On the theory of the turbulent boundary layer

MISSING IMAGE

Material Information

Title:
On the theory of the turbulent boundary layer
Physical Description:
50 p. : ill. ; 27 cm.
Language:
English
Creator:
Rotta, Julius C., 1912-
United States -- National Advisory Committee for Aeronautics
Publisher:
National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aeronautics   ( lcsh )
Turbulent boundary layer   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Turbulent energy, dissipation, and momentum relations are discussed. A procedure is given for computation of turbulent skin friction in boundary-layer flow with pressure gradients. The boundary layer is divided into a region near the wall where viscosity and surface roughness are important, an outer region which is dependent on friction coefficient and pressure gradient, and an intermediate zone between these two which is unaffected by wall roughness, viscosity, and the outer flow. Analytical confirmation is obtained for the empirical fact that turbulent boundary layers are able to overcome a greater pressure rise than laminar ones.
Bibliography:
Includes bibliographic references (p. 40-41).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by J. Rotta.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003778607
oclc - 24079820
sobekcm - AA00006174_00001
System ID:
AA00006174:00001

Full Text
Ult k74-1Wt











NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1344


ON THE THEORY OF THE TURBULENT BOUNDARY LAYER

SBy J. Rotta


: INTRODUCTION
; (By A. Betz)

In view of the high specialization of scientific research many papers,
-. basically important for further progress, are of interest only for a rela-
tively small circle of close colleagues. In normal times, such reports
nevertheless could be published in scientific periodicals without diffi-
culty. The periodicals published papers from various fields and thus
offered to a relatively large circle of readers sufficiently valuable
material. Today, this procedure is faced with considerable difficulties
which can be traced back to two main reasons: Scientific work has devel-
oped enormously so that periodicals had to be greatly increased in number
and volume. Thus, on one hand, it takes the reader a great deal of time
to follow the literature of his special field; on the other hand, sub-
scription to periodicals represents a heavy financial burden. In addi-
tion, almost all scientists, especially in Germany, are greatly impover-
-ished and can no longer carry the increasing financial load; the sale of
periodicals is thereby reduced and the costs rise still further.

Looking for a way out of this difficulty I thought it desirable to
relieve the periodicals, first of all, of reports which address only a
relatively small circle of interested parties and yet, -to be understand-
able, have to be somewhat extensive. For such reports the considerable
cost expenditure required for issue of a good periodical does not pay;
such expenditure is in order only in case of a correspondingly large
circulation. In order to acquaint the few specialists with such reports
and to make those reports accessible for later need, one can economically
recommend only a reproduction method which is relatively cheap also in
case of small circulation. On the basis of these considerations, I have
decided to print such reports which originate in the Max-Planck-Institute
for flow research and, also, a few older reports from this field, which
are no longer available by Rota printing method, and to edit them in
informal sequence as communications of the institute. Herewith, I give
to our colleagues the first issue of these communications. May it fulfill
the tasks described.

Gottingen, March 1950.
Albert Betz

"Uber die Theorie der turbulenten Grenzschichten." Mitteilungen aus
.dem Max-Planck-Institut fUr Stromungsforschung (Gottingen), No. 1, 1950.





NACA TM 1344


OUTLINE


SURVEY

SYMBOLS

1. INTRODUCTION

2. ENERGY BALANCE OF TURBULENT FLOWS

3. EQUATIONS OF THE TURBULENT BOUNDARY LAYERS
3.1. Fundamental Equations
3.2. Momentum and Energy Theorem
3.3. Supplementary Considerations Regarding Boundary-Layer Turbulence

4. THE UNIVERSAL TURBULENT BOUNDARY-LAYER FLOW

5. EXISTENCE OF SIMILAR SOLUTIONS
5.1. Differential Equations and Boundary Conditions
5.2. Properties of the Similar Solutions

6. EMPIRICAL BOUNDARY-LAYER PROFILES
6.1. Velocity Profiles
6.2. Turbulence Profiles
6.3. Dissipation Function

7. NUMERICAL ESTIMATION OF THE SIMILAR SOLUTIONS

8. SUMMARY

9. REFERENCES


SURVEY


As a rule, a division of the turbulent boundary layer is admissible:
a division into a part near the wall, where the flow is governed only by
the wall effects, and into an outer part, where the wall roughness and
the viscosity of the flow medium affects only the wall shearing stress
occurring as boundary condition but does not exert any other influence
on the flow. Both parts may be investigated to a large extent independ-
ently. Under certain presuppositions there result for the outer part
"similar" solutions. The theoretical considerations give a cue how to
set up, by appropriate experiments and their evaluation, generally valid
connections which are required for the approximate calculation of the
turbulent boundary layer according to the momentum and energy theorem.






