS ITS PURPOSE THE ESTABLISHING OF THE
C LOCATIONS OF THOSE POINTS WHERE THE RECTANGULAR LATTICE SYSTEM
C INTERSECTS THE HEMISPHERICAL SURFACES, BOTH IN TERMS OF AN X
C VALUE AND IN TERMS OF THE LATTICE NUMBER. THE VALUES ARE
C XINIT ANC XEND FOR THE FIRST AND SECOND HEMISPHERES
C RESPECTIVELY. THE LATTICE LOCATION FOLLOWS
C AS IBDRY1 ANC IBDRY2.
C
COMMON ALFAOt 21,21,2 ) ,ALFA7(21,21,2),ALFA02(21 ,21,2) ,
1 ALFA72(21,21,2),XEND{21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CCONl(21,21),CC0N2I21,21),THETA1(21,21),
3 THETA2(21,21),IBDRY1(21,21),IBDRY2(21,21),PZ(36,2i),
4 TI P(100),TOPI 11,11 ),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1S1AS2S2AS3S3A,BING,BIPB0GIG1IG2,IJK,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,NO(36,21,21)
COMMON DIV(2500),ALFA(2500,5)PHI(36*21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FL0
DO 300 K= 1KLOW
AK = K
Z = (AK1.0)*H
DO 300J=1*JMAX
AJ = J
Y(AJl.0)H
KEY = 0
DO 301 1=1,IMAX
157
AI = I
X=(AI1.0)*H
IF (KEY.EQ.l) GO TO 302
IF (J .EQ. JMAX .ANO. K
JIPI = 15
JIP2 = 15 v
IF UMAX .EQ. 21) JIPI =
(JMAX .EQ. 21) JIP2 *
(JMAX .EQ. 11) JIPI =
(JMAX .EQ. 11) JIP2 =
(J .EG. JIP2 .AND. K
(J .EQ. JIPI .AND. K
IF
IF
IF
IF
IF
IF (J .EG. 1 .AND.
SEARCH FOR SURFACE OF
THE VALUE FOR IBDRY1,
.EQ. 1) GO TO 303
13
17
7
9
.EQ.
.EQ.
JIPI)
JIP2)
GO
GO
K .EQ. KLOW) GO TO
SPHERE. WHEN SUMR
THE INDEX FOR THE
TO 303
TO 303
303
EXCEEDS
BOUNDARY
1 THEN SET
POINT IN THE
THE HEMISPHERE
THIS VALUE IS
C
C
C XDIRECTION, AND CALCULATE THE VALUE OF X ON
C WHICH CORRESPONDS TO THAT VALUE OF Y AND Z.
C CALLED X INIT(J.K ) .
SUMR = X**2.0 + Y**2.0 + Z**2.0
IF (SUMR.LT.1.0) GO TO 301
IF( SUMR ,GE. 1.00 .AND. I .EQ. 1) GO TO 303
IF (SUMR.GT. 1.0) IBDRYKJ.K) = 11
IF (SUMR.EQ.1.0) IBDRY1(JK) = I
IF ((Y2.0 + Z**2 0) .GT. 1.0) WRITE (6,1001) SUMR,X,Y,Z,H
XINIT(JfK) = SQRTd.O Y**2.0 Z**2.0)
IF (NCAS .EQ. 2) I=IMAX
IF (NCAS .EQ. 2) GO TO 301
KEY = 1
GO TO 302
303 IBDRYKJ.K) = 1
XINIT(JfK) = 0.00
KEY a 1
C POINTS ON SURFACE OF SECOND HEMISPHERE. SUMR1 SERVES THE SAME
C ROLE AS SUMR DID FOR THE FIRST HEMISPHERE. XEND(J,K) IS THE
C VALUE OF X ON THE SECOND HEMISPHERE CORRESPONDING TO THE Y
C AND Z REPRESENTED BY J AND K. IBDRY2(J,K) IS THE SUBSCRIPT
C INDEX FCR THIS SAME POINT.
302 SUMR 1 = (XSQRK3.0) )**2 + (Y1.0)**2 + Z**2.0
IF {SUMR1 .GT. 1.00 .AND. I .EQ. IMAX) GO TO 304
IF (SUMR1.GT.1.0) GO TO 301
IF (SUMR1.LE.1.0) IBDRY2(J,K ) = I
XEND(J.K) = SQRT ( 3.0) SQRTd.O (Yl.0)**2 Z**2.0)
GO TG 300
304 IBDRY2(J,K) = IMAX
XEND(J.K) = SQRT(3.00 )
301 CONTINUE
IF (IBDRYKJ.K) .EQ. IBDRY2U.K)) IBDRY2(J,K) =
1 IBDRYKJ.K) +.1
300 CONTINUE
IF (JMAX .GT. 7) GO TO 369
C SPECIAL SECTION USED ONLY WITH A/6 RUN.
non
158
IBCRY1(3,5) = 5
I BDRY1(5*3) = 5
1BDRY1(5,5) = 3
1001 FORMAT (5F10.5)
369 RETURN
DEBUG SUBCHK
END
SUBROUTINE ALFINT
C
C SUBROUTINE TO DETERMINE INTERPOLATION FACTORS TO ENABLE
C DEALING WITH LATTICE POINTS FOR WHICH ALL OR SOME OF THE SIX
C NEEDED ADJACENT POINTS ARE AT SOME DISTANCE OTHER THAN THE
C NORMAL MESH LENGTH H FROM THE GIVEN POINT. THESE VALUES ARE
C STORED IN THE ARRAY ALFA(N,L ), WHERE N REPRESENTS THE NUMBER
C OF THE POINT AS STORED IN THE ARRAY NO(I,J,K), WHICH IS ALSO
C ESTABLISHED IN THIS SUBROUTINE. THIS SUBROUTINE ALSO SETS THE
C INITIAL VALUES FOR PHI(I,J,K) FOR THE FIRST ITERATION IN THE
C SOLUTION.
C
COMNGN ALFAO(21,21,2 ),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA 1(21,21) ,
3 THETA2(2l,21),IBDRY1(21,21),IBDRY2{21,21),PZ(36,21),
4 TIP(100),T0P(11,11)H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID MARK,NCASKL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1,IG2I*J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,N0(36,21,21)
COMMON DIVC2500),ALFA(2500,5),PH 1(36,21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FL0
C EINSTEINEL SAMNI ARRANGEMENT
C INITIALIZE ARRAYS NO(I,J,K) AND PHI(I,J,K)
DO 100 1=1,IMAX
DO 100 J = 1 JMAX
CO 100 K=1,KMAX
IF (K .GT. KLOW) GO TO 100
NQ(I,J,K ) = 0
100 PHK I,J,K) = 0.00
NP = 0
C SET Z VALUE
DO 102 K=1,KMAX
IF (K .GT. KLOW) GO TO 110
AK = K
Z = (AK1.0)*H
C SET Y VALUE
DO 102 J=1 JMAX
AJ J
Y = (AJ1.0)*H
ID1 = IBDRYK J,K )
159
IF INCAS .EQ. 1) 102 = IB0RY2(JK)
IF INCAS .EQ. 2) 102 12
C BEGIN TRAVERSE IN XOIRECTION
DO 101 I =101,102
106 NP = NP 4 1
NO(I,J,K ) = NP
KIP = NP
AI = I
X = I A I1.0)*H
IF II.EQ.ID1) X=XINIT(J,K)
IF II.EQ.I02) X=XENDIJ,K)
C INITIALIZE VALUE IN PHIII,J,K) CORRESPONDING TO POTENTIAL FOR
C A UNIFORM STREAM
PHI(I,JtK) = 100.0(100.0/SQRT3.00))*X
C COMPUTE ALFA VALUES AND STORE
C HANDLE POINTS WHICH LIE' ON HEMISPHERICAL SURFACES (101,102) BY
C SENDING TO SERIES 150 OR 160 FUR COMPUTATIONS
IF l I .EQ. 101) GO TO 150
IF I I .EQ. ID2) GO TO 160
C COMPUTE ALFA IN NEGATIVE XDIRECTION
ALFA INP1) = IXXINITIJ,K))/H
IF IALFA(NP,1).GT.1.0) ALFA!NP,1) = 1.00
C COMPUTE ALFA IN POSITIVE X DIRECTION
125 ALFA(NP,2) = 1.00
ALFA(NP,2) = (XEND(J,K)X)/H
IF (ALFA(NP,2) .GT. 1.00) ALFA(NP2) = 1.00
C COMPUTE ALFA IN NEGATIVE Y DIRECTION
107 ALFA(NP,3) = 1.00
MIS = IBCRY1C1,1)
IF (I .GT. MIS .AND. I .EQ. ID2) GO TO 101
IF (I .GT. MIS) ALFA(NP,3) = 1.00
IF (I .GT. MIS) GO TO 108
YINIT = XINITt I,K )
ALFA(NP3) = (YYINIT)ZH
IF (ALFA(NP,3).GT.1.00) ALFA(NP,3) = 1.00
IF (I .EC. ID2) GO TO 101
C COMPUTE ALFA IN POSITIVE Y DIRECTION
108 ALFA(NP4) = 1.00
IF (I .EQ. IMAX .AND. K .EQ. KLOWJ GO TO 104
IF (1.0 1**2 (XSQRT(3.CO))*2) 104,104,134
134 IF (((SCRT(1.0Z**2 (XSQRT(3.0 ))*2 )+1.0Y)/H).LT.1.00)
1 ALFA(NP,4) = (SQRT(1.0Z**2 (XSQRT(3.0 ) )**2 )
2 +1.0YJ/H
IF (I .EG. ID 1) GO TO 101
104 IF (J.EC.l) ALFA(NP,3 ) = ALFA(NP,4)
IF (I .EQ. ID 11 GO TO 101
IF (J .EQ. JMAX) ALFA(NP,4)= ALFA(NP,3)
C COMPUTE ALFA FOR NEGATIVE Z DIRECTION
ALFA(NP 5) = 1.00
IF (K.EQ.l) ALFA{NP,5) = 1.00
IF (K .EQ. 1) GO TO 1011
160
IF (I .GT. IBDRY1(J,1)) GO TO 153
ZINIT = X IN IT(J I )
ALFA(NP 5) = (ZZINIT )/H
IF ( ALFA(NP5) .GT. 1.00) ALFA(NP,5) = 1.00
GO TO 1011
153 ALFA(NP,5) = 1.00
IF (I .EG. IMAX .AND. J .EQ. 1) GO TO 101
135 IF (1.0 (XSQRTO.OO) )**2 (Y1.0)**2) 1011 1011 142
142 ALFAlNP 5) = (Z SQRTI1.0(XSQRTI3.00)) **2
1 (Yl.0)*2 ))/H
IF (ALFA(NP 5) .GT. 1.00) ALFA(NP*5) = 1.00
GO TG 1011
C POINT ON FIRST HEMISPHERE
150 DO 151 L= 1,5
151 ALFA(NP L ) = 1.00
IF (((XEND(J K) X)/H) .LT. 1.00) ALFA(NP,2)=
1 (XEND(J,K)X)/H
GO TO 108
C POINT ON SECOND HEMISPHERE
160 DO 161 L= 15
161 ALFA(NP,L) = 1.00
IF U(X X IN I T( J K ) )/H) .LT. 1.00) ALFA(NP,1)=
1 (XXIN IT(J K))/H
GO TO 107
C CHECK TO SEE IF ANY OF THE ALFA VALUES FOR THE CURRENT POINT
C POINT ARE LESS THAN ONE. IF SO, PROCEED. IF NOT, THEN SET
C THE VALUE OF NO(I,J,K) = 0, THUS KEYING IT AS A POINT WHERE
C ALL ALFA VALUES ARE EQUAL TO 1.000
1011 KIR x 0
DO 1305 L= 15
IF (ALFA(NPL) .LT. 1.0) KIR = KIR + 1
1305 CONTINUE
IF U .EQ. ID1 .OR. I .EQ. ID2) GO TO 101
IF (KIR .EQ. 0) NP = NP 1
IF (KIR .EQ. 0) N0(I,J,K) = 0
101 CONTINUE
GO TO 102
C INITIALIZE PHI(I,J,K) AS UNIFORM STREAM POTENTIAL FOR THOSE
C POINTS ABOVE THE TOPS OF THE HEMISPHERES
110 DO 111 J1=1,JMAX
GO 111 1=1,IMAX
AI = I
X = (AI1.0)*H
IF (I .EQ. IMAX) X = SQRT(3.CO)
111 PHI(I J1,K) = 100.0 (1C0.0/SQRT3.0))*X
102 CONTINUE
C INITIALIZE, AS A UNIFORM STREAM POTENTIAL, THE ARRAY PZ(I,J),
C WHICH IS TO BE USED IN SUBROUTINE ZBOUND.
1306 DO 5 1=1,IMAX
AI = I
X = (AI1.0)*H
o o o
161
DQ 5 J=1,JMAX
5PZ(I,J) = 100.0 ( 1CO.O/SQRT(3.00))*X
SET VALUES FOR KEYED POINT TO BE USED FOR ANY POINT WHERE ALL
ALFA VALUES EQUAL
ALFA(MARK 11) =
ALFA(MARK,2) =
ALFA(MARK* 3) =
ALFAt MARK4) =
ALFA(MARK 5) =
169 RETURN
END
1.000, AS
l.COOCOO
1.000000
l.OOOCCO
1.000000
1.000000
DENOTED BY NO(IJK) = 0
SUBROUTINE ALFBDR
C
C SUBROUTINE ALFBDR IS USED TO CALCULATE INTERPOLATION FACTORS
C TO BE USED TO FINDFIELD VALUES IN TERMS OF NEARBY POINTS IN
C THE LATTICE THE VALUES THUS OBTAINED WILL BE USED IN SERIES
C EXPANSION EXPRESSIONS FOR DERIVATIVES AT THE HEMISPHERICAL
C SURFACES. THIS SUBROUTINE ALSO KEYS THE SURFACE POINTS,
C THROUGH AN ARRAY MIG1(J,K) FOR HEMISPHERE 1 ANO MIG2(J,K) FOR
C THE SECOND. IN THIS ARRAY THE THREE POSSIBLE VALUES HAVE
C THE FOLLOWING MEANINGS. MIGl = 10 TWO FIELD POINTS ARE
C AVAILABLE FOR THE SERIES EXPANSION FOR THE DERIVATIVE IN THAT
C DIRECTION MIG = 20 TWO POINTS ARE AVAILABLE, BUT THE
C SECOND LIES ON THE OTHER HEMISPHERE, AND A LINEAR INTERPCLA
C TION WILL BE EMPLOYED FCR THE VALUE OF THE FIRST POINT
C MIG = 30 ONLY ONE POINT IS AVAILABLE, AND IT LIES ON THE
C OTHER HEMISPHERE, WHERE ARC INTERPOLATION WILL BE USED TO
C OBTAIN A VALUE
C
COMMON ALFAOI21,21,2 ) ,ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETAl(21,21),
3 THETA2(21,21),IBDRYl(21,2l),IBDRY2(21,21),PZ(36,21),
4 TIP(100),TOP{11,11),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEYi,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,B0G,IG1*IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,ZTAG,OMEGAXPBETAKIPKEY,NO(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
CCMMCN IMIG(21),PIG,KIN,KIN1,FL0
C EINSTEINEL SAMNI ARRANGEMENT
INTEGER S
KL0W1 = KLOW 1
KL0W2 = KLOW 2
JMIT = JMAX 1
JMIT2 = JMAX 2s
C INITIALIZE ALL ARRAYS TO BE SET HEREIN
DO 310 J=1,JMAX
DO 310 K=1,KLCW
162
NIG1(J,K) =0
MIG2 (J,K) = 0
THETA 1(J #K) = 0.00
THETA2(J K) = 0.00
CCON1(J K) = 0.00
310 CC0N2(J K ) = 0.00
C SET Z VALUE
DO 200 K= 1KLOW
AK = K
Z = (AK1.0)*H
C SET Y VALUE
CO 200 J= 1 JMAX
AJ = J
Y = (AJ1.0)*H
IB1 = IBDRYl(J K)
IF (NCAS. EQ. 1) IB2 = IBDRY2(J,K)
IF INCAS .EQ. 2) IB2 = IB1
S= 1
C TRAVERSE IN X DIRECTION
CO 201 1= IB1 IB2
AI = I
AH = (XINlT(JfK) (AI1.0)*H)/H
NK = N0(I,J,K)
IF ( I.EG.IB1) X = XINITIJ,K)
IF (NCAS .EQ. 2) GO TO 1201
IF (I .EC. IB 1 .AND. I .EQ. 1) GO TO 211
IF (I .EQ. IB2) X = XEND(J tK)
IF (I .EG. IB2 .AND. I .EQ. IMAX) GO TO 231
IF (I.GT.IB1 .AND. I.LT.IB2) GO TO 201
IF (S.EQ.2) GO TO 21C
C CALCULATE INTERPOLATION FACTORS ON FIRST SPHERE
C Z DIRECTION FACTORS
C FACTOR FCR POINT PK
1201 ALFA7(J,K,1) = 1.00
IF (K .GT. KL0W1) ALFA7(J,K,1) = AH
IF (K .GT. KLOWl) GO TO 205
AG = (X X IN IT ( J Kf 1) ) /H
ALFA7{J K,1) = AN IN 1(AG# AH)
IF (ALFA7(J,K,1).GT.1.00) ALFA7(J,K,1) = 1.00
C FACTOR FCR POINT PK1
205 ALFA7(J,K,2) = 1.00
IF (K .GT. KL0W2) ALFA7(J,K#2) = AH
IF (K .GT. KL0W2) GO TO 206
AG = (X XINIT(J.K+2))/H
ALFA7 ( JK 2 ) = AMINKAG.AH)
IF (ALFA7(J,K,2).GT.1.00) ALFA7(JK2)=1.00
C CHECK TO SEE WHICH OF THREE SITUATIONS Y DIRECTION FITS FOR
C THE CURRENT POINT MIG1 =1#2, OR 3.
206 BET 1.0 Z**2.0
IF (NCAS .EQ. 2) GO TO 1220
IF (J .EQ. JMIT) GO TO 1220
163
BET1 = ( X IN IT ( J, K ) SQRTI3.0) )**2
IF (( 1**2 BET 1) .GE. 1.00) Y2= 1.00
IF (( Z**2 + BET 1) .GE. 1.00) GO TO 220
Y2 = 1.0 SORT{BET BET 1)
220 IF (ABS((Y2Y)/H).LT.1.0) GO TO 1121
IF (ADS C(Y2YJ/H).GT.1.0 .ANO. ABS((Y2Y)/H).LT.2.0)
1 GO TO 1122
C FACTOR FCR PJ
1220 ALFAO(J,K,1) = 1.00
IF (J .GT. JMIT) ALFAO(J,K1) = AH
IF (J .GT. JM IT) GO TO 207
AG = (X XINIT(J+1,K))/H
ALFAO(JK11) = AMIN1(AG, AH)
IF (ALFAO(J,K, D.GT.1.00) ALFA0(J,K,1) = 1.00
C FACTOR FCR PJ1
207
ALFAO(J K f
2) =
1.00
IF (J .GT.
JMIT2) AL
FAO (
J,K,2) =
IF (J .GT.
JM I
T2) GO
TO
105
AG = (X 
XINITIJ+2
K) )/H
ALFAO(JK,
2) =
AMIN1
(AG,
AH)
IF (ALFAO(
JK,
2).GT.
1.00
) ALFAOI
105
MIGl(JyK)
= 1
IF (NCAS
. EQ.
2) GO
TO
201
GO TO 130
211
ALFA7(J K
1) =
1.00
ALFA7(J K y
2) =
1.00
212
ALFAO(JKy
1) =
1.00
ALFAOIJyK,
2) =
1.00
GO TO 105
C ARC INTERPOLATION
C IN THIS INTERPOLATION PHI VALUES NEEDED FOR POINTS ON A
C SPHERICAL SURFACE ARE EXPRESSED BY LINEAR INTERPOLATION IN
C TERMS CF ARC LENGTHS BETWEEN TWO KNOWN POINTS ON THE SURFACE.
1121 A = SORT(3.00) X
ALFA0(J,K,1) = 1.00
ALFAO(J,K y 2) = 1.00
B = SORT(3.00) XENCIJ+ltK)
C = 1.0 Y ALFA INK,4)*H
D = 1.0 Y H
E = SORT(3.00) XEND(JtK)
G = 1.0 Y
THET A1(J K) = (ATAN2(C,A) ATAN2(DB))/
1 (ATAN2(GE) ATAN2(D B))
M IG1(J K ) = 2
GC TO 130
C IN THIS INTERPOLATION NEEDED VALUES ARE EXPRESSED BY LINEAR
C INTERPOLATION BETWEEN TWO KNOWN POINTS WHETHER THOSE POINTS
C OCCUR ON A BOUNDARY SURFACE OR A LATTICEPOINT.
1122 AIB = IBDRYl(JfK) 1
ALFAO(JK 1) = 1.00
ALFAO(JK 2) = 1.00
164
IF (IBDRYKJ,K).GT.IECRY1(J+1K)) Cl = XINIT(J,K) AIB*H
IF (IBDRYKJ,K).EQ.IBDRY1
1 XINIT(J+lfK)
IF ((XENCtJ+1,K) AIB*H).GT.H) C2 = (AIB+1.0)*HXINIT(J,K)
IF ((XENC(J+1,K) AIBH).LT.H) C2 = XENDJ+i*K)XINIT(J,K)
CCDN1(JK) = Cl/lCl + C2)
HIGl(JfK) =3
GO TO 130
C INTERPOLATION FACTORS ON SECOND SPHERE
C COMMENTS SIMILAR TO THOSE FOR FIRST SPHERE
210 ALFA721J,K,1) = 1.00
AH = (
IF (K .GT. KL0W1) ALFA72(J,K,1) = AH
IF (K .GT. KLUW1) GO TO 208
AG = (XEND(J K + l) X)/H
ALFA72 ( J K 1) = AMINK AG AH)
IF (ALFA72(JKtl).GT.1.00) ALFA72(JK1) = 1.00
208 ALFA721JK,2) = 1.00
IF (K .GT. KL0W2) ALFA72(J,K,2) = AH
IF (K .GT. KL0W2) Go TO 209
AG = { XEND(J K+2) X)/H
ALFA72(J K,2) = AMINKAG.AH)
IF (ALFA721J,K,2).GT.l.CO) ALFA72(JK,2) = 1.00
209 BET = 1.0 Z**2.0
BET2 = XENC(J,K)2.C
IF IJ .EG. 2) GO TO 1221
IF ((BET2 + Z**2) .GE. 1.00) Y2 = 0.00
IF ((BET2 + Z**2) .GE. 1.00) GO TO 221
Y2 = SORT(BET BET2)
221 IF (ABSUY2YJ/H) .LT.1.0) GO TO 121
IF (ABSl(Y2YJ/H).GT.1.0 .AND. ABS((Y2YJ/H).LT.2.0)
1 GO TO 122
1221 ALFA02(J K,1) = 1.00
IF (J .LT. 2) ALFA02(JK* 1) = AH
IF U .LT. 2) GO TO 215
AG = (XENO(JlfK) X)/H
ALFAO2(J tK11) = AMINl(AGtAH)
IF (ALFA021J,K,1).GT.1.00) ALFA02(JK1) = 1.00
215 ALFA02(J K,2) = 1.00
IF (J .LT. 3) ALFA021J,K,2) = AH
IF (J .LT. 3) GO TO 115
AG = (XEND(J2 *Ki X)/H
ALFA0 2 ( J K, 2 ) = AMINK AG AH)
IF (ALFA02(J,K,2).GT.1.00) ALFA02(J.K2) 1.00
115 MIG2(JK ) = 1
GO TO 201
231 ALFA72(J K,1) = 1.00
ALFA72(J,K,2) = 1.00
232 ALFA02(JKy1) = 1.00
ALFA02(J,K,2) = 1.00
GO TO 115
o o o o o non
165
C ARC INTERPOLATION FACTORS
121 FI Y H
ALFA02(J K,1) = 100
ALFA02(JK2) = 1.00
A = Y ALFA(NK,3)*H
THET A2(J,K) = (ATAN2(Y,XINIT(J,KMATAN2(A,X))/
1(ATAN2(Y,XINIT(J,K))ATAN2(F1,XINITCJl,KJ))
MIG2(J *K) = 2
GO TO 201
C LINEAR INTERPOLATION FACTORS
122 AIB = IBCRY2(J K) 1
ALFA02( J K, 1) = 1.00
ALFA02(J,K,2) = 1.00
IF ( IBDRY2(J,K).LT.IEDRY2(J1,K)) Cl = AIB*H XEND(J,KI
IF ( IBDRY2(J,K).EQ.IBDRY2(J1,K)) Cl = XEND(J1,K>
1 XEND(JK)
IF ((AIBH XINIT(J1K)).GT.H) C2 XEND(J,K) 
1 (A IBl .G)H
IF ((AIB*H XINIT(J~1,K)J.LT.H) C2 = XEN0(J,K) 
1 XINIT(J1K)
CCCN2(J K) = C1/(C1 + C2)
MIG2(J,K) = 3
130 S = 2
201 CONTINUE
200 CONTINUE
RETURN
END
SUBROUTINE OIVIDE
SUBROUTINE DIVIDE CALCULATES THE DIVISORS TO BE USED AT EACH
POINT IN THE LATTICE FOR THE SIXADJACENTPOINT
REPRESENTATION OF THE LAPLACIAN.
COMMON ALFA0(21,21,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72I21t21,2)tXENDl21,21),XINIT(21,21),MIG1(2121),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA1(2121),
3 THETA2(21,21),IBORY1(21,21),IBDRY2(21,21),PZ(36,21),
4 TIP(100)T0P(1111),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2JMI0MARKNCAS,KL0W1IMIDTAB1,TAB2*
6 TAE3,S1,S1A,S2S2A,S3S3A,BING,BIPB0G,IG1,1G2IJK,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,NO(36,21,21)
COMMON DIV(2500),ALFAl2500,5),PHI(36,21,32)
COMMON IMIG(21)PIG,KIN,KIN1,FL0
C EINSTEINEL SAMNI ARRANGEMENT
DIV(MARK) = 3.000000
DO 400 N=1,KIP
DIV(N) = 1.0/(ALFA(N,1)*(ALFA(N,1) f ALFA(N,2)J)
1 + 1.0/(ALFA N,2)*(ALFA(N,1) + ALFA(N,2)))
o o o
166
2 + 1.0/(ALFA(N,3)*(ALFA(N,3) + ALFA(N4)))
3 + 1.0/(ALFA(N,4)(ALFA(N,3) ALFA(N,4)))
4 + 1.0/(ALFA(N,5)*(ALFA(N,5) 1.0))
5 + 1.0/(1.O +ALFA(N,5))
400 CONTINUE
469 RETURN
END
SUBROUTINE BELOW
C
C SUBROUTINE BELOW TREATS THOSE POINTS IN THE FIELD ON AND
C BELOW THE ZPLANE FORMING A TANGENT TO THE UPPERMOST POINTS
C OF THE TWO HEMISPHERES IN THE SOLUTION SPACE. THE LAPLACIAN
C DIFFERENCE EXPRESSION IS UTILIZED AND THE VARIOUS BOUNDARY
C CONDITIONS ARE IMPOSED. SUBROUTINE BELOW CALLS THE FOLLOWING
C SUBROUTINES TO HANDLE THE DPHI/DN = 0 CONDITION ON THE HEMI
C SPHERESO BDDER,BCDER2,YBOUND,ZBOUND,ZB02.
C
COMMON ALFAO(21,21,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21 ) ,XINIT(21,21),MIG1(21,21),
2 MIG2(21.21),CCON1(21,21),CCON2(21,21),THETA1(21,21),
3 THETA2(21,21),I BORYl(21,21).IBDRY2(21,21),PZ(36,21),
4 TIP(100),T0P(11,11),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1,IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,NO(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
CCMMCN IMIG(21),'PIG,KIN,KIN1, FLO
C EINSTEINEL SAMNI ARRANGEMENT
KEG = 0
C SET ZVALUE
DO 800 K=1KLOW
L3 = K 1
IF (K .EC. 1) L3 = K 1
AK = K
Z = (AK1.0)*H
C SET Y VALUE
DO 800 J=1,JMAX
C STEP TO SKIP MIDDLE SINGULAR POINT, WHERE PHI IS KNOWN.
IF (K .EG. 1 .AND. J .EQ. JMID) GO TO 800
AJ = J
Y = (AJ1.0)*H
L4 = J + 1
L5 = J 1
C CHECK FOR IMPOSITION OF DPHI/DY =0 BOUNDARY CONDITION FOR
C Y=0 (J=l) OR Y=A (J=JMAX)
IF (J .EQ. JMAX) L4 = Jl
IF (J .EQ. 1) L5 =J+1
IG1 = IBDRYK J,K )
167
IG2 = IB CRY 2(JK)
C BEGIN TRAVERSE IN XOIRECTION.
DO 801 1=IG1IG2
IMID =0
KEG =0
MPK = 0
N = NO (I JK)
C GO TO STANDARD LAPLACIAN DIFFERENCE EQUATION WHERE AbL
C ADJACENT POINTS ARE H AWAY.
IF (N .EG. 0) GO TO 841
AI = I
X = (A I1 .0)*H
IF (I .EG. IG1) X = XINITtJ,K)
IF (I .EG. IG2) X = XEND(J,K)
IF (I .EQ. 1 .OR. I .EQ. IMAX) GO TO 801
IF ( I .EQ. IG1 .OR. I .EQ. IG2) GO TO 850
C COMPUTE XDIRECTION FACTORS FOR DIFFERENCE EXPRESSION FOR
C LAPLACIAN.
PI = PHU I1, J,K)/(ALFA(N,1)*(ALFA(N,1) + ALFA(N,2)))
P2 = PH IC 1 + 1,J,K)/(ALFA(N,2)*(ALFA(N,1) + ALFA(N,2)))
C COMPUTE YDIRECTION FACTORS FOR DIFFERENCE EXPRESSION FOR
C LAPLACIAN. THESE FACTORS ARE P3,P4, AND THE ZDIRECTION
C COMPONENTS ARE P5,P6. FOR THOSE CASES WHERE THE ADJACENT
C POINTS LIE ON A HEMISPHERICAL SURFACE, THE PHI VALUE AT THAT
C SURFACE POINT IS FOUND FROM ONE OF THE SUBROUTINES LISTED
C ABOVE WHICH TREAT THE DPHI/DN = 0 CONDITION. IN CASES WHERE
C THE PROXIMITY OF THE HEMISPHERES CREATES PROBLEMS FOR THIS
C PROCEDURE, THE PHI VALUE AT THE SURFACE POINT IS EXPRESSED
C BY AN INTERPCLATION BETWEEN NEARBY POINTS ON THE HEMISPHERE.
C CHECK TO SEE IF EITHER (OR BOTH) Y POINTS LIE ON A HEMISPHERE
IF (ALFA{N3) .EQ. 1.00 .AND. ALFA(N,4) .LT.1.00) GO TO 805
IF (ALFA(N,3) .EQ. 1.00 .AND. ALFA(N,4) .EQ.1.00) GO TO 810
IF (ALFA(N,3) .LT. 1.00 .AND. ALFA(N,4) .LT.1.00) GO TO 810
C CNE POINT LIES ON FIRST HEMISPHERE
IMID = 1
1806 CALL YBOUND
P3=BING /(ALFA(N,3)*(ALFA(N,3)+ALFA(N,4)))
811 P4 = PHH I,L4,K)/(ALFA(N,4)* { ALFA(N, 3 i+ALFAfN *4)))
C LOOK AT ADJACENT ZPOINT.
813 IF (ALFA(N,5) .EQ. 1.0) GO TO 820
IMID = 1
IF (X .GT. XIN IT(J,1) ) IMID = 2
IF (IMID .EQ. 1) GO TO 833
C Z POINT LIES ON SECOND HEMISPHERE
CALL ZBC2
BIM = BING
L3 = K1
IF (K .EQ. 1) L3=K+1
IF (ALFA(N,5) .EQ. 1.0) BIM = PHI(I,J,L3)
P5 = BIM/(ALFA(N,5)*(1.0 ALFAIN.5)))
821 P6 = PHKI,J,K+1)/(1.0+ALFA(N,5))
168
GO TO 825
C 2 POINT AT LEAST H AWAY FROM SURFACE HERE.
820 L3 = KI
IF (K .EC. 1) L3=K+1
P5 = PHI(ItJL3)/(ALFA(Nf5)*(1.0+ALFA(N t 5)))
GO TO 821
810 IF (ALFA(N,3) .LT. 1.00 .AND. ALFA(N,4) .LT.1.00) GO TO 812
C BOTH Y POINTS A FULL H AWAY
P3 = PHI(I,L5,K)/IALFA(N,3J*CALFA(N3)+ALFA(N.4n)
GO TO 811
C Z POINT ON FIRST HEMISPHERE
833 CALL ZBCUNC
P5 = PZ(IJ)/(ALFA(N5)*{1.0+ALFA(Nt5I))
GO TO 821
C ARC INTERPOLATION FOR POINTS ON SPHERES
812 F3 = Y ALFA(Nt 3)*H
IF {J .EC. 1) GO TO 830
IF (J .EQ. JMAX) GO TO 840
FI = Y H
A = SGRT(3.00) X
B = SGRT(3.00) XENC(L4 K)
C = 1.0 Y ALFA(N,4)*H
D = 1.0 Y H
E = SGRT(3.00 ) XENC{JK)
G = 1.0 Y
THM = (ATAN2ICA ) ATAN2(D,B))/
P4 = PH I(I,L4 K) THM*(PHI(I,L4 ,K) PHI(I + 1,J,KJ)
THM = (ATAN2(YfXINIT{JK) ) ATAN2IF3 X)) /
1 (ATAN2(Y,XINIT{JK)) ATAN2(F1,XINIT(L5,K) ))
P3 = PHI(I1,J,K) THM(PHI(I1,J,K) PHI(I,L5fK)J
P4 = P4/(ALFA(Nt4)*(ALFA(N,3) 4 ALFAIN*4)))
P3 = P3/lALFAtN3)*(ALFAiN3) 4 ALFAIN4))>
GO TO 813
C SPECIAL SECTION FOR Y = 0
830 IMID =2
CALL YBCUND
P4 = BING/(ALFA(N,4)*(ALFA(N,3) 4 ALFA(N4)) )
P3 = P4
GO TC 813
C SPECIAL SECTION FOR Y = A
840 IMID = 1
CALL YBOUND
P3 = BING/(ALFA(N,3)IALFA(N3) 4 ALFA(N,4)U
P4 = P3
GO TC 813
C STANDARD LAPLACIAN DIFFERENCE EGUATION
841 TAM = (PHIU4ltJ,K) 4 PHI(I1,J,K) 4 PHI(I,L4K) 4
1 PHI(I,L5,K) 4 PHI( IJ L3) 4 PHI(IJK+l))/6.0
IF (ABSlTAMPHKIfJtK)) .GT. EPS) EPS= ABS(TAMPHI(I,J,K))
PHI(11Jt K) = OMEGA*TAM 4 TAG*PHI(I,J,K)
GO TG 801
non
169
C GNE Y POINT CN SECOND HEMISPHERE
805 P3 = PH I ( I,L5 ,K)/(ALFA(N,3)*(ALFA(N,3) + ALFA(N4)))
IMID = 2
1805 CALL YBCUND
P4 = BING /(ALFA(N,4)(ALFA(N,3)+ALFA(N,4)))
GO TC 813
C EVALUATION OF NEW PHI VALUE AT THE GIVEN POINT.
825 TAM = ( PI + P2 + P3 P4 + P5 + P6 )
IF (KEG .EQ. 0) DIP = DIV(N)
TAM = TAM/DIP
IF (ABS(TAMPHI(I,J,KM.GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI (I,J,K) = 0MEGA*TAM TAG*PHI(I,J,K)
GO TG 801
850 CALL BCCER
801 CONTINUE
800 CONTINUE
RETURN
END
C
C
C
C
C
C
C
C
C
C
SUBROUTINE BDCER
SPECIAL SECTION FOR THOSE POINTS ON THE BOUNDARIES OF THE
TWO HEMISPHERES, WHERE THE BOUNDARY CONDITION OF A ZERO
NORMAL DERIVATIVE MUST BE SATISFIED. THE DERIVATIVES IN ALL
THREE (X,Y, AND Z) DIRECTIONS ARE COMPUTED AND THEN
COMBINED TO FORM AN EXPRESSION FOR THE VALUE OF PHI
CN THE BOUNDARY POINT.
COMMON ALFAO(21,212),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XIN IT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA1(21,21),
3 THETA2121,21),I BORY1(21,21),IBDRY2(21,21),PZ(36,21),
4 TIP(100),TOP(11,11),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1,IG2,1,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA KIP,KEY,NO(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
COMMON IMIG(2i),PIG,KIN,KINl,FLO
EINSTEINEL SAMNI ARRANGEMENT
JMIT = JMAX l
KL0W1 = KLOW 1
PREPARE FOR CERTAIN
JIPI = 15
IF (JMAX
IF (JMAX
JIP2 = 15
IF (JMAX
IF UMAX
IF (NCAS
POINTS FALLING EXACTLY ON HEMISPHERE
EQ.
11)
JIPI
= 7
EQ.
21)
JIPI
= 13
s
EQ.
11)
JIP2
= 9
EQ.
21)
JIP2
= 17
EQ.
3)
IG2 =
IMAX
170
C FOR SURFACE POINTS LYING ON OTHER HEMISPHERE, CALL BDDER2.
IF (I .EC. IG2) GO TC 9950
C CONSIDER DERIVATIVE IN XDIRECTION.
XP = A IH
AX = (XEND(J,K) XIMT(JtK) )/H
C CHECK FOR NEXT POINT ON NEXT HEMISPHERE.
IF (XP .GE. XEND(J,K}) GO TO 860
BX = (XP XINIT(JK))/H
C CHECK FOR NEXT POINT A LATTICE POINT, BUT FOLLOWING ONE IS ON
C THE NEXT HEMISPHERE.
IF (AX .LT. (1.0 + BXH GO TO 861
NT = N0(I+1JK)
IF (NT .EQ. 0) NT = MARK
SI = (X)*(2.0*PHI(141,J,K) + {1.02.0*ALFA(NT,1))*
1 PHK 1 + 2, J,K )/( 1.0 + ALFA (NT, 1) ) )
S1A = CX)*(3.0/(1.O+ALFA(NT,1)))
C MIG1(J,K) IS A KEY WHICH IDENTIFIES EACH POINT ACCORDING TO
C ITS RELATION TO THE POINTS AROUND IT. IF 2 POINTS ARE AVAIL
C ABLE IN THE YDIRECT ION FOR A SERIES EXPRESSION OF DPHI/DY,
C THEN MIG 1 = 1. IF ONLY ONE LATTICE POINT IS AVAILABLE AND IT
C MUST BE DETERMINED BY A LINEAR INTERPOLATION BETWEEN TWO FIELD
C POINTS, THEN MIG1 = 2. IF THE ONLY AVAILABLE POINT LIES ON
C THE NEXT HEMISPHERE, THIS VALUE IS DETERMINED BY ARC INTER
C POLATICN, AND MIG1 = 3. VALUES FOR MIG1 ARE COMPUTED IN
C SUBROUTINE ALFBDR
MAN = MIG1(J,K)
GO TC (851,852,853),MAN
C CASE WHERE NEXT POINT IS ON THE OTHER HEMISPHERICAL SURFACE.
860 SI = (PHI(1+1,J,K)/ALFA(N,2))*(!.0)*X
IF (J .EQ. JMID .AND. Z .LE. 0.30) GO TO 8860
S1A = (1.0)*XINIT(J,K)/ALFA(N,2)
MAN = MIG1(J,K)
GO TO (851,852,853),MAN
C SPECIAL DIFFERENCE EQUATION FOR POINTS NEAR MIDDLE SINGULAR
C POINT, TAKING ADVANTAGE OF ONE LINE OF KNOWN PHI.
8860 SI = X *(PH 1(1 + 1,J*K) 200.0)/ALFA(N,2)
S1A = (X)*(3.0/ALFA(N,2))
MAN = MIG 1 (J,K)
GO TG (851,852,853),MAN
C CASE WHERE ONLY ONE LATTICE POINT EXISTS BETWEEN
C THE TWO HEMISPHERES.
861 SI = (PHKT+l,J,K)/ALFA(N+1,1))*(1.0)*X
IF (J .EC. JMID .AND. Z .LE. 0.30) GO TO 8860
SI A = (i.O)*XINIT(J,K)/ALFA(N+l,l)
MAN = MIG 1(J,K)
GO TO (851,852,853),MAN
C YDIRECTION POINTS.
C FIRST, FOR BOTH POINTS IN YOIRECTION AVAILABLE FOR EXPANSION.
C MUST ALSO CHECK AT EACH Y POINT TO SEE IF THE NEEDED X POINTS
C ARE AVAILABLE FOR INTERPOLATION OR IF A LINEAR INTERPOLATION
C MUST BE EMPLOYED WITH THOSE X POINTS AVAILABLE.
171
851 IF (J .EC. JMAX) GO TO 1851
NT = NO(I+1,J+1,K)
IF (NT .EQ. 0) NT = MARK
XP = AIH
IF (J .EC. 1) GO TO 680
IF (K .EC. 1 .AND. J .EQ. JIPI) GO TO 882
IF (K .EQ. 1 .AND. J .EQ. JIP2) GO TO 882
C CHECK TO SEE IF LINEAR INTERPOLATION, RATHER THAN SERIES
C EVALUATION, SHALL BE MACE FOR PJ.
IF (XP .GE. XEND(J+1K)) GO TO 854
BETA = (XP X )/H
C EXPAND ABOUT NEARER POINT FOR EVALUATING PJ.
IF (ALFAO(J,K,1) .LT. BETA) GO TO 2851
ALSUM = ALFA(NT,2) + BETA + ALFA0(J,K,1)
C PJ IS THE VALUE TO BE USED FOR THE FIRST POINT IN THE SERIES
C EXPANSION USED IN EVALUATION OF DPHI/DY AT THE SURFACE. PJ
C IS ITSELF FOUND FROM OTHER FIELD VALUES BY SERIES EXPANSION.
PJ = PHI ( I+1,J+1, K) BETAMPHK 1+2, J + 1,K) PHI (I,J+1,K) )/
1 ALSUM + (BETA**2)*(PHI(I+2,J+l,K)/ALFA(NT,2) +
2 PHI(I,J+1K)/(BETA+ALFA0(JK,1)) PHI(1+1,J+l,K)*
3 (1.0/ALFA(NT,2) + 1.0/(ALFAO(J,K,1)+BETA)))/ALSUM
IF (J .EC. JMIT) GO TO 1850
856 NT 1 = NO( 1 + 1,J+2,K)
IF (NT1 .EQ. 0) NT1 = MARK
C CHECK.ON USE OF LINEAR INTERPOLATION FOR PJi DUE TO PROXIMITY
C OF THE HEMISPHERES, RATHER THAN USING SERIES EXPANSION.
IF (XP .GE. XEND(J+2,K)) GO TO 857
C EXPAND ABOUT NEARER POINT FOR EVALUATING PJi.
IF (ALFAO(J,K,2) .LT. BETA) GO TO 2853
ALSUM = ALFA(NTl,2) + BETA + ALFA0(J,K,2)
PJI = PHI(1+1J+2,K) BETA*(PHI(1+2,J+2,K) 
1 PH I(I,J + 2K))/ALSUM + (BETA**2)*(PHI(1 + 2,J + 2,K)/
2 ALFA(NT 1,2) + PHI(I,J + 2,K)/(BETA+ALFAO(J,K,2)) 
3 PHI(I+1,J+2,K)*(1.0/ALFA(NTl,2) +
4 1.0/(ALFA0(J,K,2)+BETA)))/ALSUM
8701 S2 = (1.0)*Y*(2.0*PJ PJ1/2.0)
S2A = (3.0)*Y/2.0
GO TO 870
C CASE FOR DPHI/DY = 0 ON Y = 0.
880 S2 = 0.0
S2A = 0.0
GO TO 870
882 PJ = PH I(I,J+l,K)
PJI = PH 1(1,J + 2,K)
GO TO 87C1
C LINEAR INTERPOLATION FOR PJ WHEN PJI EXISTS.
854 BETA = (XEND(J+1,K) X)/H
NTE = NO( IJ+1,K)
IF (NTE .EC. 0) NTE = MARK
PJ = PHI(IJ+1K) + (PHI(I+1,J+1,K) PHI(I,J+1K))*
1 (ALFA(NTE,2)BETA)/ALFA(NTE,2)
172
GC TG 856
C LINEAR INTERPOLATION FOR PJ1.
857 BETA = (XEND(J42,K) X)/H
NTE = NCI IJ+2K)
IF INTE .EQ. 0) NTE = HARK
PJ1 = PHIII,J42,K) + (PHI(I+1J+2K)PHI(IJ+2K))*
1IALFA(NTE,2)BETA)/ALFA(NTE*2)
S2 = I1.0)*Y*(2.0*PJ PJ1/2.0)
S2A = (3.0)*Y/2.0
GO TO 870
1851 PH II I, J, K ) = 100.0
GO TO 869
1850 S2 = (1.0)Y*PJ*2.0
S2A = {1.0 )*Y*20
GO TO 870
C 2851 AND 2853 ARE EXPRESSIONS FOR PJ AND PJ1 WHEN THEY LIE
C NEARER PHIII) THAN PHIII+l).
2851 Cl = ALFAOIJ,K,1) + BETA
PJ = (PHI!I,J41,K) + ALFAOIJ,K,l)*IPHI(I+lfJ+1.K
1 PHIII,J+1K))/C1 (ALFAOIJ,K,1)**2)*
2 I PH 11 1 +1 yJ+1K)/BETA + PHI I IJ+1,K)/
3 ALFAOIJ,K,1))/Cl )/11.0 ALFAOIJ,K,1)/BETA)
IF (J .EG. JMIT) GO TO 1850
GO TO 856
2853 Cl = ALFAO(J,K 2 ) + BETA
PJ1 = (P HI (IJ+2 K) 4 ALFAOIJ,K2)*(PHI(l4lfJ+2,K)
1 PHIII,J42,KJ/C1 (ALFAOIJ,K,2)**2)*
2 I PH I ( I + 1,J+2,K)/BETA + PHI I I,J42,K)/
3 ALFAOIJ,K,2))/Cl )/tl.O ALFAO(J,K,2)/BETA I
GO TO 8701
C Y POINTS OBTAINED BY ARC INTERPOLATION.
852 II = IBDRY2IJ41,K)
12 = IBDRY2IJ,K)
PJ = PHI IIl,j4lfK) THETA1IJK)*
1 IPHI(I1,J41,K) PHI(I2,J,K))
S2 = I1.0)*YPJ/ALFA(N,4)
S2A = I1.0)*Y/ALFAINA)
GO TO 870
C Y POINTS BY LINEAR INTERPOLATION.
853 PJ = PHIII,J41,K) CC0N1(J,K)*(PHI(IJ+1K) 
1 PHI I 141,J41,K))
S2 = (10)*Y*PJ
S2A = I1.0)*Y
IF (J .EG. JMIT) S2 = 2.0*S2
IF (J .EG. JMIT) S2A = 2.0*S2A
C FACTORS FOR Z DERIVATIVE.
870 IF IK .EC. KL0W1) GO TO 4870
NT2 = NO!141,J,K41)
IF INT2 .EQ. 0) NT2 = MARK
NT3 = NO!141,J,K42)
IF INT3 .EQ. 0) NT3 = MARK
173
C CHECK FCR OBTAINING PK BY LINEAR INTERPOLATION WHEN PROXIMITY
C OF TWO HEMISPHERES MAKES SERIES EXPANSION DIFFICULT
IF (XP .GE. XEND(J,K+1)) GO TO 871
BETA = (XPXJ/H
ALSUM = ALFA(NT2,2) 4 ALFA7(J,K,1) + BETA
IF (K .EQ. 1) GO TO 881
IF (J .EQ. 1 .AND. K .EQ. JIPI) GO TO 883
IF (J .EQ. 1 .AND. K .EQ. JIP2) GO TO 883
C CHOOSE NEARER POINT FOR EXPANSION TO FIND PK.
IF
C SERIES EXPANSION TO EVALUATE PK
PK = PH 11I4lfJ,K+1) BETA*! PHK 1+2, J,K+l)
1 PHI(I, J,K+1))/(ALSUM)
2 4 (BETA*2)*(PHI(I+2,J,K+1)/ALFA(NT2,2) 4 PHI(I,J,K4l)/
3 (BETA+ALFA7(J,K,1)) PH I (I 41,J,K+l)*(1.O/ALFACNT2,2)
4 1.0/(ALFA7(J,K,1)+BETA)))/ALSUM
C CHECK FCR LINEAR INTERPOLATION FOR PK1.
872 IF (XP .GE. XEND(J,K+2)) GO TO 873
BETA = (XPXJ/H
ALSUM = ALFA(NT32) 4 ALFA7(J,K,2) + BETA
C CHOOSE NEARER POINT (TO PK1) FOR EXPANSION TO EVALUATE PK1.
IF (ALFA7J,K,2) .LT. BETA) GO TO 2857
PK1 = PHKI + l, J,K + 2) BETAMPHKI + 2, J,K+2) PHI(I,J,K+2))/
1 (ALSUM) 4 (BETA**2)*(PHI(I+2,J,K+2)/ ALFA(NT32) 4
2 PHI(I,J,K+2)/(BETA+ALFA7(J,K,2)) PHI(1 + 1 JK+2)*
3 (1.0/ALFA(NT32) 4 1.0/(ALFA7J,K,2)+BETA)))/ALSUM
GO TO 875
C SERIES 4070 AND 4872 ARE EXACTLY ANALOGOUS TO 870 AND 872
C ABOVE, BUT WERE INCLUDEC AS NO VALUE IS STORED FOR ALFA FOR
C THOSE POINTS ABOVE K = KLOW. THIS VALUE IS THE 1.0 IN THE
C ALSUM EXPRESSIONS.
4870 BETA = (XPXJ/H
ALSUM = 1.0 4 BETA 4 ALFA7(J,K1)
IF (ALFA7(J,K,1) .LT. BETA) GO TO 2855
PK = PHK 141, J,K4l) BETA*(PHI(I42,J,K41) PHI (I J,K+l))
1 /ALSUM 4 (BETA*2)*(PHI(I42,J,K+l) +PHI(I,J,K+1)/
2 (BETA+ALFA7(J,K,1)) PHI (1 + 1,J,K+1)*(1.0 +
3 1.0/(ALFA7(J,K,1)+BETA)))/ALSUM
4872 ALSUM = 1.0 + BETA + ALFA7(J,K,2)
IF (ALFA7(J ,K,2) .LT. BETA) GO TO 2857
PK1 = PH 1(1 + 1,JK + 2) BETA*(PHI(I+2*JK+2) PHI(I,J,K+2))/
1ALSUM + (BETA*2)*(PHI(I+2JK+2) + PHI(I,J,K+2)/
2 (BETA+ALFA7(JK2)) PHI(I+1,J,K+2*(1.0 +
3 1.0/(ALFA7(J,K,2) +BETA)))/ALSUM
GO TO 875
C FOR DPHI/DZ = 0 AT Z = 0.
881 S3 = 0.0
S3A = 0.0
GO TO 890
883 PK = PHK I,J,K + l)
PK 1 = PHK I,J,K+2)
o o o o o
174
GO TO 875
C 2855 AND 2857 ARE FOR CASES WHERE PK AND PK1 ARE NEARER TO
C PHI(I) THAN TO PHIU + l).
2855 Cl = ALFA7lJ,K,1) 4 BETA
PK = (P H IC IyJtK+1) 4 ALFA7(J*K1)*(PHI{I+ 11JtK+1)~
1 PHIII,J,K41))/Cl IALFA71J,K,1)**2I*
2 ( PHIU + 1, J,K4l J/BETA 4 PHIUtJK+l)/
3 ALFA7(JK1))/Cl )/<1.0 ALFA7IJ,K,1)/BETAJ
IF (K .EG. KLOW1) GO TO 4872
GO TO 872
2857 Cl = ALFA7(J K 2) 4 BETA
PK1 = (PHIU.J.K42) 4 ALFA7IJ,K,2)*IPHI(I41,JfK42)
1 PHI(I,J,K42 ) )/C1 IALFA7IJ,K2)**2)*
2 IPHII141,J,K42)/BETA 4 PHI(I,J,K+2)/
3 ALFA7IJ,K,2))/C1 )/11.0 ALFA7{J,K,2)/BETA)
GO TO 875
C LINEAR INTERPOLATION FOR PK.
871 BETA = (XEND(J,K41) X)/H
PK = PH I(11J K41) 4 {PHI(l4ltJ,K4l) PHI(I,J,K4l))*
1 ALFA7(J,K,1)/(ALFA7(J,K,1J4BETA)
GO TO 872
C LINEAR INTERPOLATION FOR PK1.
873 BETA = {XEND(J,K42) X)/H
PK 1 = PHIU,J,K42) 4 (PHH I41,J,K42) PHIII t JK42)) *
1 ALFA7IJ,K,2)/IALFA7IJ,K,2)4BETA)
875 S3 = (2.0*PK PK1/2.0)*(1.C)*Z
S3A = (3 .0 )*Z/2.0
C THE S TERMS REPRESENT THE CONTRIBUTION TO DPHI/DN = 0 FROM THE
C THREE DIRECTIONAL PHIDERIVATIVES THE TERMS OF THE FORM SN
C ( N AN INTEGER ) REPRESENT ALL TERMS BUT THOSE CONTAINING
PHIB, THE BOUNDARY PHI VALUE. THE TERMS OF FORM SNA ARE THE
C COEFFICIENTS FOR PHIB. N=10 X, N=20 Y* N=30 Z.
890 SUMI = SI 4 S2 4 S3
IF INCAS .EQ. 3) GO TO 869
SUM2 = S1A 4 S2A 4 S3A
TAM = SUM1/SUM2
IF (ABSITAMPHIII,J,KH.GT.EPS) EPS=ABSITAMPHIIIJK))
PHIII,J,K) = OMEGA*TAM 4 TAG*PHI11,J,K)
GO TO 869
C COMPUTATIONS FOR BOUNDARY SURFACE POINTS ON SECOND HEMISPHERE.
9950 CALL BDDER2
869 RETURN
END
SUBROUTINE BDDER2 _
SUBROUTINE BDDER2 HAS THE FUNCTION OF APPLYING THE NORMAL DERI
VATIVE
C BOUNDARY CONDITION FOR THOSE POINTS LYING ON THE SECOND HEMISP
175
HERE.
C THE STEPS IN FORM AND ORCER ARE QUITE SIMILAR TO THOSE IN
C, SUBROUTINE BCDER. FOR THIS REASON NO EXTENSIVE COMMENT STATE
C KENTS ARE INCLUDED IN THIS SUBROUTINE (BDDER2). FOR CLARI
C FICATION, CONSULT THE STATEMENTS IN BDDER. THE STATEMENT
C NUMBERS ARE GENERALLY THE SAME AS THOSE OF BDDER WITH THE
C INITIAL 8 OF BDDER REPLACED BY AN INITIAL 9 IN BCDER2.
C
COMMON ALFA0(21,21,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CCONl(21,21),CCGN2(21,21),THETA 1(21,21),
3 THETA2(21,21),IBDRY1{21,21),IBDRY2(21,21),PZ(36,21),
4 TIP{100),TOPHI,11),H,EPS,IMAX,JMAXKMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1,IG2.I,J,K,
7 AI,AJ,AK.N.X.Y.Z,TAG,OMEGA,XP,BETA,KIP,KEY,NO*36,21,21)
COMMON DIV(2500),ALFA(2500,5),PH1(36,21,32)
COMMON IMIG(21)PIG,KINKIN1FL0
C EINSTEINEL SAMNI ARRANGEMENT
C CONSIDER FACTORS FOR XCERIVATIVE.
950 XP = (AI~2.0)H
AX = (XEND(J,K) XINITlJ,K))/H
IF (XP .LE. XINIT(J,K ) ) GO TO 960
BX = (XEND(J,K ) XP)/H
IF (AX .LT. (1.0+BX)) GO TO 961
NT = N0(I1,J,K)
IF (NT .EG. 0) NT = MARK
SI = (1.0)*(XSQR T(2.0))*(PHI(I2,J,K)*
1 ((2.0*ALFA(NT ,2)1.0)/(1.O+ALFA(NT,2))) 
2 2.0PHI(Il,JtK))
S1A = (XSQRT(3.00))(3.0/(1.O+ALFA(NT,2)))
MAN = MIG2(J,K)
GO TO (951,952,953),MAN
C NEXT ADJACENT XPOINT ON HEMISPHERICAL BOUNDARY.
960 SI = (XSQRT(3.00))*PHI(I1,J,K)/ALFA(N,1)
IF (J .EG. JMIC .AND. Z .LE. 0.20) GO TO 8960
S1A = (XSQRT(3.00))/ALFA(N,1)
MAN s MIG2(J,K)
GO TO (951,952,953),MAN
8960 SI = (XSQRT3.00))#(200.0 PHI(11,J,K))/ALFA(N,1)
S1A = (XSCRT(3.00))3.0/ALFA(N,1)
MAN = MIG2(J,K)
GO TO (951,952,953),MAN
C ONLY ONE LATTICE POINT AVAILABLE BETWEEN HEMISPHERES.
961 SI = (XSQRT(3.C0))*PHI(11,J,K)/ALFA(Nl,2)
SIA = (XSQRT(3.00))/ALFA(Nl2)
MAN = MIG2(J,K )
GO TO (951,952,953),MAN
C BOTH YPCINTS AVAILABLE FOR EXPANSION.
951 IF (J .EG. 1) GO TO 1951
NT = N0(IlvJltK)
v
176
t
IF (NT .EQ. 0) NT = MARK
XP = (AI2.0)*H
IF (J .EC. JMAX) GO TO 980
IF (XP .LT. XINIT(JlK)) GO TO 954
BETA = (XXPI/H
AlSUM = ALFA(NTtl) + BETA + ALFA02(J,K,1)
IF (ALFA02(J,K1) .LT. BETA) GO TO 3851
PJ = PH I(11 Jl,K) + BETA*(PHKIJ1K)PHI(12 JlK))
1 /ALSUM 4 (BETA**2)*(PH I (12JltK)/{ALSUM*ALFA(NT,1))
2 4PHKI,Jl,K)/(ALSIM(BETA4ALFA02(
3 PHI(I1,J1,K)/{ALFA(NT,1){BETA4ALFA02(J,K,1))))
IF (J .EQ. 2) GO TO 1950
956 NT 1 = NO(11,J2,K)
IF (NT1 .EQ. 0) NT1 = MARK
IF (XP .LT. XI NIT(J2 K) ) GO TO 957
ALSUM = ALFA(NT1,1) 4 BETA 4 ALFA02(J,K,2)
IF (ALFA02(Jt Kt 2) .LT. BETA) GO TO 3852
PJ1 = PHI(11,J2,K) 4 BETA*(PHI(IJ2,K)
1 PHI (I2,J2,K))/ALSUM
2 4 (BETA*2)*(PHI(I2tJ2tK)/(ALSUM*ALFA(NTl,i))
3 4 PHKIJ2,K)/(ALSUM*(BETA4ALFA02(J,K,2)))
4 4 PFI( I1,J2K)/(ALFA(NT1,1)*(BETA+ALFA02(J*K2S)))
9701 S2 = {1.0)*(Y1.0)*(PJl/2.0 2.0*PJ)
S2A = 3.0*(Y1.0)/2.0
GO TO 970
3851 Cl = ALFA02(J,K,1) 4 BETA
PJ =(PHI(IJ1K) ALFA021J,K,1)*
1 (PHK I,Ji,K) PHK 11, J1,K) )
2 /Cl (ALFA021J,K,1)*2)*(PHKI,Jl,K)/ALFA02(J,K,1) 4
3 PHI(I1,J1,K)/BETA)/C1 )/(1.0 ALFA02(J,K,1)/BETA)
IF (J .EC. 2) GO TO 1950
GO TC 956
3852 Cl = ALFA02(J,K,2) 4 BETA
PJ1 =(PHI(IJ2 K) ALFAC2(J,K,2)*
1 (PHI( It J2tK) ~ PHK I1,J2K) )
2 /Cl (ALFA02J,K,2)**2)*(PHKIJ2,K)/ALFA02(JK,2) 4
3 PHI(11J2K)/BETA)/Cl )/(1.0 ALFA02(J,K,2)/BETA)
GO TO 9701
980 S2 =0.0
S2A =0.0
GO TO 970
C LINEAR XINTERPOLAT I ON FOR YDERIVATIVE POINTS.
954 BETA = {XXIN IT(J1,K))/H
PJ = PHI(11 J1K) 4 (BETA/(ALFA02(JtK,l)4BETA))*
1 (PHKIiJl.K) PH I (11 J1 K) )
GO TO 956
957 BETA = (X XINIT(J2,K))/H
PJ1 = PHK 111 J2 K ) + BETA*(PHKIJ2,K) PHK 111 J2 ,K) )
1 /(BETA 4 ALFA02(J,K,2) )
S2 = (i.0)*(Y1.0)*(PJl/2.02.0*PJ)
S2A = 3.0*(Y1.0)/2.0
177
GO TO 970
1951 PHI(ItJ(K) = 0.00
GO TO 869
1950 S2 = (Yl.0)*PJ*2.0
S2A = (Y1.0)*2.0
GO TO 970
C POINTS FOR YDERIVATIVE BY ARC INTERPOLATION.
952 II = IBDRYKJ1K)
12 = IBCRYlUtK)
PJ = PHI(12 tJ K) THETA2(JK)*
1 ( P HI (12 JK) PHI(II,J1,K))
S2 = PJ*(Y1.0)/ALFA(N,3)
S2A = (Y1.0)/ALFA(N,3)
GO TO 970
C LINEAR INTERPOLATION (BETWEEN KNOWN XPOINTS) FOR YPOINTS.
953 PJ = PHKItJlfK) + CCCN2(JK) *
1C PH ICI1J1K)PHI(IJ1K))
S2 = (Yl.0)*PJ
S2A = Y 1.0
IF (J .EQ. 2) S2 = 2.0*S2
IF (J .EC. 2) S2A = 2.0*S2A
C DERIVATIVES
970 IF (K .EQ. KL0W1) GO TO 4970
NT2 = NC{11, JK + 1)
IF (NT2 .EC. 0) NT2 = MARK
NT3 = NO(11 Jt K+2)
IF (NT3 .EQ. 0) NT3 = MARK
IF (XP .LT. XINIT(J,K + 1) ) GO TO 971
BETA = (X XP)/H
ALSUM = ALFA(NT21) + ALFA72(JtK,1) + BETA
IF (K .EQ. 1) GO TO 981
IF (ALFA72( J K11) .LT. BETA) GO TO 3853
PK = PHI ( 11 JK+1) BETAMPHK I2,J,K+1)
1 PHI(I,JK+L))/ALSUM +
2 (BETA**2)(PHI(I2JK+1)/ALFA(NT21) PHI{I,J,K+1)/
3 (ALFA721J,K,D+BETA) PHI(11,J,K+1)*(1.O/ALFA(NT2,1)
4 1.0/(ALFA72(J,K,1)+EETA)>)/ALSUM
972 IF (XP .LT. XINIT(J(K + 2) ) GO TO 973
BETA = (XXPJ/H
ALSUM = ALFA(NT3 1) + ALFA72(J,K,2 ) + BETA
IF (ALFA72(J> K,2) .LT. BETA) GO TO 3854
PK1 = PH I(I 1 J K + 2 ) BETA*(PHI(I2JK+2)
1 PHI(IvJtK42)J/ALSUM
2 + (BETA**2)*(PHI(12,J,K+2)/ALFANT3,1) + PHI(I,J,K+2)/
3 (ALFA72(J,K,2)+BETA) PH I(11,J,K + 2)*(1.O/ALFA(NT3,1) +
A 1.0/(ALFA72(JtK,2)+BETA))J/ALSUM
GO TO 975
4970 BETA = (XXPJ/H
ALSUM = 1.0 + ALFA72(JtK1) + BETA
IF (ALFA72(JK1) .LT. BETA) GO TO 3853
PK PHI(11 JK+1) BETA*(PHI(I2,J,K+1)
o o o
178
1 PH I ( I, J,K+1) ) /ALSUM +
2 (BETA**2)MPHI( 12, J,K4D 4 PHI(I,J,K+1)/
3
4 1.0/(ALFA72(J,K,D+BETA))/ALSUM
4972 ALSUM = 1.0 + ALFA72(J,K,2 ) + BETA
IF (ALFA72IJ,K,2) .LT. BETA) GO TO 3854
PK1 = PHI(Il,J,K+2) BETAMPHIII2J,K+2)
1 PHI(I,J,K + 2))/ALSUM
2 + (BETA**2)*(PHI(I2,J,K+2) PHI(I,J,K+2)/
3 (ALFA72(J,K,2)+BETA) PHI(11,J,K+2)*(1.0 +
4 1.0/(ALFA72(J,K,2)+BETA)))/ALSUM
GO TC 975
3853 Cl = ALFA72IJ,K,1) + BETA
PK =(PHI(I,J,K+1) ALFA72IJ,K,1 )*
1 (PHKIyJfK+l) PHI(I1,J,K + 1))
2 /C1 (ALFA72(J,K,l)**2)*(PHI(I,J K+D/ALFA72 ( J ,K, 1) +
3 PHI(Il,J,K+l)/BETA)/Cl )/C10 ALFA72(JK,1)/BETA)
IF (K .EG. KL0W1) GO TO 4972
GO TO 972
3854 Cl = ALFA72(J,K,2) + BETA
PK1 =(PH ICI,J,K+2) ALFA72IJ,K,2)*
1 tPHK I, J,K+2) PHH Il,J,K+2) i
2 /Cl (ALFA721J,K,2)**2)(PH I(I,J,K+2)/ALFA72IJ,K,2) +
3 PHI(Il,J,K+2)/BETA)/Cl J/C1.0 ALFA72(J,K,2)/BETA)
GO TO 975
981 S3 = 0.0
S3A = 0.0
GO TO 990
971 BETA = {XXIN IT(J,K+1))/H
PK = PHIt11,J,K+1) + (PH I(I,J,K +1) PHI(I1*JK+1))BETA/
1 (BETA4ALFA72(J,K,1) )
GO TO 972
973 BETA = (XXIN IT(J,K42))/H
PK1 = PHI(11,JK42 ) 4 (PHKI,J,K42) 
1 PHIlI1,J,K42))*BETA/(BETA+ALFA72(J,K,2)i
975 S3 = Z*(PK1/2.0^2.CPK)
S3A = (3 .0 )*Z/2.0
990 SUMI = SI 4 S2 4 S3
SUM2 = S1A 4 S2A 4 S3A
TAM = SUM1/SUM2
IF (ABSITAMPHItI,J,K)).GT.EPS) EPS=ABSITAMPHHI,J,K))
PHI(IJ,K) = OMEGATAM + TAGPHI(I,J,K)
869 RETURN
ENO
SUBROUTINE ZBOUND
C
C SUBROUTINE ZBOUND IS FOR HANDLING THOSE POINTS ON THE HEMI
C SPHERICAL SURFACE ENCOUNTERED AS ONE OF THE ADJACENT ZPOINTS
179
C IN THE SIXADJACENTPOINT LAPLACIAN EXPRESSION. THE NOFLOW
C BOUNDARY CONDITION IS APPLIED HERE.
C
COMMON ALFAOt 21,21,2 ),ALFA7(21,21,2), ALFA02(21,21,2),
1 ALFA72(21,21,2),XENC(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CCONl(21,21),CCON2(21,21),THETAl(21,21),
3 THETA2(21,21), IBDRYK 21,21 ),IBDRY2( 21,21), PZ( 36,21) ,
4 TIP(1QO)TOP(1111)H,EPSIMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KLOWl,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,B0G,IG1,IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,N0(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,2132)
CGMMCN IMIG(21),PIG,KIN,KINl,FLO 
C EINSTEINEL SAMNI ARRANGEMENT
JMIT = JMAX 1
MULE = 0
Z1 = Z ALFA(N,5)*H
C CALCULATE CONTRIBUTIONS FROM DPHI/DZ
S3 = (Z1)C2*0PHI(I,J,K) + (1.02.0*ALFA(N,5))*
1 PHI(I,J,K+1)/(1.0+ILFA(N,5)))
S3A = (Z1) *<3.0/(l.C+ALFA(N,5)))
MUT = 1
C THE SAME EQUATION IS USED FOR CALCULATION (BY SERIES EXPANSION
C ABOUT A FIELD POINT ) OF ALL FOUR POINTS NEEDED FOR THE EX
C PANS IONS TO EVALUATE DPHI/DX AND DPHI/DY. THE ONLY CHANGES
NEEDED ARE IN THE IDENTIFICATION OF THE SURROUNDING POINTS.
C THIS END IS ACCOMPLISHED BY FIRST GOING TO ONE OF THE SERIES
C 601,602,603,604 FOR THE NEEDED IDENTIFICATIONS AND THEN
C RETURNING TO THE SERIES BEGINNING AT 610 FOR THE ACTUAL
C CALCULATION.
605 GO TO (601,602,603,604),MUT
610 G = ALFA(NT,5) ALFA(N,5)
PC = PHI(LIL2K1)
IF (ALFA(NT,5) .LT. 1.0) PC = PZ(L1,L2)
IF (ALFAIN,5) .LT. G) GO TO 606
GAG = 1.0 (G* 2) *(1 .C/ALFA(N,5) + 1.0/G)/ALFA(NT,5)
GIG = PC + G*(PHI(L1,L2,K)PC)/ALFA(NT,5)
GUF = (G**2)*(PHI(L1,L2,K)/ALFA(N,5) + PC/G)/ALFA(NT,5)
GO TO 607
606 GAG = 1.0 (ALFA(N,5)*2)*(1.0/ALFA(N,5)+1.0/G)/ALFA(NT,5)
GIG = PHI(L1,L2,K) ALFA(N,5)(PHI(L1,L2,K)PC)/ALFA(NT,5)
GUF = (ALFA(N,5)*2)(PHI(LiL2,K)/ALFA(N,5) 4
1 PC/G)/ALFA(NT,5)
607 TUM = (GIG GUF)/GAG
GO TO (611,612,613,614),MUT
601 NT = N0(I41,J,K)
IF (NT .EQ. 0) NT = MARK
IF (((XENDtJ,K1) XJ/H) .LT. 1.0) GO TO 673
LI = 141
L2 = J
GO TO 610
180
NT = N0(I + 2J K)
IF (NT .EC. 0) NT = PARK
IF (((XENC(J,K1) X)/H> .LT. 2.0) GO TO 674
LI = 1+2
L2 = J
GO TO 610
NT = NG(IfJ+lfK)
IF (NT .EQ. 0) NT = PARK
IF ( ( (ZH )2 + (XSCRT(3.00))**2) .GT. 1.0) Y2 = 1.00
IF (((ZH)**2 + (XSCRT3.00))**2) .GT. 1.0) GO TO 643
Y2 = 1.0 SQRT(1.0 (ZH)**2 XSQRT(3.00))**2)
IF (J .EQ. JM IT) GO TO 743
IF (((Y2Y)/H) .LT. 1.0) GO TO 663
LI = I
L2 = J+l
GO TO 610
NT = NO(I> J+2 K)
IF (NT .EG. 0) NT = PARK
IF (((Y2Y)/H) .LT. 2.0) GO TO 664
LI = I
L2 = J+2
GO TO 610
C ONLY ONE XPOINT AVAILABLE.
674 SI = (1.0)*X*PI
S1A = (1.0)*X
PULE = 1
PUT = 3
GO TO 605
C ARC INTERPOLATION
673 18 = IBDRY2{J K)
19 = IBDRY2 (J K1 )
XT = SQRT(3.00) SQRT(1.0 (Y1.0)*2 21**2)
XDEL = (XT X)/H
A = SCRT(3.00) XENC(JtKl)
B = SQRT(3.00) XENDIJ.K)
C = SQRT(3.00) XT
C = Z H
THP = (ATAN2(21C) ATAN2(0,A))/(ATAN2CZ,B) ATAN2(DA))
PI = PHI(I9JK1) + THM*(PHI(I8JK) PHI(19,J#K1))
SI = (X)*PI/XOEL
S1A = (XJ/XDEL
PULE =1
PUT = 3
GO TO 605
C ONLY ONE POINT AVAILABLE IN YDIRECTION
664 S2 = (Y)PJ
S2A = (1.0)*Y
IF (J .EQ. JMIT) S2 = 2.0*S2
IF (J .EQ. JMIT) S2A = 2.0*S2A
GO TO 622
C ARC INTERPOLATION FOR PZ(I,J) WHEN PROXIMITY OF SPHERES
602
603
643
743
604
o o o
181
C PREVENTS GETTING EFFECTIVE YDIRECTION VALUE.
663 16 = IBDRY1 (J K)
17 = IBCRY1(J K1)
A = ZALFA(N,5)*H
8 = Z~H
THM = (ATAN2(Z,XINIT(J,K)) ATAN2(A,X))/
1(ATAN2(ZXINIT(JfK)) ATAN2(B,XINIT(JKl)))
TAM = PHI (16,J,K) + THM*(PHI(I7,J,K1) PHIU6,J,K))
PZ(ItJ) = TAMOMEGA TAG*PZ(I,J)
GO TO 669
611 PI = TUR
MUT = 2
GO TO 605
612 PI1 = TUM
MUT = 3
GO TO 605
613 PJ = TUM
MUT = A
GO TO 605
614 PJ1 = TUM
620 S2 = (Y)(2.0*PJ PJl/2.0)
S2A = (3.0)*Y/2O
622 IF (MULE .EQ. 1) GO TO 621
SI = (X)*(2.0*PI P11/2.0)
S1A = (~3.0)*X/2O
621 IF (NCAS .EQ. 3) GO TO 669
SUMI = SI + S2 + S3
SUM2 = S1A + S2A + S3A
C FINAL CALCULATION OF NEW PHI VALUE FOR BOUNDARY SURFACE POINT.
623 PZ(I,J> = (SUM1/SUM2)OMEGA TAG*PZ(I,J)
669 RETURN
END
SUBROUTINE ZB02
C
C SUBROUTINE ZB02 HAS THE SAME FUNCTION AS SUBROUTINE ZBOUND,
C EXCEPT THAT IT TREATS THOSE POINTS ENCOUNTERED ON THE SECOND
C COMMENTS FOR SUBROUTINE ZB02 ARE ALL VERY SIMILAR TO THOSE FOR
C SUBROUTINE ZBCUNO. FOR THIS REASON, REFER TO THAT SUBROUTINE
C WHERE CLARIFICATION IS NEEDED. THE STATEMENT NUMBERS ARE
C GENERALLY SUCH THAT THE NUMBERS OF ZB02 ARE THE SAME AS THOSE
C OF ZBOUND WITH AN INITIAL 1 ADDED.
C HEMISPHERE.
C
COMMON ALFA0(21,2i,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21t21,2),XEND(21,21)XINIT(2121)MIGl(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(2121),THETA 1(21,21) ,
3 THETA221,21),IBORYll21,21),IBDRY2(21,21),PZ(36,21),
4 TIP(IOO),TOP(11,11),H,EPS,IMAX,JMAXKMAX,KLOW,KGRADE,
182
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1*IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP#BETA,KIP,KEY,N0(36,21,21)
COMMON CIV(2500)ALFA(25005)PHI(36,2132)
COWMEN IMIG(21)PIG,KINKINlFLO
DIMENSION PZ1(37,21)
INTEGER S
C EINSTEINEL SAMNI ARRANGEMENT
JMIT = JMAX 1
MULE =0
IF lKEY2 .EQ. 1) GO TO 640
DO 677 II = ltIMAX
All = II
XI = (All1.0)*H
IF (II .EG. IMAX) XI = SORT(3.00)
DO 677 J1 = 1,JMAX
677 PZl(IltJl) = 100.0 (100.O/SQRT(3.00))*X1
KEY2 =1
640 Z1 = Z ALFA(N 5)*H
S3 = lZl)*(2.0*PHI(I,J,K) + (1.02.0*ALFA(N5))*
1 PHI(I,J,K + l )/(1.0 + ALFA(N,5)))
S3A = (Z1)*(3.0/(1.C+ALFA(N,5)))
MUT = 1
1605 GO TO (1601,1602,1603,1604),MUT
1610 G = ALFA(NT,5 ) ALFA(N,5)
PC = PH I (L1L2K1)
IF (ALFA(NT,5) .LT. 1.0) PC = PZ1(L1,L2)
IF (ALFA(N,5) .LT. G) GO TO 1606
GAG = 1.0 G/ALFA(N,5)
GIG = PC + G*(PHI(L1L2K) PC)/ALFA(NT,5)
GUF = (G**2)*(PHI(L1,L2,K)/ALFA(N,5) + PC/G)/ALFA(NT,5)
GO TO 1607
1606 GAG = 1.0 ALFA(N,5)/G
GIG = PHI (LI,L2K) ALFA(N,5)*(PHItLI,L2,K)PC)/ALFA(NT,5)
GUF = (ALFA(N,5)2)(PHI(L1,L2,K)/ALFA(N,5)
1 PC/G)/ALFA(NT,5)
1607 TUM = (GIG GUFJ/GAG
GO TO (1611,1612,1613,1614),MUT
1601 NT = N0(I1,J,K)
IF (NT .EQ. 0) NT = MARK
IF (((X XINIT(J,K1))/H) .LT. 1.0) GO TO 1673
LI = 11
L2 = J
GO TO 1610
1602 NT = NO(I~2,J,K)
IF (NT .EG. 0) NT = MARK
IF (((X XINIT(J,K1))/H) .LT. 2.0) GO TO 1674
LI = 12
L2 = J
GO TO 1610
1603 NT NG(I,JiK)
183
IF
(NT .EC. 0) NT =
MARK
IF
(MZH)*2 X**2) .GT. 1.0) Y2 0.00
IF
((t ZH) *2 X**2) .GT. 1.0) GO TO 1643
Y2
= SORT(1.0 (Z
H)2 X*2)
1643
IF
(J .EC. 2) GO TO
1743
IF
(((YY2/H) .LT.
1.0) GO TO 1663
1743
LI
= I
L2
= Jl
GO
TO 1610
1604
NT
= NO(I J2,K)
IF
(NT .EQ. 0) NT =
MARK
IF
(((YY2 ) /H ) .LT.