NACA TM 1344


SYMBOLS


x,y,z coordinates (x, z parallel to the wall; y perpendicular
distance from the wall)

p density
p of the flow medium
V kinematic viscosity of the flow medium

Velocities and pressures, stresses:

U,V,W velocity components of basic flow (average values in
time) (U in x-direction, V in y-direction,
W in z-direction)

U1 flow velocity in x-direction, outside of the boundary layer


u,v,w

p



Txy, xzyz,

TxyT xzyTyz


components of the turbulent fluctuation velocities

mean value in time of the static pressure

fluctuations of the static pressure

normal stresses
(mean values in time)
shearing stresses
(in section 3 following Tx is written Tx = T)


TO wall shearing stress

v* = FOp shearing-stress velocity

cf' = 2(v*/U) local friction coefficient

Turbulence quantities:

E kinetic turbulence energy (per unit mass)

S energy dissipation (per unit mass)

pQ x,PQy pQ components of the energy diffusion (energy
flow per unit time and unit area) (in sec-
tion 3 following Qy is written Q = Q)


D =J S dy dissipation function


>mean values
in time





NACA TM 1344


1 characteristic length of the large turbulence elements
(statement 1 = KY; K 0.4)

k,kq,c dimensionless factors according to equations (3.16),
(3.17), (3.18). (Characterized for the universal
boundary-layer flow in the range 5w 5 y << by the
index "O")

Thicknesses of the boundary layer:

5 total thickness

6y thickness of the sublayer directly affected by the
viscosity and the wall roughness


81 = (1 U/ l)dy disp
0


82 =f (U/Ul)(1 U/Ul)dy



3 = (u/ul) (U/U) dY


lacement thickness



momentum-loss thickness



energy-loss thickness


H12 1/52
Form parameters
H32 53/8 2J


Similar solutions:


I = y/x

m

Ulx
Re = -
x V


stream function

dimensionless coordinate

exponent of the law prescribed at the outer edge (eq. (5.4))


Reynolds number formed with the coordinate x






NACA TM 1344


Empirical boundary-layer profiles:





> form parameters

12. p d(yvo
12 0 bJ ii


Rel -
V


B U 1 In
V* K


constant of the velocity profile near the wall (eq. (4.8))

constant of the outer velocity profile (eq. (6.8))


Reynolds number formed with the displacement thickness 61


Re = C + K


A form parameter in equation (6.12)


G = D/v3 In Re1



1. INTRODUCTION


For evaluation of the flow conditions about a body and in particular
for estimation of its flow drag, the behavior of the flow layer bordering
on the body, which may be either laminar or turbulent, is of very high
importance. Whereas, for the laminar boundary layers, the physical rela-
tions have been clarified and the mathematical problems, too, have been
worked out sufficiently to have calculation methods at disposal which are
serviceable in practice, there are, for the turbulent boundary layers,
above all still problems of a physical kind to be solved.

From the basic hydrodynamic equations, one may derive relations for
the time averages of the flow quantities in turbulent boundary layers
which are similar to those valid for laminar boundary layers. Further-
more, a relation for the energy balance of the turbulent movement is at
disposal which was given first by L. Prandtl (ref. 1). In spite of
these equations, an exact calculation of the turbulent boundary layers






6 NACA TM 1344


is not possible since one has not yet succeeded in setting up formulas
for essential processes in the mechanism of the turbulent movement. Thus
the question arises whether there is, under such circumstances, any sense
in a discussion of the boundary-layer equations. Actually, however, a
few statements are possible on the basis of the means at disposal, if one
considers the following two empirical facts which may be regarded as
absolutely certain today:

(1) The total processes are affected by the kinematic viscosity v
and the geometrical properties of the wall (wall roughness) only in a
very thin layer in the neighborhood of the wall; in the remaining domain
of the boundary layer, the flow appears to be practically independent of
the viscosity and the wall roughness. If the thickness of the layer in
which these influences are effective is called 6, and the thickness of
the boundary layer 5, one has therefore, as a rule, 6 << 6.