2.0) GO TO 1664
LI
= I
L2
= J2
GC
TO 1610
C CNLY ONE POINT AVAILABLE IN XDIRECTION
1674 SI = (XSCRT(3.00))*PI
S1A = XSGRTl3.00)
MULE =1
MUT = 3
GO TO 1605
C ARC INTERPOLATION FOR XDIRECTION POINT
1673 18 = IBDRY1(J K)
19 = IBDRY1(J K1)
XT = SGRTd.O Y*2 Z 1**21
XDEL = (X XTJ/H
A = ZH
THM = (ATAN2(ZXINIT(JK)) ATAN2(Z1XT))/
1 (ATAN2(Z,XINIT(J,K)) ATAN2(A,XINIT(JKl)) )
PI = PHI (18 J K) + THMMPHK 19 JK1) PHI{18,J,K))
SI = (XSCRT(3.00))*PI/XDEL
S1A = (XSQRTI3.00))/XDEL
MULE =1
MUT = 3
GO TO 1605
C CNLY ONE POINT AVAILABLE IN YOIRECTION
1664 S2 = (Yl.0)*PJ
S2A = Y 1.0
IF (J .EQ. 2) S2 = 2.0*S2
IF (J .EG. 2) S2A = 2.0*S2A
GO TO 1622
C ARC INTERPOLATION FOR PZ1(I,J) NECESSITATED BY PROXIMITY OF OT
HER
C HEMISPHERE.
1663 16 = IBRY2(J K)
17 = IBDRY2(J,K1)
A = Z ALFA(N,5)*H
B = Z H
C = SGRTC3.00) XENC(JK)
D = SGRT (3.00 ) XENCUtK1)
E = SORT(3.00) X
o o o o o no o
184
THM = (ATAN2(Z,C) ATAN2{AE ))/(ATAN2(Z.C) ATAN2
TAM = PHI(I6,J,K) THM*(PHI(I7,J,K1) PHIII6,J,K))
PZ1(IJ) = TAM*OMEGA PZ1(I,J)*TAG
GG TO 1668
1611 PI = TUM
NUT = 2
GO TC 1605
1612 PI1 = TUM
NUT = 3
GO TO 1605
1613 PJ = TUN
NUT = A
GO TO 1605
1614 PJi = TUN
1620 S2 = ( 1.0)CY1.0)*(PJl/2.0 2.0*PJ)
S2A = 3.0*(Y1.0)/2.0
1622 IF (NULE .EQ. 1) GO TO 1621
SI = (l.C)(XSQRT(3.00))*C PI 1/2.0 2.0*PI)
SI A = (XSCRT13.00))3.0/2.0
1621 SUN1 = SI S2 S3
SUM2 = S1A S2A + S3A
1623 PZl(ItJ) = (SUMI/SUM 2 ) *OMEGA + TAG*PZUI,J)
1668 BING = PZ1(I J)
1669 RETURN
1009 FGRNAT (1IF 10.5/10F10.5)
ENO
SUBROUTINE YBOUND
SUBROUTINE YBOUND SERVES THE FUNCTION OF SOLVING THE NOFLOW
BOUNDARY CONDITION ON A HEMISPHERICAL SURFACE WHEN ONE OF THE
TWO ADJACENT POINTS IN THE YDIRECTION FALLS ON THE HEMISPHERE
CONMGN ALFA0(21,21,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,212),XEND{21,21),XINIT(21,21),MIG1( 21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA 1(21,21),
3 THETA2(21,21),IBDRY1(21,21),IBDRY2(21,21),PZ(36,21),
4 TIP(100),TOP(11,11 ),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JNID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,NOC36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FLO
C EINSTEINEL SAMNI ARRANGEMENT
DIMENSION PY(37,21),PY1(37,21)
C THE ARRAY PY CONTAINS THE CURRENT VALUES FOR THOSE POINTS
C LYING ON THE FIRST HEMISPHERE, WHILE PY1 IS FOR POINTS ON THE
C SECOND HEMISPHERE.
KL0W1 KLOW l
185
KAB = O
MULE = O
C INITIALIZE PY AND PY1 THE FIRST TIME ENTERING YBOUND.
IF (KEY1 .EG. 1) GO TO 376
DO 377 Kl=l,KLOW
DO 377 11=1,IMAX '
All = II
XI = l A111.0)H
IF (II.EC. IMAX) XI = SQRT(3.00)
PYKIl.Kl) = 100.0 (100.0/SQRT(3.00))*X1
377 PY(11K1) = 100.0 (100.0/SQRT(3.0))XI
KEY1 = 1
C IF IMID EQUALS 2, THIS IS A KEY TO GO TO THE PORTION OF
C YBOUND DEALING WITH THE SECOND HEMISPHERE.
376 IF (IMID .EQ. 2) GO TO 379
C POINTS ON FIRST HEMISPHERE.
IF ( J .EC. JMAX) GO TO 1376
C VALUES FOR DPHI/DY
S2 = (l.O)MYALFA(N,3)*H)*(2.0*PHI(I,J,K) + PHI(I,J+1,K)
1 *(1.02.0*ALFA(N3))/(1.0+ALFA(N3) ))
S2A = (1.0)*(YALFA(N3)*H)*(3.0/(1. O+ALFA (N,3 ) ) )
1377 IF (((XEND(J K) X)/H) .LT. 1.0) GO TO 374
C EVALUATE PI.PI1 (FOR DPHI/DX) AND PK,PK1 (FOR DPHI/OZ) ALL BY
C LINEAR INTERPOLATION BETWEEN OTHER POINTS IN THE FIELD.
C THESE FOUR VALUES ARE THEN USED LATER IN SERIES EXPANSIONS TO
C EVALUATE THE INDICATED DERIVATIVES.
PB = PH I(1 + 1,J1,K)
NT = NO(1+1,J,K)
IF (NT .EC. 0) NT = MARK
IF (ALFA(NT,3) .LT. 1.0) PB = PY(I+1,K)
PI = PHI(I+1,JK) (ALFA(N,3)/ALFA(NT ,3))*
1 (PH I(1 +1,J,K)PB)
IF (((XEND(J,K) X)/H) .LT. 2.0) GO TO 373
PB1 = PHI(1+2,J1,K)
NT = N0(1+2,JK)
IF (NT.EC. 0) NT = MARK
IF (ALFA(NT,3) .LT. 1.0) PB1 = PY(I + 2,K)
PI1 = PH 1(1+2,J,K) (ALFA(N,3)/ALFA(NT,3 ) )*
1 (PH I(1 + 2,J,K )PBl)
372 PB = PHK I, J1,K + 1)
NB = NO(I,J,K+1)
IF (NB .EQ. 0) NB = MARK
IF (ALFA(NB,3 ) .LT. 1.0) PB = PY(I,K+1)
PK = PHI(I,JK+1) (ALFA(N,3)/ALFA(NB,3))*
1 (PHK I, J.K+DPB)
PB1 = PHI(I,JlK + 2 )
IF (K .EC. KL0W1) NB = MARK
IF (K .EQ. KL0W1) GO TO 375
NB NO(I,J,K+2)
IF (NB .EQ. 0) NB = MARK
IF (ALFA(NB,3) .LT. 1.0) PB1= PYU.K+2)
186
375 PK1 = PH I (If J t K+2) (ALFA(N,3 I/ALFA(NB#3 ) I*
1 (PHIU, J,K+2)PBl)
IF (MULE .EQ. 1) GO TO 1372
SI = (X)*(2.0*PI PI1/2.0)
S1A = (3.0)*X/2.0
1372 S3 = (20*PK PK1/2 .0 I*(1.0 I*Z
S3A = (3.0)*Z/2.0
GO TO 378
C FOR POINTS LOCATED NEAR Y = A, WHERE THE SECOND POINT (IN THE
C YDIRECTION) LIES ON ANOTHER HEMISPHERE OUTSIDE THE SOLUTION
C SPACE.
1376 S2 = (1.0)*(YALFA(N3)*H)*PHI(I,JK)*2.0/ALFA(N3)
S2A = (1.0)*(YALFA(N,3)*H)*2.0/ALFA(N,3I
GO TO 1377
C CNE POINT AVAILABLE IN XDIRECTION.
373 SI = PIX*(1.0)
S1A = (1.0)*X
MULE = 1
GO TO 372
C ARC INTERPOLATION FOR NEXT XPOINT.
374 XT = SQRT(3.00) SQRTd.O (YALFA(N,3)*Hl.0)**2 Z**2)
17 = IBDRY2{J1 K I
0 = 1.0 Y + H
E = SORT(3.00 ) XENC(JK)
G = 1.0 Y
B = SORT(3.00) XENCtJ~lfKJ
A = SORT(3.00) XT
C = 1.0Y+ALFA(N,3J*F
THM = (ATAN2(C,A) ATAN2(G E11/(ATAN21DB) ATAN2(GE)I
PI = PHI(I+1,J,K) + THMMPHI ( I7,J1,K) PHI(I+1,J,K1)
XDEL = (XT Xl/H
51 = (XI*PI/XDEL
S1A = (XI/XDEL
MULE = 1
GO TO 372
C SECTION FOR POINTS ON BOUNDARY OF SECOND HEMISPHERE.
C COMMENTS FOR STEPS ON SECOND HEMISPHERE ARE SIMILAR TO THOSE
C INDICATED ALREADY FOR EARLIER STEPS FOR SPHERE 1.
379 IF (J .EG. 1) GO TO 1379
52 = (1.0)*(Y+ALFA(N4)*H 1.01*(PH I(I,Jl,KJ*
1 (<2.0*ALFA(N,4)l.C)/(1 .O+ALFA(N4)1 I 2.0*PHI(I,J,K11
S2A = (Y+ALFA(N,4I*H1.0J*3.0/(1.0+ALFA(N4II
1378 IF (((X XINIT(J,K) l/H) .LT. 1.0) GO TO 364
PB = PHI( I1*J + lfK)
NT = NO( I1,J,K)
IF (NT.EC. 0) NT = MARK
IF (ALFA(NT4I .LT. 1.0) PB = PY1(I1,K)
PI = PHI(I1,JK) (ALFA(N,4)/ALFA(NT,4)I*
1 (PB PHI(IltJK)l
IF (((X XINIT(J,K) l/H) .LT. 2.0) GO TO 363
PB1 = PHI(12*J+1#K)
187
NT = NO(12 J K)
IF (NTEG 0) NT = NARK
IF (ALFA{NT4) .LT. 1.0) PB1 = PYl(I2tK)
PIl = PHI(I2JK) (ALFA(Nf4)/ALFA(NT4))*
l (P81PHK I2J*K) )
362 PB = PHH ItJ+lfK + 1)
NB = N0(ItJ,K+1)
IF (NB .EC. 0) NB = PARK
IF (ALFA(NB*4) .LT. 1.0) PB = PY1(I,K+1)
PK = PHI( 11Jt K + l) +(ALFA(N,4)/ALFA(NB,4))(PBPHI(IfJfK+1))
PB1 = PH I(If J +1,K + 2)
IF (K .EC. KL0W1) NB = MARK
IF (K .EG. KL0W1) GO TO 365
NB = N0(IfJfK+2)
IF (NB .EQ. 0) NB = MARK
IF (ALFA(NB f4) .LT. 1.0) PB1= PYl(ltK+2)
365 PK1 = PHI(I,J,K+2)+(ALFA(N,4)/ALFA(NB,4))*
1 (PB1PH I (It JtK + 2))
IF (MULE .EQ. 1) GO TO 1362
SI = (l.C)*(XSQRT(3.0))*(PIl/2.0 2.0*PI)
SiA = (XSQRT(3.0))*(3.0/2.0)
1362 S3 = Z*(PK1/2.0 PK2.0)
S3A = (3.0 )*Z/2.0
GO TO 378
1379 S2 = 2.0*(Y+ALFA(N,4)*H1.0)*PHI(I,J,K)/ALFA(Nf4)
S2A = 2.0*(Y+ALFA(Nf4)*H1.0)/ALFA(N,4)
GO TO 1378
C CNE POINT AVAILABLE IN XDIRECTION.
363 SI = (XSQRT13.00))*PI
SIA = X SQRT(3.00)
MULE =1
GO TO 362
C ARC INTERPOLATION FOR XPOINT.
364 XT = SQRT(1.0 Z**2 (Y+ALFA(N,4)*H)**2)
17 = IBDRY1(J + 1 K)
A = Y + H
B =Y + ALFA(N,4)*H
THM = (ATAN2(B,XT) ATAN2(Y,XINIT(JtK)))/
1 (ATAN2(A,XINIT(J+l.K)) ATAN2(YfXlNIT(J,K)))
PI = PHI(IlfJ,K) + THM*(PHI(I7J+1,K) PHI(I1,J,K))
XDEL = (X XTJ/H
SI = (XSQRT(3.00))*PI/X0EL
SIA = (XSQRT(3.00))/XDEL
MULE = 1
GO TO 362
378 SUMI = SI + S2 + S3
SUM2 = SIA + S2A + S3A
C FINAL CALCULATION OF A NEW VALUE FOR PHI AT THE BOUNDARY
C SURFACE POINT.
IF {IK ID .EQ. 2) PYl(IfK) = (SUM1/SUM2)*0MEGA +TAG*PY1(I,K)
IF (IK ID .EQ. 2) BING = PY1(I,K)
o o o o no o
188
IF (IMID .EQ. 2) GO TO 2369
PY(IK) = (SUM1/SUM2)*0MEGA 4 TAG*PY(I,K)
BING= PY(I,K)
2369 RETURN
1009 FORMAT (1IF 10.5/L0F1C.5)
END
SUBROUTINE BETWN
SUBROUTINE BETWN HANDLES THOSE LATTICE POINTS BETWEEN THE TOPS
OF THE HEMISPHERES ANO THE SECTION WHERE THE NET IS GRADED.
COMMON ALFAO(21,21,2),ALFA7(21,21,2),ALFA02( 21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA 1(21,21),
3 THETA2(21,21),IBDRY1(21,21),IBDRY2(21,21),PZ(36,21),
4 TIP(100),TOP(11,11 ),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1KEY2,JMID,MARK,NCAS,KL0W1,IMIDTAB1TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1,IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,N0(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FL0
C THIS PORTION OF THE ITERATION PROGRAM IS FOR THE REGION WHERE
C THE SIMPLE SIXPOINT FORMULA FOR THE LAPLACIAN EXISTS.
C COMPUTE CONSTANTS NEEDED DUE TO LAST UNEVEN XSPACING.
IMIT = IMAX 1
AIMAX = IMAX
TAB7 = (SCRT(3.00) (AIMAX2.0)*H)/H
TAB1 = 4.0 + 2.0/(1.0+TAB7) 42.0/(1.04TAB7)*TAB7)
TAB2 = 2.0/(TAB7*(1.04TAB7))
TAB3 = 2.0/(1.0+TAB7)
KB 1 = KLCW + 1
KB2 = KGRADE 1
IF (KEYTAG .EQ. 1) KB2 = KMAX 1
TRAVERSE
IN Z
DIRECT
ION
DO
700
K = KB
i 1,KB2
TRAVERSE
IN Y
DIRECT
ION
DO
701
J =
1,JMAX
TRAVERSE
IN X
DIRECT
ION
DO
701
I =
2,IMIT
IF
(J
.EQ.
1) GO
TO
702
IF
(J
.EQ.
JMAX)
GO
TO 703
IF
( I
.EQ.
IMIT)
GO
TO 705
C STANDARD FORMULA FOR THE LAPLACIAN USING SIX ADJACENT POINTS
C ALL AT A DISTANCE H AWAY FROM THE OBJECT POINT.
TAM = (PHI(11JK) 4 PH I(1 + 1J,K) 4 PHI(I,J4l,K) 4
1 PHI(I,J1,K) 4 PHI(I,J,K+1) 4 PHI(IJK1)J/6.0
C SECTION 710 COMPARES THE CHANGE IN THE PHI VALUE AT THE
C PRESENT POINT WITH THE MAXIMUM CHANGE THUS FAR ENCOUNTERED
o o o o n o o
189
C IN THE CURRENT ITERATION. THE NEW VALUE OF PH1(1,J,K) FOR
C THE GIVEN POINT IS ALSO CALCULATED FROM THE OVERRELAXATION
C EQUATION, USING OMEGA.
710 IF (AES (TAMPH I( I,J,KH.GT.EPS) EPS=ABS t TAMPHI ( I J ,K))
PHI(IJ,K) = QMEGA*T AM + TAG*PHI(I,J,K)
GO TO 701
C SPECIAL SECTION FOR FINAL XSPACING LESS THAN H
705 TAM = (PHKI, J + 1,K) 4 PHI(I,J1,K)
1 + PHI(I,JK+1) 4 PHI(I,J ,K1) 4
2 PHI ( I4l,J,K)TAB2 4 PHI(I1,J,K)*TAB3)/TAB1
GO TO 710
C SECTION FOR Y=0 WHERE DPHI/OY EQUALS ZERO.
702 IF (I .EG. IMIT) GO TO 715
TAM = (PH I ( I1JK) 4 PHI(I41,J,K) 4 PHI(I,J,K4l) 4
1 PHI(I,J,K1) + 2.0PHI(I,J+1,K))/6.0
GO TO 710
C SPECIAL SECTION FOR Y=A, WHERE DPHI/OY EQUALS ZERO
703 IF (I .EQ. IMIT) GO TO 725
TAM = (PHI(I1,J,K) 4 PHI(1+1J,K) 4 PHI(I,J,K4l) 4
1 PH I (I,J,K1) 4 2.0*PHI( I,J1,K))/6.0
GO TO 710
C SECTIONS 715 AND 725 ARE FOR THE COMBINED CASES OF DPHI/DY =
C ZERO AND THE UNEQUAL XSPACING NEAR X = SQRTO.GO).
715 TAM = (PHI(I1,J,K)*TAB3 4 PH I(141,J,K)*TAB2 4
1 PHI(I,J,K41) 4 PHI(I,J,Ki) 4 2.0*PHI(I,J41,K))/TAB1
GO TO 710
725 TAM = (PHI(I1,J,K)*TAB3 4 PHI{141,J,K)*TAB2 4
1 PH I (I,J K41) 4 PH I(I,J,K1) 4 2.0*PHI(IJ1,K))/TAB1
GO TO 710
701 CONTINUE
7C0 CONTINUE
RETURN
END
SUBROUTINE GRADE
THIS SUBROUTINE MAKES THE TRANSITION FROM A GIVEN MESH SIZE,
H, TO CNE WHICH IS TWICE AS LARGE ( 2H).
COMMON ALFAO(21,21,2),ALFA7<21,21,2),ALFA02{21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA 1(21,21),
3 THETA2(21,21),IBDRY1(21,21),IBDRY2(21,21),PZ(36,21),
A TIP(100),TOP(11,11),H,EPS,I MAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JKID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,'
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,B0G,IG1,IG2,I,J,K,
7 AI,AJ*AKNXYZTAG0MEGAXPBETA,KIPKEY*N0(3621,21)
COMMON DIV(2500),ALFA(2500,5),PH 1(36,21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FLO
190
C EINSTEINEL SAMNI ARRANGEMENT
C COMPUTE CONSTANTS NEEDED FOR CALCULATIONS NEAR X SQRT(3.00)
C DUE TO UNEQUAL XSPACING THERE.
IMIT = IMAX 1
IMIT2 = IMAX 2
JMIT = JMAX 1
AI MAX = IMAX
TAB7 = (SQRTO.OO) tAIMAX2.0)*H)/H
BUG = 1.0 TAB72
BUG1 = TAB7*(1.0+TAB7)
BIP = (3.0+TAB7)/8.0
BIP1 = 1.0 + TAB7
BING = 4.0 + (0.5 + 1.0/(1.0+TAB7))/<0.125*<3.0+TAB7))
C MAKE COMPUTATIONS FOR LAST FULL ZVALUE WHERE POINTS ARE
C LOCATED A DISTANCE H APART IN HE XYPLANE.
C SERIES THROUGH 760 HANDLES THOSE POINTS WHICH HAVE SIX
C ADJACENT POINTS ALL LYING H AWAY.
DO 760 J = 2 ,JMIT,2
DO 760 I = 2 IMIT
IF (I .EC. IMIT) GO TO 751
TAM = (PH I(I 11J K) + PHI(I+1,J,K) 4 PHI(I,J1,K) 4
l PH KI J+lK) + PHI(IJ,K1) PHI(I,J,K + 1))/6.0
752 IF (ABS{TAMPHI(IJ K )).GT.EPS) EPS=ABS( TAMPHK IJ,K))
PHI(ItJfK) = OMEGA*TAM 4 TAG*PHI(I,J,K)
GO TO 760
C STEP 751 IS FOR THE UNEQUAL XSPACINGS AT X=SQRT(3.00).
751 TAM = (PHKI,J+1,K) + PHI(I,J1,K) 4
1 PHIUfJtK + L) + PHI(IJ,Ki) 4
2 PHI (I + 1JK)*TAB2 4 PH I(I1.J,K)*TAB3)/TAB 1
GO TO 752
760 CONTINUE
C THE SERIES OF STEPS THROUGH 770 ARE FOR THOSE POINTS (STILL
C IN THE SAME ZPLANE) WHICH HAVE (ALTERNATING FROM POINT TO
C POINT) ALL SIX ACJACENT POINTS EITHER AT A DISTANCE H OR 2H.
DO 770 J= 1 JMAX 2
00 770 1 = 2,IMIT,2
IF ( J .EC. 1) GO TO 762
IF (J .EC. JMAX) GO TO 763
IF (I .EQ. IMIT) GO TO 761
C POINTS WITH SIX ACJACENT POINTS H AWAY.
TAM = (PHKI1,J,K) 4 PHI(I41,J,K) 4 PHI(I,J1,K) 4
1 PHI(I,J41,K) 4 PHK I, J ,K1) 4 PHK I,J,K41) )/6.0
IF (ABS(TAMPHI(I,J,K)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(I,J,K) = OMEGATAM 4 TAGPHKI,J,K)
C POINTS WITH SIX ADJACENT POINTS AT A DISTANCE 2H.
TAM = (PHK I lJK) + PHI(I + 3JK) 4 PHI (1.41, J42.K) 4
1 PHI(I41,J2,K) 4 PHI(141,J,K+2) 4 PHI(I + l,JK2))/6.0'
IF (ABS(TAMPHI(141,J,K)).GT.EPS) EPS=ABS(TAMPHI(141,J,K))
PHKI+1J*K) = TAM*OMEGA 4 TAG*PHI(141,J,K)
GO TO 770
C SERIES 7610 FOR UNEQUAL XSPACING NEAR X = SQRTO.O)
191
761 TAM = (PHI ( I J + 1,K) 4 PHI(ItJitK) +
1 PHI(I,J,K4l) 4 PHI(IfJ,Kl) 4
2 PHI(I+l,J,K)TAB2 4 PH I (I1,J,K)TAB3)/TAB 1
IF (ABS(TAMPHI(I,J,K ) J.GT.EPS) EPS=ABS(TAMPHI
PHI(IfJfK) = OMEGA*TAM 4 TAG*PHI(I,J,K)
GG TO 770
C SERIES 7620 FOR Y = 0, WHERE DPHI/DY = 0.
762 IF (I .EC. IMIT) GO TO 764
TAM = (PH I(I1fJK) 4 PHI(I+1,J,K) 4 PHI(I,J,K4l) 4
1 PHI{I,J,K1) 4 2.0*PHI(I,J+1,K))/6.0
IF (ABS(TAMPHI(ItJK)J.GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(IfJfK) = OMEGATAM 4 TAG*PHI(I,J,K)
TAM = (PHI(IltJtK) 4 PHI(I43,J,K) 4 PHI(1+1J,K+2)
1 PHI(I+ltJK2) + 2.0*PHI(I+lJ+2K))/6.0
IF (ABS(TAMPHI(I+lfJfK)J.GT.EPS) EPS=ABS(TAMPHI(1+1,J,Ki)
PHKI + lfJfK) = T AM*OMEGA + TAGPHI(I+1,J,K)
GO TO 770
C SERIES 7630 FOR Y=A, WHERE DPHI/DY = 0.
763 IF (I .EC. IMIT) GO TO 765
TAM = (PHKIlfJfK) 4 PH I(1+1J,K ) + PHI(I,J,K+1) +
1 PHI(I,J,K1) + 2.0PHI(I,J1K)J/6.0
IF (ABS(TAMPHI(I,JK)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(IfJfK) = OMEGATAM + TAG*PHI(IJtK)
TAM = (PHKI1,J,K) + PHI ( 1 + 3JK) 4 PHI(1 + 1JK+2) +
1 PHI(1+1,J,K2) + 2.0*PHI(I+1,J2,K))/6.0
IF (ABStTAKPHKI+l,J,K)J.GT.EPS) EPS=ABS(TAMPHI(1+1,J,K))
PH I(I +1 J K) = TAM*OMEGA + TAG*PHI(1 + 1,J,K)
GO TO 770
C SERIES 764 AND 765 ARE FOR POINTS WHERE BOTH DPHI/DY = 0 AND
C THE UNEGUAL XSPACING OCCURS.
764 TAM = (PHK11,J,K)*TAB3 + PH I (1 + 1,J,K)*TAB2 +
1 PHI(IJ,K + 1) + PHI(I,d K1) + 2.0*PHI(I,J+1,K)J/TAB1
IF (ABS(TAMPHI{IJK)J.GT.EPS) EPS = ABS(TAMPHI(I,J,K))
PH I (I,J,K) = OMEGATAM + TAG*PHI(I,J,K)
GO TO 770
765 TAM = (PHKI1,J,K)*TAB3 + PHI(1+1,J,K)*TAB2 +PHI(I,J,K+1)+
1 PHI(IJ,Kl) + 2.0PHI(I,Jl,K))/6.0
IF (ABS(TAMPHI(I,J,KJJ.GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(IfJiK) = OMEGATAM + TAG+PHI(IJK)
770 CONTINUE
C INTERMEDIATE ROW BETWEEN NETWORK OF SIZE H AND THAT OF SIZE 2H
K= KGRADE + 1
C THROUGH 780, TRAVERSE ODD JVALUES, ALL XVALUES.
CO 780 J=1JMAX,2
DO 780 1=2,IMIT
IF (J .EQ. 1 .OR. J .EQ. JMAX) GO TO 772
IF (I .EQ. IMIT) GO TO 771
C DIFFERENCE EQUATION BASED ON INVARIANCE OF LAPLACIAN WITH AXIS
C ROTATION.
TAM = (2.0*(PHI(I,J1,K) + PHI(I,J+l,Kn + PHI(1+1JKI)
192
1+ PHKI + l, JtK+1) + PHI ( 11 J,K1) + PHI ( I1,J,K+1))/8.0
IF (ABS(TAMPHI(IJK)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PH KI J K) = OMEGA*TAM TAG*PHI(I*J*K)
GO TO 780
C SERIES 771 IS FOR THE UNEQUAL XSPACING NEAR X = SQRK3.00).
C PVC IS A VALUE INTERPOLATED LINEARLY BETWEEN TWO LATTICE
C VALUES. THE ADDED TERMS IN THE TAM EXPRESSION REPRESENT
C THE FIRST AND SECOND DERIVATIVE OF PHI IN THE XDIRECTION.
771 PVC = {PHI!I1,J,K+1 ) + PHK I1JK1)J/2.0
TAM = (2.0*(PHI( IJ1VK) + PHI ( I J + l VK) ) + PH I (I+ 1, J, Kl)
1 + PH I {1 + 1, J,K+1) + PHK Ii, J,K1) + PHKI1, J,K+l))/8.0
2 +ITAB71.0)* PVC/(4.C*BIP1) + BUG*{PVC/BI PI) /4.0
TAM = TAM/(1.0 + BUG/(A .0*TAB7))
IF (ABS{TAM PHI(I JK) ) .GT. EPS) TAM = TAB7*PVC/BIPl
IF (ABS(TAMPHI(IJK)I.GT.PS) EPS=ABS(TAMPHI(I,J,K))
PHI( IJ K) = CMEGA*TAM + TAG*PHI{I,J,K)
GO TO 780
C SERIES 772 IS FOR THE DPHI/DY = 0 CONDITION AT Y = 0 OR Y A.
772 IF (J .EQ. 1) L = J + l
IF (J .EC. JMAX) L = Jl
IF { I .EC. IMIT) GO TO 774
TAM = (4.0*PHI{ILK) + PH I (I + 1,J,K1) PHK 1 + 1, J ,K+1)
1 + PH I (11 J Kl) + PHKIl,J,K+l))/8.0
IF (A8S(TAMPHI(I,J,KK.GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(ItJtK) = OMEGA*TAM + TAG*PHI(I,J,K)
GO TO 780
C SERIES 774 IS FOR UNEQUAL XSPACING IN ADDITION TO THE OTHER
C CONDITIONS OF SERIES 772
774 PVC = (PHIUl, J,K+1) + PHK I1J Kl) J/2.0
TAM = (4.0*PHI(I,L,K) + PHKI+1,J,K1) + PHI(I+1,JK+1)
1 + PHI(IliJ.Kl) + PHK11,J.K+l))/8.0
2 +(TAB71.0)*PVC/(4.C*BIP1) + BUG*(PVC/BI PI)/4.0
TAM = TAM/{1.0 + BUG/(40*TAB 7))
IF (ABS(TAM PH I(I J K)) .GT. EPS) TAM = TAB7*PVC/BIPI
IF (ABS(TAMPHKI,J,KJJ.GT.EPS) EPS=ABS( TAMPHI (I ,J,K))
PH I(I,JK) = OMEGA*TAM + TAG*PHI(IJK)
780 CONTINUE
C TRAVERSE EVEN JVALUES, WITH EVEN VALUES OF I (X).
DO 790 J=2 JMIT 2
DO 790 1 = 2IMIT 2
IF (I .EG. IMIT) GO TO 776
C DIFFERENCE EQUATION FROM A TRIPLESERIES EXPANSION.
TAM = (PHK 11 Jl, Kl) + PH 1(11,J + l, Kl) +
1 PHI(I1,J1,K+1) + PHI(11,J + lK + 1) +
2 PHKI + 1 J1K 1) + PHK 1 + 1, J + l, Kl) +
3 PHKI + 1,Jl,K+l) + PHK 1 + 1,J + 1,K+1) )/8.0
IF (ABS(TAMPHI( I,JK)).GT.EPS) EPS=ABS( TAMPHK I J,K ))'
PHKI,J,K) = OMEGA*T AM + TAG*PHI(I,J,K)
C DIFFERENCE EQUATION BASED ON INVARIANCE OF LAPLACIAN.
TAM = (2.0*(PHI(I+2,JK) + PHI(I,J,K)) PHKI+l,J+l*Kl) +
1 PHKI + 1, J1K1) +' PHI( I + l,J+1*K+1)
193
2 PHI(I+l,Jl,K+l))/8.0
IF (ABS(TAMPHI(1+1,J,K)J.GT.EPS) EPS*ABS(TAMPHIU + l,J,K))
PHI(I +1, J K ) = T AM*OMEGA + TAG*PHI(1+1,J,K)
GO TO 790
C SERIES 7760 FOR UNEQUAL XSPACING.
776 BUG7 = 0.5MTAB71.0 )/(1.0+TAB7)
TAM = (PHIt I1,J1,K1) + PH I {I1 J + l,K1)
1 + PHI(I1,J1,K+1) +
2 PH I(I1 J + l,K+1) + PHI(1+1,J1,K1) + PH1(1+1,J+l,Kl) +
3 PHI ( 1 + 1 J1K+1} + PHIU + 1, J+l,K+l) )/8.0 BUG7
4 *(PHI(1 + 1J K PHI(I1,J,K)) + BUG*(PHHI+1,JK)/BUG1
5 PHKI1, J KJ/BIP1J/2.0
TAM = TAM/(l.O+BUG/t 2.0*TAB7))
IF ( ABSTAMPHK I, J,K) ) .GT. EPS) TAM=TAB7*
1 PHI (11,J,K J/BIPl
IF (ABS(TAMPHIU,J,K)J.GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PH I(IJ K) a QMEGA*T AM + TAG*PHI(I,J,K)
790 CONTINUE
C FIRST ROW AT NEW LEVEL WITH MESH LENGTH 2H.
K = KGRADE +2
00 795 J=1, JMAX 2
DO 795 1=3IMIT22
IF (J .EG. 1 .OR. J .EQ. JMAX) GO TO 777
IF ( I .EQ. IMIT2) GO TO 779
C DIFFERENCE EQUATION BASED ON SIX ADJACENT POINTS
C A DISTANCE 2H AWAY FROM OBJECT POINT.
TAM = (PHI(I2,J,K) + PHI( 1 + 2,J K) + PHI(IJ,K2) +
1 PH I (I J K+l) + PH I (I, J21K ) + PHK IJ + 2,K ) ) /6.0
IF (ABS(TAMPHI(IJK)).GT.EPS) EPS=ABS(TAMPHI(IJ,K))
PH I(I,J,K) = OMEGA*T AM + TAG*PHI(I,JK)
GO TO 795
C SERIES 7770 FOR DPHI/DY = 0 CONDITION (Y=0 OR Y=A).
777 IF (J .EG. 1) L =J+2
IF (J .EG. JMAX) L=J2
IF (I .EG. IMIT2) GO TO 781
TAM = (PHI(I2J,K) + PHI(I+2JK) + PHI(I,J,K2) +
1 PH I(I J K + l) + 2.0*PHI(I,L,K))/6.0
IF (ABS(TAMPHI(IJK)).GT.EPS) EPS=ABS(TAMPHI(I,JK))
PHIt I J K) = OMEGA*T AM + TAG*PHI(I,J,K)
GC TO 795
C POINTS WITH UNEQUAL XSPACING NEAR X = SQRT(3.00).
779 TAM = (PHI(I,J,K2) + PHI(IJ,K+1) + PHI(I,J2,K) +
1 PHI(I,J + 2,K) + PH 1(1 + 2,J,K)/(BIP*(1.0+TAB7))
2 + PHI(I2,J,K)/(2.C*BIP))/BING
IF (ABS(TAMPHI(IJ,K)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PH I ( I,J,K) = OMEGA*TAM + TAG*PHI(I,J,K)
GO TO 795
C POINTS WITH COMBINED EFFECTS OF 777 AND 779
781 TAM = (PHI(I2,J,K)/(2.0*BIP) +
1 PHI(I+2,J,K)/(BIP*(1.0+TAB7)) 4
2 PH I(I J K2) + PHI(I,J K + 1) 4 2.0*PHI(ILK)I/BING
o o o
194
IF (AES(TAMPHI(I ,J,K)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PH I (I, J, K ) = OMEGA*TAM TAGPHI(I,J,K)
795 CONTINUE
C THE FOLLOWING STEPS CARRY THE ITERATIVE PROCESS FROM THE
C FIRST ROW WITH ALL SPACINGS OF 2H TO THE UPPER LIMITS OF THE
C SOLUTION SPACE.
K3 = KGRACE +3
DO 799 K=K3KMAX
DO 799 J=ItJMAX2
DO 799 1 = 3,IMIT2 2
KMIT = KMAX 1
C STEPS FOR CONDITION DPHI/DZ = 0 ON UPPER LIMIT OF SOLUTION
C SPACE.
IF (K .LE. KMIT) BOG = PHI(I,J,K4l)
IF (K .EC. KMAX) BOG = PHI(I,J,K1)
IF (J .EC. I .OR. J .EQ. JMAX) GO TO 783
IF ( I .EQ. IMIT2) GC TO 784
C ORDINARY DIFFERENCE FORMULA.
TAM = (PHI(I + 2,J,K) 4 PHI ( 12 J*K) 4 PHI (I J42 K)
1 PH I (I, J2 ,K ) + BOG + PHK I, J,K1) )/6.0
IF (ABS(TAMPHI(I,JK))GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(I,J,K) = OMEGA*TAM 4 TAGPHI (I, J,K)
GO TO 799
783 IF{J .EC. 1) L=J+2
IF (J .EQ. JMAX) L=J2
IF (I .EQ. IMIT2) GO TO 785
C FOR TREATMENT OF DPHI/DY = 0 CONDITION.
TAM = (PH 1(142,J,K) 4 PH I (12 J, K ) 4 BOG 4 PHI(I,J,K1) 4
1 2.0*PHI(I,L,K))/6.0
IF (ABS(TAMPHKI,J,K)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PH I(I,J,K) = CMEGA*T AM 4 TAG*PHI(I,J,K)
GO TO 799
C UNEQUAL XSPACING
784 TAM = (PHIlI2,J,K)/(2.0*BIP) 4 PHI(142,J,K)/(BIP*BI Pi) 4
1 PHI(I,JK1) 4 BOG 4 PH I(I,J42,K) 4 PHI(I,J2,K))/BING
IF (ABS(TAMPHI (I,J,K)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI{I,J,K) = OMEGA*TAM 4 TAG*PHI(IJ,K)
GO TC 799
C COMBINATION OF EFFECTS IN 783 AND 784.
785 TAM = (PHI(I2,J,K)/(2.0*BIP) 4 PHI(142*JK)/(BIP*BIP1) 4
1PHKI,J,K1)4 BOG 4 2.0*PHI(I,L,K)J/BING
IF (ABS(TAMPHI(I,J,K)).GT.EPS) EPS=ABS(TAMPHI{I,J,K))
PHI(I,J,K) = OMEGA*TAM 4 TAG*PHI(I,J,K)
799 CONTINUE
RETURN
END
SUBROUTINE VELOC
C
195
C SUBROUTINE VELOC COMPUTES THE VELOCITIES AT SURFACE POINTS,
C USING RELATIONS AND VALLES OBTAINED FROM SUBROUTINES SUCH AS
C BDDER AND ZBOUND, THE SUBROUTINES FOR HANDLING THE NORMAL
C DERIVATIVE BOUNDARY CONDITION AT THE HEMISPHERES. WITH THE
C VELOCITIES COMPUTED OVER THE HEMISPHERICAL SURFACE, THE SQUARE
C OF THE VELOCITY IS THEN INTEGRATED OVER THE SURFACE TO OBTAIN
C A VALUE FROM WHICH IS THEN SUBTRACTED THE PRESSURE CONTRIBU
C TICN FROM THE VELOCITY ALONG THE BASE OF THE HEMISPHERE. THE
C FINAL RESULT IS A LIFT COEFFICIENT, CL.
C
COMMON ALFA0(21,21,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),X IN ITI 21,21),MIG1C21,21),
2 MIG2(2121),CCONl(21,21),CCON2(21,21),THETA1(21,21),
3 THETA2(2121),IBCRY1(2121)IBDRY2(2121),PZ(36,21),
A TIP(100)TOP(11,11),H,EPSIMAX,JMAX,KMAX,KLOWKGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KLOWl,IMID.TABI,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3S3ABINGBIP,B0G,IG1IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIPKEY,NQ(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,2l,32)
COMMON IMIG(21),PIG,KIN,KIN1,FLO
DIMENSION CSQR(35,21),VEL(35,21),PRESi35,21)
DIMENSION UBASE2I21)
C EINSTEINEL SAMNI ARRANGEMENT
JMIT = JMAX 1
C COMPUTE CONSTANTS NEEDED FOR LATER COMPUTATION OF A VELOCITY
C AT THE HEMISPHERE BASE NEAR THE MIDDLE SINGULAR POINT.
PD1 = 0.50 + H
PD2 = 0.50 H
JP01 = JMID + 1
JPD2 = JMID 1
THG = (ATAN2(XINIT(JPD2,1),PD2) 
1 ATAN2(XINIT(JPD1,1),PD1))/H
SUMLT = 0.00
KEY = 0
NCAS = 3
C BEGIN TRAVERSING ALONG YCONSTANT SECTIONS OF HEMISPHERE.
DO 640 J =1,JM IT
C CALCULATE WIDTH OF PROJECTED AREA ELEMENT FOR GIVEN POINT.
W = H
IF (J .EG. 1) W = H/2.0
L = 0
AJ = J
Y = (AJl.OMH
C BEGIN AT Z VALUE JUST ABOVE FIRST POINT ON THE HEMISPHERE AT
C THE GIVEN YSECTICN I AT X = 0 ). THIS Z IS ESTABLISHED BY
C IMIG(J), WHICH IS READ IN EARLIER IN THE PROGRAM.
K = I MIG CJ)
AK = K
Z = (AK1.0)*H
XB = 0.00
IT = IBDRY1(J,l)
196
C TRAVERSE IN XDIRECTION ALONG THE GIVEN YSECTION.
DO
650
11 = 1,
IT
All
=
11
XI
= (
All1.0)
H
JIP
1 =
15
IF
(JMAX .EQ.
11)
JIPI
= 7
IF
UMAX .EQ.
21)
JIPI
= 13
JIP2 =
15
IF
UMAX .EQ.
11)
JIP2
= 9
IF
UMAX .EQ.
21)
JIP2
= 17
IF
( 11
.EG. 1
.AND
. J
.EQ.
1
)
I
= 11
IF
(11
.EG. 1
.AND
. J
.EG.
1
)
GO
TO
1650
IF
(11
.EG. 1
.AND
. J
.EQ.
J
I
PI)
I =
11
IF
(11
.EC. 1
AND
. J
.EQ.
J
I
PI)
GO
TO 1650
IF
(11
.EQ. 1
.AND
. J
.EQ.
J
I
P2)
I =
11
IF
(11
.EC. 1
.AND
. J
.EQ.
J
IP2)
GO
TO 1650
IF
(II
.EQ. 1)
GO
TO 650
645 IF
(Z
.LT. 0.6
) GO
TO
1645
IF
(XI
.LT. XINITUtK
D) GO
TO
641
C CALCULATION GF VELOCITIES FOR POINTS HANOLED BY SUBROUTINE
C BDDER.
1645 KEY = 1
I = IBDRYK J Kl)
AI = I
K = Kl
AK = K
Z = (AK1.0)*H
X = XINIT(JyK)
IF (K .EG. 1) GO TO 639
N = NG(IJK)
CALL BDDER
L = L + l
PIG = PHI { I yJ K)
GO TO 642
C CALCULATION OF VELOCITIES FOR SURFACE POINTS HANDLEO BY
C SUBROUTINE ZBOUND.
641 I = II
AI = I '
X = X1
IF (I .EG. JIPI .AND. J .EQ. 1) GO TO 650
IF (I .EQ. JIP2 .AND. J .EQ. 1) GO TO 1645
AK = K
Z = I AK1.0)*H
N = NC(IyJyK)
KEY = 0
CALL ZBOUND
L = L + l
PIG = PZ(I,J)
GO TO 642
C CALCULATION OF VELOCITIES FOR X 0* WHERE DPHI/DZ AND DPHI/DY
C BOTH EQUAL ZERO.
197
C CPX IS SHORT FOR DPHI/DX SIMILARLY FOR DPY,DPZ
1650 DPX = PHI(I+l,J,K) 100.0
KEY = 0
DPY = 0.00
OPZ = O.CO
X = 0.00
L = 1
GO
TO
646
1643
DPY
=
O.CO
GO
TO
644
1644
DPZ
=
0.00
GO
TO
646
642
IF
( I
.EC.
1) GO
TO
1650
DPX
=
(SI 
S1A*
PIG)
/X
643
I F (
J
.EQ. 1
.OR.
J .
EQ.
DPY
=
(S2 
S2A*
PIG)
/Y
644
IF
(K
.EC.
1) C0
TO
1644
Z1
Z
IF
(K
EY .EQ
. 0)
Z1 =
Z 
CPZ
=
(S3 
S3A*
PIG)
/Z1
JMAX) GO TO 1643
 ALFA(N, 5)*H
[ = (S3 S3A*PIG)/Z1
C CSQRO SCUARE OF SURFACE VELOCITY AT GIVEN POINT.
C CTSQRO SCUARE OF VELOCITY AT UPPERMOST POINT OF HEMISPHERE.
C VELO SURFACE VELOCITY AS A FRACTION OF THE VELOCITY AT THE
C TOP OF THE HEMISPHERE.
C PRESO SCUARE OF SURFACE VELOCITY AS A FRACTION OF QTSQR.
646 QSQR(L J) = DPX**2 + DPY**2 + DPZ**2
IF (L .EC. 1 .ANO. J .EQ. 1) CTSQR = QSQR(L,J)
VEL(LJ) = SQRT(QSQR(L,J)/QTSQR)
PRES(LJ ) = QSQR(L,JJ/CTSQR
DPC = SQRT((DPX**2 + DPZ*2)/QTSQR)
CPX = DPX/SQRT(QT SQR)
DPY = DPY/SGRT(QTSQR)
OPZ = DPZ/SQRT(QTSQR)
IF (L .EC. 1) WRITE (6,2010)
WRITE (6,2009) OPX,DPY,OPZ,DPC
IF (J .EG. JMIT) GO TO 660
C CALCULATE PROJECTED AREA BENEATH GIVEN SURFACE POINT.
AI = I
D1 AIH
D2 = XINIT(J,Kl)
XF = AM IN1(DI,02)
DA = W*(XFXB)/2.0
IF (X .EQ. XINIT(J,2)) DA = DA +
C COMPUTE CONTRIBUTION TO INTEGRAL OF
C HEMISPHERE
661 SUMLT =
XB = X
IF (KEY .EQ
GO TO 645
660 DA = 0.00
PEP = 0.00
(XINIT(J,1) XF)*H
SURFACE PRESSURES OVER
THE
AND ADD IT TC THE CUMULATIVE TOTAL.
SUMLT DA*PRES(L,J)
0) GO TC 650
c
c
GO TO 661
650 CONTINUE
639 WRITE (6,2002)
WRITE (6,2004)
WRITE (6,2005)
WRITE (6,2003)
WRITE (6,2006)
WRITE (6,2003)
640 CONTINUE
CL = SUMLT
WRITE (6,2000)
WRITE (6,2001)
J
Y
(VEL(L7,J),L7=1,L)
(PRES(L7*J)L7=1L)
CL
SECTION FOR CALCULATING PRESSURES ALONG THE BASE OF THE
HEMISPHERE AND INTEGRATION OF THIS PRESSURE.
00 670 J = 2 JM IT
IF (J .EQ. JMID) GO TO 671
K = 1
I = IBORY1(J,K )
AI = I
X = X INIT(J,K )
AJ J
Y = (AJ1.0)*H
AK = K
Z = 0.0
N = NO(I,J,K)
CALL BODER
DPX = (Si S1A*PFI(1,J,K))/X
DPY = (S2 S2A*PHI(I,J,K))/Y
UBASE2 ( J) = (CPX**2 DPY*2)/QTSQR
GO TO 670
C CALCULATION FOR MIDDLE SINGULAR POINT.
671 13 = IBDRY1(J+1*K)
14 = IBDRY1(J1,K )
THP = (PHK I3,J+1,K) PHHI4,J1,K) l/THG
UBASE2(J) = THP**2/QTSQR
670 CONTINUE
WRITE (6,2007)
WRITE (6,2003) (UBASE2(J2), J2 = 2,JMITl
SIG =0.0
K = 1
DO 672 J=2,JMIT
IF (J .EQ. JM IT) GO TO 673
SIG = SIG + H*XINIT(J,K)*UBASE2(J)
GO TO 672
673 SIG = SIG + UBASE2J)PEP
672 CONTINUE
C CALCULATION OF FINAL DESIRED LIFT COEFFICIENT, CL.
CL = CL SIG
WRITE (6,2008)
WRITE (6,2001) CL
2000 FORMAT(1H1)
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
3000
C
C
C
199
FORMAT (20H LIFT COEFFICIENT 1F12.6)
FORMAT (15)
FORMAT (10F10.5)
FORMAT (5H Y = F5.2)
FORMAT (11H VELOCITIES)
FORMAT (10H PRESSURES)
FORMAT (27H SQUARE OF BASAL VELOCITIES)
FORMAT (46H LIFT INTEGRATION CORRECTED FOR BASAL PRESSURE)
FORMAT (4F12.4)
FORMAT (20H VELOCITY COMPONENTS)
FORMAT (1115)
RETURN
END
NOTES ON EQUIVALENT GRAIN SIZE
Presented here is a brief derivation of equation (610) of
the text, subject to the conditions stated there. The following addi
tional symbols are used.
t
Yg = unit weight of soil particles
G = buoyant weight of particles
A^ = exposed equivalent area
A = exposed natural area
n
First, calculate the exposed area as the product of the area
for one grain and the number of grains. The latter i obtained by
dividing G, the total buoyant weight, by the buoyant toeight of one
particle.
A IT 2 G 3 G
A = d
e 4e tt ,3, 2 (y v)d
TT d (y y) Ts e
6 e Ts Y
A^ must be calculated from the grain size distribution curve*
o G dy
1A tt 2 as
n 4 d tT~3
6d '8
1
6 d (v8y)
A = f1 I
n 2 YSY o d
Equating A^ and A^ yields (610) in the text.
200
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New York: John Wiley and Sons, Inc., 1966, 139 pp.
59. Dingle, A. N., and Young, Charles. Computer Applications in the
Atmospheric Sciences. College of Engineering* University of
Michigan, 1965, 256 pp.
60. Allen, D. N. deG. Relaxation Methods in Engineering and Science.
New York: McGrawHill, Inc., 1954, 257 pp.
61. Collatz, Luther. The Numerical Treatment of Differential Equations
New York: SpringerVerlag New York, Inc., 1966, 568 pp.
62. Allen, D. N. deG., and Dennis, S. C. R. "The Application of
Relaxation Methods to the Solution of Differential Equations
in Three Dimensions. I. Boundary Value Problems," Quarterly
Journal of Mechanics and Applied Mathematics, vol. 4, 1951,
pp. 199208.
63. Fox, L. An Introduction to Numerical Linear Algebra. New York:
Oxford University Press, 1965, 327 pp.
64. Southwell, R. V. Relaxation Methods in Theoretical Physics.
Oxford University Press, 1946.
65. Shaw, F. S. An Introduction to Relaxation Methods. New York:
Dover Publications, Inc., 1953, 396 pp.
66. Fox, L. Numerical Solution or Ordinary and Partial Differential
Equations. Reading, Massachusetts: AddisonWesley Publishing
Company, Inc., 1967, 509 pp.
67. Forsythe, G. E., and Wasow, W. R. FiniteDifference Methods for
Partial Differential Equations. New York: John Wiley and Sons,
Inc., 1960, 444 pp.
206
68. Allen, D. N. deG., and Dennis, S. C. R. "The Application of
Relaxation Methods to the Solution of Differential Equations
in Three Dimensions. II. Potential Flow Round Aerofoils,"
Quarterly Journal of Mechanics and Applied Mathematics,
vo1. 6, 1953, pp. 81100.
69. Olmsted, John M. H. Advanced Calculus, New York: Appleton
CenturyCrofts, Inc., 1961, 706 pp.
70. Prandtl, L. Essentials of Fluid Dynamics. New York: Hafner
Publishing Company, 1952, 452 pp.
7
71.' Michaels, Paul. "Ideal Flow Along a Row of Spheres," The Physics
of Fluids, vol. 8, No. 7, July 1965, pp. 12631266.
72. Koloseus, H. J., and Davidian, J. "RoughnessConcentration Effects
on Flow Over Hydrodynamicslly Rough Surfaces," U.S. Geological
Survey WaterSupply Paper 1592D. Washington, D.C.: U.S. Govern
ment Printing Office, 1966, 21 pp.
73. Rouse, H. "Critical Analysis of OpenChannel Resistance,"
Journal of the Hydraulics Division, Proceedings, ASCE, vol. 91,
No. HY4, July 1965, pp. 125.
74. Christensen, B. A. "Theory of Sediment Transport," Unpublished
Lecture Notes, University of Florida, 1967.
BIOGRAPHICAL SKETCH
Barry Arden Benedict was born February 7, 1942, at Wauchula,
Florida. In June, 1960, he was graduated from DeLand High School,
DeLand, Florida. In April, 1965, he received the degree of Bachelor
of Civil Engineering from the University of Florida. At that time
he enrolled in the Graduate School of the University of Florida.
He obtained the degree of Master of Science in Engineering in June,
1967, and has since then pursued his work toward the degree of
Doctor of Philosophy.
Barry Arden Benedict is married to the former Betty Janell
Truluck. He is a member of Phi Kappa Phi, Tau Beta Pi, Sigma Tau,
Phi Eta Sigma, Florida Blue Key, the American Society of Civil
Engineers, and Pi Kappa Phi.
207
This dissertation was prepared under the direction of the
chairman of the candidates supervisory committee and has been
approved by all members of that committee. It was submitted to the
Dean of the College of Engineering and to the Graduate Council, and
was approved as partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
December, 1968
Dean, Graduate School
Supervisory Committee:
Chairman
1/
Internet Distribution Consent Agreement
In reference to the following dissertation:
AUTHOR: Benedict, Barry
TITLE: Hydrodynamic lift in sediment transport, (record number: 953375)
PUBLICATION DATE: 1968
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155
IF (KIP .GE. 30 .ANO. EPS .GT. 0.5) GO TO 2
IF (KIP .LE. 50) GO TO 2069
C STEPS TO WRITE OUT POTENTIAL FIELD AND CALL SUBROUTINE
C VELOC TO CALCULATE VELOCITIES.
2 WRITE (6,1012)
IF UMAX .ECU 11) GO TO 201
IF UMAX .EQ. 21) GO TO 202
WRITE (6,1007) tl(PHKI,JfK),J=l,JMAX),I=lIMAX),K=
22 CALL VELOC
IF (EPS .LE. EPSMIN) KIN1 = 1
IF (EPS .GT. 15.0) KIN1 = 1
IF (KIP .GE. 30 .AND. EPS .GT. 0.5) KIN1 = 1
GO TO 2069
201 WRITE (6,1000) (((PH I(I,J,K),J=1,JMAX),1 = 1,IMAX),K =
GO TO 22
202 WRITE (6,1009) (((PH I(I,J,K),J = 1,JMAX),I = 1,I MAX),K=
K2 = KLOW +1
WRITE (6,1000) (((PHI(I,J,K)*J=ltJMAX,2),I=l,IMIT2
GO TO 22
C READ IN INITIAL DATA
20 READ (5,1002) H
READ (5,1002) EPSMIN
READ (5,1002) OMEGA
READ (5,1005) NCAS,JMAX,IMAX
READ (5,1005) KLOW,KGRADE,KEYTAG
READ (5,1005) KMAX,KEY1,KEY2
READ (5,1005) JMID,MARK,KIP
READ (5,1001) (IMIG(L),L=1,JMAX)
TAG = 1.0 OMEGA
GO TO 2069
C SPECIAL SECTION FOR A/20 RUN. READ IN DATA FOR Z = 1.
C AND Z = 1.10, WHICH WILL BE USED AS FIXED DATA FOR
C THE SOLUTION INSIDE THE REDUCED SOLUTION SPACE.
21 JMIT = JMAX 1
IMIT = IMAX 1
READ (5,1008) (((PH I(I,J,K),J = 1,JMAX,2),1 = 1,IMIT,2)
1 K=22,23 )
C INTERPOLATE BETWEEN POINTS FROM DATA GIVEN FROM A/IO
C SOLUTION
211 DO 4 K=22,KMAX
DO 4 J = 2,JMIT,2
DO 14 1=1,IMIT,2
14 PHI(I,J,K) = 0.5*(PHI(I,J1,K) + PHI(I,J+1,K))
PHK IMAX, J,K) = 0.0000
4 CONTINUE
DO 6 K=22,KMAX
DO 6 J=1,JMAX
DO 6 1=2,IMAX,2
IF (I .EC. IMAX) PHI(IJK) = 0.0000
IF (I .EQ. IMAX) GO TO 6
1,KMAX)
1,KMAX)
l,KLOW)
),K=K2,K
MAX,2)
05
He found the amount by which the lift exceeded the weight of the
enclosed liquid, and thence wrote the following condition for move
ment of the cylinder:
13
(22)
where
o = density of particle
g = acceleration due to gravity
a = cylinder radius
p = mass density of water
This can be written as
(23)
Jeffreys noted that J. S. Owens [28] had earlier measured
the velocity required to move pebbles, finding
U2 = 1.65 ga
(24)
The motion observed by Owens was not, however, a jumping or
lifting motion, but rather a rolling motion, yielding a difference
from values predicted by theory. However, as Jeffreys says, "The
proportionality of U to the square root of the linear dimensions is in
agreement with theory" [8, p. 276].
Use of a value for <7 of 2.7 times p in Jeffreys* equation yields
the following:
U2 > 1.19 ga
(25)
121
the hemisphere than at the base. This increase is generally about
10 per cent. Ordinarily a type of stagnation zone would be expected
to exist in this region, with only slight pressure decrease. The
slight discrepancy here may well be due to the effects of the gravel
bed surrounding the hemispheres. These particles near the base of the
hemisphere likely create disturbances and shed vortices which could
7
affect conditions at the base, while possibly not having influence up
at thirty degrees. Such a situation could result in a pressure decrease
at the basal point due to motion while not decreasing the higher point.
Chepil considered three surface zones, relative to positions of
his pressure measurements. Averaging the pressure on two planes over
this zone, he then projected this pressure vertically downward to get
the lift. Figure 42 shows a plan view of the zones, A, B, and C.
The pressure PP^ represents the pressure difference between the
bottom and zone A, with PP and PP being similarly defined. Equa
tion (68) is then used for the lift per unit area occupied by the
hemisphere.
\ = 0.062 (PPA) + 0.394 (PP ) + 0.544 (PP,,) (68)
u A B C
Application of (68) was made, using pressures as shown in earlier
figures, with the results shown in Table 5. Also shown in the table
are values of lift based on equation (577) with C 0.405. In
liU
CHAPTER VII
CONCLUSIONS AND FUTURE WORK
The goal of this dissertation was to demonstrate that potential
flow theory could be used to predict hydrodynamic lift in sediment
transport. Such predictions are made by relating the pressure distri
bution from potential flow theory to known flow characteristics.
Two bed configurations were studied analytically and related to avail
able experimental observations. The work of Chepil [36] was treated
first. He used metal hemispheres, three diameters apart center to
center. Due to a gravel floor, the test situation differed from the
one treated analytically. For three cases with Chepil*s largest hemi
sphere, where the theoretical and natural conditions were most similar,
the predicted lift was 19 per cent, 13 per cent, and 8 per cent above
the corresponding measured values.
Next, a hexagonal pattern of closely packed hemispheres was
treated. Theory yielded a lift 16 per cent higher than that measured
by Einstein and El Samni [33,34]. If allowances are made for possible
sidewall effects of the narrow flume used in their work, the discrep
ancy decreases to perhaps less than 10 per cent.
Einstein and El Samni also made measurements for a gravel bed.
This author made use of an Approximate gravel gradation curve to
establish an equivalent bed of equal hemispheres. Use of the lift
146
197
C CPX IS SHORT FOR DPHI/DX SIMILARLY FOR DPY,DPZ
1650 DPX = PHI(I+l,J,K) 100.0
KEY = 0
DPY = 0.00
OPZ = O.CO
X = 0.00
L = 1
GO
TO
646
1643
DPY
=
O.CO
GO
TO
644
1644
DPZ
=
0.00
GO
TO
646
642
IF
( I
.EC.
1) GO
TO
1650
DPX
=
(SI 
S1A*
PIG)
/X
643
I F (
J
.EQ. 1
.OR.
J .
EQ.
DPY
=
(S2 
S2A*
PIG)
/Y
644
IF
(K
.EC.
1) C0
TO
1644
Z1
Z
IF
(K
EY .EQ
. 0)
Z1 =
Z 
CPZ
=
(S3 
S3A*
PIG)
/Z1
JMAX) GO TO 1643
 ALFA(N, 5)*H
[ = (S3 S3A*PIG)/Z1
C CSQRO SCUARE OF SURFACE VELOCITY AT GIVEN POINT.
C CTSQRO SCUARE OF VELOCITY AT UPPERMOST POINT OF HEMISPHERE.
C VELO SURFACE VELOCITY AS A FRACTION OF THE VELOCITY AT THE
C TOP OF THE HEMISPHERE.
C PRESO SCUARE OF SURFACE VELOCITY AS A FRACTION OF QTSQR.
646 QSQR(L J) = DPX**2 + DPY**2 + DPZ**2
IF (L .EC. 1 .ANO. J .EQ. 1) CTSQR = QSQR(L,J)
VEL(LJ) = SQRT(QSQR(L,J)/QTSQR)
PRES(LJ ) = QSQR(L,JJ/CTSQR
DPC = SQRT((DPX**2 + DPZ*2)/QTSQR)
CPX = DPX/SQRT(QT SQR)
DPY = DPY/SGRT(QTSQR)
OPZ = DPZ/SQRT(QTSQR)
IF (L .EC. 1) WRITE (6,2010)
WRITE (6,2009) OPX,DPY,OPZ,DPC
IF (J .EG. JMIT) GO TO 660
C CALCULATE PROJECTED AREA BENEATH GIVEN SURFACE POINT.
AI = I
D1 AIH
D2 = XINIT(J,Kl)
XF = AM IN1(DI,02)
DA = W*(XFXB)/2.0
IF (X .EQ. XINIT(J,2)) DA = DA +
C COMPUTE CONTRIBUTION TO INTEGRAL OF
C HEMISPHERE
661 SUMLT =
XB = X
IF (KEY .EQ
GO TO 645
660 DA = 0.00
PEP = 0.00
(XINIT(J,1) XF)*H
SURFACE PRESSURES OVER
THE
AND ADD IT TC THE CUMULATIVE TOTAL.
SUMLT DA*PRES(L,J)
0) GO TC 650
14
Differences from theory are primarily due to two causes.
First, the motion measured was rolling rather than totally lifting,
which (25) is based on. Second, the measured particles were three
dimensional, contacting the bed at only a small number of points,
while Jeffreys' cylinder made contact over the whole length of its
body. These effects are compounded by the fact that the Uvalues used
by the two men are not the same. Jeffreys employed a potential flow
field, but Owens made his measurements in a flow field exhibiting a
logarithmic velocity distribution. The differences involved here will
be discussed in Chapters III and IV. Despite the discrepancies, the
theory seems to provide a much better starting point than might first
be thought possible.
2.4.6 Flow around a single sphere.The results from measure
ments on flow around a single sphere will be presented because the shape
relates to this study and the results reveal some factors influencing
the actual flow pattern.
The flow involved is that around a single sphere suspended in
an otherwise uniform flow. For this case the potential can be
expressed [29] as
Ua3
cp = cos 0 + Ur cos 0 (26)
2r
where a = radius of sphere
r = radial distance
0 = angle measured from horizontal and
through sphere center
U = free stream velocity at 00
CHAPTER II
REIATED BACKGROUND ON LIFT FORCE STUDIES
2.1 Historical interest
For centuries engineers have been interested in the movement of
sediment due to flowing water, which Rouse and Ince call "A class of
flow phenomena inherently hydraulic in nature . [3, p. 246].
Concerned with problems of scour, deposition of material, stable
channels, and the like, engineers have for years attempted to increase
their knowledge of sediment motion. Both empirical and theoretical
means have been employed in these attempts.
Domenico Guglielmini [3, p. 70] was perhaps representative of
the whole Italian school interested in flow resistance in open channels.
(
His work in the seventeenth century made some qualitative observations
which were very accurate, though his analytical work was more faulty.
Later, Pierre Louis Georges Du Buat [3, p. 129] collected a
vast array of experimental results, including extensive data on the
beginning of sediment movement. His eighteenth century works over
shadowed that of other hydraulicians for about a century.
Work continued through various periods until the work of Grove
Karl Gilbert in the years around 1910. His tests on initial sediment
movement and various phases of transport covered a wide range. It has
been noted that "... the results he presented in U.S.G.S. Professional
4
168
GO TO 825
C 2 POINT AT LEAST H AWAY FROM SURFACE HERE.
820 L3 = KI
IF (K .EC. 1) L3=K+1
P5 = PHI(ItJL3)/(ALFA(Nf5)*(1.0+ALFA(N t 5)))
GO TO 821
810 IF (ALFA(N,3) .LT. 1.00 .AND. ALFA(N,4) .LT.1.00) GO TO 812
C BOTH Y POINTS A FULL H AWAY
P3 = PHI(I,L5,K)/IALFA(N,3J*CALFA(N3)+ALFA(N.4n)
GO TO 811
C Z POINT ON FIRST HEMISPHERE
833 CALL ZBCUNC
P5 = PZ(IJ)/(ALFA(N5)*{1.0+ALFA(Nt5I))
GO TO 821
C ARC INTERPOLATION FOR POINTS ON SPHERES
812 F3 = Y ALFA(Nt 3)*H
IF {J .EC. 1) GO TO 830
IF (J .EQ. JMAX) GO TO 840
FI = Y H
A = SGRT(3.00) X
B = SGRT(3.00) XENC(L4 K)
C = 1.0 Y ALFA(N,4)*H
D = 1.0 Y H
E = SGRT(3.00 ) XENC{JK)
G = 1.0 Y
THM = (ATAN2ICA ) ATAN2(D,B))/
P4 = PH I(I,L4 K) THM*(PHI(I,L4 ,K) PHI(I + 1,J,KJ)
THM = (ATAN2(YfXINIT{JK) ) ATAN2IF3 X)) /
1 (ATAN2(Y,XINIT{JK)) ATAN2(F1,XINIT(L5,K) ))
P3 = PHI(I1,J,K) THM(PHI(I1,J,K) PHI(I,L5fK)J
P4 = P4/(ALFA(Nt4)*(ALFA(N,3) 4 ALFAIN*4)))
P3 = P3/lALFAtN3)*(ALFAiN3) 4 ALFAIN4))>
GO TO 813
C SPECIAL SECTION FOR Y = 0
830 IMID =2
CALL YBCUND
P4 = BING/(ALFA(N,4)*(ALFA(N,3) 4 ALFA(N4)) )
P3 = P4
GO TC 813
C SPECIAL SECTION FOR Y = A
840 IMID = 1
CALL YBOUND
P3 = BING/(ALFA(N,3)IALFA(N3) 4 ALFA(N,4)U
P4 = P3
GO TC 813
C STANDARD LAPLACIAN DIFFERENCE EGUATION
841 TAM = (PHIU4ltJ,K) 4 PHI(I1,J,K) 4 PHI(I,L4K) 4
1 PHI(I,L5,K) 4 PHI( IJ L3) 4 PHI(IJK+l))/6.0
IF (ABSlTAMPHKIfJtK)) .GT. EPS) EPS= ABS(TAMPHI(I,J,K))
PHI(11Jt K) = OMEGA*TAM 4 TAG*PHI(I,J,K)
GO TG 801
119
Flow
Chepils measured values
Values from theory
Pressure scale:
2
1" = 400 dynes/cm
Avpragp of Ghepil's
front and back values
2.54 cm radius
hemisphere
Pressures plotted radially as difference between given point
and uppermost point of hemisphere. Above circle, positive;
below circle, negative.
Figure 40. Measured and theoretical pressure distributions:
v = 128 cm/sec.
r *.
138
6.5 Comparison with EinsteinEl Samni observations
6.5.1 Physical details of experiments.The experiments of
interest now [33,34] used plastic hemispheres placed on the flume bed
in a hexagonal pattern. This is essentially the same as the theoretical
case studied. The flows studied were typified by wall Reynolds numbers
Cv k/v) in the range 3330 to 5580. These are very definitely in the
^ v
1
hydrodynamically rough range, where the laminar effects are negligible.
Hence, it is to be expected that the lift would not be a function of
the Reynolds number.
The theoretical model for which a solution has been made treats
a bed of infinite extent. This would be approximated Well over an
entire flume length in the experimental flow. However, some differences
might be expected to occur due to sidewall influences in a narrow flume.
Also, the bed had a finite width of perhaps five elements, as the flume
used in [33,34] was 12 inches wide. Section 3.5 was presented with an
eye to evaluating this effect approximately.
The flume had a depth of 36 inches. Section 3.5, equation (325)
shows that the ratio between sidewall and bottom roughness is needed in
order to evaluate o, the angle through the "corners" of the idealized
isovels. The bottom equivalent sand roughness, k^, equals the diameter
of the plastic hemispheres, or 0.225 feet. It is much more difficult
to estimate the sidewall roughness, although a picture in [34] indicates
a sideboard which is rougher than most This observation prompted a
first choice of k^ = 0.001 feet, or about 0.3 mm. Use of this value in
equation (325) enables evaluation of Qi to be 14.6 degrees. The line
at angle Qf will meet the center line at about 23 inches above the bed,
39
Introducing y/k = Tj enables the writing of
FC^) = ^ F(T]2)
(321)
where
F(T) = [In (29.7 Tj) ] v'T
The curve shown in Figure 9 indicates that the Ffunction can
be approximated by a function of the form
F(T) = atf
(322)
This would enable the writing of
T
'2
\^Â¡i
l/b
(ylAl) yl k2
(y2A2) y2 kl
(323)
Using the fact that = tan it can be written that
l(l/2b)
tan or = ()
k2
<324)
Evaluation from Figure 9 reveals that bcequals about 0.664. Thus,
, 0.248
tan o, = ()
k2
(325)
The last equation provides an opportunity for an
approximate evaluation of the sidewall's effect on the isovels and
hence on the center line profile. The likelihood of simulating two
dimensional flow in the center region can then be estimated.
89
First, points met in the zdirection will be discussed, begin
ning with those points at a sufficient distance from the other hemisphere
that all needed surrounding values are available. Figure 29 portrays
the situation here, where the plane indicated could be either one of
constant x or constant y.
Constant yplane
Figure 29. Zpoint for normal derivative condition.
In this case, z becomes analogous to x in the previous discus
sions, as it lies on a lattice line. Hence, it can be expanded in
terms of lattice points, and, with appropriate replacements, equa
tion (549) can be used for the derivative, &p/dz. Then, points PI
and PI1 (PJ and PJ1) must be found to evaluate Scp/dx and cXp/dy.
Because the situations are similar, one expansion can serve all four
needed points. Emphasis will again be placed on the point nearer the
desired value. In this case, however, an expansion like that of (557)
will be used for both cases,.not just the point nearer the boundary
(C in this case). Note also that C might itself lie on the boundary.
Consider the case for PI, using G = oc^.
91
The equations derived for Sphere 1 can also be used for points
on Sphere 2, since these equations are for interpolations in the same
direction. Here C still represents the point at the lower zelevation,
either on the surface or a lattice point, and A depicts the point at
the higher zelevation, similar to Figure 29.
Just as was the case earlier, there will be areas where not all
of. the x or ypoints needed for equation (565) or equation (567)
will be available. Then it will be necessary to use a oriepoint, one
way derivative approximation as shown in the xdirection discussions.
As in that case, the single point is obtained for use in an equation
such as (551) by one of two means. Points on the other hemisphere
can be evaluated by the arc interpolation procedure, while other points
are expressed by interpolation between lattice points.
For the points met in the ydirection, the problems and methods
are the same as for those in the x and zdirections. Hence, no dis
cussion will be given here except to state that equations similar to
those for the other cases will be written and used. The difference
expressions will be utilized to form the normal derivative condition
just as indicated in the next section.
5.5.6 Final boundary formulation.In Section 5.5.1, expres
sions were shown for dcp/dn in terms of the three derivatives with
respect to coordinate axes. With these three derivatives expressed in
difference form a difference equation for the normal derivative is
available. Each of the three derivatives has been shown to contain
9, the function value at the surface point being treated, and some
B
66
Difference expressions can be formed for this case.
Forward difference:
6I+1,J,K ~ ^
Q^h
I,J,K
Backward difference:
&Â£ = yi.J.K ~ ^Il.J.K
Ax o^h
(516)
(517)
From these the following equation for the second derivative can be
obtained.
9.
1+1,J,K
" ,J,K ~ ^Il.J.K
Tsra5 
V
ff2h
(Oii+Q'2)h
(518)
or
JL2. ~ ,J,K ^I.J.R ~ ^Il.J.K
ax'
a (q? +a )
2 1 2J
(519)
Similar expressions can be found for the other directions,
1,2 a2cp CPI.J+1,K ^X.J.K CpItJ,K ^I.Jl.K
2 *ay2 WV WV
(520)
L2 = ^I^K+l ~ ^1 ,J,K ~ ^I.J.Kl
2 az2 WV WV
(521)
Adding these three expressions yields an approximation for
the Laplacian.
87
cases are derived in the same manner as (553) and (554), the only
difference arising in the distance between points PJ and PJ1. These
equations are included below.
Sphere 1 (y = a):
frp PJ cp' 2CpB 2PJ
3y 2cv3h a3 cv3
(561)
Sphere 2 (y = 0):
, &Â£_ cp7 PJ 2PJ
5y 2a4h " C*4
(562)
The preceding subsection has described the means for obtain
ing a value for dcp/dy at the hemispherical boundaries. It remains
now to compute dcp/dz..
5.5.4 Zdirection derivative.The computation of the value of
dcp/dz is in many instances virtually the same as the case for dcp/dy.
In fact, Figure 27a serves as a representation of the situation, except
that the plane is now one of constant y, rather than constant z, and
the points PJ and PJ1 will be replaced by PK and PK1. Expansion as
in (552) enables removal of the fictitious cp7value and expressions
for the derivatives.
Sphere 1 and Sphere 2
h Sr~ hiÂ£Â£irE} i%*2PK (563)
The values of the points PK and PK1 can be expressed just as
indicated in Section 5.5.3 for the yderivative, with expansion in the
ACKNOWLEDGMENTS
The author wishes to express his sincere appreciation to
Dr. B. A. Christensen, Dr. J. H. Schraertmann, Dr. E. A. Farber,
and Dr. T. 0. Moore for their service on his supervisory committee
and their interest in his work. The author is especially indebted
to the committee chairman, Dr. B. A. Christensen, whose guidance,
encouragement, and personal interest and enthusiasm were truly
valuable.
Appreciation is extended to the National Science Foundation,
under whose traineeship the author.has been working.
The author also wishes to express his gratitude to the
University of Florida Computing Center for the computing facilities,
services, and aid extended by the Center.
Finally, the author wishes to extend thanks for the aid and
encouragement provided by his wife, who has persevered through many
trying, times.
11
192
1+ PHKI + l, JtK+1) + PHI ( 11 J,K1) + PHI ( I1,J,K+1))/8.0
IF (ABS(TAMPHI(IJK)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PH KI J K) = OMEGA*TAM TAG*PHI(I*J*K)
GO TO 780
C SERIES 771 IS FOR THE UNEQUAL XSPACING NEAR X = SQRK3.00).
C PVC IS A VALUE INTERPOLATED LINEARLY BETWEEN TWO LATTICE
C VALUES. THE ADDED TERMS IN THE TAM EXPRESSION REPRESENT
C THE FIRST AND SECOND DERIVATIVE OF PHI IN THE XDIRECTION.
771 PVC = {PHI!I1,J,K+1 ) + PHK I1JK1)J/2.0
TAM = (2.0*(PHI( IJ1VK) + PHI ( I J + l VK) ) + PH I (I+ 1, J, Kl)
1 + PH I {1 + 1, J,K+1) + PHK Ii, J,K1) + PHKI1, J,K+l))/8.0
2 +ITAB71.0)* PVC/(4.C*BIP1) + BUG*{PVC/BI PI) /4.0
TAM = TAM/(1.0 + BUG/(A .0*TAB7))
IF (ABS{TAM PHI(I JK) ) .GT. EPS) TAM = TAB7*PVC/BIPl
IF (ABS(TAMPHI(IJK)I.GT.PS) EPS=ABS(TAMPHI(I,J,K))
PHI( IJ K) = CMEGA*TAM + TAG*PHI{I,J,K)
GO TO 780
C SERIES 772 IS FOR THE DPHI/DY = 0 CONDITION AT Y = 0 OR Y A.
772 IF (J .EQ. 1) L = J + l
IF (J .EC. JMAX) L = Jl
IF { I .EC. IMIT) GO TO 774
TAM = (4.0*PHI{ILK) + PH I (I + 1,J,K1) PHK 1 + 1, J ,K+1)
1 + PH I (11 J Kl) + PHKIl,J,K+l))/8.0
IF (A8S(TAMPHI(I,J,KK.GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(ItJtK) = OMEGA*TAM + TAG*PHI(I,J,K)
GO TO 780
C SERIES 774 IS FOR UNEQUAL XSPACING IN ADDITION TO THE OTHER
C CONDITIONS OF SERIES 772
774 PVC = (PHIUl, J,K+1) + PHK I1J Kl) J/2.0
TAM = (4.0*PHI(I,L,K) + PHKI+1,J,K1) + PHI(I+1,JK+1)
1 + PHI(IliJ.Kl) + PHK11,J.K+l))/8.0
2 +(TAB71.0)*PVC/(4.C*BIP1) + BUG*(PVC/BI PI)/4.0
TAM = TAM/{1.0 + BUG/(40*TAB 7))
IF (ABS(TAM PH I(I J K)) .GT. EPS) TAM = TAB7*PVC/BIPI
IF (ABS(TAMPHKI,J,KJJ.GT.EPS) EPS=ABS( TAMPHI (I ,J,K))
PH I(I,JK) = OMEGA*TAM + TAG*PHI(IJK)
780 CONTINUE
C TRAVERSE EVEN JVALUES, WITH EVEN VALUES OF I (X).
DO 790 J=2 JMIT 2
DO 790 1 = 2IMIT 2
IF (I .EG. IMIT) GO TO 776
C DIFFERENCE EQUATION FROM A TRIPLESERIES EXPANSION.
TAM = (PHK 11 Jl, Kl) + PH 1(11,J + l, Kl) +
1 PHI(I1,J1,K+1) + PHI(11,J + lK + 1) +
2 PHKI + 1 J1K 1) + PHK 1 + 1, J + l, Kl) +
3 PHKI + 1,Jl,K+l) + PHK 1 + 1,J + 1,K+1) )/8.0
IF (ABS(TAMPHI( I,JK)).GT.EPS) EPS=ABS( TAMPHK I J,K ))'
PHKI,J,K) = OMEGA*T AM + TAG*PHI(I,J,K)
C DIFFERENCE EQUATION BASED ON INVARIANCE OF LAPLACIAN.
TAM = (2.0*(PHI(I+2,JK) + PHI(I,J,K)) PHKI+l,J+l*Kl) +
1 PHKI + 1, J1K1) +' PHI( I + l,J+1*K+1)
o o o o n o o
189
C IN THE CURRENT ITERATION. THE NEW VALUE OF PH1(1,J,K) FOR
C THE GIVEN POINT IS ALSO CALCULATED FROM THE OVERRELAXATION
C EQUATION, USING OMEGA.
710 IF (AES (TAMPH I( I,J,KH.GT.EPS) EPS=ABS t TAMPHI ( I J ,K))
PHI(IJ,K) = QMEGA*T AM + TAG*PHI(I,J,K)
GO TO 701
C SPECIAL SECTION FOR FINAL XSPACING LESS THAN H
705 TAM = (PHKI, J + 1,K) 4 PHI(I,J1,K)
1 + PHI(I,JK+1) 4 PHI(I,J ,K1) 4
2 PHI ( I4l,J,K)TAB2 4 PHI(I1,J,K)*TAB3)/TAB1
GO TO 710
C SECTION FOR Y=0 WHERE DPHI/OY EQUALS ZERO.