(2) Because of w << 5 one may expect the conditions for wall
distances y < 6, to be independent of the flow conditions at the outer
edge of the boundary layer (y --- 5). Inside of the layer by there
exists, therefore, a velocity law affected solely by the geometrical
properties of the wall; the wall shearing stress TO represents the
essential parameter.

With these assumptions, one may separate the influence of the kine-
matic viscosity and that of the wall properties from the other influences.
Thereby, it becomes not only possible to discuss various properties of the
turbulent boundary layers but also to determine empirically, with the aid
of similarity relations, from suitable measurements, the quantities needed
for the development of approximation methods for calculation of turbulent
boundary layers. Such series of measurements with all required quantities
are not available in a desirable form at present; however, it is possible
to perform them with today's test techniques.

In the following sections only flows of an incompressible fluid are
considered which are steady on the average.


2. ENERGY BALANCE OF TURBULENT FLOWS


Since following the motion to the last detail is not possible in
turbulent flows, a statistical treatment must be applied. The flow
quantities will be expressed by arithmetical mean values, and the aver-
aging in time will be simplest where one deals on the average with
stationary flows. The time-averaged velocity with the components U, V,
and W in x-, y-, and z-direction is called basic flow. Superimposed






NACA TM 1344


on it is the disordered turbulence motion fluctuating with time with the
components u, v, and w, which is always three-dimensional even when
the basic flow may be regarded as two-dimensional. The velocity fluctu-
ations cause fluctuations of the static pressure which will be denoted
by p, whereas the time average of the static pressure is expressed
by p.

If one sets up, for the purpose of theoretical treatment, the
Navier-Stokes equations of motion for turbulent flows and performs the
time averaging, a few mean-value expressions formed from the fluctu-
ation velocities, which are to be regarded as new unknowns, remain in
the equations. In the search for further relations, in order to estab-
lish a mutual connection between these unknowns as well as a connection
with the other flow quantities, one can derive from the Navier-Stokes
differential equations further equations which partly convey very inter-
esting insight into the turbulent flow phenomena. It is not the purpose
of this report to discuss this more closely; but an important equation
among those mentioned above, which describes the balance of the kinetic
energy contained in the velocity fluctuations and for that reason is
physically the most graphic one, is utilized subsequently to a very great
extent. Since it is not yet to be found in exact form in literature, it
will be given herein for general three-dimensional flows.

With the occurrence of shearing and tensile or compressive stresses,
kinetic energy is withdrawn from a basic flow which partly reappears as
kinetic energy of the disordered turbulence motion. Let the time average
of this turbulence energy, referred to the unit mass, be



u2 ,+2 + 2
E + + (2.1)
2


with the bars signifying the time averaging. Owing to the viscosity of
the fluid, kinetic energy is withdrawn continually from the basic flow
as well as from the turbulence motion and is converted into heat (dissi-
pation S); moreover, because of the turbulence motion in general, an
exchange of turbulence energy takes place between various points of the
flow space. If one deals with nonhomogeneous turbulence, which is mostly
the case, these exchange processes do not balance and an energy trans-
port pQ occurs which one could compare to a diffusion process. An
energy balance for the coordination of these single effects (as first
formulated in this manner by L. Prandtl (ref. 1)) expresses that the sum
of these contributions equals the total (substantial) variation of the
turbulence energy. For a three-dimensional basic flow with the





NACA TM 1344


components U, V, and W this energy balance reads in the stationary
case quite generally:



PU L-E+ V -- + W T-
o x oy b


Total variation of the turbulence energy


-
A X


ay
oy


SZ + T ( + +
dz x y y 3xy


+yz


Energy withdrawn from the basic flow


-pS


Qy
+ -- +
dy


(2.2)


Dissipation


Ox = P 2V




a, = (2V



az = p (2


Energy diffusion


- 2)