702 IF (I .EG. IMIT) GO TO 715
TAM = (PH I ( I1JK) 4 PHI(I41,J,K) 4 PHI(I,J,K4l) 4
1 PHI(I,J,K1) + 2.0PHI(I,J+1,K))/6.0
GO TO 710
C SPECIAL SECTION FOR Y=A, WHERE DPHI/OY EQUALS ZERO
703 IF (I .EQ. IMIT) GO TO 725
TAM = (PHI(I1,J,K) 4 PHI(1+1J,K) 4 PHI(I,J,K4l) 4
1 PH I (I,J,K1) 4 2.0*PHI( I,J1,K))/6.0
GO TO 710
C SECTIONS 715 AND 725 ARE FOR THE COMBINED CASES OF DPHI/DY =
C ZERO AND THE UNEQUAL XSPACING NEAR X = SQRTO.GO).
715 TAM = (PHI(I1,J,K)*TAB3 4 PH I(141,J,K)*TAB2 4
1 PHI(I,J,K41) 4 PHI(I,J,Ki) 4 2.0*PHI(I,J41,K))/TAB1
GO TO 710
725 TAM = (PHI(I1,J,K)*TAB3 4 PHI{141,J,K)*TAB2 4
1 PH I (I,J K41) 4 PH I(I,J,K1) 4 2.0*PHI(IJ1,K))/TAB1
GO TO 710
701 CONTINUE
7C0 CONTINUE
RETURN
END
SUBROUTINE GRADE
THIS SUBROUTINE MAKES THE TRANSITION FROM A GIVEN MESH SIZE,
H, TO CNE WHICH IS TWICE AS LARGE ( 2H).
COMMON ALFAO(21,21,2),ALFA7<21,21,2),ALFA02{21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA 1(21,21),
3 THETA2(21,21),IBDRY1(21,21),IBDRY2(21,21),PZ(36,21),
A TIP(100),TOP(11,11),H,EPS,I MAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JKID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,'
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,B0G,IG1,IG2,I,J,K,
7 AI,AJ*AKNXYZTAG0MEGAXPBETA,KIPKEY*N0(3621,21)
COMMON DIV(2500),ALFA(2500,5),PH 1(36,21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FLO
163
BET1 = ( X IN IT ( J, K ) SQRTI3.0) )**2
IF (( 1**2 BET 1) .GE. 1.00) Y2= 1.00
IF (( Z**2 + BET 1) .GE. 1.00) GO TO 220
Y2 = 1.0 SORT{BET BET 1)
220 IF (ABS((Y2Y)/H).LT.1.0) GO TO 1121
IF (ADS C(Y2YJ/H).GT.1.0 .ANO. ABS((Y2Y)/H).LT.2.0)
1 GO TO 1122
C FACTOR FCR PJ
1220 ALFAO(J,K,1) = 1.00
IF (J .GT. JMIT) ALFAO(J,K1) = AH
IF (J .GT. JM IT) GO TO 207
AG = (X XINIT(J+1,K))/H
ALFAO(JK11) = AMIN1(AG, AH)
IF (ALFAO(J,K, D.GT.1.00) ALFA0(J,K,1) = 1.00
C FACTOR FCR PJ1
207
ALFAO(J K f
2) =
1.00
IF (J .GT.
JMIT2) AL
FAO (
J,K,2) =
IF (J .GT.
JM I
T2) GO
TO
105
AG = (X 
XINITIJ+2
K) )/H
ALFAO(JK,
2) =
AMIN1
(AG,
AH)
IF (ALFAO(
JK,
2).GT.
1.00
) ALFAOI
105
MIGl(JyK)
= 1
IF (NCAS
. EQ.
2) GO
TO
201
GO TO 130
211
ALFA7(J K
1) =
1.00
ALFA7(J K y
2) =
1.00
212
ALFAO(JKy
1) =
1.00
ALFAOIJyK,
2) =
1.00
GO TO 105
C ARC INTERPOLATION
C IN THIS INTERPOLATION PHI VALUES NEEDED FOR POINTS ON A
C SPHERICAL SURFACE ARE EXPRESSED BY LINEAR INTERPOLATION IN
C TERMS CF ARC LENGTHS BETWEEN TWO KNOWN POINTS ON THE SURFACE.
1121 A = SORT(3.00) X
ALFA0(J,K,1) = 1.00
ALFAO(J,K y 2) = 1.00
B = SORT(3.00) XENCIJ+ltK)
C = 1.0 Y ALFA INK,4)*H
D = 1.0 Y H
E = SORT(3.00) XEND(JtK)
G = 1.0 Y
THET A1(J K) = (ATAN2(C,A) ATAN2(DB))/
1 (ATAN2(GE) ATAN2(D B))
M IG1(J K ) = 2
GC TO 130
C IN THIS INTERPOLATION NEEDED VALUES ARE EXPRESSED BY LINEAR
C INTERPOLATION BETWEEN TWO KNOWN POINTS WHETHER THOSE POINTS
C OCCUR ON A BOUNDARY SURFACE OR A LATTICEPOINT.
1122 AIB = IBDRYl(JfK) 1
ALFAO(JK 1) = 1.00
ALFAO(JK 2) = 1.00
22
with his drag and velocity measurements to attempt a correlation with
the case of closely packed hemispheres and found a C of 0.0680.
Li
Chepil concludes by noting that "This study shows that lift on
hemispherical surface projections, similar to soil grains resting on
a surface in a windstream, is substantial. Therefore, lift must be
recognized together with drag in determining an equilibrium or crit
ical condition between the soil grains and the moving fluid at the
threshold of movement of the grains" [36, p. 403].
2.5.3 Chao and Sandborn.Chao and Sandborn [39] also performed
experiments on spheres, but, by means of a transducer, they actually
measured the pressure distribution on the upper half of an element.
However, the type of flow they used bore no real relation to that in
nature's streams, and no attempts were made at analyzing velocities and
the like. Einstein used the flow of water in a channel with a measured
logarithmic velocity distribution. On the other hand, Chao and Sandborn
placed lead shot on a flat surface and blew a stream of air down onto
the particles, the air diverting horizontally at the flat surface.
Their conclusions included, "The present experimental results are of
primary interest in demonstrating that a problem exists. . More
extensive research is needed before there can exist a better understand
ing of the mechanism" [39, p. 203].
2.6 Shapes of bodies studied
With the exception of the vast amounts of work done on wings,
struts, and related areas, most effort has beenaimed at circular
cylinders and spheres, especially in work related to the sediment
t
BIOGRAPHICAL SKETCH
Barry Arden Benedict was born February 7, 1942, at Wauchula,
Florida. In June, 1960, he was graduated from DeLand High School,
DeLand, Florida. In April, 1965, he received the degree of Bachelor
of Civil Engineering from the University of Florida. At that time
he enrolled in the Graduate School of the University of Florida.
He obtained the degree of Master of Science in Engineering in June,
1967, and has since then pursued his work toward the degree of
Doctor of Philosophy.
Barry Arden Benedict is married to the former Betty Janell
Truluck. He is a member of Phi Kappa Phi, Tau Beta Pi, Sigma Tau,
Phi Eta Sigma, Florida Blue Key, the American Society of Civil
Engineers, and Pi Kappa Phi.
207
97
dA = area in xyplane
Figure 30. Pressures on area of hemisphere.
Evaluation of the lift on the particles must take into account
the finite flow occurring along the basal diameter of the hemisphere
in the xyplane. This velocity also reduces the pressure at the base
below pQ. Application of (572) shows that the pressure, p, at any
point on the surface is found as below.
(573)
159
IF INCAS .EQ. 1) 102 = IB0RY2(JK)
IF INCAS .EQ. 2) 102 12
C BEGIN TRAVERSE IN XOIRECTION
DO 101 I =101,102
106 NP = NP 4 1
NO(I,J,K ) = NP
KIP = NP
AI = I
X = I A I1.0)*H
IF II.EQ.ID1) X=XINIT(J,K)
IF II.EQ.I02) X=XENDIJ,K)
C INITIALIZE VALUE IN PHIII,J,K) CORRESPONDING TO POTENTIAL FOR
C A UNIFORM STREAM
PHI(I,JtK) = 100.0(100.0/SQRT3.00))*X
C COMPUTE ALFA VALUES AND STORE
C HANDLE POINTS WHICH LIE' ON HEMISPHERICAL SURFACES (101,102) BY
C SENDING TO SERIES 150 OR 160 FUR COMPUTATIONS
IF l I .EQ. 101) GO TO 150
IF I I .EQ. ID2) GO TO 160
C COMPUTE ALFA IN NEGATIVE XDIRECTION
ALFA INP1) = IXXINITIJ,K))/H
IF IALFA(NP,1).GT.1.0) ALFA!NP,1) = 1.00
C COMPUTE ALFA IN POSITIVE X DIRECTION
125 ALFA(NP,2) = 1.00
ALFA(NP,2) = (XEND(J,K)X)/H
IF (ALFA(NP,2) .GT. 1.00) ALFA(NP2) = 1.00
C COMPUTE ALFA IN NEGATIVE Y DIRECTION
107 ALFA(NP,3) = 1.00
MIS = IBCRY1C1,1)
IF (I .GT. MIS .AND. I .EQ. ID2) GO TO 101
IF (I .GT. MIS) ALFA(NP,3) = 1.00
IF (I .GT. MIS) GO TO 108
YINIT = XINITt I,K )
ALFA(NP3) = (YYINIT)ZH
IF (ALFA(NP,3).GT.1.00) ALFA(NP,3) = 1.00
IF (I .EC. ID2) GO TO 101
C COMPUTE ALFA IN POSITIVE Y DIRECTION
108 ALFA(NP4) = 1.00
IF (I .EQ. IMAX .AND. K .EQ. KLOWJ GO TO 104
IF (1.0 1**2 (XSQRT(3.CO))*2) 104,104,134
134 IF (((SCRT(1.0Z**2 (XSQRT(3.0 ))*2 )+1.0Y)/H).LT.1.00)
1 ALFA(NP,4) = (SQRT(1.0Z**2 (XSQRT(3.0 ) )**2 )
2 +1.0YJ/H
IF (I .EG. ID 1) GO TO 101
104 IF (J.EC.l) ALFA(NP,3 ) = ALFA(NP,4)
IF (I .EQ. ID 11 GO TO 101
IF (J .EQ. JMAX) ALFA(NP,4)= ALFA(NP,3)
C COMPUTE ALFA FOR NEGATIVE Z DIRECTION
ALFA(NP 5) = 1.00
IF (K.EQ.l) ALFA{NP,5) = 1.00
IF (K .EQ. 1) GO TO 1011
177
GO TO 970
1951 PHI(ItJ(K) = 0.00
GO TO 869
1950 S2 = (Yl.0)*PJ*2.0
S2A = (Y1.0)*2.0
GO TO 970
C POINTS FOR YDERIVATIVE BY ARC INTERPOLATION.
952 II = IBDRYKJ1K)
12 = IBCRYlUtK)
PJ = PHI(12 tJ K) THETA2(JK)*
1 ( P HI (12 JK) PHI(II,J1,K))
S2 = PJ*(Y1.0)/ALFA(N,3)
S2A = (Y1.0)/ALFA(N,3)
GO TO 970
C LINEAR INTERPOLATION (BETWEEN KNOWN XPOINTS) FOR YPOINTS.
953 PJ = PHKItJlfK) + CCCN2(JK) *
1C PH ICI1J1K)PHI(IJ1K))
S2 = (Yl.0)*PJ
S2A = Y 1.0
IF (J .EQ. 2) S2 = 2.0*S2
IF (J .EC. 2) S2A = 2.0*S2A
C DERIVATIVES
970 IF (K .EQ. KL0W1) GO TO 4970
NT2 = NC{11, JK + 1)
IF (NT2 .EC. 0) NT2 = MARK
NT3 = NO(11 Jt K+2)
IF (NT3 .EQ. 0) NT3 = MARK
IF (XP .LT. XINIT(J,K + 1) ) GO TO 971
BETA = (X XP)/H
ALSUM = ALFA(NT21) + ALFA72(JtK,1) + BETA
IF (K .EQ. 1) GO TO 981
IF (ALFA72( J K11) .LT. BETA) GO TO 3853
PK = PHI ( 11 JK+1) BETAMPHK I2,J,K+1)
1 PHI(I,JK+L))/ALSUM +
2 (BETA**2)(PHI(I2JK+1)/ALFA(NT21) PHI{I,J,K+1)/
3 (ALFA721J,K,D+BETA) PHI(11,J,K+1)*(1.O/ALFA(NT2,1)
4 1.0/(ALFA72(J,K,1)+EETA)>)/ALSUM
972 IF (XP .LT. XINIT(J(K + 2) ) GO TO 973
BETA = (XXPJ/H
ALSUM = ALFA(NT3 1) + ALFA72(J,K,2 ) + BETA
IF (ALFA72(J> K,2) .LT. BETA) GO TO 3854
PK1 = PH I(I 1 J K + 2 ) BETA*(PHI(I2JK+2)
1 PHI(IvJtK42)J/ALSUM
2 + (BETA**2)*(PHI(12,J,K+2)/ALFANT3,1) + PHI(I,J,K+2)/
3 (ALFA72(J,K,2)+BETA) PH I(11,J,K + 2)*(1.O/ALFA(NT3,1) +
A 1.0/(ALFA72(JtK,2)+BETA))J/ALSUM
GO TO 975
4970 BETA = (XXPJ/H
ALSUM = 1.0 + ALFA72(JtK1) + BETA
IF (ALFA72(JK1) .LT. BETA) GO TO 3853
PK PHI(11 JK+1) BETA*(PHI(I2,J,K+1)
75
In the case of points such as V, rewrite (529) for the case where G
is at a distance h and A at a point a^h away.
2h
2 SjÂ£
r t9 v 4
^2 U,ar2) i2 V
(533)
Adding this to (530) and equating to zero produces the desired result.
 4 {' + 9v = 0 (534>
fc ^ p M
2 2
Points such as G require more attention. The triple McLaurin
expansion used to obtain (532) is such that, due to symmetry, all
oddorder terms (first derivative, third derivative, and so on) cancel
between the terms. However, an unequal xspacing causes those terms
involving a &x to remain in the equation, though those oddorder terms
3
in the other directions still disappear. Here, the 0(h ) terms will
be neglected so that of the following form can be written for cp(x,y,z)
at some point 6x, 6y, and 6z from cp^.
cp(x,y,z)
= 9,
dcÂ£
dz
+
(6x)
+ 6x
2 + (6y)2 I + (dz)
dx dy
Sy
dcp
dxdy
+
6x
6z
d2cp
dxdz
~2
S_Â£
dz2
6y 6z
>+ o(h3)
d2cp
dydz,
(535)
where all derivatives are taken at point 0. Expansions similar to this
are discussed by Olmsted [69],
147
value from theory for the equivalent bed yielded a lift less than 12
per cent higher than that measured.
The results above indicate satisfactory agreement and strongly
support the validity of using potential flow theory to study hydro
dynamic lift. All the measurements were from flows in the hydrodynam
ics lly rough range.' It is in this range that the theoretical approach
is expected to hold valid, for viscous effects are negligible. Use
of such methods in flows not in the rough range may be quite risky.
Extensive experimental work is needed for studying lift.
Related to this is a need for establishing knowledge of the velocity
distribution as it varies from point to point over the bed. Emphasis
should also be placed on defining an appropriate theoretical bed.
Studies should consider the use of idealized beds to replace natural
ones along with what characteristics this idealized bed should possess
to most adequately portray the natural conditions.
The approach of this dissertation provides now an analytical
means of studying many bed forms and patterns. These forms may arise
on a macro scale (dunes and ripples) or on a micro scale (individual
sediment particles). It is to be expected that any arrangement studied
will have relation to some particular aspect of sediment movement.
The analytical results would provide a basis for comparison with
experimental results and reasons for those results, Also, in some
instances, analysis might aid in evaluation of which cases to treat
experimentally and what to look for.
175
HERE.
C THE STEPS IN FORM AND ORCER ARE QUITE SIMILAR TO THOSE IN
C, SUBROUTINE BCDER. FOR THIS REASON NO EXTENSIVE COMMENT STATE
C KENTS ARE INCLUDED IN THIS SUBROUTINE (BDDER2). FOR CLARI
C FICATION, CONSULT THE STATEMENTS IN BDDER. THE STATEMENT
C NUMBERS ARE GENERALLY THE SAME AS THOSE OF BDDER WITH THE
C INITIAL 8 OF BDDER REPLACED BY AN INITIAL 9 IN BCDER2.
C
COMMON ALFA0(21,21,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CCONl(21,21),CCGN2(21,21),THETA 1(21,21),
3 THETA2(21,21),IBDRY1{21,21),IBDRY2(21,21),PZ(36,21),
4 TIP{100),TOPHI,11),H,EPS,IMAX,JMAXKMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1,IG2.I,J,K,
7 AI,AJ,AK.N.X.Y.Z,TAG,OMEGA,XP,BETA,KIP,KEY,NO*36,21,21)
COMMON DIV(2500),ALFA(2500,5),PH1(36,21,32)
COMMON IMIG(21)PIG,KINKIN1FL0
C EINSTEINEL SAMNI ARRANGEMENT
C CONSIDER FACTORS FOR XCERIVATIVE.
950 XP = (AI~2.0)H
AX = (XEND(J,K) XINITlJ,K))/H
IF (XP .LE. XINIT(J,K ) ) GO TO 960
BX = (XEND(J,K ) XP)/H
IF (AX .LT. (1.0+BX)) GO TO 961
NT = N0(I1,J,K)
IF (NT .EG. 0) NT = MARK
SI = (1.0)*(XSQR T(2.0))*(PHI(I2,J,K)*
1 ((2.0*ALFA(NT ,2)1.0)/(1.O+ALFA(NT,2))) 
2 2.0PHI(Il,JtK))
S1A = (XSQRT(3.00))(3.0/(1.O+ALFA(NT,2)))
MAN = MIG2(J,K)
GO TO (951,952,953),MAN
C NEXT ADJACENT XPOINT ON HEMISPHERICAL BOUNDARY.
960 SI = (XSQRT(3.00))*PHI(I1,J,K)/ALFA(N,1)
IF (J .EG. JMIC .AND. Z .LE. 0.20) GO TO 8960
S1A = (XSQRT(3.00))/ALFA(N,1)
MAN s MIG2(J,K)
GO TO (951,952,953),MAN
8960 SI = (XSQRT3.00))#(200.0 PHI(11,J,K))/ALFA(N,1)
S1A = (XSCRT(3.00))3.0/ALFA(N,1)
MAN = MIG2(J,K)
GO TO (951,952,953),MAN
C ONLY ONE LATTICE POINT AVAILABLE BETWEEN HEMISPHERES.
961 SI = (XSQRT(3.C0))*PHI(11,J,K)/ALFA(Nl,2)
SIA = (XSQRT(3.00))/ALFA(Nl2)
MAN = MIG2(J,K )
GO TO (951,952,953),MAN
C BOTH YPCINTS AVAILABLE FOR EXPANSION.
951 IF (J .EG. 1) GO TO 1951
NT = N0(IlvJltK)
v
77
In plane DEKJ:
32cq  a2cp Id Pe + fj %
dx2 dz2 (jjl h)2
The resulting difference equation is
(540)
^Wl5 + 9D + + + 'Pj 8CPN = C5"41)
Treatment of points such as N becomes somewhat different when the
xintervals are not equal. A McLaurin expansion will also be applied
in this instance, and,as for point G, the first and second derivatives
will respect to x remain in the difference equation, as shown below.
2
2
 2h(o1) Â§ h2(cl) = 0 (542)
2 ox 2^2
dx
Evaluation of these two derivatives requires values at a point where
no value is being computed, as seen in Figure 25. Thus, such a
value is obtained by a linear interpolation between two known points,
such as D and E.
The foregoing completes the discussion on graded nets and some
special problems they create. Application of the grading process will
take place away from the hemispherical elements, as a finer lattice
is desired there.
94
A second concern is the possibility of error in the differences
solution for regions very near the singular points. To detect any
propagation of such errors and also to aid in the extrapolation to a
finite velocity, solutions will be made for varying values of h, the
lattice spacing. The solutions thus obtained can be compared to eval
uate any problems near the various singular points.
In order to avoid certain problems about the tangent point
midway in the solution space, advantage was taken of certain symmetries.
The value of cp at this midpoint for any zvalue is the same. It equals
the average of the values at the ends of the space. Hence, in some of
the lower regions where no lattice point exists between hemispheres,
the midpoint cpvalue can be used as a point in the difference equa
tions. The equations are quite similar to others developed. Inclu
sion of such an approach in the differences solution aided convergence
in the region of this particular singularity.
5.6 Velocity and lift calculations
Once the potential (9) field has been sufficiently determined
through the differences solution described, it is then possible to
calculate the velocities created by that potential field. Of interest
here are the velocities on the surfaces of the hemispherical elements,
which will enable computation of pressures and lift forces. The
velocities can be calculated from the potential as shown in (570),
where u, v, and w represent velocity components in the x, y, and z
directions, respectively.
Ill
above the top of the hemispheres. Chepil defines the limits of the
expanding boundary layer in the wind tunnel as the greatest distance
y to which the equation (62) is still valid.
The work done by Chepil on the three sizes mentioned falls in
the hydrodynamicslly rough range, with Reynolds numbers of form
(v^a/v) from 517 to 6840, where a is the hemispherical radius. This
is similar to the wall Reynolds number, differing by the ratio (a/k).
Hence it is to be expected that the lift is independent of the
Reynolds number. If this is true, some consistency should be expected
in relating lift to a form of some lift coefficient times a reference
velocity head.
Some investigation was necessary to provide a relatively con
sistent picture. Generally, the velocity picture here will be related
with (62) to remain consistent with Chepil. Chepil determined a
value for yQ in (62) by extrapolating his velocity measurements to the
point of zero velocity. Table 2 shows values which Chepil found,
including yQ for the three sizes. Using these values of yQ for the
equation (62) and choosing as a velocity that at the elevation of
the hemisphere tops resulted in lift coefficients varying from
0.169 to 0.374, or over a twofold difference. Such a variance is not
reasonable. Some of the factors involved in the experimental work
needed to be studied. First, recall that in Chapter III, a value of
0.0670a was found for the value of y^ for the given pattern. This was
derived on a volume basis, considering a smooth surface between hemi
spheres. However, Chepil*s work had a rough gravel surface. Consider
Roughness
Idealized
Isovels
/
/
/
\
\
\
I
\
h
/
!
I
' Roughness:
\
J:
k
2
Figure 8. Sketches for sidewall effect.
206
68. Allen, D. N. deG., and Dennis, S. C. R. "The Application of
Relaxation Methods to the Solution of Differential Equations
in Three Dimensions. II. Potential Flow Round Aerofoils,"
Quarterly Journal of Mechanics and Applied Mathematics,
vo1. 6, 1953, pp. 81100.
69. Olmsted, John M. H. Advanced Calculus, New York: Appleton
CenturyCrofts, Inc., 1961, 706 pp.
70. Prandtl, L. Essentials of Fluid Dynamics. New York: Hafner
Publishing Company, 1952, 452 pp.
7
71.' Michaels, Paul. "Ideal Flow Along a Row of Spheres," The Physics
of Fluids, vol. 8, No. 7, July 1965, pp. 12631266.
72. Koloseus, H. J., and Davidian, J. "RoughnessConcentration Effects
on Flow Over Hydrodynamicslly Rough Surfaces," U.S. Geological
Survey WaterSupply Paper 1592D. Washington, D.C.: U.S. Govern
ment Printing Office, 1966, 21 pp.
73. Rouse, H. "Critical Analysis of OpenChannel Resistance,"
Journal of the Hydraulics Division, Proceedings, ASCE, vol. 91,
No. HY4, July 1965, pp. 125.
74. Christensen, B. A. "Theory of Sediment Transport," Unpublished
Lecture Notes, University of Florida, 1967.
non
158
IBCRY1(3,5) = 5
I BDRY1(5*3) = 5
1BDRY1(5,5) = 3
1001 FORMAT (5F10.5)
369 RETURN
DEBUG SUBCHK
END
SUBROUTINE ALFINT
C
C SUBROUTINE TO DETERMINE INTERPOLATION FACTORS TO ENABLE
C DEALING WITH LATTICE POINTS FOR WHICH ALL OR SOME OF THE SIX
C NEEDED ADJACENT POINTS ARE AT SOME DISTANCE OTHER THAN THE
C NORMAL MESH LENGTH H FROM THE GIVEN POINT. THESE VALUES ARE
C STORED IN THE ARRAY ALFA(N,L ), WHERE N REPRESENTS THE NUMBER
C OF THE POINT AS STORED IN THE ARRAY NO(I,J,K), WHICH IS ALSO
C ESTABLISHED IN THIS SUBROUTINE. THIS SUBROUTINE ALSO SETS THE
C INITIAL VALUES FOR PHI(I,J,K) FOR THE FIRST ITERATION IN THE
C SOLUTION.
C
COMNGN ALFAO(21,21,2 ),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA 1(21,21) ,
3 THETA2(2l,21),IBDRY1(21,21),IBDRY2{21,21),PZ(36,21),
4 TIP(100),T0P(11,11)H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID MARK,NCASKL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1,IG2I*J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,N0(36,21,21)
COMMON DIVC2500),ALFA(2500,5),PH 1(36,21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FL0
C EINSTEINEL SAMNI ARRANGEMENT
C INITIALIZE ARRAYS NO(I,J,K) AND PHI(I,J,K)
DO 100 1=1,IMAX
DO 100 J = 1 JMAX
CO 100 K=1,KMAX
IF (K .GT. KLOW) GO TO 100
NQ(I,J,K ) = 0
100 PHK I,J,K) = 0.00
NP = 0
C SET Z VALUE
DO 102 K=1,KMAX
IF (K .GT. KLOW) GO TO 110
AK = K
Z = (AK1.0)*H
C SET Y VALUE
DO 102 J=1 JMAX
AJ J
Y = (AJ1.0)*H
ID1 = IBDRYK J,K )
179
C IN THE SIXADJACENTPOINT LAPLACIAN EXPRESSION. THE NOFLOW
C BOUNDARY CONDITION IS APPLIED HERE.
C
COMMON ALFAOt 21,21,2 ),ALFA7(21,21,2), ALFA02(21,21,2),
1 ALFA72(21,21,2),XENC(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CCONl(21,21),CCON2(21,21),THETAl(21,21),
3 THETA2(21,21), IBDRYK 21,21 ),IBDRY2( 21,21), PZ( 36,21) ,
4 TIP(1QO)TOP(1111)H,EPSIMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KLOWl,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,B0G,IG1,IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,N0(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,2132)
CGMMCN IMIG(21),PIG,KIN,KINl,FLO 
C EINSTEINEL SAMNI ARRANGEMENT
JMIT = JMAX 1
MULE = 0
Z1 = Z ALFA(N,5)*H
C CALCULATE CONTRIBUTIONS FROM DPHI/DZ
S3 = (Z1)C2*0PHI(I,J,K) + (1.02.0*ALFA(N,5))*
1 PHI(I,J,K+1)/(1.0+ILFA(N,5)))
S3A = (Z1) *<3.0/(l.C+ALFA(N,5)))
MUT = 1
C THE SAME EQUATION IS USED FOR CALCULATION (BY SERIES EXPANSION
C ABOUT A FIELD POINT ) OF ALL FOUR POINTS NEEDED FOR THE EX
C PANS IONS TO EVALUATE DPHI/DX AND DPHI/DY. THE ONLY CHANGES
NEEDED ARE IN THE IDENTIFICATION OF THE SURROUNDING POINTS.
C THIS END IS ACCOMPLISHED BY FIRST GOING TO ONE OF THE SERIES
C 601,602,603,604 FOR THE NEEDED IDENTIFICATIONS AND THEN
C RETURNING TO THE SERIES BEGINNING AT 610 FOR THE ACTUAL
C CALCULATION.
605 GO TO (601,602,603,604),MUT
610 G = ALFA(NT,5) ALFA(N,5)
PC = PHI(LIL2K1)
IF (ALFA(NT,5) .LT. 1.0) PC = PZ(L1,L2)
IF (ALFAIN,5) .LT. G) GO TO 606
GAG = 1.0 (G* 2) *(1 .C/ALFA(N,5) + 1.0/G)/ALFA(NT,5)
GIG = PC + G*(PHI(L1,L2,K)PC)/ALFA(NT,5)
GUF = (G**2)*(PHI(L1,L2,K)/ALFA(N,5) + PC/G)/ALFA(NT,5)
GO TO 607
606 GAG = 1.0 (ALFA(N,5)*2)*(1.0/ALFA(N,5)+1.0/G)/ALFA(NT,5)
GIG = PHI(L1,L2,K) ALFA(N,5)(PHI(L1,L2,K)PC)/ALFA(NT,5)
GUF = (ALFA(N,5)*2)(PHI(LiL2,K)/ALFA(N,5) 4
1 PC/G)/ALFA(NT,5)
607 TUM = (GIG GUF)/GAG
GO TO (611,612,613,614),MUT
601 NT = N0(I41,J,K)
IF (NT .EQ. 0) NT = MARK
IF (((XENDtJ,K1) XJ/H) .LT. 1.0) GO TO 673
LI = 141
L2 = J
GO TO 610
9
Reasoning such as Prandtl's has through the years prompted
many potential flow solutions to the lift problem, probably the most
famous of which are those by Kutta and Joukowsky.
2.4.3 Work of Kutta and Joukowsky.Generally, an understand
ing of the work of Kutta and Joukowsky begins with flow around a cir
cular cylinder. It is well known that no lift or drag forces are
predicted by any transformation of the symmetrical flow around a cylin
der. The addition of a vortex in the center of the cylinder produces
a streamline pattern, the effect of which is to yield a lift force.
This force is due to the circulation's tendency to incrase the velocity
above the cylinder and decrease it below, thus causing a pressure dif
ference and a consequent lifting action.
Kutta [18] first applied the methods of conformal transforma
tion to transform the cylinder with circulation into a line inclined
to the flow. This produces a force acting perpendicular to the veloc
ity at infinity. This force is, of course, the one originally exerted
on the cylinder.
Kutta's work and use of transformations prompted further work.
Joukowsky [19] wanted to avoid difficulties at the sharp leading edge
of Kutta's plane, and he employed a mapping function by which a curvi
linear profile very similar to actual airfoil shapes was developed.
Flow around a Joukowsky profile is shown in Figure 1.
Numerous investigations of Joukowsky profiles have been carried
out. Joukowsky [20] himself performed experiments in 1912, and
Blumenthal [2l] calculated the pressure distribution from theory
31
3.3 Use of proposed adjusted velocity distribution
Christensen [52] is suggesting use of a slightly different
velocity distribution which he has developed. The new expression is
as follows:
= 8.48 + 2.5 In (f + 0.0338)
v_ k
(39)
Calculation will reveal one very desirable feature of (39),
that being prediction of a zero velocity for y equal to zero. The
added constant factor will continue to have a large effect very near
the wall, but as y/k increases, the effect of the additional term will
quickly become negligible, yielding essentially the same expression
as (32). This is correct, since (32) has proved adequate away from
the wall.
3.3.1 Comparison of distributions at wall.In addition to
the effect of zero velocity at the wall as compared with the infinite
value predicted by (32), other features can be noted by looking at
the change of velocity with distance from the wall, or dv/dy shown
below.
Former: ~ =2.5
dy y
dy
(310)
For y * 0 dv/dy * 00
y dv/dy * 0
QV
Proposed: = 2.5 v
(311)
For y = 0, dv/dy = 74.0 v^/k Finite
y  , dv/dy 0
132
x/a
Figure 48. Closely packed hemispheres: velocities on y = 0.8a.
z
a
a.
V
e
H
k
k
U
V
P
CT
T
O
'Pb
U)
fraction finer than (in soil gradation curve)
Cartesian coordinate; also elevation above datum
angle through "corners" of isovels
ratio of given lattice leg length to full increment, h
unit weight of fluid
spherical coordinate; also angle for arc interpolation
von Karman's constant
lift per unit area (total bed area)
lift per unit area (only projected area of grain)
kinematic viscosity of fluid
mass density of fluid
density of particle in Jeffreys analysis
bed shear stress
potential function
value of potential function on boundary
overrelaxation factor
xii
167
IG2 = IB CRY 2(JK)
C BEGIN TRAVERSE IN XOIRECTION.
DO 801 1=IG1IG2
IMID =0
KEG =0
MPK = 0
N = NO (I JK)
C GO TO STANDARD LAPLACIAN DIFFERENCE EQUATION WHERE AbL
C ADJACENT POINTS ARE H AWAY.
IF (N .EG. 0) GO TO 841
AI = I
X = (A I1 .0)*H
IF (I .EG. IG1) X = XINITtJ,K)
IF (I .EG. IG2) X = XEND(J,K)
IF (I .EQ. 1 .OR. I .EQ. IMAX) GO TO 801
IF ( I .EQ. IG1 .OR. I .EQ. IG2) GO TO 850
C COMPUTE XDIRECTION FACTORS FOR DIFFERENCE EXPRESSION FOR
C LAPLACIAN.
PI = PHU I1, J,K)/(ALFA(N,1)*(ALFA(N,1) + ALFA(N,2)))
P2 = PH IC 1 + 1,J,K)/(ALFA(N,2)*(ALFA(N,1) + ALFA(N,2)))
C COMPUTE YDIRECTION FACTORS FOR DIFFERENCE EXPRESSION FOR
C LAPLACIAN. THESE FACTORS ARE P3,P4, AND THE ZDIRECTION
C COMPONENTS ARE P5,P6. FOR THOSE CASES WHERE THE ADJACENT
C POINTS LIE ON A HEMISPHERICAL SURFACE, THE PHI VALUE AT THAT
C SURFACE POINT IS FOUND FROM ONE OF THE SUBROUTINES LISTED
C ABOVE WHICH TREAT THE DPHI/DN = 0 CONDITION. IN CASES WHERE
C THE PROXIMITY OF THE HEMISPHERES CREATES PROBLEMS FOR THIS
C PROCEDURE, THE PHI VALUE AT THE SURFACE POINT IS EXPRESSED
C BY AN INTERPCLATION BETWEEN NEARBY POINTS ON THE HEMISPHERE.
C CHECK TO SEE IF EITHER (OR BOTH) Y POINTS LIE ON A HEMISPHERE
IF (ALFA{N3) .EQ. 1.00 .AND. ALFA(N,4) .LT.1.00) GO TO 805
IF (ALFA(N,3) .EQ. 1.00 .AND. ALFA(N,4) .EQ.1.00) GO TO 810
IF (ALFA(N,3) .LT. 1.00 .AND. ALFA(N,4) .LT.1.00) GO TO 810
C CNE POINT LIES ON FIRST HEMISPHERE
IMID = 1
1806 CALL YBOUND
P3=BING /(ALFA(N,3)*(ALFA(N,3)+ALFA(N,4)))
811 P4 = PHH I,L4,K)/(ALFA(N,4)* { ALFA(N, 3 i+ALFAfN *4)))
C LOOK AT ADJACENT ZPOINT.
813 IF (ALFA(N,5) .EQ. 1.0) GO TO 820
IMID = 1
IF (X .GT. XIN IT(J,1) ) IMID = 2
IF (IMID .EQ. 1) GO TO 833
C Z POINT LIES ON SECOND HEMISPHERE
CALL ZBC2
BIM = BING
L3 = K1
IF (K .EQ. 1) L3=K+1
IF (ALFA(N,5) .EQ. 1.0) BIM = PHI(I,J,L3)
P5 = BIM/(ALFA(N,5)*(1.0 ALFAIN.5)))
821 P6 = PHKI,J,K+1)/(1.0+ALFA(N,5))
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
HYDRODYNAMIC LIFT IN SEDIMENT TRANSPORT
by
Barry Arden Benedict
December, 1968
Chairman: Dr. B. A. Christensen
Major Department: Civil Engineering
Hydrodynamic lift is a force often neglected in studies of
sediment movement, despite being of the same order of magnitude as
the drag force. The goal of this dissertation is to demonstrate that
potential flow theory can be used to predict hydrodynamic lift in
sediment transport. It is known that concentration of streamlines,
rather than viscous forces, contribute most to lift. This is rein
forced by many early airfoil works using potential flow theory, which
yield good values for lift, even though exhibiting zero drag.
The theoretical methods used treat mean, steady flows with no
free surface or sidewall effects. Since potential flow involves an
inviscid fluid, the method considered is only applicable to studying
cases of real flow in the hydrodynamically rough range, where viscous
forces are negligible. The two sets of experimental work studied
involve flows in the rough range over beds of hemispheres arranged in
hexagonal patterns. One work uses hemispheres three diameters apart
center to center; the other uses touching hemispheres.
xiii
129
0.0 0.2 0.4 0.6 0.8 1.0
x/a '
Figure 45. Closely packed hemispheres: velocities on y = 0.4a.
21
Chepil measured pressures over the surface of one metal
hemisphere. This was done by placing pressure taps, starting at the
base of the hemisphere, 30 degrees apart along one line running parallel
with and another normal to the wind direction. The negative pressure
end of the manometer was connected to a tap on top of the hemisphere.
The remainder of the hemisphere pattern consisted of gravel hemispheres.
The hemispheres occupied 11 per cent of the total floor area. The lift
and drag forces on the hemisphere were determined by integrating the
measured pressure distributions and also, as a check, by means of two
torsion balances measuring the forces directly. Hemispheres of three
different sizes were used, and also some measurements were made on
relatively small sand and gravel mounds. Measurements were also made
at different points downstream in the tunnel to note the effects as the
air boundary layer developed to its full extent.
Chepil found that increasing the depth of the fluid boundary
layer, after a certain limiting depth is reached, has little effect on
the lift to drag ratio, though the depth of boundary layer had a
profound effect on the magnitude of both lift and drag. For the study,
it was found that
LIFT = 0.85 Drag (216)
For this study it was also found that the pressure difference between
the top and the bottom of the hemisphere is about 2.85 times the lift
per unit bed area directly under the hemispheres. Using the latter
finding, Chepil surmised that the C from Einstein and El Samni [33]
should equal 0.178/2.85, or 0.0624. He then used equation (216)
LIST OF FIGURES (Continued)
Figure Page
41 Measured and theoretical pressure distributions:
= 159 cm/sec 120
42 Chepil's hemisphere 122
43 Closely packed hemispheres: velocities on y = 0 127
44 Closely packed hemispheres: velocities on y = 0.2a . 128
45 Closely packed hemispheres: velocities on y = 0.4a . 129
46 Closely packed hemispheres: velocities on y = 0.5a ... 130
47 Closely packed hemispheres: velocities on y = 0.6a ... 131
48 Closely packed hemispheres: velocities on y = 0.8a . 132
49 Closely packed hemispheres: velocities on x = 0 133
50 Closely packed hemispheres: trace of some equipotential
surfaces in plane y = 0 134
51 Closely packed hemispheres: trace of some equipotential
surfaces in plane y = 0.5a 135
52 Closely packed hemispheres: flow pattern on surface
viewed toward xyplane . 136
53 Closely packed hemispheres: flow pattern on surface
viewed toward yzplane 137
54 Velocity suppression near sidewall 140
*
ix
68
The next few sections will cover these cases as well as the grading of
the lattice. For clarity, the point cp ^ will be termed the object
point, while the six surrounding points will be called adjacent points.
The case for an adjacent point, lying on a regular lattice point or on
a hemispherical surface point where 9 is being calculated, will be
omitted, as these simply involve the substitution of the present value
of cp for that adjacent point. The same is true when an adjacent point
lies on one of the equipotential planes.