(2.3)


are the time-averaged normal stresses and


Therein


X u + aw+
TXZ + O
^oz 6x/


aw\
+N
4yj


- I6






NACA TM 1344


TLU 6V --
pny -- + v
cy mx/ j




y z 6x)z -


are the time-averaged
expression


shearing stresses. For the dissipation the


(2 2
S =v2 ( +2 (
ox 6y


z + LxW
6z 6x)


+ \V


2
12 W _w


2
+ --
oy/


2
+2 \Z/
dUz


2
+ +


2
avy
Oy)


I2
oz( 0" )


(b6w + 2
+y oz
'dy 6z


+lu +w W2
(6Z ax)


is valid, and the components
have the following form


Qx= -E
\6x







Qz = -v (
z \z


ax


dvu
+ x-
6x


+ 6
6x


Qx' Qy, and Qz


u+ OV + u
Oy -z /


sv2
Tby


+
dz


of the energy diffusion


J(2 + 2 2
2 P/


+ v(2 + 2 + +
2


+ Vu2 +i 2 P +
2 P/


(2.4)


vdx
dx


2
y)


(2.5)


(2.6)






10 NACA TM 1344


Equation (2.2), the derivation of which would be too lengthy here, is
attained by addition of the three Navier-Stokes equations of motion
after they have been multiplied by u, v, and w, respectively. By
application of the continuity equation and several transformations one
finally obtains, after having formed the mean values, the form (2.2)
with the expressions (2.5) and (2.6).

After this explanation which applies quite generally for turbulent
flows, we shall deal with the special problem of turbulent boundary-
layer flow.


3. EQUATIONS OF THE TURBULENT BOUNDARY LAYERS

3.1. Basic equations


In the following, the x-axis is assumed to lie parallel to the wall
and y to be the vertical distance from the wall. The three-dimensional
turbulence motion with the components u, v, w is assumed to be super-
imposed on the components U and V of the two-dimensional basic flow
in x- and y-direction. The boundary-layer equation-for a two-dimensional
flow along a plane wail resulting from the Navier-Stokes equations of
motion then reads with the simplifications introduced by L. Prandtl1


pU + V y -- -- (3.1)
\ x oy 6x dy



The theory developed by L. Prandtl in 1904 at first for laminar
boundary layers starts from the fact that the processes producing the
friction drag take place in a very thin layer on the body. Accordingly,
one may assume, for simplification of the problem, that the velocity
component V normal to the wall is small compared with the component U
acting parallel to the wall; furthermore, the static pressure p may be
assumed to be independent of the wall distance. An estimation of the
order of magnitude then indicates which terms in the equations may be
neglected. In case of turbulent boundary layers the mean value p is
influenced by the velocity fluctuations (p = pO pv2; p = p for y = 0)
in the derivation with respect to x this slight influence is partly com-
pensated by the term -u2/6x neglected in equation (3.1) so that 6p/dx
may be regarded as independent of y.






NACA TM 1344


Furthermore, one uses the continuity equation for the basic flow

dU Vy
--+ o
bx oy


11





(3.2)


The continuity condition must of course be satisfied also by the fluctu-
ating motion u, v, w which is already taken into consideration in the
following formulas. In case of turbulent boundary-layer flows, there
applies for the shearing stress T in the xy-plane the expression


T = p Lu -
( dy U


(3.3)


Herein uv
components
expression


is the time mean value of the product of the
u and v acting vertically to one another.
-puv is also denoted as Reynolds stress.


fluctuation
The


We now include into our considerations as a further equation the
energy balance of the turbulent flows given in section 2. For steady
two-dimensional boundary layers expressions (2.2), (2.5), and (2.6) are
simplified, under the custom
simplified, under the customary assumptions to


(3.4)


pU E + V E = T pS p
\ ox oy) -y oy


+2 L2
+2 -


S O 2


2
+2 )
\oy /


+ 2 z


2
wy vdz
dy dz)


2

\Oz 6x


+ (V- 2
dax 3y


u2 +2 +2 W
v +
2 P


(3.5)


(3.6)


2
See also footnote 1. T is put equal to Txy and Q is put equal
to Qy. The indices may be omitted here as before in equation (3.1) and
later on, since a confusion is quite impossible.


dy


Q (E y





12 NACA 4M 1344 :J


The boundary-layer equations (3.1), (3.2), and (3.4) have to satisfy
the following boundary conditions:


y = O: U = 0; V = 0; u2 =v2 w2 = 0


y -> 5: U ->U; u2 -- 0; v2--- 0; w2 ->0 (3.7)


U1 is the velocity outside of the boundary layer which is assumed to be
prescribed as a function of x.

These relations are valid under the assumption of a sufficiently
smooth wall. For uneven walls the formulation is considerably more
complicated. In section 4 we shall revert to the treatment of rough
walls where unevennesses of a certain mean magnitude are statistically
distributed over the surface.


3.2. Momentum and Energy Theorem

If one introduces the quantities


Displacement thickness 51 = (1 -U)dy (3.8)



Momentum-loss thickness 52 = tu-/ -UJ)dy (3-9)



and wall shearing stress T0 = T for y = 0, one may derive from equa-
tions (3.1) and (3.2) the momentum equation


(U1 2 + U161 dU1 (3.10)
dx. 1 2) 1 1 dx P


which has proved to be advantageous for the approximated integration of
the boundary-layer equation, and which has the same form for turbulent
as for laminar boundary layers.





** .





NACA M4 1344


By integration over y one may develop from equation (3.4) a corre-
sponding energy equation. The partial integration of the left side of
equation (3.4) yields, with use of equation (3.2)


O'(U E +V 6E dy f
S x ay dx0J,


UE dy


(3.11)


Furthermore there applies for the basic flow the relation to be derived
from equation (3.1) (cf. the reports by K. Wieghardt, ref. 2)


J0O


T -dy = p
ay xJo


-Ul2 U2)dy


(3.12)


The diffusion term in equation (3.4) disappears
dy
since the components u, v, w tend toward zero for
for y --- After introduction of the energy-loss



1 dy
"3 =. u1


and of the dissipation function


D = S dy
"0


in the integration
y = 0 as well as
thickness


(3.13)





(3.14)


one then obtains as the energy theorem


1 d (U 3
2 dTx_ 53)


0d_ UE dy
dx


(3.15)


'Energy flow loss of Dissipation' (Increase of the
the basic flow per turbulent energy
unit length flow per unit
length

This equation is significant for the behavior of turbulent boundary
layers. The energy losses of the basic flow essentially are first con-
verted into kinetic turbulent energy which in turn is transformed into





14 NACA TO 1344 4


heat by friction; however, the conversion of the basic flow energy into
turbulent energy, and the transformation of the turbulent energy into
heat need not take place at the same location and at the same time. This
state of affairs is expressed in equation (3.15). Very frequently the
increase of the turbulence energy flow contributes only slightly to
equation (3.15) thus, for instance, in case of ordinary plate flow
without pressure gradient however, there are also cases where this
term gains more significant influence besides the dissipation function.


3.3. Supplementary Considerations Regarding Boundary-Layer Turbulence

By qualitative considerations one obtains a survey of the connec-;
tions still lacking between the quantities E, uv, S, and Q occurring
in the energy balance (eq. (3.4)). We shall limit ourselves especially
to the region y > 5 where the viscosity of the flow medium may be
regarded as arbitrarily small. First of all, the terms with V in equa-
tions (3.3) and (3.6) are hereby eliminated; whereas, in equation (3.5)
the contribution v(6U/dy)2 of the basic flow to the dissipation
becomes negligibly small.

The amount E of the kinetic energy of the turbulence, which is a
quantity suitable for dimensional considerations, is composed of the
contributions of a very large number of turbulence elements of many
different orders of magnitude; however, there exist for turbulence two
characteristic lengths. One is the characteristic length I for the
dimensions of the large turbulence elements, which for boundary-layer
flows is given approximately by the pertinent distance from the wall.
The second is the diameter iR of the smallest turbulence elements,
which is determined by the quotient of the kinematic viscosity and the
mean fluctuation velocity thus, approximately by v/\/K. For the fol-
lowing considerations, the fact is important that the kinetic energy E
is contained chiefly in the large elements and that, therefore, the
momentum and energy exchange phenomena are essentially caused by the
large elements. If the viscosity is sufficiently small or, more accu-

rately, if the Reynolds number of the turbulence is sufficiently
V
large, which is the case in the region y = 85w, we need in our consider-
ations only to refer to the one characteristic length I which corre-
sponds to the dimensions of the large elements to make the total effects
of turbulence independent of the viscosity.