5.4.2 Object point on planar noflow boundary.The tern
planar is intended to exclude the hemispherical boundaries, which will
be treated in Section 5.5. Therefore, the planar surfaces indicated
are the constant ybounding planes and the constant zbounding planes,
as illustrated in Figures 18 and 19. If the object point lies on one
or two of such planes, one or more of the adjacent points will lie
outside the solution space, and it must be replaced by an equivalent
expression involving points on the interior of the space. For this
purpose, use is made of the fact that the derivative of cp normal to the
planes is zero. Then the value at such an exterior point can be
expressed as equaling the corresponding point inside the space. Fig
ure 23 illustrates what is meant. Here, as later, cp represents the
cpvalue at a surface point.
o o o
156
PHI{IJ,K) = 0.5*(PHI(I1,J,K) PHI(I+1,J,K))
6 CONTINUE
1000
FORMAT
( IX, 11F10.5 )
1001
FORMAT
(1115)
1002
FORMAT
(IF 10.3 )
1003
FORMAT
( 15,2F10.6,15)
1004
FORMAT
(6F10.6)
1005
FORMAT
(315)
1006
FORMAT
(1110)
1007
FORMAT
(7F10.5)
1008
FORMAT
(6F10.4/5F10.4)
1009
FORMAT
(11F10.5/10F10.5)
1010
FORMAT
(6H LOOP 13)
1011
FORMAT
(9H EPSILON F8.3)
1012
FORMAT(
16H POTENTIAL FIELO)
1013
FORMAT
(115)
1014
FORMAT
(5F10.6)
1015
2069
FORMAT(8F10.5)
RETURN
END
SUBROUTINE XLIMIT
C
C SUBROUTINE XLIMIT HAS AS ITS PURPOSE THE ESTABLISHING OF THE
C LOCATIONS OF THOSE POINTS WHERE THE RECTANGULAR LATTICE SYSTEM
C INTERSECTS THE HEMISPHERICAL SURFACES, BOTH IN TERMS OF AN X
C VALUE AND IN TERMS OF THE LATTICE NUMBER. THE VALUES ARE
C XINIT ANC XEND FOR THE FIRST AND SECOND HEMISPHERES
C RESPECTIVELY. THE LATTICE LOCATION FOLLOWS
C AS IBDRY1 ANC IBDRY2.
C
COMMON ALFAOt 21,21,2 ) ,ALFA7(21,21,2),ALFA02(21 ,21,2) ,
1 ALFA72(21,21,2),XEND{21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CCONl(21,21),CC0N2I21,21),THETA1(21,21),
3 THETA2(21,21),IBDRY1(21,21),IBDRY2(21,21),PZ(36,2i),
4 TI P(100),TOPI 11,11 ),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1S1AS2S2AS3S3A,BING,BIPB0GIG1IG2,IJK,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,NO(36,21,21)
COMMON DIV(2500),ALFA(2500,5)PHI(36*21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FL0
DO 300 K= 1KLOW
AK = K
Z = (AK1.0)*H
DO 300J=1*JMAX
AJ = J
Y(AJl.0)H
KEY = 0
DO 301 1=1,IMAX
1.0
I
u
Ut
or
()2
Ut
4
i

\ \
v u
ut
(JL)2 \
ut
i Flow
r
\j
0.0 0.2 0.4
0.6 0*8 1.0
x/a
Figure 43. Closely packed hemispheres: velocities on y 0.
133
Figure 494 Closely packed hemispheres: velocities on x = 0.
This dissertation was prepared under the direction of the
chairman of the candidates supervisory committee and has been
approved by all members of that committee. It was submitted to the
Dean of the College of Engineering and to the Graduate Council, and
was approved as partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
December, 1968
Dean, Graduate School
Supervisory Committee:
Chairman
1/
38
TI = y A
Figure 9. Evaluation of constant for sidewall analysis.
95
(570)
The means for calculating these three directional derivatives at the
surface have already been discussed in Section 5.5. Therefore, let it
simply be said that the surface velocities will be calculated using
the surface derivative expressions already available. The directional
components of velocity can be combined to yield the magnitude of the
velocity, q, at any point through (571)
(571)
Hence, the distribution of velocities over the surface can
be found. From these, it is desired to obtain the pressures acting
on the surface in accordance with these velocities. The general means
for all the relations needed herein were presented in a discussion of
earlier twodimensional works in Chapter IV. First, the system of
particles or elements treated will be considered as spheres whose lower
half is immersed in fluid, which is stagnant, being therefore subject
to be calculated flow only on the upper surfaces of the elements.
This situation is analogous to the case which is assumed as an approx
imation for the natural condition, with only the upper portion of the
grains exposed to flow.
To relate velocities to pressures, Bernoulli's principle will
be employed. This is developed for flow along a streamline by, among
many, Prandtl [70]. Irrotational flow is not assumed for such a deri
vation, and thus the surface velocities related to it need not be from
191
761 TAM = (PHI ( I J + 1,K) 4 PHI(ItJitK) +
1 PHI(I,J,K4l) 4 PHI(IfJ,Kl) 4
2 PHI(I+l,J,K)TAB2 4 PH I (I1,J,K)TAB3)/TAB 1
IF (ABS(TAMPHI(I,J,K ) J.GT.EPS) EPS=ABS(TAMPHI
PHI(IfJfK) = OMEGA*TAM 4 TAG*PHI(I,J,K)
GG TO 770
C SERIES 7620 FOR Y = 0, WHERE DPHI/DY = 0.
762 IF (I .EC. IMIT) GO TO 764
TAM = (PH I(I1fJK) 4 PHI(I+1,J,K) 4 PHI(I,J,K4l) 4
1 PHI{I,J,K1) 4 2.0*PHI(I,J+1,K))/6.0
IF (ABS(TAMPHI(ItJK)J.GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(IfJfK) = OMEGATAM 4 TAG*PHI(I,J,K)
TAM = (PHI(IltJtK) 4 PHI(I43,J,K) 4 PHI(1+1J,K+2)
1 PHI(I+ltJK2) + 2.0*PHI(I+lJ+2K))/6.0
IF (ABS(TAMPHI(I+lfJfK)J.GT.EPS) EPS=ABS(TAMPHI(1+1,J,Ki)
PHKI + lfJfK) = T AM*OMEGA + TAGPHI(I+1,J,K)
GO TO 770
C SERIES 7630 FOR Y=A, WHERE DPHI/DY = 0.
763 IF (I .EC. IMIT) GO TO 765
TAM = (PHKIlfJfK) 4 PH I(1+1J,K ) + PHI(I,J,K+1) +
1 PHI(I,J,K1) + 2.0PHI(I,J1K)J/6.0
IF (ABS(TAMPHI(I,JK)).GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(IfJfK) = OMEGATAM + TAG*PHI(IJtK)
TAM = (PHKI1,J,K) + PHI ( 1 + 3JK) 4 PHI(1 + 1JK+2) +
1 PHI(1+1,J,K2) + 2.0*PHI(I+1,J2,K))/6.0
IF (ABStTAKPHKI+l,J,K)J.GT.EPS) EPS=ABS(TAMPHI(1+1,J,K))
PH I(I +1 J K) = TAM*OMEGA + TAG*PHI(1 + 1,J,K)
GO TO 770
C SERIES 764 AND 765 ARE FOR POINTS WHERE BOTH DPHI/DY = 0 AND
C THE UNEGUAL XSPACING OCCURS.
764 TAM = (PHK11,J,K)*TAB3 + PH I (1 + 1,J,K)*TAB2 +
1 PHI(IJ,K + 1) + PHI(I,d K1) + 2.0*PHI(I,J+1,K)J/TAB1
IF (ABS(TAMPHI{IJK)J.GT.EPS) EPS = ABS(TAMPHI(I,J,K))
PH I (I,J,K) = OMEGATAM + TAG*PHI(I,J,K)
GO TO 770
765 TAM = (PHKI1,J,K)*TAB3 + PHI(1+1,J,K)*TAB2 +PHI(I,J,K+1)+
1 PHI(IJ,Kl) + 2.0PHI(I,Jl,K))/6.0
IF (ABS(TAMPHI(I,J,KJJ.GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI(IfJiK) = OMEGATAM + TAG+PHI(IJK)
770 CONTINUE
C INTERMEDIATE ROW BETWEEN NETWORK OF SIZE H AND THAT OF SIZE 2H
K= KGRADE + 1
C THROUGH 780, TRAVERSE ODD JVALUES, ALL XVALUES.
CO 780 J=1JMAX,2
DO 780 1=2,IMIT
IF (J .EQ. 1 .OR. J .EQ. JMAX) GO TO 772
IF (I .EQ. IMIT) GO TO 771
C DIFFERENCE EQUATION BASED ON INVARIANCE OF LAPLACIAN WITH AXIS
C ROTATION.
TAM = (2.0*(PHI(I,J1,K) + PHI(I,J+l,Kn + PHI(1+1JKI)
93
Other equations could be formed and would hold in the special
situations discussed earlier for which other means were used to obtain
a derivative. These will not be cataloged here, however.
These equations above thus complete the considerations of the
finite difference equations for the solutions herein. The next step
will involve the use of the final solution in calculating surface
velocities.
5.5.7 Singular points.Many of the preceding equations are
necessary only for the case of closely packed hemispheres, as certain
situations do not arise in the Chepil arrangement. Each case has a
singularity at the stagnation point, where the velocity equals zero.
This point presents no real problem for the differences solution, as
the value of the velocity is still finite.
In the closely packed hemispheres, however, three other singular
points exist within the solution space. These are the points where the
hemispheres touch. Mathematically, the velocities would approach infin
ite values at these points because of the convergence of streamlines
at those points. This presents no problem as to the potential value
at these points, for this is known already. There are, however, two
considerations which must be made. First, for evaluating the lift on
the hemisphere, a more meaningful velocity at the base can be found by
extrapolating to a finite value from the distribution over the rest of
the surface. This avoids the obviously incorrect negative infinity
pressure predicted mathematically at the singular points.
57
differences solution, however, this was not done. Two alternative
approaches were available, but neither seemed as practical as simply
including two hemispherical portions in the solution space. One of
these alternatives was assuming an equipotential surface between the
two hemispherical portions and making finite differences solutions in
which this assumed potential surface was to be checked and itself
adjusted and treated as a variable in a solution. However, this would
have involved a complete solution of the problem for each assumed sur
face and would have required, therefore, exorbitant computation time.
The other alternative was to choose a plane between the two hemispheres
and use it along with conditions of "symmetry" to reduce by onehalf
the number of points involved in the differences solution. In the case
of Figure 18, the proximity of the hemispheres introduces complexities
which offset the benefits gained in computation time. For the Chepil
case, however, the latter approach could be used easily as indicated
in Figure 19.
The constant xplanes passing through the hemispheres are
equipotential surfaces. All other boundaries in the solution spaces
represent surfaces across which there is no flow, with the exception
of the added symmetry boundary shown in Figure 19. This means that the
derivative of the potential function normal to the given surface is
equal to zero. Thus, the following conditions hold.
On constant yplanes through hemispheres:
^=0 (56)
By
On upper zsurface:
(57)
Figure 16. Definition sketch for experimental application.
LIST OF FIGURES
Figure Page
1 Flow around a Joukowsky profile 10
2 Calculated and measured pressure distribution around
a Joukowsky profile . . 10
3 Pressure distribution around a Fuhrmann body 11
4 Jeffreys' cylinder 12
5 Pressures on a single sphere 17
6 Arrangement of spheres and theoretical bed in
EinsteinEl Samni work 28
7 Arrangement of spheres and theoretical bed in
Chepil's work 30
8 Sketches for sidewall effect ........ 36
9 Evaluation of constant for sidewall analysis 38
10 Grains placed in rough bed configuration 42
11 Pressure distribution on twodimensional grain 43
12 Piezometric head distribution on twodimensional grain 43
13 Distributions of surface hydrodynamic pressure decreases. 45
14 Lift coefficient C . 46
Li
15 Theoretical bed used in relating logarithmic and
potential velocity profiles 48
16 Definition sketch for experimental application ..... 50
17 Comparison of theoretical and measured values 51
18 Solution space for closely packed hemispheres 58
19 Solution space for Chepil's arrangement .... 59
vii
90
Sphere 1
PI nearer A:
PI nearer C:
(564)
(565)
(566)
(567)
The equations above can be used to express each of the desired
four interpolated values and hence allow computation of dcp/dx and
Scp/dy from an equation which is the same as (553). The derivatives
thus obtained can therefore be utilized in the equations of Section
5.5.1 just as indicated in the following section.
71
and the interpolation is made by the arc length between the three
points. These lengths along the arc are proportional to the angles
shown in Figure 24, and the ratio of lengths can be replaced by the
ratio, of the angles, written in (528).
Constant zplane
Figure 24. Arc interpolation for surface value.
1
= ^1 1,J,K + i+2 fal.Jl.K ^IljJ,!^
(528)
A similar expression can be written for the point on the other
hemisphere, and the two values are then inserted into the general
sevenpoint equation.
It would have been possible to use the difference expressions
at these ypoints. It was felt, however, that in the regions where
both points lie on a hemisphere, the accuracy of the arc interpolation
202
13. Bagnold, R. A. "The Movement of Desert Sand/ Proceedings, Royal
Society of London, Series A, No. 892, vol. 157, 1936, pp. 594
620.
14. Velikanow, M. A. "Dynamics of Alluvial Streams, vol. II,"
(SolidsTransport and the Riverbed), Moscow, 1956.
15. Goldstein, S. (editor). Modern Developments in Fluid Dynamics.
New York: Dover Publications, Inc., 1965.
16. von Karman, Theodore. Aerodynamics: Selected Topics in the Light
of Their Historical Development. Ithaca, New York: Cornell
University Press, 1954, 203 pp.
17. Prandtl, L., and Tietjens, 0. G. Applied Hydro and Aeromechanics
New York: Dover Publications, Inc., 1957, 311 pp.
18. Kutta, W. "Lift Forces in Flowing Fluids," Ill. Aeronaut. Mitt.,
1902.
19. Joukowsky, N. "On the Profiles of Airfoils," Zeitschrift fur
Flugtechnische und Motorluftschiffahrt, vol. 1, 1910, p. 281;
and vol. 3, 1912, p. 81. (German)
20. Joukowsky, N. "Aerodynamics," Paris, 1916, p. 145. (French)
21. Blumenthal, 0. "On the Pressure Distribution along Joukowsky
Profiles," Zeitschrift fur Flugtechnische und Motorluftschif
fahrt, vol. 4, 1913, p. 125. (German)
22. Betz, A. "Investigation of a Joukowsky Airfoil," Zeitschrift fur
Flugtechnische und Motorluftschiffahrt, vol. 6, 1915, pp. 173
179. (German)
23. Prandtl, L., and Tietjens, 0. G. Fundamentals of Hydro and
Aeromechanics, New York: Dover Publications, Inc., 1957, 311 pp
24. Fuhrmann, G. "Theoretical and Experimental Investigations on
Balloon Models," Dissertation, Gottingen, 1912. (German)
25. Rodgers, E. J. "Vorticity Generation of a Body of Revolution at
an Angle of Attack," Transactions, American Society of Mechan
ical Engineering, Journal of Basic Engineering, vol. 86,
Series DNo. 4, December 1964, pp. 845850.
26. Lamb, Horace. Hydrodynamics. New York: Dover Publications, Inc.,
1945, 738 pp.
27. MilneThomson, L. M. Theoretical Hydrodynamics, 4th edition.
New York: The Macmillan Company, 1958.
106
0.0 0.2 0.4 0.6 0.8 1.0
x/a
Figure 34. Chepil's case: velocities on y 0.6a.
186
375 PK1 = PH I (If J t K+2) (ALFA(N,3 I/ALFA(NB#3 ) I*
1 (PHIU, J,K+2)PBl)
IF (MULE .EQ. 1) GO TO 1372
SI = (X)*(2.0*PI PI1/2.0)
S1A = (3.0)*X/2.0
1372 S3 = (20*PK PK1/2 .0 I*(1.0 I*Z
S3A = (3.0)*Z/2.0
GO TO 378
C FOR POINTS LOCATED NEAR Y = A, WHERE THE SECOND POINT (IN THE
C YDIRECTION) LIES ON ANOTHER HEMISPHERE OUTSIDE THE SOLUTION
C SPACE.
1376 S2 = (1.0)*(YALFA(N3)*H)*PHI(I,JK)*2.0/ALFA(N3)
S2A = (1.0)*(YALFA(N,3)*H)*2.0/ALFA(N,3I
GO TO 1377
C CNE POINT AVAILABLE IN XDIRECTION.
373 SI = PIX*(1.0)
S1A = (1.0)*X
MULE = 1
GO TO 372
C ARC INTERPOLATION FOR NEXT XPOINT.
374 XT = SQRT(3.00) SQRTd.O (YALFA(N,3)*Hl.0)**2 Z**2)
17 = IBDRY2{J1 K I
0 = 1.0 Y + H
E = SORT(3.00 ) XENC(JK)
G = 1.0 Y
B = SORT(3.00) XENCtJ~lfKJ
A = SORT(3.00) XT
C = 1.0Y+ALFA(N,3J*F
THM = (ATAN2(C,A) ATAN2(G E11/(ATAN21DB) ATAN2(GE)I
PI = PHI(I+1,J,K) + THMMPHI ( I7,J1,K) PHI(I+1,J,K1)
XDEL = (XT Xl/H
51 = (XI*PI/XDEL
S1A = (XI/XDEL
MULE = 1
GO TO 372
C SECTION FOR POINTS ON BOUNDARY OF SECOND HEMISPHERE.
C COMMENTS FOR STEPS ON SECOND HEMISPHERE ARE SIMILAR TO THOSE
C INDICATED ALREADY FOR EARLIER STEPS FOR SPHERE 1.
379 IF (J .EG. 1) GO TO 1379
52 = (1.0)*(Y+ALFA(N4)*H 1.01*(PH I(I,Jl,KJ*
1 (<2.0*ALFA(N,4)l.C)/(1 .O+ALFA(N4)1 I 2.0*PHI(I,J,K11
S2A = (Y+ALFA(N,4I*H1.0J*3.0/(1.0+ALFA(N4II
1378 IF (((X XINIT(J,K) l/H) .LT. 1.0) GO TO 364
PB = PHI( I1*J + lfK)
NT = NO( I1,J,K)
IF (NT.EC. 0) NT = MARK
IF (ALFA(NT4I .LT. 1.0) PB = PY1(I1,K)
PI = PHI(I1,JK) (ALFA(N,4)/ALFA(NT,4)I*
1 (PB PHI(IltJK)l
IF (((X XINIT(J,K) l/H) .LT. 2.0) GO TO 363
PB1 = PHI(12*J+1#K)
64
9
o
0)
6
(1 cu) cp'
O
(515)
where C{/ indicates the value from the preceding iteration. Obviously,
as in equation (513), the value of cpQ from (515) will show little
change as convergence is neared.
If the value of cu is equal to one, the method is called the
Liebmann method, while the name extrapolated Liebmann is applied for
cases where U) is not equal to unity. The latter case will be employed
here. Much work has been done, at least in twodimensional cases, in
finding an optimum cu to give most rapid convergence [59]. Approximate
indications are that the optimum covalue for the cases herein, lies
below about 1.80. However, no great amount of work will be done to
refine this value, as such work could easily involve very extensive
t ime.
The solutions of this dissertation will therefore be carried
out using a sevenpoint finite difference scheme in rectangular Cartesian
coordinates, applying the extrapolated Liebmann iterative method. The
ensuing sections will develop the needed relationships for use in
I
special situations.
5.4 Finite differences equations: interior space
This section will present the equations needed to handle all
the different situations which arise in the solutions. While many
applications of finite differences have been made in two 'dimensions,
few cases other than problems involving simple cubes, and the like,
0.1 0.2 0.5 1.0 5.0 b/a
Figure 14. Lift coefficient C .
F
o\
o o o
181
C PREVENTS GETTING EFFECTIVE YDIRECTION VALUE.
663 16 = IBDRY1 (J K)
17 = IBCRY1(J K1)
A = ZALFA(N,5)*H
8 = Z~H
THM = (ATAN2(Z,XINIT(J,K)) ATAN2(A,X))/
1(ATAN2(ZXINIT(JfK)) ATAN2(B,XINIT(JKl)))
TAM = PHI (16,J,K) + THM*(PHI(I7,J,K1) PHIU6,J,K))
PZ(ItJ) = TAMOMEGA TAG*PZ(I,J)
GO TO 669
611 PI = TUR
MUT = 2
GO TO 605
612 PI1 = TUM
MUT = 3
GO TO 605
613 PJ = TUM
MUT = A
GO TO 605
614 PJ1 = TUM
620 S2 = (Y)(2.0*PJ PJl/2.0)
S2A = (3.0)*Y/2O
622 IF (MULE .EQ. 1) GO TO 621
SI = (X)*(2.0*PI P11/2.0)
S1A = (~3.0)*X/2O
621 IF (NCAS .EQ. 3) GO TO 669
SUMI = SI + S2 + S3
SUM2 = S1A + S2A + S3A
C FINAL CALCULATION OF NEW PHI VALUE FOR BOUNDARY SURFACE POINT.
623 PZ(I,J> = (SUM1/SUM2)OMEGA TAG*PZ(I,J)
669 RETURN
END
SUBROUTINE ZB02
C
C SUBROUTINE ZB02 HAS THE SAME FUNCTION AS SUBROUTINE ZBOUND,
C EXCEPT THAT IT TREATS THOSE POINTS ENCOUNTERED ON THE SECOND
C COMMENTS FOR SUBROUTINE ZB02 ARE ALL VERY SIMILAR TO THOSE FOR
C SUBROUTINE ZBCUNO. FOR THIS REASON, REFER TO THAT SUBROUTINE
C WHERE CLARIFICATION IS NEEDED. THE STATEMENT NUMBERS ARE
C GENERALLY SUCH THAT THE NUMBERS OF ZB02 ARE THE SAME AS THOSE
C OF ZBOUND WITH AN INITIAL 1 ADDED.
C HEMISPHERE.
C
COMMON ALFA0(21,2i,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21t21,2),XEND(21,21)XINIT(2121)MIGl(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(2121),THETA 1(21,21) ,
3 THETA221,21),IBORYll21,21),IBDRY2(21,21),PZ(36,21),
4 TIP(IOO),TOP(11,11),H,EPS,IMAX,JMAXKMAX,KLOW,KGRADE,
157
AI = I
X=(AI1.0)*H
IF (KEY.EQ.l) GO TO 302
IF (J .EQ. JMAX .ANO. K
JIPI = 15
JIP2 = 15 v
IF UMAX .EQ. 21) JIPI =
(JMAX .EQ. 21) JIP2 *
(JMAX .EQ. 11) JIPI =
(JMAX .EQ. 11) JIP2 =
(J .EG. JIP2 .AND. K
(J .EQ. JIPI .AND. K
IF
IF
IF
IF
IF
IF (J .EG. 1 .AND.
SEARCH FOR SURFACE OF
THE VALUE FOR IBDRY1,
.EQ. 1) GO TO 303
13
17
7
9
.EQ.
.EQ.
JIPI)
JIP2)
GO
GO
K .EQ. KLOW) GO TO
SPHERE. WHEN SUMR
THE INDEX FOR THE
TO 303
TO 303
303
EXCEEDS
BOUNDARY
1 THEN SET
POINT IN THE
THE HEMISPHERE
THIS VALUE IS
C
C
C XDIRECTION, AND CALCULATE THE VALUE OF X ON
C WHICH CORRESPONDS TO THAT VALUE OF Y AND Z.
C CALLED X INIT(J.K ) .
SUMR = X**2.0 + Y**2.0 + Z**2.0
IF (SUMR.LT.1.0) GO TO 301
IF( SUMR ,GE. 1.00 .AND. I .EQ. 1) GO TO 303
IF (SUMR.GT. 1.0) IBDRYKJ.K) = 11
IF (SUMR.EQ.1.0) IBDRY1(JK) = I
IF ((Y2.0 + Z**2 0) .GT. 1.0) WRITE (6,1001) SUMR,X,Y,Z,H
XINIT(JfK) = SQRTd.O Y**2.0 Z**2.0)
IF (NCAS .EQ. 2) I=IMAX
IF (NCAS .EQ. 2) GO TO 301
KEY = 1
GO TO 302
303 IBDRYKJ.K) = 1
XINIT(JfK) = 0.00
KEY a 1
C POINTS ON SURFACE OF SECOND HEMISPHERE. SUMR1 SERVES THE SAME
C ROLE AS SUMR DID FOR THE FIRST HEMISPHERE. XEND(J,K) IS THE
C VALUE OF X ON THE SECOND HEMISPHERE CORRESPONDING TO THE Y
C AND Z REPRESENTED BY J AND K. IBDRY2(J,K) IS THE SUBSCRIPT
C INDEX FCR THIS SAME POINT.
302 SUMR 1 = (XSQRK3.0) )**2 + (Y1.0)**2 + Z**2.0
IF {SUMR1 .GT. 1.00 .AND. I .EQ. IMAX) GO TO 304
IF (SUMR1.GT.1.0) GO TO 301
IF (SUMR1.LE.1.0) IBDRY2(J,K ) = I
XEND(J.K) = SQRT ( 3.0) SQRTd.O (Yl.0)**2 Z**2.0)
GO TG 300
304 IBDRY2(J,K) = IMAX
XEND(J.K) = SQRT(3.00 )
301 CONTINUE
IF (IBDRYKJ.K) .EQ. IBDRY2U.K)) IBDRY2(J,K) =
1 IBDRYKJ.K) +.1
300 CONTINUE
IF (JMAX .GT. 7) GO TO 369
C SPECIAL SECTION USED ONLY WITH A/6 RUN.
non
169
C GNE Y POINT CN SECOND HEMISPHERE
805 P3 = PH I ( I,L5 ,K)/(ALFA(N,3)*(ALFA(N,3) + ALFA(N4)))
IMID = 2
1805 CALL YBCUND
P4 = BING /(ALFA(N,4)(ALFA(N,3)+ALFA(N,4)))
GO TC 813
C EVALUATION OF NEW PHI VALUE AT THE GIVEN POINT.
825 TAM = ( PI + P2 + P3 P4 + P5 + P6 )
IF (KEG .EQ. 0) DIP = DIV(N)
TAM = TAM/DIP
IF (ABS(TAMPHI(I,J,KM.GT.EPS) EPS=ABS(TAMPHI(I,J,K))
PHI (I,J,K) = 0MEGA*TAM TAG*PHI(I,J,K)
GO TG 801
850 CALL BCCER
801 CONTINUE
800 CONTINUE
RETURN
END
C
C
C
C
C
C
C
C
C
C
SUBROUTINE BDCER
SPECIAL SECTION FOR THOSE POINTS ON THE BOUNDARIES OF THE
TWO HEMISPHERES, WHERE THE BOUNDARY CONDITION OF A ZERO
NORMAL DERIVATIVE MUST BE SATISFIED. THE DERIVATIVES IN ALL
THREE (X,Y, AND Z) DIRECTIONS ARE COMPUTED AND THEN
COMBINED TO FORM AN EXPRESSION FOR THE VALUE OF PHI
CN THE BOUNDARY POINT.
COMMON ALFAO(21,212),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XIN IT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA1(21,21),
3 THETA2121,21),I BORY1(21,21),IBDRY2(21,21),PZ(36,21),
4 TIP(100),TOP(11,11),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1,IG2,1,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA KIP,KEY,NO(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
COMMON IMIG(2i),PIG,KIN,KINl,FLO
EINSTEINEL SAMNI ARRANGEMENT
JMIT = JMAX l
KL0W1 = KLOW 1
PREPARE FOR CERTAIN
JIPI = 15
IF (JMAX
IF (JMAX
JIP2 = 15
IF (JMAX
IF UMAX
IF (NCAS
POINTS FALLING EXACTLY ON HEMISPHERE
EQ.
11)
JIPI
= 7
EQ.
21)
JIPI
= 13
s
EQ.
11)
JIP2
= 9
EQ.
21)
JIP2
= 17
EQ.
3)
IG2 =
IMAX
35
3.5 Effect of sidewalls
As will be seen later, the experimental flume to be studied is
rather narrow. For this reason, possible effects of the sidewalls on
the velocity will be discussed here.
Sayre and Albertson mentioned this problem, when, in reference
to their analysis, they indicated that one assumption which they made
was . that the channel is of sufficient width, or that the bed
roughness is so great relative to the sidewall effect" [53, p. 124].
Rouse [54, pp. 276277] points out the effect that sidewalls have,
through secondary flows, in varying the isovels and in depressing the
region of maximum velocity below the surface of the flow.
In order to provide some quantitative means of evaluating the
sidewall effect, aside from the qualitative evaluation of experimen
tally obtained velocity profiles and isovels, the following approx
imate analysis is presented. Figure 8 indicates the assumptions util
ized listed below.
Assumptions:
1. Idealized isovel picture, enabling passing of
line through corners.
2. Idealized pressure distribution (varying linearly,
as indicated in Figure 8).
3. proportional to y^ in some way; proportional
to y in the same way.
4. Logarithmic velocity distribution in center line
profile.
19
At this point it seems appropriate to define certain lift
coefficients to be employed in this dissertation. The difference
lies in the area over which the force is considered. The subscript,
u, will denote those cases where the area considered is only that
directly beneath the body being considered, the projected area of the
grain. Thus, \ denotes lift per unit area based on the total bed
area, while is based on the area of the grain projected onto the
bed. Similarly, the coefficients and C are used with the corre
sponding ^*s.
As indicated, most work done on lift has dealt with single
bodies. Attention will now be turned to systems with more than one
particle.
2.5 Multiparticle studies
2.5.1 Einstein and El Samni.It was natural that multi
particle studies should arise, as these begin to approach the sediment
conditions found in nature. Unfortunately, however, work in this area
has been limited. Some values for the lift force came from the work
by Einstein and El Samni [33,34]. Using the upper onehalf of
plastic spherical balls 0.225 feet in diameter placed in a hexagonal
pattern, they measured the lift force as a pressure difference. They
made the following statement.
The procedure in making such measurements was as follows:
if a lift force is exerted on the top layer of a stream bed,
the solid support of the sediment particles is relieved of part
of their load and this load is transmitted hydrostatically to
the fluid between the solid bed particles. Thus, it must be
possible to detect and measure this lift as a general pressure
increase of the pore fluid in the bed. [33, p. 52l]
REFERENCES
1. Brown, Carl B. "Sediment Transportation," Chapter XII,
Engineering Hydraulics, ed. Hunter Rotise. New York: John
Wiley and Sons, Inc., 1950, 1039 pp.
2. Langbein, Walter B., and Leopold, Luna B. "A Primer on Water,"
United States Geological Survey. Washington, D.C.: Government
Printing Office, 1960, 50 pp.
' 3. Rouse, Hunter., and Ince, Simon. History'of Hydraulics. New York
Dover Publications, Inc., 1963, 269 pp.
4. Leliavsky, Serge. An Introduction to Fluvial Hydraulics. New
York: Dover Publications Inc., 1966, 257 pp.
5. White, C. M. "The Equilibrium of Grains on the Bed of a Stream,"
Proceedings of the Royal Society of London, Series A, vol. 174
1940, pp. 322338.
6. Young, Donald F. "Drag and Lift on Spheres within Cylindrical
Tubes," Journal of the Hydraulics Division, Proceedings, ASCE,
vol. 86, No. HY6, June 1960, pp. 4757.
7. Lane, E. W. "Design of Stable Channels," Transactions, American
Society of Civil Engineers, vol. 120, 1955, p. 1234.
8. Jeffreys, Harold. "On the Transport of Sediment by Streams,"
Proceedings of,the Cambridge Philosophical Society, vol. 25, .
1929, p. 272.
9. Fage, A. Philosophical Magazine, vol. 7, No. 21, 1936, p. 80.
10. Chang, Y. L. "Laboratory Investigation of Flume Traction and
Transportation," Transactions, ASCE, vol. 104, 1939, pp. 1246
1313.
11. Yalin, M. Selim. "An Expression for BedLoad Transportation,"
Journal of the Hydraulics Division, Proceedings, ASCE, vol. 89
No. HY3, May 1962, pp. 221250.
12. Einstein, H. A. "Formulas for the Transport of Bed Load,"
Paper 2140, Transactions, ASCE, vol. 107, 1942.
201
34
Combination of these equations and elimination of v^ yield
k =
29.7 pd
<Â£>
(v )/(v V )
p p q
(314)
Elimination of k yields
v
f
v v
p q
2.5 In (Â£)
q
(315)
Frequently the depths chosen are for p = 0.90 and q = 0.15.
Use of the proposed expression in (39) for equations such as
(312) and (313) above, does not lend itself as readily to direct
elimination of variables and solution for k and v^. Therefore, some
consideration must be given to the relative values of the terms.
As noted earlier, for terms where y/k is greater than one, the effect
of the added 0.0338 is less than 1 per cent and can presumably be
neglected. If indeed p = 0.90 and q = 0.15 were used, for most
practical cases y/k would be far greater than one for both depths,
and the expressions for k and v^ would be precisely those found in
(314) and (315).
In any case, the mechanism is available for computation of k
and Vg through observations. These parameters can then be used in
computing forces, velocities, and the like according to equations
to be subsequently derived.
92
other values from the field. Therefore, replacement of 3cp/dx, dcp/dy,
and &p/Sz by their difference forms enables a subsequent equation for
cd The latter equation becomes the one to be employed at that point
to impose the noflow condition to the solution. There are numerous
expressions for the three derivatives, developed in Sections 5.5.1
through 5.5.4. It is not intended here to list all possible combina
tions, but rather to illustrate by defining the equation for the cases
where all points such as PJ are available. Using these points with
equations (549), (553), and (563) and cancelling a common h, enables
writing from (551), after rearranging, the following:
r ^i i
X12CPI + 1,J,K 1+0^ CPI+2,J,KJ
+ y{2PJ + z{2PK (568)
Similarly, on the second sphere, use of equations (550),
(554), and (563) allows an expression for co on the second
D
hemisphere.
Sphere 2:
2or2l
1,J,K T^T
+
12 ,J
(569)
96
an irrotational flow. In this case they are, but the purpose is to
discover if such predicted velocities form a good representation of the
actual. If they do, then since rotational flow is allowed in this form
of Bernoulli's theorem, the pressures thus predicted would be similarly
representative. The equation follows.
2
z +Â£ + Â§= constant (572)
V 2g
where p = pressure
z = elevation
Notice that this constant is a constant only for the given
streamline and would differ from one streamline to another. However,
in this case, each streamline on the hemispherical surface will pass
through the stagnation point on the hemisphere. This is located at
coordinates (1, 0, 0) on Sphere 1.
It can be seen that at a stagnation point, with z = 0 and
q = 0, the constant of (572) becomes the static pressure at that
point, pQ. This, therefore, is the constant to be used in employing
(572) to determine surface pressure. This simply means that all sur
face pressures will be found relative to pQ. This static pressure
acting alone would give rise to a lift known as the buoyant, or hydro
static lift. The addition of a flow will produce the added lift
called hydrodynamic lift, which is of interest here. Figure 30 illus
trates an elemental surface area on the hemisphere.
CHAPTER V
THREEDIMENSIONAL NUMERICAL SOLUTIONS
5.1 General
i
Previously it was indicated that irrotational flow theory would
be utilized to study velocities on a threedimensional bed surface
toward the end of making an analytical evaluation of hydrodynamic lift
on such a surface. The results of some previous similar twodimensional
studies were presented in Chapter IV. In this dissertation emphasis is
placed on a comparison of the analytical results obtained by the writer
with those obtained by Einstein and El Samni [33,34] and Chepil [36].
Evaluation of lift in these cases by the proposed method will require
solution of the potential flow equation for the two arrangements of
hemispheres as indicated in Figures 6 and 7.
A potentil flow solution implies the solution of Laplace's
equation (51) in the given flow space.
I = 0 (51)
cbc dy dz
Laplace's equation is of elliptic type and hence its solution
is fully determined by conditions on the boundary enclosing the solu
tion space. On the boundary must be specified either values of the
potential function, 9 or its derivative normal to the boundary.
52
NOTES ON EQUIVALENT GRAIN SIZE
Presented here is a brief derivation of equation (610) of
the text, subject to the conditions stated there. The following addi
tional symbols are used.
t
Yg = unit weight of soil particles
G = buoyant weight of particles
A^ = exposed equivalent area
A = exposed natural area
n
First, calculate the exposed area as the product of the area
for one grain and the number of grains. The latter i obtained by
dividing G, the total buoyant weight, by the buoyant toeight of one
particle.
A IT 2 G 3 G
A = d
e 4e tt ,3, 2 (y v)d
TT d (y y) Ts e
6 e Ts Y
A^ must be calculated from the grain size distribution curve*
o G dy
1A tt 2 as
n 4 d tT~3
6d '8
1
6 d (v8y)
A = f1 I
n 2 YSY o d
Equating A^ and A^ yields (610) in the text.
200
148
All the analytical and experimental work mentioned has a
common goalthe development of improved design methods in sediment
transport. It is hoped that the use of potential theory to study
hydrodynamic lift, advocated and shown valid in this dissertation,
will aid in reaching these ultimate design goals. .
i
\
\
LIST OF FIGURES (Continued)
Figure Page
20 Foldedsymmetry boundary 60
21 Sevenpoint finite difference scheme .... 62
22 General lattice point 65
23 Examples for object point on planar boundary 69
24 Arc interpolation for surface value 71
25 Grading the lattice 73
26 Xderivative condition at boundary 79
27 Yderivatives 82
28 Special points for yderivative 86
29 Zpoint for normal derivative condition 89
30 Pressures on area of hemisphere 97
31 Chepil's case: velocities on y = 0 103
32 Chepil's case: velocities on y = 0.2a 104
33 Chepils case: velocities on y = 0.4a 105
34 Chepil's case: velocities on y = 0.6a 106
35 Chepil's case: velocities on y = 0.8a 107
36 Chepil's case: velocities on x = 0 108
37 Chepil's case: trace of some equipotential surfaces
in plane y = 0 109
38 Measured and theoretical pressure distributions:
v^ = 68 cm/sec 117
39 Measured and theoretical pressure distributions:
v^ = 91 cra/sec 118
40 Measured and theoretical pressure distributions:
v^ = 128 cm/sec 119
viii
116
The decrease in r^, especially for those values from (66),
indicates a very definite contribution from the substantial roughness
of the gravel used for the bed. This same gravel was used to form the
other bodies in the hexagonal pattern. As the size of hemisphere
grows larger, the gravel plays less and less of a role in determining
the flow.
6.3,2 Comparison of lift forces.Values of lift will be com
puted from equation (576), using Chepil's velocity distribution and
the lift coefficient from Section 6.3.1.
These values will be reported later. First, comparison will be
made with the pressure distributions that Chepil gives which are for
only the 2.54 cm hemisphere. The four figures on ensuing pages will
show his measured distributions, this author's predicted distributions,
and distributions corresponding to an average of front and back pres
sures for Chepil. The predicted distributions follow the numerical
results shown earlier in this chapter. The maximum pressure differ
1 2
ence is taken for this prediction as p u^, where u^_ is evaluated
from (62).
The values show some discrepancies, especially for the lower
pressure runs.. However, the agreement seems good for the case of
highest v^. Therefore, it was decided to compute lift on the hemi
spheres by the same means used by Chepil and compare those values with
those computed by Chepil from his measurements.