The apparent shearing stress puv caused by the turbulent fluctu-
ation velocities may be traced back to a momentum transport which can be
expressed by the form used by L. Prandtl (ref. 1)

-Tv = kEl d (3.16)
p oy






NACA TM 1344


wherein k is a dimensionless factor. The product k/EZ represents
an apparent kinetic viscosity (cm2/sec) or, respectively, an exchange
quantity concepts first introduced by J. Boussinesq and W. Schmidt.

By considerations similar to those on which equation (3.16) is
based, one arrives for the energy transport Q caused by exchange phe-
ncmena at the expression introduced by L. Prandtl (ref. 1)


Q = -klEz E
dy

Since, however, the turbulence elements of different order of magnitude
have different energy density, an energy diffusion takes place, not only
when an energy gradient is present but is obviously possible also when
the turbulence at adjacent locations differs only by the linear dimensions
or by the structure (for instance, the energy spectrum). With this inter-
pretation the expression

SkqiE3/2)
4Q- (3.17)
ay


is justified,3 which originates by taking the exchange quantity kfEZ
for the energy transport under the differential sign. Prandtl's form
is contained in expression (3.17) as a special case. Here again kq
is a dimensionless factor which like k in equation (3.16) is chiefly
determined by the structure of the large turbulence elements.

In contrast, the energy dissipation is caused mainly by the small
elements. If one combines again the influences dependent on the struc-
ture (that is, the spectrum) in a dimensionless factor cl, one obtains

according to equation (3.5), with omission of the contribution v(dU/Py)",
the relation

E
S = vc1 -


3The expressions (3.16) and (3.17) cannot yet lay claim to full
general validity. A further discussion of these questions is not pos-
sible within the scope of this report and will, therefore, be the subject
of another publication. For the present problem, the expressions (3.16)
and (3.17) are, at any rate, sufficient.





16 NACA TM 13441


Herein cl is determined chiefly by the small turbulence elements and is
independent of the Reynolds number only for very small Cl (cf. ref. 3).
V
For sufficiently large Reynolds numbers, we may express the dissipation
process as a wandering of the kinetic energy (taking its course inde-
pendently of the viscosity) from larger to smaller elements whereby,
however, the energy content of the turbulence motion does not change.
The extent of transformation into heat, occurring almost exclusively in
the smallest turbulence elements, is guided by the amount of energy
supplied to them by the larger elements (cf. the reports by C. F.
v. Weizsicker (ref. 4) and W. Heisenberg (ref. 5)). Thus for large
Reynolds numbers, one may replace in the given expression for S the
kinematic viscosity by an apparent kinematic viscosity of the dimension
EZ one then obtains the relation

E3/2
S = c (3.18)


which is valid for y > 6w. The magnitude of the factor c appearing
therein is determined mainly by the large elements.

The dimensionless quantities k, kq, and c, (appearing in equa-
tions (3.16), (3.17), and (3.18)) which depend on the structure of the
turbulence, will generally assume amounts differing from point to point;
however, for sufficiently large Reynolds numbers, they are independent
of the kinematic viscosity. Their calculation presupposes complete
theoretical mastery of the statistic behavior of the turbulence motion.
This goal, for turbulence research, however, is still far remote. For
the following investigations, k, kq, and c are therefore introduced
formally as functions of the location although without selection of
special statements. However, it will be possible to assume offhand that
they are continuous functions and do not become infinite at any point.

For the characteristic length I in equations (3.16), (3.17), and
(3.18) for the large turbulence elements, there is, with consideration
of the regions near the wall, the expression

I = sy (3.19)

of advantage where K is a general constant. In this form, the length I
is in the region 6w < y << 6 identical with the mixing length introduced
by L. Prandtl (ref. 6). For the sake of simplicity, the expression (3.19)
is used for the entire boundary layer, although the dimensions of the
large turbulence elements for larger distances from the wall no longer
increase in proportion to y. The deviations between the actual dimen-
sions of the large elements and the expression (3.19) one may assume as
taken into consideration in the factors k, kq, and c.






NACA TM 1344 17


By introduction of the relations (3.16), (3.17), (3.18), and (3.19)
into the boundary-layer equations, one may obtain a few statements
regarding the behavior of the solutions; no limitation of the general
validity seems to be connected with it.


4. THE UNIVERSAL TURBULENT BOUNDARY-LAYER FLOW


For wall distances which are small compared with the boundary-layer
thickness 5, the shearing stress does not noticeably deviate from the
value TO of the wall shearing stress and the flow conditions are here
practically independent of the pressure gradient dp/6x. It has already
been mentioned in the introduction that the viscosity and the wall rough-
ness exert an immediate influence on the flow phenomena only in a layer
of the thickness 6 adjacent to the wall. If this thickness & is
sufficiently small, there will certainly exist wall distances y larger
than 6w, yet so small compared with the boundary-layer thickness 6 that
in this region a universal boundary-layer flow results for which all flow
quantities are determined by only two quantities which have physical
dimensions, namely, the shearing stress velocity


v* =-T (4.1)


and the absolute distance y from a plane of reference which practically
coincides with the wall surface. This flow is influenced by the viscosity,
the wall properties, and the pressure gradient 8p/ix only insofar as
they determine the magnitude of v*. Aside from this influence, the flow
in this region is not affected by either the outer boundary conditions or
the wall properties and the viscosity. The velocity variation of the
basic flow is prescribed for 5w < y << 5 by the known relation (ref. 6).


oy

Therein K 0.4 is a universal constant which is identical with the
value K in equation (3.19).

Not only the velocity variation is known, however, but also important
statements are possible concerning the turbulence energy and dissipation.
Within the validity range of the universal boundary-layer flow no value
of" y is in any way distinguished. The structure of turbulence (energy
spectrum, etc.) is therefore similar in all sections parallel to the wall4

With the exception of the smallest turbulence elements.





NACA TM 1344


Owing to this similarity and the equality of the shearing stress TO = -puv,
the energy E has a value independent of the wall distance y so that the
turbulence at different wall distances of this region differs only in its
linear dimensions.

The factors k, kq, c in expressions (3.16), (3.17), and (3.18)
become for bw 5 y << 5 generally valid constants which we shall emphasize

by the subscript "0" (ko, kqo, Co). With -uv = v*2 and I = ny there
follows from the relations (3.16) and (4.2)


E = (43)
(ko/

With this value, the expression (3.17) then yields for the energy trans-
port Q caused by exchange a value also independent of y


k=- _v* (4.4)
ko


which obviously corresponds to an energy flow in the direction toward the
wall. Since, furthermore, under the presuppositions named, the terms on
the left side of equation (3.4) become, in first approximation, small com-
pared with the expressions on the right side and finally (because Q = Const.
the last term on the right side disappears, the dissipation is, for
y << 5, equal to the energy withdrawn from the basic flow:


0 2 d (4.5)
S = U v (4.5)
P Cy oy


Hence results with expressions (3.18), (3.19), (4.2), and (4.3) the
relation


cO = k3 (4.6)


which like equation (4.3) was found by L. Prandtl (ref. 1); on the basis
of measurements, the value of kO was estimated to be kO = 0.56.






NACA TM 1344


The existence of the universal boundary-layer flow in the region
w < y << 5 suggests the division of the boundary-layer flow into a
part near the wall (0 S y << 5) and an outer part (y > 6w). The flows
of both parts merge asymptotically into the universal boundary-layer
flow: The flows of the first part with increasing y, those of the
second with decreasing y. The advantage attained by this division is
that one is now able to investigate the flow phenomena in each part
separately with reduction of the influencing quantities (experimentally
or theoretically) and to combine both parts, as occasion demands, since
both have the same asymptotic variation at the point of junction.