It is interesting to note that in all four instances shown,
Chepil's measurements yield a higher pressure at thirty degrees up
0.2
reference pressure
0.4
0.6 0.8 1.0
Bed Location
u : mean velocity
m
Figure 17. Comparison of theoretical and measured values.
182
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1*IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP#BETA,KIP,KEY,N0(36,21,21)
COMMON CIV(2500)ALFA(25005)PHI(36,2132)
COWMEN IMIG(21)PIG,KINKINlFLO
DIMENSION PZ1(37,21)
INTEGER S
C EINSTEINEL SAMNI ARRANGEMENT
JMIT = JMAX 1
MULE =0
IF lKEY2 .EQ. 1) GO TO 640
DO 677 II = ltIMAX
All = II
XI = (All1.0)*H
IF (II .EG. IMAX) XI = SORT(3.00)
DO 677 J1 = 1,JMAX
677 PZl(IltJl) = 100.0 (100.O/SQRT(3.00))*X1
KEY2 =1
640 Z1 = Z ALFA(N 5)*H
S3 = lZl)*(2.0*PHI(I,J,K) + (1.02.0*ALFA(N5))*
1 PHI(I,J,K + l )/(1.0 + ALFA(N,5)))
S3A = (Z1)*(3.0/(1.C+ALFA(N,5)))
MUT = 1
1605 GO TO (1601,1602,1603,1604),MUT
1610 G = ALFA(NT,5 ) ALFA(N,5)
PC = PH I (L1L2K1)
IF (ALFA(NT,5) .LT. 1.0) PC = PZ1(L1,L2)
IF (ALFA(N,5) .LT. G) GO TO 1606
GAG = 1.0 G/ALFA(N,5)
GIG = PC + G*(PHI(L1L2K) PC)/ALFA(NT,5)
GUF = (G**2)*(PHI(L1,L2,K)/ALFA(N,5) + PC/G)/ALFA(NT,5)
GO TO 1607
1606 GAG = 1.0 ALFA(N,5)/G
GIG = PHI (LI,L2K) ALFA(N,5)*(PHItLI,L2,K)PC)/ALFA(NT,5)
GUF = (ALFA(N,5)2)(PHI(L1,L2,K)/ALFA(N,5)
1 PC/G)/ALFA(NT,5)
1607 TUM = (GIG GUFJ/GAG
GO TO (1611,1612,1613,1614),MUT
1601 NT = N0(I1,J,K)
IF (NT .EQ. 0) NT = MARK
IF (((X XINIT(J,K1))/H) .LT. 1.0) GO TO 1673
LI = 11
L2 = J
GO TO 1610
1602 NT = NO(I~2,J,K)
IF (NT .EG. 0) NT = MARK
IF (((X XINIT(J,K1))/H) .LT. 2.0) GO TO 1674
LI = 12
L2 = J
GO TO 1610
1603 NT NG(I,JiK)
Main Stream Flow
JSJ
Figure 10.
Grains placed in rough bed configuration.
p
fO
124
Replacement of the pressure differences in (68) by their
values from the numerical results will result in CT = 0.348. This
value is somewhat lower than the 0.405 found by integrating over the
entire surface. Computation of X^ using each C^u is included in
Table 5. The value of u^ is also chosen for this table in two ways.
First, Chepils values for yQ are chosen, and then, for comparison,
values based on (66) are used. Therefore, four X values based on
u
theoretical results are included.
A number of statements may be made concerning the results in
Table 5. First, there is not too good agreement for the smaller sizes,
with u^ based on Chepil's yQ. However, for the last three values in
the table, the agreement is much better, being high by 19, 13, and
8 per cent. It is for this largest hemisphere that the theoretical
model is most closely simulated. The differences between the two cases
lie in the gravel floor and gravel hemispheres in Chepil's work versus
a smooth floor and identical hemispheres in the theory.
One possible source of error exists in v^, the friction veloc
ity. Chepil obtained this by extrapolating his velocity curves and
taking the slope of the uln y plot. This is related to possible
error in y found by extrapolating velocity measurements to zero
velocity. Such extrapolations from elevations, where Chepil apparently
ended his velocity measurements, could have been faulty without affect
ing his work. Any change of y however, can be seen to cause quite
a change in the value of X^ through indicating a different ut
6
lift is often neglected is in stable channel design, where the widely
used method of E. W. Lane [7], as well as other methods, does not
include a lift force. It should be noted, however, that other methods
are beginning to include lift effects. This neglect of the lift force
is on the safe. side but is economically costly.
2.3 Indications of significance of lift
As Leliavsky noted, contrary to White's results, Jeffreys [8]
and Fage [9] produced results which definitely indicate lift as an
important factor. Fage's experimental work provided evidence, and
Jeffreys' gave a theoretical approach. Studying a cylinder resting on
the flat bed of a deep stream and applying the principles of classical
hydrodynamics, Jeffreys found a relation for the lift and then for
a scour criterion. Use of realistic sizes indicated that lift alone
should be capable of dislodging particles.
Chang [10], considering the lift on particles as due to the
pressure of a velocity head, worked from the simple fact that the
particle would tend to lift when the vertical lifting force (or hydro
dynamic lift) plus the buoyant force equalled the particle's weight.
He also did work on drag and made the following comparison. "Theo
retically, the force required to lift a particle from the bottom of
a stream is about 40 per cent greater than that required to move it
along the bed" [10, p. 1282]. This certainly indicates the same order
of magnitude for drag and lift.
Further validity is given to the value of studying lift by
Yalin [11] who says "A consideration of the paths of saltating particles
145
It therefore seems likely that where sufficient information
is available about a natural bed and its material, values of lift may
be predicted from analysis based on some idealized model. Relation
ships between natural beds and idealized counterparts will have to be
developed more fully as future work occurs.
139
so that any portion of the flow above that depth is calculable
directly from the sidewall. It should be noted that there is some
possibility that El Samni's wotfk had even greater sidewall influence.
He noted that velocities taken in horizontal traverses through the
flow plotted as straight .lines versus distance from the sidewall on
semilogarithmic paper. No allowance was needed for a theoretical bed
because of the small roughness of the sidewall.
The concern here is how much effect will be created down at
the level of flow near the top of the hemispheres or grains, for the
lift is largely determined by these velocities. The amount of any
effect will give an idea of how much the finite bed might differ from
the infinite bed studied theoretically. The following figure indicates
how the velocity decreases at u^_ near the wall. From the values indi
cated, an average velocity across the flume can be computed at the
level of the tops of the hemispheres. This velocity indicates an aver
2 2
age u^ about 94.2 per cent of the u^ at the center of the flume. This
then is a very approximate measure of the decrease of lift below that
for a very wide bed. If the value above is applied to C = 0.178 for
the flume bed average, a value of = 0.189 is indicated for a very
wide bed. A check reveals that 10 per cent variation corresponds to
a = 24.5 degrees and a sidewall roughness k^ of about 0.009 feet, or
nine times that used and probably too large. Hence, the original value
chosen will be used as a gauge. In any event, 5 per cent is not a very
great influence and indicates reasonable similarity between the theo
retical and actual situations.
15
Note that this is written for one meridian plane, since the
case is axisymmetric. The tangential surface velocity on r = a
can be found as
1 ocp 3 TT a
"7^=2Usin0=q C2"7)
Using Bernoulli's principle,'with a pressure of p^ at the
forward stagnation point, yields the following expression for the
surface pressure, p.
PQ P = Yz + 2 q2 (28)
where z = elevation
Y = unit weight of fluid
What is of interest here is to study only the vertical force
component over the upper half of the hemisphere. This will involve
integrating (28) over the upper surface. Integration of the first
term, z, would yield a hydrostatic lift (buoyancy) which would, in
fact, not even be measured, as the difference in piezometric head is
the measured value. Hence, integration of the last term in (28)
will yield the desired force component. It should be noted that this
would actually be the theoretically predicted valtle for the case
of a single hemisphere on a flat bed. However, the measurements herein
used are for a suspended sphere. The total vertical force can be
found as follows:
F = f f* T U2 sin26 dA
v 2 4
o
(29)
102
6.2 Numerical results for Chepil arrangement
The arrangement considered in this section is that of hemi
spheres placed three diameters apart (center to center) in a hexag
onal pattern on a horizontal bed. The values of interest from the
finite differences solution are the pressures and velocities as dis
tributed over the surface. It is possible to present the results in
numerous ways, but the threedimensional character of the solution
makes clarity sometimes difficult. Here* the results will be shown on
traces of the hemisphere found in planes parallel to the zxplane or
the yzplane. Some of the plots may look unusual as the velocity at '
the end of the trace doesn't equal zero. This is because the veloc
ity shown is the total velocity at each point. There is a velocity
along the basal circumference of the sphere. These distributions are
. \
shown on the following pages.
Integration of the pressures over the surface as discussed
in Chapter V yields a lift coefficient, based on the lift force per
unit area directly under the hemisphere as follows.
Chepil's arrangement:
C = 0.405 (61)
Lu
This coefficient is used with equation (577) to evaluate the lift.
The value of C = 0.405 compares with that of 0.50 for
a single hemisphere shown in Section 2.4.6. This is not unexpected,
as the threediameter separation approaches the singleelement case.
Work done by Michaels .[7l], who analytically treated a single row
137
1
u/ut
1.0
Vectors plotted are resultants of
dcp/dy and dcp/dz at the points
of the tails of the vectors
Figure 53. Closly packed hemispheres: flow pattern on surface
viewed toward yzplane.
28. Owens, J. S. Geographic Journal, vol. 31, 1908, p. 418.
29. Streeter, V. L. Fluid Dynamics. New York: McGrawHill Book
Company, Inc., 1948, 263 pp.
30. Flachsbart, 0. "Neuere Untersuchungen uber den Luftwiderstand
von Kugeln," Physik Zeitschrift, vol. 28, 1927, pp. 461469.
31. Schlichting, H. BoundaryLayer Theory. New York: McGrawHill
Book Company, 1968, 747 pp.
32. Fage, A. A. R. C. Reports and Memoranda, No. 1766, 1937.
33. Einstein, H. A., and El Samni, E. A. "Hydrodynamic Forces on
a Rough Wall," Reviews of Modern Physics, vol. 21, No. 3,
1949, pp. 520524.
34. El Samni, E. A. "Hydrodynamic Forces Acting on the Surface
Particles of a Stream Bed," Ph.D. Dissertation, Department
of Engineering, University of California, 1949.
35. Engelund, F., and Hansen, E. "A Monograph on Sediment Transport
in Alluvial Streams," Copenhagen, Denmark: Teknish For lag,
1967, 62 pp.
36. Chepil, W. S. "The Use of Evenly Spaced Hemispheres to Evaluate
Aerodynamic Forces on a Soil Surface," Transactions, American
Geophysical Union, vol. 39, No. 3, June 1958, pp. 397404.
37. Zingg, A. W. "Wind Tunnel Studies of the Movement of Sedimentary
Material," Proceedings, 5th Annual Hydraulics Conference,
Iowa City, Iowa, May 1952.
38. Kadib, AbdelLatif A. "Mechanism of Sand Movement on Coastal
Dunes," Journal of the Waterways and Harbors Division, Proceed
ings, ASCE, vol. 92, No. WW2, May 1966, pp. 2744.
39. Chao, J. L., and Sandborn, V. A. "Study of Static Pressure Along
a Rough Boundary," Journal of the Hydraulics Division, Proceed
ings, ASCE, vol. 91, No. HY2, March 1965, pp. 193204.
40. Rouse, Hunter (editor). Engineering Hydraulics. New York: John
Wiley and Sons, Inc., 1950, 1039 pp.
41. Krumbein, W. C. "SettlingVelocity and Flume Behavior of Non
Spherical Particles," Transactions, American Geophysical Union,
vol. 23, August 1942. Published by National Research Council
of the American Academy of Sciences, Washington, D.G.
APPENDIX
183
IF
(NT .EC. 0) NT =
MARK
IF
(MZH)*2 X**2) .GT. 1.0) Y2 0.00
IF
((t ZH) *2 X**2) .GT. 1.0) GO TO 1643
Y2
= SORT(1.0 (Z
H)2 X*2)
1643
IF
(J .EC. 2) GO TO
1743
IF
(((YY2/H) .LT.
1.0) GO TO 1663
1743
LI
= I
L2
= Jl
GO
TO 1610
1604
NT
= NO(I J2,K)
IF
(NT .EQ. 0) NT =
MARK
IF
(((YY2 ) /H ) .LT.
2.0) GO TO 1664
LI
= I
L2
= J2
GC
TO 1610
C CNLY ONE POINT AVAILABLE IN XDIRECTION
1674 SI = (XSCRT(3.00))*PI
S1A = XSGRTl3.00)
MULE =1
MUT = 3
GO TO 1605
C ARC INTERPOLATION FOR XDIRECTION POINT
1673 18 = IBDRY1(J K)
19 = IBDRY1(J K1)
XT = SGRTd.O Y*2 Z 1**21
XDEL = (X XTJ/H
A = ZH
THM = (ATAN2(ZXINIT(JK)) ATAN2(Z1XT))/
1 (ATAN2(Z,XINIT(J,K)) ATAN2(A,XINIT(JKl)) )
PI = PHI (18 J K) + THMMPHK 19 JK1) PHI{18,J,K))
SI = (XSCRT(3.00))*PI/XDEL
S1A = (XSQRTI3.00))/XDEL
MULE =1
MUT = 3
GO TO 1605
C CNLY ONE POINT AVAILABLE IN YOIRECTION
1664 S2 = (Yl.0)*PJ
S2A = Y 1.0
IF (J .EQ. 2) S2 = 2.0*S2
IF (J .EG. 2) S2A = 2.0*S2A
GO TO 1622
C ARC INTERPOLATION FOR PZ1(I,J) NECESSITATED BY PROXIMITY OF OT
HER
C HEMISPHERE.
1663 16 = IBRY2(J K)
17 = IBDRY2(J,K1)
A = Z ALFA(N,5)*H
B = Z H
C = SGRTC3.00) XENC(JK)
D = SGRT (3.00 ) XENCUtK1)
E = SORT(3.00) X
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
3000
C
C
C
199
FORMAT (20H LIFT COEFFICIENT 1F12.6)
FORMAT (15)
FORMAT (10F10.5)
FORMAT (5H Y = F5.2)
FORMAT (11H VELOCITIES)
FORMAT (10H PRESSURES)
FORMAT (27H SQUARE OF BASAL VELOCITIES)
FORMAT (46H LIFT INTEGRATION CORRECTED FOR BASAL PRESSURE)
FORMAT (4F12.4)
FORMAT (20H VELOCITY COMPONENTS)
FORMAT (1115)
RETURN
END
Figure 15. Theoretical bed used in relating logarithmic and potential velocity profiles.
p
oo
5
Paper No. 86 in 1914 still continue to be those most often quoted of
any in the field" [3, p. 225],
From the time of Gilbert's work, laboratory facilities increased,
enabling further, broader studies. As various governmental agencies
began to attack the problems in the United States, more and more quan
titative information became available. Additionally, interest in analyt
ical approaches to sediment problems grew with the advent of more
theories on movement. Consideration of the forces actually acting on
particles exposed to flow received increasing emphasis.
2.2 Neglect of lift force
As Leliavsky [4] indicated, the drag component of forces acting
on particles received the bulk of work earlier in this century. Indic
ative of this was the work by C. M. White [5] in 1940. White gave
great care to study of the drag, while the lift was treated only briefly,
with a guarded conclusion that lift did not exist. Leliavsky noted
in 1955 that there still exists ". .an almost unexplored aspect of
the problem, viz., the vertical component of the resultant of the
hydraulic forces applied to the grain, i.e., 'the lift'" [4, pp. 6465].
Similarly, Young stated, "Although some attention has been given
to the effect of the drag components on the behavior of suspensions,
little work has been done in connection with the determination of the
lift component" [6, p. 47].
This disregard for the lift force has resulted in a lack of
understanding of this force, with probably attendant shortcomings in
full understanding of drag. One area of very practical interest where
20
Their results enabled them to write
2
Ap = CL p Â¡y (215)
where Ap = pressure difference
CL = lift coefficient
p = fluid density
u = velocity
They found a constant 0.178 if u was taken as the velocity 0.35
sphere diameters above the theoretical bed, determined by experiment
as 0.20 sphere diameters below the sphere tops. Further studies
which they made on natural gravel yielded the same expression for Ap
with some redefinition of u along lines consistent with Einstein's
earlier work [12 ]. This work forms the only example of a lift force
essentially integrated over a number of particles, though the distri
bution over individual particles was not ascertained. Support for the
rationale of measuring lift as a pressure difference is given by
Engelund and Hansen [35, p. 19] in discussing variations from hydro
static pressure due to streamline curvature.
2.5.2 Chepil.Chepil [36] performed experiments in a wind
tunnel on hemispherical elements placed on a plane bed in a hexagonal
pattern three diameters apart. He chose this spacing based on work
by Zingg [37], which indicated this is the average spacing between
particles erodible from a sand bed.' The processes of erosion of sand
by wind and by water involve essentially the same factors, as indicated
by, among others, Kadib [38].
2
a particle being moved from the stream bed. A need for a better defin
ition of the fundamental forces acting on such particles has prompted
the author's study of hydrodynamic lift in sediment transport, with
observations of flow characteristics in the vicinity of a rough bed.
Lift has been an often neglected force in fluvial hydraulics
despite the fact that it has the same order of magnitude as the bed
shear stress. This is reflected in the many stable channel design
procedures which neglect lift. These procedures may result in a design
which is too costly. Hence, studies of the lift force take on further
economic significance.
Most work on lift has been on single particles, with no con
sideration for the particle interaction existent in the actual stream.
Those works which have included interaction have generally measured
lift forces over a given area rather than in the form of pressure
distributions on the individual particles. The author intends to study
interaction effects, trying to provide a base for analytical studies
of lift forces.
First, potential flow theory will be used as a guide to provide
a means of studying the flow near the surface of a grain or bed shape
and then relating that flow to other flow characteristics. The poten
tial flow velocity profile will be related, for this work, to the
logarithmic velocity profile, furnishing a means for predicting the
pressure distribution and total lift force on single particles within
a series. It should be noted that forms of velocity profile other than
the logarithmic might be used if they are characteristic of the flow.
7
by R. A. Bagnold reveals that saltation begins with a motion directed
'upward*; . However, if this is true, then, as has already been
maintained by various authors, the lift force must be the cause
of the detachment" [11, p. 229]. Support for Yalin's statement is
offered by the work of Einstein [123> Bagnold [13], and Velikanow [14].
Young, doing work in 1960 on spheres in a cylindrical tube,
found the lift to be of the order of onehalf the drag for experiments
with a Reynolds number based on the pipe diameter and the mean veloc
ity in the range 3601115 (in the laminar range). He summarized,
"It is thus apparent that the lift force should not be overlooked in
studies related to the incipient motion of particles resting on stream
beds or pipe walls" [6, p. 57].
It seems apparent that sufficient evidence exists to prompt
efforts to increase understanding of the lift phenomena. The author
will now consider earlier works on lift.
2.4 Use of potential theory in lift studies on single bodies
2.4.1 General.Much of the analytical work done on lift has
dealt with single bodies or particles, especially spheres and circular
cylinders, isolated from any other bodies. An example of this approach
is the work done by Jeffreys [8] mentioned earlier.
The overwhelming amount of work on lift has been done in the
realm of aerospace engineering. While the principles thus developed
are applicable to hydraulics, the work concerns single bodies only,
often airfoils, struts, and the like. One of many works giving dis
cussions of several of these approaches is the famous work edited by
non
154
IF (KEYTAG .EQ. 1) GO TO 10
K = KGRADE
C CALL ROUTINE FOR GRADING NET, OR CHANGING SIZE OF H.
750 CALL GRADE
10 KIP = KIP +1
KIN = 2
C CALL INOUT TO WRITE ITERATION NUMBER, MAXIMUM FIELD
C CHANGE, AND TO CALL FOR VELOCITY CALCULATIONS.
CALL INCUT
IF (KINI .EQ. 1) GO TO 69
IF (KIP .LT. 50) GO TO 1
1006 FORMAT (1110)
1009 FORMAT (11F10.5/10F10.5)
1011 FORMAT (9H EPSILON F10.3)
69 STOP
END
SUBROUTINE INOUT
C
C SUBROUTINE INOUT IS TO READ IN ANY DATA REQUIRED FOR THE
C SOLUTION AND TO PRINT OUT ANY RESULTS AS THEY ARE DESIRED
C THIS SUBROUTINE ALSO CALLS SUBROUTINE VELOC TO COMPUTE VELO
C CITIES AND CALCULATE A LIFT COEFFICIENT*
C
COMMCN ALFAO(21,21,2),ALFA7(21,21,Z),ALFA02(21,21,2),
1 ALFA72(21,21,2),XENC(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA 1(21,21),
3 THETA2(21,21),IBDRY1(21,21),IBDRY2(21,21),PZ(36,21),
4 TIPI 100),TOP(11,11),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1iSlA,S2,S2A,S3,S3A,BING,BIP,BOGtlGl,IG2,I,J,K,
7 AI ,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,N0{36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FLO
KL0W1 = KLOW 1
IF (KIN .EQ. 0) GO TO 20
IF (KIN .EQ. 1) GO TO 21
C WRITE ITERATION NUMBER AND MAXIMUM FIELD CHANGE.
WRITE (6,1010) KIP
WRITE (6,1011) EPS
C CHECK TO SEE WHETHER TO CALCULATE VELOCITIES.
IF
(KIP .EG.
30)
GO
TO
2
IF
(KIP .EQ.
40)
GO
TO
2
IF
(KIP .EQ.
50)
GO
TO
2
C EPSMIN = ESTABLISHED MINIMUM FIELD CHANGE WHICH WILL
C SIGNIFY SUFFICIENT CONVERGENCE.
IF (EPS .LE. EPSMIN) GO TO 2
C CHECKS TO STOP IF SOLUTION NOT CONVERGING PROPERLY.
IF (EPS .GT. 15.0) GO TO 2
67
2 V2 CpI+I>J;K ~ Cf'1>J>K ~ ^11,3,K
2h ^ a (o +a ) a (o? +a )
lv 1 2J 21 2
^J+^K ^I^jK ^I,J1,K
+ c (a +a ) ~ a (o +a, )
3v3r 4V34
^I^K+l
a_(Qf +of )
3 5 6
~ ^I^Kl
a*(0lTi + a&')
OJO
= O
(522)
Since the desire at a given point is to solve for the potential, or cp
value there, the equation above must be solved for cp(IjJ>K).
/ *I+
1+1,J,K ^11,3,K + yI,J+1,K 1
0il(ai+a2') + a2('ai+0i2^ Q'3(Q3+Qi4)
^1, Jl, K ^I^K+l ^I^Kl
+ a (a+a ) + or (a +a ) + a* (Qv+Q,<) )
v 4 3 4 556 656'
9.
I,J,K
(523)
, a/,ia5+a/,) (a +al (or+of,) 7
' 434 556 656
Note that if all legs are of length h, which implies that all arvalues
are 1.0, equation (523) reduces to (513), as it should.
Equation (523) is applicable for all interior points of the
solution space. The six points may contain among them points that
lie on a regular lattice point, on a hemispherical surface point where
cp is being calculated, on or beyond a noflow boundary, on or beyond
the "foldedsymmetry" boundary of Figure 19, or on one of 'the two
equipotential planes at the xextremities of the solution space.
40
Essentially, the work above is an attempt to check beforehand
the probable validity of all the previous discussion on logarithmic
distributions in this chapter. The chapter, in general, was intended
to point out phases of the velocity distribution applicable to the
present work.
108
Figure 36* Chepil's case: velocities on x = 0
18
This allows the surface pressure distribution to more nearly approach
the theoretical. For even lower Reynolds numbers than those indicated,
the viscous forces would play an even greater role, thereby causing
further deviation in the flow from theoretical. These ideas are of
importance in relation to the limitations of work in this dissertation.
The forces evaluated in this section were only those in the
vertical direction on the upper surface of the sphere, with an eye to
noting agreement with theory.
For later comparison a value of C^u will be determined here
for the theoretical case of a single hemisphere on a flat bed. This
will entail integrating the pressure at the hemispherical base, found
from (28) and subtracting it from the force of (211) to give a
resultant vertical lift. Integration here occurs in a direction normal
to that in (29). Using 0^ for this integration yields a relation
cos 0j = sin 0. Therefore, the ratio of the basal velocity (u^) to the
top velocity (u^_) equals cos 9^. The pressure decrease along the
bottom can be found as below.
o J7 9 9
F = p t (2a ) f sin 9 cos 0, d
vb 2 Jo 1 1 ]
u.
(213)
The lift per unit area is therefore
\
u
F F ...
v vb
TTa
(214)
This lift coefficient of 0.50 should form an upper bound for the
work to be done later.
73
Figure 25. Grading the lattice.
o o o o no o
188
IF (IMID .EQ. 2) GO TO 2369
PY(IK) = (SUM1/SUM2)*0MEGA 4 TAG*PY(I,K)
BING= PY(I,K)
2369 RETURN
1009 FORMAT (1IF 10.5/L0F1C.5)
END
SUBROUTINE BETWN
SUBROUTINE BETWN HANDLES THOSE LATTICE POINTS BETWEEN THE TOPS
OF THE HEMISPHERES ANO THE SECTION WHERE THE NET IS GRADED.
COMMON ALFAO(21,21,2),ALFA7(21,21,2),ALFA02( 21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2(21,21),CC0N1(21,21),CC0N2(21,21),THETA 1(21,21),
3 THETA2(21,21),IBDRY1(21,21),IBDRY2(21,21),PZ(36,21),
4 TIP(100),TOP(11,11 ),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1KEY2,JMID,MARK,NCAS,KL0W1,IMIDTAB1TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1,IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,N0(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
COMMON IMIG(21),PIG,KIN,KIN1,FL0
C THIS PORTION OF THE ITERATION PROGRAM IS FOR THE REGION WHERE
C THE SIMPLE SIXPOINT FORMULA FOR THE LAPLACIAN EXISTS.
C COMPUTE CONSTANTS NEEDED DUE TO LAST UNEVEN XSPACING.
IMIT = IMAX 1
AIMAX = IMAX
TAB7 = (SCRT(3.00) (AIMAX2.0)*H)/H
TAB1 = 4.0 + 2.0/(1.0+TAB7) 42.0/(1.04TAB7)*TAB7)
TAB2 = 2.0/(TAB7*(1.04TAB7))
TAB3 = 2.0/(1.0+TAB7)
KB 1 = KLCW + 1
KB2 = KGRADE 1
IF (KEYTAG .EQ. 1) KB2 = KMAX 1
TRAVERSE
IN Z
DIRECT
ION
DO
700
K = KB
i 1,KB2
TRAVERSE
IN Y
DIRECT
ION
DO
701
J =
1,JMAX
TRAVERSE
IN X
DIRECT
ION
DO
701
I =
2,IMIT
IF
(J
.EQ.
1) GO
TO
702
IF
(J
.EQ.
JMAX)
GO
TO 703
IF
( I
.EQ.
IMIT)
GO
TO 705
C STANDARD FORMULA FOR THE LAPLACIAN USING SIX ADJACENT POINTS
C ALL AT A DISTANCE H AWAY FROM THE OBJECT POINT.
TAM = (PHI(11JK) 4 PH I(1 + 1J,K) 4 PHI(I,J4l,K) 4
1 PHI(I,J1,K) 4 PHI(I,J,K+1) 4 PHI(IJK1)J/6.0
C SECTION 710 COMPARES THE CHANGE IN THE PHI VALUE AT THE
C PRESENT POINT WITH THE MAXIMUM CHANGE THUS FAR ENCOUNTERED
Figure 51. Closely packed hemispheres: trace of some
equipotential surfaces in plane y = 0.5a.
54
hemispherical surface being treated here. While the latter method
showed promise, there were numerous problems involved, such as a choice
of distribution and types of singularities and possible stability
problems in the numerical process required to vary these factors and
approach a solution. Having considered these other alternatives, the
author finally chose the method of finite differences for the solution
since this method enables obtaining the desired accuracy, while, at
the same time, providing a mathematically stable numerical analysis.
The potential flow case being considered is one in which the
depth of flow is essentially infinite. However, the choice of finite
differences method of solution makes treatment of such an infinite flow
space impossible. Therefore, the geometry of flow patterns of other
cases was reviewed with an eye to choosing a depth of flow which would
enable solution of the problem without adversely affecting the desired
results.
5.2.2 Depth of flow space.Work in two dimensions reported in
Chapter IV indicated that a depth of four times the height of the bed
element was adequate to approach the straight streamlines associated
with free stream flow. To further check this, consider the flow around
a single sphere, where, from [29]
Ua3
cp =  cos 0 + Ur cos 0 (52)
2r
This is the same case described in Section 2.4.6. The expression for
the tangential velocity component is shown below, along with its value
when 0 = tt/2, or points immediately above the center of the sphere.
NOTES ON FORTRAN IV COMPUTER PROGRAM
The program listed on the following pages was written in
Fortran IV and used for the solution of the flow over the closely
packed hemispheres. The program used for the other case had a few
differences for the author's convenience, but it differed essentially
only in that many features of the listed program were unnecessary.
As a result, only a listing of the more general program is included
here.
Numerous comment cards are included within the various sub
routines to define usage and describe steps taken. However, to provide
a summary, brief descriptions of the functions of each of the fifteen
subroutines are offered below.
The MAIN program is primarily a control section, calling into
play the routines used in the iterative process and calling for input
and output.
Subroutine INOUT reads in initial data, and, in the a/20 run,
interpolated cpvalues are calculated. Output for iteration number
and maximum field change, as well as the potential field itself comes
from here. INOUT also calls subroutine VEUOC to calculate a lift
coefficient.
Subroutine XLIMIT calculates xvalues on hemispherical surfaces
s
corresponding to lattice lines and also assigns lattice line numbers
to those points.
150
76
The crossderivative terms cancel due to symmetry in directions
other than x, as do the first derivatives except for the xderivative,
which can be approximated by
&p
(l+C^h Sx
(536)
and the second xderivative, which can be approximated by
ih2JÂ£
Sx
9,
a2(l+a2)
1+n
(537)
Then (535) yields, when applied successively to all eight of the
adjacent points used by object point G, the following:
2 cp. 8cp 4h2(V2cp) 2(Qi2l)h2
L G G 2 5x2
4(a1)
" Tk? {cpv V = 0 <5"38>
where Â£ cp^ indicates the sum of the eight adjacent points. Setting
the Laplacian equal to zero yields an expression the same as (532),
except for including the last terms in the equation above.
One further point, such as N, is slightly different from V,
though the ideas are similar.
By
9.
9
Nl
 29
N
(539)
where (Nl) is the point lying a distance h from N and not shown
in Figure 25.
Figure 27. Yderivatives.
c
c
GO TO 661
650 CONTINUE
639 WRITE (6,2002)
WRITE (6,2004)
WRITE (6,2005)
WRITE (6,2003)
WRITE (6,2006)
WRITE (6,2003)
640 CONTINUE
CL = SUMLT
WRITE (6,2000)
WRITE (6,2001)
J
Y
(VEL(L7,J),L7=1,L)
(PRES(L7*J)L7=1L)
CL
SECTION FOR CALCULATING PRESSURES ALONG THE BASE OF THE
HEMISPHERE AND INTEGRATION OF THIS PRESSURE.
00 670 J = 2 JM IT
IF (J .EQ. JMID) GO TO 671
K = 1
I = IBORY1(J,K )
AI = I
X = X INIT(J,K )
AJ J
Y = (AJ1.0)*H
AK = K
Z = 0.0
N = NO(I,J,K)
CALL BODER
DPX = (Si S1A*PFI(1,J,K))/X
DPY = (S2 S2A*PHI(I,J,K))/Y
UBASE2 ( J) = (CPX**2 DPY*2)/QTSQR
GO TO 670
C CALCULATION FOR MIDDLE SINGULAR POINT.
671 13 = IBDRY1(J+1*K)
14 = IBDRY1(J1,K )
THP = (PHK I3,J+1,K) PHHI4,J1,K) l/THG
UBASE2(J) = THP**2/QTSQR
670 CONTINUE
WRITE (6,2007)
WRITE (6,2003) (UBASE2(J2), J2 = 2,JMITl
SIG =0.0
K = 1
DO 672 J=2,JMIT
IF (J .EQ. JM IT) GO TO 673
SIG = SIG + H*XINIT(J,K)*UBASE2(J)
GO TO 672
673 SIG = SIG + UBASE2J)PEP
672 CONTINUE
C CALCULATION OF FINAL DESIRED LIFT COEFFICIENT, CL.
CL = CL SIG
WRITE (6,2008)
WRITE (6,2001) CL
2000 FORMAT(1H1)
195
C SUBROUTINE VELOC COMPUTES THE VELOCITIES AT SURFACE POINTS,
C USING RELATIONS AND VALLES OBTAINED FROM SUBROUTINES SUCH AS
C BDDER AND ZBOUND, THE SUBROUTINES FOR HANDLING THE NORMAL
C DERIVATIVE BOUNDARY CONDITION AT THE HEMISPHERES. WITH THE
C VELOCITIES COMPUTED OVER THE HEMISPHERICAL SURFACE, THE SQUARE
C OF THE VELOCITY IS THEN INTEGRATED OVER THE SURFACE TO OBTAIN
C A VALUE FROM WHICH IS THEN SUBTRACTED THE PRESSURE CONTRIBU
C TICN FROM THE VELOCITY ALONG THE BASE OF THE HEMISPHERE. THE
C FINAL RESULT IS A LIFT COEFFICIENT, CL.
C
COMMON ALFA0(21,21,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),X IN ITI 21,21),MIG1C21,21),
2 MIG2(2121),CCONl(21,21),CCON2(21,21),THETA1(21,21),
3 THETA2(2121),IBCRY1(2121)IBDRY2(2121),PZ(36,21),
A TIP(100)TOP(11,11),H,EPSIMAX,JMAX,KMAX,KLOWKGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KLOWl,IMID.TABI,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3S3ABINGBIP,B0G,IG1IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIPKEY,NQ(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,2l,32)
COMMON IMIG(21),PIG,KIN,KIN1,FLO
DIMENSION CSQR(35,21),VEL(35,21),PRESi35,21)
DIMENSION UBASE2I21)
C EINSTEINEL SAMNI ARRANGEMENT
JMIT = JMAX 1
C COMPUTE CONSTANTS NEEDED FOR LATER COMPUTATION OF A VELOCITY
C AT THE HEMISPHERE BASE NEAR THE MIDDLE SINGULAR POINT.
PD1 = 0.50 + H
PD2 = 0.50 H
JP01 = JMID + 1
JPD2 = JMID 1
THG = (ATAN2(XINIT(JPD2,1),PD2) 
1 ATAN2(XINIT(JPD1,1),PD1))/H
SUMLT = 0.00
KEY = 0
NCAS = 3
C BEGIN TRAVERSING ALONG YCONSTANT SECTIONS OF HEMISPHERE.
DO 640 J =1,JM IT
C CALCULATE WIDTH OF PROJECTED AREA ELEMENT FOR GIVEN POINT.
W = H
IF (J .EG. 1) W = H/2.0
L = 0
AJ = J
Y = (AJl.OMH
C BEGIN AT Z VALUE JUST ABOVE FIRST POINT ON THE HEMISPHERE AT
C THE GIVEN YSECTICN I AT X = 0 ). THIS Z IS ESTABLISHED BY
C IMIG(J), WHICH IS READ IN EARLIER IN THE PROGRAM.
K = I MIG CJ)
AK = K
Z = (AK1.0)*H
XB = 0.00
IT = IBDRY1(J,l)
170
C FOR SURFACE POINTS LYING ON OTHER HEMISPHERE, CALL BDDER2.
IF (I .EC. IG2) GO TC 9950
C CONSIDER DERIVATIVE IN XDIRECTION.
XP = A IH
AX = (XEND(J,K) XIMT(JtK) )/H
C CHECK FOR NEXT POINT ON NEXT HEMISPHERE.
IF (XP .GE. XEND(J,K}) GO TO 860
BX = (XP XINIT(JK))/H
C CHECK FOR NEXT POINT A LATTICE POINT, BUT FOLLOWING ONE IS ON
C THE NEXT HEMISPHERE.
IF (AX .LT. (1.0 + BXH GO TO 861
NT = N0(I+1JK)
IF (NT .EQ. 0) NT = MARK
SI = (X)*(2.0*PHI(141,J,K) + {1.02.0*ALFA(NT,1))*
1 PHK 1 + 2, J,K )/( 1.0 + ALFA (NT, 1) ) )
S1A = CX)*(3.0/(1.O+ALFA(NT,1)))
C MIG1(J,K) IS A KEY WHICH IDENTIFIES EACH POINT ACCORDING TO
C ITS RELATION TO THE POINTS AROUND IT. IF 2 POINTS ARE AVAIL
C ABLE IN THE YDIRECT ION FOR A SERIES EXPRESSION OF DPHI/DY,
C THEN MIG 1 = 1. IF ONLY ONE LATTICE POINT IS AVAILABLE AND IT
C MUST BE DETERMINED BY A LINEAR INTERPOLATION BETWEEN TWO FIELD
C POINTS, THEN MIG1 = 2. IF THE ONLY AVAILABLE POINT LIES ON
C THE NEXT HEMISPHERE, THIS VALUE IS DETERMINED BY ARC INTER
C POLATICN, AND MIG1 = 3. VALUES FOR MIG1 ARE COMPUTED IN
C SUBROUTINE ALFBDR
MAN = MIG1(J,K)
GO TC (851,852,853),MAN
C CASE WHERE NEXT POINT IS ON THE OTHER HEMISPHERICAL SURFACE.
860 SI = (PHI(1+1,J,K)/ALFA(N,2))*(!.0)*X
IF (J .EQ. JMID .AND. Z .LE. 0.30) GO TO 8860
S1A = (1.0)*XINIT(J,K)/ALFA(N,2)
MAN = MIG1(J,K)
GO TO (851,852,853),MAN
C SPECIAL DIFFERENCE EQUATION FOR POINTS NEAR MIDDLE SINGULAR
C POINT, TAKING ADVANTAGE OF ONE LINE OF KNOWN PHI.
8860 SI = X *(PH 1(1 + 1,J*K) 200.0)/ALFA(N,2)
S1A = (X)*(3.0/ALFA(N,2))
MAN = MIG 1 (J,K)
GO TG (851,852,853),MAN
C CASE WHERE ONLY ONE LATTICE POINT EXISTS BETWEEN
C THE TWO HEMISPHERES.
861 SI = (PHKT+l,J,K)/ALFA(N+1,1))*(1.0)*X
IF (J .EC. JMID .AND. Z .LE. 0.30) GO TO 8860
SI A = (i.O)*XINIT(J,K)/ALFA(N+l,l)
MAN = MIG 1(J,K)
GO TO (851,852,853),MAN
C YDIRECTION POINTS.
C FIRST, FOR BOTH POINTS IN YOIRECTION AVAILABLE FOR EXPANSION.
C MUST ALSO CHECK AT EACH Y POINT TO SEE IF THE NEEDED X POINTS
C ARE AVAILABLE FOR INTERPOLATION OR IF A LINEAR INTERPOLATION
C MUST BE EMPLOYED WITH THOSE X POINTS AVAILABLE.
o o o
166
2 + 1.0/(ALFA(N,3)*(ALFA(N,3) + ALFA(N4)))
3 + 1.0/(ALFA(N,4)(ALFA(N,3) ALFA(N,4)))
4 + 1.0/(ALFA(N,5)*(ALFA(N,5) 1.0))
5 + 1.0/(1.O +ALFA(N,5))
400 CONTINUE
469 RETURN
END
SUBROUTINE BELOW
C
C SUBROUTINE BELOW TREATS THOSE POINTS IN THE FIELD ON AND
C BELOW THE ZPLANE FORMING A TANGENT TO THE UPPERMOST POINTS
C OF THE TWO HEMISPHERES IN THE SOLUTION SPACE. THE LAPLACIAN
C DIFFERENCE EXPRESSION IS UTILIZED AND THE VARIOUS BOUNDARY
C CONDITIONS ARE IMPOSED. SUBROUTINE BELOW CALLS THE FOLLOWING
C SUBROUTINES TO HANDLE THE DPHI/DN = 0 CONDITION ON THE HEMI
C SPHERESO BDDER,BCDER2,YBOUND,ZBOUND,ZB02.
C
COMMON ALFAO(21,21,2),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21 ) ,XINIT(21,21),MIG1(21,21),
2 MIG2(21.21),CCON1(21,21),CCON2(21,21),THETA1(21,21),
3 THETA2(21,21),I BORYl(21,21).IBDRY2(21,21),PZ(36,21),
4 TIP(100),T0P(11,11),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCAS,KL0W1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,BOG,IG1,IG2,I,J,K,
7 AI,AJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,NO(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,21,32)
CCMMCN IMIG(21),'PIG,KIN,KIN1, FLO
C EINSTEINEL SAMNI ARRANGEMENT
KEG = 0
C SET ZVALUE
DO 800 K=1KLOW
L3 = K 1
IF (K .EC. 1) L3 = K 1
AK = K
Z = (AK1.0)*H
C SET Y VALUE
DO 800 J=1,JMAX
C STEP TO SKIP MIDDLE SINGULAR POINT, WHERE PHI IS KNOWN.
IF (K .EG. 1 .AND. J .EQ. JMID) GO TO 800
AJ = J
Y = (AJ1.0)*H
L4 = J + 1
L5 = J 1
C CHECK FOR IMPOSITION OF DPHI/DY =0 BOUNDARY CONDITION FOR
C Y=0 (J=l) OR Y=A (J=JMAX)
IF (J .EQ. JMAX) L4 = Jl
IF (J .EQ. 1) L5 =J+1
IG1 = IBDRYK J,K )
114
to the increased possibility of experimental discrepancies in the
smaller hemisphere. Chepil himself directly indicates a discrepancy
by showing two distinct values of yQ for the same boundary surface.
The term yQ should not be dependent upon the flow conditions for a
given boundary arrangement. As a means of evaluating the lift coeffi
cient, the values obtained by (66) will be used in (62). Here the
2
lift equation can be rewritten, using T0 = P vf an<* u = v^.
This yields (67).
C
Lu
(67)
Using (67) and Chepil's data from Table 2, values. for. C^./are Calcu
lated as shown below in Table 3, based on u = it ^ .
t top
TABLE 3
CT FOR CHEPIL'S WORK
Lu
1.27 cm,
y from
Jo
(66)
2.54 cm,
y from
Jo
(66)
5.08 cm,
yQ from
(66)
5.08 cm,
yo =
0.46 cm
0.340
0.244
0.266
0.262
0.300
0.248
0.346
0.340
0.290
0.284
0.365
0.358
0.368
0.302
0.380
0.373
130
x/a
Figure 46. Closely packed hemispheresi velocities on y 0.5a.
o o o o
153
//HYDRQ2 JOB (1432,41,020,152000),BENEDICT,B.A.
// EXEC F4GDX FORTRAN G, COMPILE, PUNCH OBJ.
//FORTSYS IN DD
MSGLE
VEL= 1
DECK, EXE
CUTE
MAIN PROGRAM FOR ITERATION SOLUTION OF LAPLACES EQUATION
OVER A BED COMPOSED OF HEMISPHERICAL ELEMENTS.
COMMON ALFAO 21,21,2 ),ALFA7(21,21,2),ALFA02(21,21,2),
1 ALFA72(21,21,2),XEND(21,21),XINIT(21,21),MIG1(21,21),
2 MIG2121,21),CC0Nl(212i)CC0N2l212i),THETA1(21,21),
3 THETA2( 21,21), IBDRY1{ 21,21) IBDRY2(21,21) PZ ( 36,21) ,
A TIP(100),TOP(11,11 ),H,EPS,IMAX,JMAX,KMAX,KLOW,KGRADE,
5 KEYTAG,KEY1,KEY2,JMID,MARK,NCASKLQW1,IMID,TAB1,TAB2,
6 TAB3,S1,S1A,S2,S2A,S3,S3A,BING,BIP,B0G,IG1,IG2,I,J,K,
7 AIAJ,AK,N,X,Y,Z,TAG,OMEGA,XP,BETA,KIP,KEY,N0(36,21,21)
COMMON DIV(2500),ALFA(2500,5),PHI(36,2132)
COMMON IMIG(21),PIG,KIN,KIN1,FL0
C OMEGA IS AN OVERRELAXATION FACTOR FOR THE ITERATION PROCESS.
KIN1 = 0
KIN = 0
C REAO INITIAL DATA
CALL INOUT
C ROUTINE TO LOCATE THE SURFACES OF THE HEMISPHERES, BOTH IN
C TERMS OF I, THE ARRAY XVARIABLE AND THE ACTUAL VALUES
C OF X ON THE SURFACE FOR Z,Y VALUES CORRESPONDING
C TO LATTICE LINES.
CALL XLIMIT
C CALCULATE REMAINING NEEDED INITIAL VALUES AND DATA.
CALL ALFINT
CALL ALFBOR
CALL DIVIDE
IF (KEYTAG .EQ. 0 .OR. KEYTAG EQ. 1) GO TO 3
C SECTION FOR A/20 RUN, BEGINNING WITH GIVEN PHI VALUES.
KIN =1
CALL INOUT
3 KIP = 0
C STATEMENT TO ALLOW STOPPING AFTER CHECKING INITIAL DATA.
IF (KEYTAG .EQ. 3) GO TO 69
C PORTION OF RELAXATION ITERATION PROGRAM FOR REGION BELOW UPPER
C POINTS OF THE HEMISPHERES.
1 NCAS =1
EPS = 0.00
WRITE (6,1011) EPS
CALL BELOW
C PORTION OF SOLUTION SPACE ABOVE HEMISPHERES BUT BEFORE
C GRADED NET BEGINS.
IF (KEYTAG .EQ. 2) GO TO 10
5 CALL BETWN
WRITE (6,1011) EPS
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF TABLES vi
LIST OF FIGURES vii
LIST OF SYMBOLS . x
ABSTRACT . xiii
CHAPTER
I INTRODUCTION ... 1
II RELATED BACKGROUND ON LIFT FORCE STUDIES 4
2.1 Historical interest 4
2.2 Neglect of lift force 5
2.3 Indications of significance of lift 6
2.4 Use of potential theory in lift studies on
single bodies 7
2.4.1 Genera 1 7
2.4.2 Applicability of potential flow 8
2.4.3 Work of Kutta and Joukowsky 9
2.4.4 Fuhrmann's work and other studies .... 11
2.4.5 Jeffreys' analysis 12
2.4.6 Flow around a single sphere ....... 14
2.5 Multiparticle studies 19
2.5.1 Einstein and El Samni 19
2.5.2 Chepil 20
2.5.3 Chao and Sandborn 22
2.6 Shapes of bodies studied 22
2.7 Relation to work of dissertation 23
III LOGARITHMIC VELOCITY DISTRIBUTION 25
3.1 Development of logarithmic velocity
distribution ..... 25
3.2 Special problems of present expressions .... 26
iii
136
y
Figure 52. Closely packed hemispheres: flow pattern on surface
viewed toward xyplane.
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TITLE: Hydrodynamic lift in sediment transport, (record number: 953375)
PUBLICATION DATE: 1968
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131
0.0 0.2 0.4 0.6 0.8 1.0
x/a
Figure 47. Closely packed hemispheres: velocities on y = 0.6a.
126
CL = 0.359 (69)
Comparison shows this value still lower than the coefficient for the
hemispheres at three diameters, which is expected. Note that this
C is based on the lift per unit area, and the velocity u The
L t
area used here was the total bed area, rather than merely that under,
the hemisphere. The difference in this case is slight, but the concept
of lift per unit area of total bed has more meaning in sediment trans
port studies.
Some items should be noted concerning the results shown.
First, the velocities for y = 0.40a and y = 0.50a are shown as com
puted using two values of h and three values of h, respectively.
Indicated on the velocity plots are the extrapolated values actually
used in the lift integrations. These values were obtained based on
consideration of the general trend of the curve and the variation due
to changing h. Extrapolations were necessary in both x and ydirec
tions near singular points as discussed in Section 5.5.7.
Values of C were obtained, based on u for all three h values
L t
a/6, a/10, and a/20. The C values were 0.301, 0.349, and 0.359,
il
respectively. The last value is used here without any attempt to
extrapolate it to h = 0, since it differs by only about 3 per cent
from the a/10 result.
Also included in the figures are two plots of flow patterns
over the hemispherical surface. One is viewed parallel to the xyplane
the other parallel to the yzplane. Note the flow near the middle
singular point, up over the point of contact.
79
5.5.2 Xdirection derivative.In this section the general
means for expressing the boundary derivatives will be indicated.
Consider the situation of Figure 26.
Figure 26. Xderivative condition at boundary.
Indicated in Figure 26 is a fictitious cp7 value located outside the
solution volume. This point will be utilized in the derivative and
eliminated by expressing it in terms of points in the field. The
difference expression for the derivative at the boundary can be written
as (547).
Sphere 1:
Vl ~ *
20? h
*Â£. =
dx
cp cp.
11
2Q2h
(547)
Sphere 2:
65
have been treated in three dimensions. This condition has caused some
new procedures to be employed in the following work.
For clarity, certain conventions will be followed in presenting
the equations and their descriptions. First, each point will be given
three subscripts representing the three coordinate directions, with I
denoting the xdirection, J denoting the ydirection, and z represented
by K. Hence, the subscript 1+1 implies the next point in the positive
xdirection beyond a point at I. Additionally, where one or more of
the variables is a constant for the investigations of a special relation
ship, the figure will be drawn in two dimensions, eliminating the third,
constant dimension.
5.4.1 General lattice point.Equation (513) was presented
as representing those cases where six adjacent points are available,
all at a distance h. However, frequently one or more points lie at
some distance other than h. For this reason, the general equation will
be developed for the case of Figure 22, where all six lengths are dif
ferent from h.
Figure.22. General lattice point*
123
TABLE 5
COMPARISON OF THEORETICAL WITH CHEPIL'S WORK
a,
cm
X
u
Chepil's
work
dynes
2
cm
y from Table 2
o
y from
'o
(66)
CLu
0.348
\T
CLu
0.405
CLu
0.348
CLu
0.405
1.27
13
23.6
27.4
13.3
15.5
1.27
22
45.4
52.9
25.6
29.8
1.27
42
73.4
85.5
50.5
59.0
1.27
83
114.0
133.0
78.5
91.3
2.54
19
27.3
31.8
27.2
31.6
2.54
37
52.1
60.6
52.0
60.5
2.54
79
97.3
113.0
96.8
113.0
2.54
130
151.0
175.0
150.0
175.0
5.08
33
43.8
51.0
43.0
50.0
5.08
88 .
90.3
105.0
88.5
103.0
5.08
165
160.0
186.0
158.0
183.0
5.08
287
268.0
311.0
262.0
306.0
60
On lower zsurface:
j*p =
dz
0
(58)
On hemispherical surfaces:
*Â£ =
dh
0
(59)
The selection of the symmetry plane for the Chepil case is
based on a type of "folded" symmetry. This is indicated in Figure 20
and equations (510) and (511).
Figure 20. Foldedsymmetry boundary.
56
For these reasons as well as earlier studies, a flow space of depth
four times the hemisphere height was chosen for the solution space
for (51).
Choosing a depth for the flow space assumes streamlines at
that elevation to be horizontal. Thus, at any greater depth than this
the indicated flow lines above that level will also be horizontal, and
the solution along a lower boundary will remain the same. It might
also be noted that the choice of such a solution space eliminates any
consideration of free surface effects, such as wave resistance. Flow
of greater depth relative to particle size may then be superimposed
on the flow space solved for with the knowledge that the solution of
distribution of velocity and pressure along the lower boundary will
remain essentially the same. The error here is determined by how
closely the stream surface at the chosen elevation approaches a flat
plane. It should also be noted that the depth chosen would form an
even better approximation in the cases of more closely packed hemi
spheres being treated here, since the stream surfaces in these cases
more rapidly approach free stream conditions.
5.2.3 Boundary conditions.Due to the periodic nature of the
hemispheres in the solutions, numerous conditions of symmetry are avail
able which enable reduction of the solution to handling a typical
repeated portion of the total flow space. Figures 18 and 19 represent
the portions chosen for the solutions of this dissertation. It can
be seen that the two drawings are different in that Figure 18 could be
divided by "symmetry" one more time. For purposes of the finite
122
Plan
Flow Elevation
^"
Figure 42. Chepil's hemisphere.
81
difference. These difference expressions simply involve taking the
difference in value between the two points and dividing it by the
distance between the two.
Sphere 1:
Sphere 2:
^1+1 ~
SCÂ£ ~ ^Il
5x Q^h
(551)
5.5.3 Ydirection derivative.For determining the derivative
in the ydirection, the same ideas exist as for the xderivatives.
However, since the points needed for an expansion in the ydirection
generally do not lie at a lattice intersection, special means must be
used to develop these points. For an indication of the problem, see
Figure 27. For simplicity, call the needed points PJ and PJl. Their
value will be discussed. First, a series expansion can be written for
the fictitious points shown.
Sphere 1:
cp' = PJ 2h {
iPJ1
4h2
fcpB + PJl 2PJ^
l 2h J
' 21
l v2 J
(552)
Rearranging yields the derivative
Sphere 1:
(553)
Similar expansion of the fictitious point at the second sphere results
in (554).
151
Subroutine ALFINT calculates the leg lengths af^ shown in
Figure 22. Also, the cpfield is set to initial values corresponding
to flow in a uniform stream.
Subroutine ALFBDR calculates the lengths shown in Figure 27,
needed for interpolation of the PJ and PJlvalues shown there, which
are used for series expansions to evaluate derivatives at the surface.
Corresponding values for the zdirection are also calculated.
Subroutine DIVIDE computes divisors for use in the sevenpoint
scheme when not all Of.values are 1.0.
i
Subroutine BELOW treats all points in the iterative process
from z = 0 to the tops of the hemispheres, applying the general differ
ence equation subject to all boundary conditions. When one of the six
adjacent points falls on a hemisphere, BELOW calls the appropriate, sub
routine: BDDER, BDDER2, ZBOUND, ZB02, or YBOUND.
Subroutine BDDER applies the normal derivative boundary condi
tion for adjacent points in the xdirection lying on the first hemi
sphere in the region.
Subroutine BDDER2, which is similar to BDDER, treats points on
the second hemisphere in the xdirection.
Subroutines ZBOUND and ZB02 treat adjacent points in the z
direction which fall on a hemisphere. ZBOUND handles those points on
hemisphere one, while ZB02 handles those on hemisphere two.
Subroutine YBOUND makes the calculations for those adjacent
points in the ydirection falling on either hemisphere.
70
5.4.3 Object point on "foldedsymmetry" boundary.This
problem arises only in the case with hemispheres spaced wider apart,
as in Figure 7. One of the points, in the sevenpoint difference
scheme lies beyond the boundary, but the symmetry of the flow pattern
makes it possible to express this value at this point by a value inside
the given space. Equations (510) and (511) show the expressions
needed. The value thus obtained for this external point is then
inserted into its proper place in the sevenpoint formula.
5.4.4 Adjacent point on hemispherical surface.In some
instances, the lattice point being treated may have as adjacent points
one, two, or as many as three points which lie on a hemispherical bound
ary. At first, the author hoped to expand some of these points in
terms of other values in the field, while still applying the normal
derivative boundary condition (59) to most surface points. It was
found, however, that this approach created stability problems in some
portions of the iterative solutions. Therefore, the application of
the difference equations for (59) was extended to obtain a value for
all adjacent points falling on a hemisphere, with a small number of
exceptions. Discussion of the noflow condition and its use will occur
in ensuing sections.
The single exception occurs when both adjacent ypoints lie on
hemispherical surfaces. In order to obtain values for these two points
a process which will be called arc interpolation will be used. This
entails interpolating a value for the point from two surface points
where a value exists. The two points selected lie in the same zplane,
100
pressures) over the hemisphere. For this reason, convergence here
would best be judged in terms of the velocities and their distribution.
The point where velocities have stabilized within reasonable limits
will then be used as a criterion for convergence here.
The program for the Chepil arrangement was run through 70 iter
ations, at which point the maximum change in the field was 0.014 for
a field of values ranging from 0.00 to 100.00. This run took slightly
less than 14 minutes on the IBM 360 computer. The value for C^u, the
lift coefficient, was within 0.2 per cent of its final value after
30 iterations. The added iterations primarily refined the velocities
and pressures.
The solution for the closely packed hemispheres was made using
three different values for h: a/6, a/10, and a/20, where a is the
radius of the hemisphere. Each run was made through 50 iterations,
with computer times of 4.5, 6.5, and 17.8 minutes, respectively. The
corresponding maximum field changes were 0.027, 0.039, and 0.050.
One note should be made concerning the h = a/20 solution. Due to
the extensive computer storage and time required, savings in these two
areas were made by limiting the height of the solution space. Values
of cp for elevation z = 1.10a were found from the h = a/10 solution
and then used as given, fixed data. The a/20 solution then proceeded
within this space.
The results obtained from the differences solutions will be
presented in Chapter VI.
1.0
Figure 35. Chepils case: Velocities on y = 0.8a.
72
method is of the same order as the expressions which would have to be
used for (59) in such cases. Therefore, arc interpolation was chosen
as a more convenient manner of obtaining values for the points which
were still consistent with the remainder of the potential field.
The evaluation of cp at the six adjacent points has been discussed,
with the exception of surface points to which the boundary condition
equation is applied, which will be covered in Section 5.5 and following.
5.4.5 Graded lattice.The greatest accuracy is desired near
the boundaries to enable velocity calculations on those surfaces. Those
points which are farther away from the hemispheres can be handled,
therefore, with a larger mesh spacing, h. This is more true the
nearer the approach to a free stream condition. An advantage is gained
in computation, as using a larger lattice cube at some distance can
greatly reduce the number of points at which a solution is sought.
In the present cases, it was decided to use a larger spacing in the
upper onehalf of the solution space. Allen and Dennis [68] used a
scheme similar to that which is shown in Figure 25 and which will be
used here.
First, consideration will be given to those points where the
distances involved are all h or 2h = or = 1). The problem can be
considered on three zplanes. First is the plane CDJH Here, letters
will be used for points rather than the usual convention, as it may
allow greater clarity. For points such as G, D,. H, or J, six adjacent
points exist at a distance 2h; for points like Q, R, S, T, and U, the
six needed points lie at a distance h. Similarly, for all points in
plane EFKL, there are six points 2h away.
CHAPTER III
LOGARITHMIC VELOCITY DISTRIBUTION
3.1 Development of logarithmic velocity distribution
Due to the use which will be made of the velocity distribution,
the author wishes to present briefly some background and to study some
specific points. Historically, three modern approaches to velocity
distributions in steady, uniform turbulent flow have arisen. The three
are the following: Prandtl [42], who introduced the concept of the
mixing length (related to the mean free path of particles) with momen
tum conserved; G. I. Taylor [43], who considered vorticity to be con
served along the mixing length; and von Karman [44], who developed a
similarity hypothesis for the problem.
Prandtl's derivation, beginning with the expression for shear
stress in fluid, is frequently cited in texts, such as [45]. It
neglects the viscous forces and considers only the socalled Reynolds
stresses, after 0. Reynolds [46]. The final expression can be written
as below.
(31)
where
B = C +  In k
k = equivalent sand roughness
25
176
t
IF (NT .EQ. 0) NT = MARK
XP = (AI2.0)*H
IF (J .EC. JMAX) GO TO 980
IF (XP .LT. XINIT(JlK)) GO TO 954
BETA = (XXPI/H
AlSUM = ALFA(NTtl) + BETA + ALFA02(J,K,1)
IF (ALFA02(J,K1) .LT. BETA) GO TO 3851
PJ = PH I(11 Jl,K) + BETA*(PHKIJ1K)PHI(12 JlK))
1 /ALSUM 4 (BETA**2)*(PH I (12JltK)/{ALSUM*ALFA(NT,1))
2 4PHKI,Jl,K)/(ALSIM(BETA4ALFA02(
3 PHI(I1,J1,K)/{ALFA(NT,1){BETA4ALFA02(J,K,1))))
IF (J .EQ. 2) GO TO 1950
956 NT 1 = NO(11,J2,K)
IF (NT1 .EQ. 0) NT1 = MARK
IF (XP .LT. XI NIT(J2 K) ) GO TO 957
ALSUM = ALFA(NT1,1) 4 BETA 4 ALFA02(J,K,2)
IF (ALFA02(Jt Kt 2) .LT. BETA) GO TO 3852
PJ1 = PHI(11,J2,K) 4 BETA*(PHI(IJ2,K)
1 PHI (I2,J2,K))/ALSUM
2 4 (BETA*2)*(PHI(I2tJ2tK)/(ALSUM*ALFA(NTl,i))
3 4 PHKIJ2,K)/(ALSUM*(BETA4ALFA02(J,K,2)))
4 4 PFI( I1,J2K)/(ALFA(NT1,1)*(BETA+ALFA02(J*K2S)))
9701 S2 = {1.0)*(Y1.0)*(PJl/2.0 2.0*PJ)
S2A = 3.0*(Y1.0)/2.0
GO TO 970
3851 Cl = ALFA02(J,K,1) 4 BETA
PJ =(PHI(IJ1K) ALFA021J,K,1)*
1 (PHK I,Ji,K) PHK 11, J1,K) )
2 /Cl (ALFA021J,K,1)*2)*(PHKI,Jl,K)/ALFA02(J,K,1) 4
3 PHI(I1,J1,K)/BETA)/C1 )/(1.0 ALFA02(J,K,1)/BETA)
IF (J .EC. 2) GO TO 1950
GO TC 956
3852 Cl = ALFA02(J,K,2) 4 BETA
PJ1 =(PHI(IJ2 K) ALFAC2(J,K,2)*
1 (PHI( It J2tK) ~ PHK I1,J2K) )
2 /Cl (ALFA02J,K,2)**2)*(PHKIJ2,K)/ALFA02(JK,2) 4
3 PHI(11J2K)/BETA)/Cl )/(1.0 ALFA02(J,K,2)/BETA)
GO TO 9701
980 S2 =0.0
S2A =0.0
GO TO 970
C LINEAR XINTERPOLAT I ON FOR YDERIVATIVE POINTS.
954 BETA = {XXIN IT(J1,K))/H
PJ = PHI(11 J1K) 4 (BETA/(ALFA02(JtK,l)4BETA))*
1 (PHKIiJl.K) PH I (11 J1 K) )
GO TO 956
957 BETA = (X XINIT(J2,K))/H
PJ1 = PHK 111 J2 K ) + BETA*(PHKIJ2,K) PHK 111 J2 ,K) )
1 /(BETA 4 ALFA02(J,K,2) )
S2 = (i.0)*(Y1.0)*(PJl/2.02.0*PJ)
S2A = 3.0*(Y1.0)/2.0
128
/
Figure 44. Closely packed hemispheres: velocities on y = 0.2a
HYDRODYNAMIC LIFT IN SEDIMENT
TRANSPORT
By
BARRY ARDEN BENEDICT
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1968
8
Goldstein [15]. Another, viewing historical development, is by
von Karman [16]. In a number of these, lift is studied by means of
potential flow.
2.4.2 Applicability of potential flow.The idea that lift
could be predicted by use of potential theory, even perhaps in those
areas where it might seem out of place, was offered by Prandtl. He
stated that any explanation of drag requires a consideration of
viscosity, "... whereas the lift can be explained entirely without
the concept of viscosity so that the wellknown methods of the clas
sical hydrodynamics of the ideal fluid are applicable" [17, p. 159].
Part of the reason for some reluctance to use such methods here is the
fact that within a boundary layer adjacent to a surface, irrotational
flow does not exist. However, in the cases of interest, the sublayer,
where viscous and inertial forces are of the same order, over the surface
is very thin. Outside this layer, up to the point of separation, the
equations of inviscid fluid flow are valid. Hence, the pressures on
the outer edge of the sublayer can be found from such equations, and
since the pressure difference from the outer edge of the layer to the
surface is assumed negligible, the normal pressures on the surface, and
thus lift, can be predicted by inviscid flow principles. Also, since
the sublayer is thin, it is possible to discuss velocities "on the
surface," though the velocities actually considered are those a small
distance away at the edge of the boundary layer. Of course, the
actual velocities at the wall in a real fluid would be zero.
84
Sphere 2:
pj =
A ^ a +0+0 J
o
+ &
p+a
3A1 f 1 1 *1
a' "Hv' P+CV
at' + p + a
(556)
where A: (I1,J1,K)
Al: (12 ,J1 ,K)
C: (I,J1,K)
Again a very similar expansion provides a value for PJ1.
To gain accuracy the expansions used are to be developed about
the point nearest the desired point. In the case where atq is less
than P, a slightly different approach becomes more convenient due to
the proximity of the spherical surface. For that reason, the point at
a distance at will be expressed by expanding about PJ (or PJ1), and
subsequently an equation obtained for the latter point. This is
illustrated below.
Sphere 1:
rn PT PT r
= PJ a e
o la + P J o
cpA PJ PJ cpr
Ot
(557)
p + a
r a i a r s TA/p
Pjjl = cp + ^ cp _cp } _
l p J Yc a +p 17a o of +
{V. *
(558)
CHAPTER VI
RESULTS AND COMPARISONS
6.1 General
The foregoing numerical work has been conducted to enable
testing a hypothesis, namely, that potential flow theory can be used
to describe the hydrodynamic lift forces experienced in sediment
transport. Several assumptions are involved in the analytical
approach and with regard to possible applications and limitations.
First, the flow studied is a mean, steady flow. There are no forces
involved from either free surfaces or sidewalls. Probably the most
telling restriction for application is that the flow should exist in
the hydrodynamicslly rough range. This is usually considered to hold
for wall Reynolds numbers (v^k/v) of 70 or greater. Essentially, in
this range the surface roughness elements completely penetrate the
laminar sublayer and this sublayer is of negligible effect.. For flows
in the ranges below the rough range, the viscous forces play a more
dominant role. In these ranges the results from ideal flow theory
would be less and less valid. A number of examples of this effect
were offered in Chapter II, and the same should hold in the present
work.
With the results of the numerical solutions in hand, attempts
will now be made to study the validity of the abovementioned hypoth
esis as a possible tool for sediment transport studies.
101
115
Also included in the table are lift coefficients based on Chepil's
yQ for the largest hemisphere. It is at this size where the value of
yQ from experiments is most likely to relate to a similar yQ for the
theoretical case studied. This is because the gravel particles on
the bed have a much smaller effect than for the smallest hemispheres.
Even the values presented in Table 3 do not provide complete consist
ency, but some trends exist. Considering the values based on Chepil's
y where the closest relation to theory is expected, the value would
be around 0.36. This value of C^ is only about. 11 per cent less than
the theoretically found value, which is very satisfactory agreement.
Some benefit may be gained from translating the velocity
distribution (62) into a rough bed form such as found in Chapter III.
If this is done, it is possible to find that k, the equivalent sand
roughness, equals 29.7 y^. Using this relation, Table 4 can be formed,
relating y k, and the ratio.of roughness to grain size, herein
denoted rc> which is k divided by the diameter of the metal hemisphere.
TABLE 4
ROUGHNESSGRAIN SIZE RATIOS
Chepil's Data
From (66)
yo
k
r
c
V
k
r
c
0.12
3.57
1.40
1 0.216
6.42
2.53
0.15
4.46
1.75


0.30
8.91
1.75
0.301
8.95
.1.76
0.46
13.67
1.34
! ,
0.471
13.99
1.37
62
Allen and Dennis [62]. Section 5.4 will give details of how this
is obtained.
6
2 cp. 6 cp =0 (512)
i=l 1
or
1
^o = g S cpL (513)
1=1
6
where 2 cp^ represents the 6 points around cpQ.
Figure 21. Sevenpoint finite difference scheme.
Similarly, if a net, or lattice, were placed over the entire
solution space, an equation could be written for each point of the net
of the same type as (512). Thus, Laplace's equation could be written
in the form of a matrix equation as
TABLE 2
CHEPIL'S EXPERIMENTAL DATA
Radius,
cm
yo
cm
R =
e
v a/v
V
dynes
2
cm
\ ,
u
dynes
2
cm
1.27
0.12
517
3.9
13
1.27
0.12
715
7.5
22
1.27
0.15
1007
14.8
42
1.27
0.15
1254
23.0
83
2.54
0.30
1232
5.5
19
2.54
0.30
1651
10.5
37
2.54
0.30
2317
19.6
79
2.54
0.30
2885
30.4
130
5.08
0.46
2772
7.0
33
5.08
0.46
3962
14.4
88
5.08
0.46
5290
25.6
165
5.08
0.46
6840
42.7
287
29
Therefore, the distance down from the upper tangent is
0.198 k in excellent agreement with the 0.2 k found by experiment,
s s
It might be noted that the value of 0.2 kg seems to be satisfactory
even when the roughness pattern is more irregular. This was shown
experimentally by Einstein and El Samni [33]. Other instances could be
reviewed, but let it suffice to state that the above definition of the
/
theoretical bed is satisfactory.
At this point the conditions for the Chepil [36] case are shown
in Figure 7. The computations for the theoretical bed are also shown,
since this will be needed later. Considering the region inside the
dotted lines, with dimensions 3K by 3K the following is obtained:
s s
Volume under theoretical bed = 9^/3* K Y, (36)
s b
13
Volume of particles = TT K (37)
6 s
Equating these and solving yields
Y = 0.0335 K (38)
b s
Therefore, for this pattern the distance down from the upper tangent to
the hemispheres is 0.467 very near the plane bed itself.
3.2.2 Conditions near the bed.Experience indicates that
expression (32) is quite valid away from the bed. However, inspection
reveals that problems occur near the wall. As y approaches zero, the
value of v/v and hence of v, approaches minus infinity. This obvious
discontinuity presents a real problem in understanding what happens at
the bed, where sediment movement begins.
152
Subroutine BETWN controls those points in the solution space
between the tops of the hemispheres and the portion where the lattice
is graded.
Subroutine GRADE treats the transition from lattice size h to
size 2h and the remaining points beyond the transition.
Subroutine VELOC computes, from the present potential field,
velocities on the surface of the hemisphere and integrates velocities
squared to yield a lift coefficient. VELOC calls BDDER and ZBOUND to
evaluate surface derivatives of cp. Desired values are also written
out from this subroutine.
TABLE OF CONTENTS (Continued)
CHAPTER Page
3.2.1 Theoretical bed 27
3.2.2 Conditions near the bed 29
3.3 Use of proposed adjusted velocity distribution 31
3.3.1 Comparison of distributions at wall ... 31
3.3.2 Comparison of distribution with
increasing y 32
3.4 Determination of k and v_ from experimental
data 33
3.5 Effect of sidewalls 35
IV TWODIMENSIONAL WORK: EARLIER RESULTS 41
4.1 Shapes studied 41
4.2 General methodsvelocity and pressure results 41
4.3 Lift integration 44
4.4 Calculation of predicted lift by use of
measured logarithmic velocity profile .... 47
4.5 Application to experimental results 49
V THREEDIMENSIONAL NUMERICAL SOLUTIONS 52
5.1 General 52
5.2 Problem formulation 53
5.2.1 Choice of solution method 53
5.2.2 Depth of flow space 54
5.2.3 Boundary conditions 56
5.3 General finite differences approach 61
5.4 Finite differences equations: interior space 64
5.4.1 General lattice point . 65
5.4.2 Object point on planar noflow boundary 68
5.4.3 Object point on "foldedsymmetry"
boundary 70
5.4.4 Adjacent point on hemispherical surface 70
5.4.5 Graded lattice 72
5.5 Finite difference equations: hemispherical
boundary 78
5.5.1 General 78
5.5.2 Xdirection derivative 79
iv
110
of spheres, indicates that for his case a spacing of three diameters
would very closely approach the case of infinite spacing. Hence, the
coefficient of (61) is within the upper bound set for it.
6.3 Comparisons with Chepil's observations
6.3.L Details of Chepil's work.The pattern Chepil studied
has already been mentioned. Tests were made in a wind tunnel. One of
the hemispheres was a metal element with holes every thirty degrees on
a plane parallel to the flow direction and a plane normal to the flow.
The remaining hemispheres were formed of gravel (ranging from 2 to
6.4 mm in diameter) as was the floor surrounding all the hemispheres.
Three radii were used for these hemispheres1.27 cm, 2.54 cm, and 5.08
cm. He then used a means of averaging the values found on those two
planes over the surface of the hemisphere and hence integrating that
supposed average pressure to determine the lift. The drag was deter
mined by similar means, using the actual differences of pressure between
front and back of the hemisphere.
The velocity profiles found for the observations were expressed
in a form as (62).
(62)
where
Vj, = friction velocity
y^ = distance at which velocity equals zero
In the only set of velocity data given for the 2.54 cm hemisphere,
Chepil shows his lowest measurement at about 3.5 cm, or about 1 cm
144
these two known points. The diameter dg can be found, aB shown by
Christensen [74] and as outlined in the Appendix, by
n 1 dy
_1 [ Js
d J0 d
e u
(610)
where d = grain diameter ,
y = fraction finer than
3 '
Evaluation of d by this means gives a diameter of 0.178 feet. This
e
equivalent diameter, d^, is based on converting the top layer of
natural material to an equivalent layer. This new layer will have the
same weight of material and the same surface area exposed to flow over
a given bed area sufficient to contain all grain sizes in the natural
sediment.
The theoretical bed, as used earlier, is 0.1662 d below the
e
grain top, or 0.0296 feet in this instance. The lift coefficient based
on u^ is still 0.359 from theory. Finding the ratio of u^ (at 0.0296
feet above the theoretical bed, y ) to the velocity at 0.058 feet above
y enables computation of C based on the latter velocity. Doing so
yields a value of 0.202 for C which compares with the 0.178 from El
Li
Samni's measurements. This value is based on the use of k = d = 0.178
e
feet. However, a better duplication of the natural conditions would be
obtained by using the roughness determined to be valid by El Samni,
0.20 feet. Using this value yields C = 0.199 for the velocity at
0.058 feet. The latter CT is only about 12 per cent above the measured
L
0.178.
180
NT = N0(I + 2J K)
IF (NT .EC. 0) NT = PARK
IF (((XENC(J,K1) X)/H> .LT. 2.0) GO TO 674
LI = 1+2
L2 = J
GO TO 610
NT = NG(IfJ+lfK)
IF (NT .EQ. 0) NT = PARK
IF ( ( (ZH )2 + (XSCRT(3.00))**2) .GT. 1.0) Y2 = 1.00
IF (((ZH)**2 + (XSCRT3.00))**2) .GT. 1.0) GO TO 643
Y2 = 1.0 SQRT(1.0 (ZH)**2 XSQRT(3.00))**2)
IF (J .EQ. JM IT) GO TO 743
IF (((Y2Y)/H) .LT. 1.0) GO TO 663
LI = I
L2 = J+l
GO TO 610
NT = NO(I> J+2 K)
IF (NT .EG. 0) NT = PARK
IF (((Y2Y)/H) .LT. 2.0) GO TO 664
LI = I
L2 = J+2
GO TO 610
C ONLY ONE XPOINT AVAILABLE.
674 SI = (1.0)*X*PI
S1A = (1.0)*X
PULE = 1
PUT = 3
GO TO 605
C ARC INTERPOLATION
673 18 = IBDRY2{J K)
19 = IBDRY2 (J K1 )
XT = SQRT(3.00) SQRT(1.0 (Y1.0)*2 21**2)
XDEL = (XT X)/H
A = SCRT(3.00) XENC(JtKl)
B = SQRT(3.00) XENDIJ.K)
C = SQRT(3.00) XT
C = Z H
THP = (ATAN2(21C) ATAN2(0,A))/(ATAN2CZ,B) ATAN2(DA))
PI = PHI(I9JK1) + THM*(PHI(I8JK) PHI(19,J#K1))
SI = (X)*PI/XOEL
S1A = (XJ/XDEL
PULE =1
PUT = 3
GO TO 605
C ONLY ONE POINT AVAILABLE IN YDIRECTION
664 S2 = (Y)PJ
S2A = (1.0)*Y
IF (J .EQ. JMIT) S2 = 2.0*S2
IF (J .EQ. JMIT) S2A = 2.0*S2A
GO TO 622
C ARC INTERPOLATION FOR PZ(I,J) WHEN PROXIMITY OF SPHERES
602
603
643
743
604
98
The lift force will require integrating the pressure differ
ences existing between pressures on the hemispheres and corresponding
pressures at the base. Call the velocity at a given point at the base
and then integrate the difference. Integration of the terms involv
ing z will yield a term y times the volume of the hemisphere, which is
the buoyant lift. The remaining terras indicate the hydrodynamic lift,
L, as shown below.
L = / / f(q2
surface &
where dA is the elemental surface area projected onto, the horizontal.
To express this in a better form, use the velocity, u^at the upper
most part of the hemisphere, as a reference.
L
_ p 2
2 Ut
where
2 2
X.
surface
dA
(575)
(576)
A more appropriate expression uses lift per unit area, with L divided
by either the total bed area or the projected area of the grain. The
'
former yields X and the latter X .
u
CL 2
u
2
t
X
u
= C
Â£
Lu 2
u
2
t
(577)
83
Sphere 2:
h
2PJ +
PJ1
2
(554)
The values indicated by PJ and PJ1 must be established by
interpolation, since they do not, in general, lie at a lattice point.
The interpolation will utilize points in the lattice with expansion
to be made about the nearer of two surrounding points in the xdirection,
shown in Figure. 27a as J.tl>Kc and
First, consider the case where f3h is less than afQh. Here, PJ
can be expressed as follows, for Sphere 1.
Sphere 1:
PJ
4
,cp
A+l
a' + $+a
a + a + B
o
where
A: (I+1,J+1,K)
A+l: (I+2,J+1,K)
C: (I,J+1,K)
(555)
A similar expression can be written by the other point needed, PJ1,
involving points one increment in the ydirection. In addition, the
same process yields an expression for points located from the second
hemisphere, indicated in Figure 27b.
12
Other works where pressure distributions have been predicted
and checked experimentally could also be shown. The important factor
is not to go into details of numerous cases, but rather to point out
that there is strong historical backing for use of potential theory
to study lift forces. It should be recalled that these earlier uses
of potential theory have employed it to describe the entire flow
pattern. Only two further cases will be studied, the first being the
one case where potential flow was used to describe the entire flow
pattern in an effort to solve the problem of sediment movement.
2.4.5 Jeffreys' analysis.Jeffreys [8] dealt with a single
long circular cylinder resting on a bed in a twodimensional study,
as shown in Figure 4.
Uniform Flow
Figure 4. Jeffreys' cylinder.
Jeffreys developed his work based on the complex potential for
this case,
TTfl
W = rraU coth ()
z
(21)