For the part near the wall (0 < y << ), there exists a velocity
law of the general form (ref. 6)


U= v*f ) (4.7)


wherein the function f is dependent not only on but, in
general, also on the wall roughness. The existing experimental results
on smooth and rough walls may be understood and represented in formulas
(ref. 7) directly up to the wall, and with aid of L. Prandtl's mixing-
length expression. Here we are interested only in the asymptotic form
for y > 6B which results from relation (4.2) by integration:


U = v* n V + (4.8)


Therein the constant C is a function of the wall roughness.

The outer part (y > 6,) has to satisfy the boundary-layer equations
given in section 3.1; using the relations named in section 3.3, one may,
however, neglect herein the kinematic viscosity. In flow problems of
the practice, the desideratum usually is the boundary-layer flow, with
the velocity at the outer edge Ul(x) and wall properties and viscosity
prescribed. For theoretical investigations, the problem may be formulated
differently: beside Ul(x), the shearing stress velocity v* is pre-
scribed as a function of x and the desideratum is the wall condition
required in order to produce this variation v*(x). Instead of the
boundary conditions indicated in section 3.1, in this problem the fol-
lowing conditions for lim y -- 0 at the wall (y = 0) must be satisfied
for the outer part (y > 5w) in order to guarantee the connection with
expression (4.8):
2
lim -V = ; E = (4.9)
y --> 0 dy Ky kO





20 NAGA K 1344



Since on one hand the value which corresponds to the local
Ul/
friction coefficient cf'


v)2 cf' (4.10o)
Ul pU2 2



may be estimated quite satisfactorily, according to existing approxi-
mation formulas (for instance, ref. 10), even without exact knowledge
of all boundary-layer details and varies only slowly with x, and since
on the other hand the velocity profile of the outer part in case of
appropriate normalization seems to be dependent on v*/U1 only to a
comparatively slight extent, as will be shown later, a treatment of the
boundary layer in this manner where the outer part is simply determined
with v*(x) and Ul(x) prescribed promises some prospect of success
also for the first-named problem of practice.

The presuppositions for the division into two mutually independent
regions are, in most cases, satisfactorily fulfilled. This is, however,
by no means self-evident and is, therefore, to be checked for the indi-
vidual case. For this purpose, we add the following orders of magnitude:
The thickness w8 is for smooth walls 6, 100 V/v*; the pertinent
Reynolds number of the turbulence for y = 6, is EZ/V ~ 100. For
pronounced wall roughness, bw is determined by the dimensions of the
roughnesses. According to the experiments of J. Nikuradse (ref. 8) on
sand-rough pipes, Ew is approximately equal to the grain size of the
roughness; the y-values are measured in this case from a plane of refer-
ence in which U, on the average, disappears.


5. EXISTENCE OF SIMILAR SOLUTIONS

5.1. Differential Equations and Boundary Conditions


It will now be shown that under certain assumptions so-called similar
solutions exist for turbulent boundary layers, too, similar to the case
of laminar boundary layers that is, solutions for which the velocity
profile along the wall is distorted only affinely. We investigate the
solution of the boundary-layer equations to be expected, with neglect of
the viscosity, in the range y >w In order to satisfy the continuity





NACA TM 1344


condition, we introduce for the basic flow the stream function from
which the components U and V are derived5 by the relations


U = y; V = -Ix


After substitution of this function into the equation of motion (3.1),
there follows

1 --
yTi lx r = -- p (uv) (5.1)
y xy x yy p x y

and the energy equation (3.4) assumes with the relations (3.17), (3.18),
and (3.19) the form


Ex xE = -u cE3/2 + (kqyE3/2)yy (5.2)


Finally, one obtains for uv according to equation (3.16)

-uv = Kk Eyijyy (5.3)


It may now be shown that for velocity distributions prescribed at
the outer edge of the boundary layer of the form

U1 = axm (5.4)

with a and m being constant quantities, and for a prescribed constant
v*/Ul there exist similar solutions for relations (5.1), (5.2), and (5.3)b.
If the flow is unaffected by the viscosity, the geometric similarity of
the flow pattern requires that the boundary-layer thickness for similar


Partial derivatives with respect to x and y in this section are
expressed by subscripts x and y.
6Constant v*/U1 signifies a constant local friction coefficient.
Under what circumstances and to what extent this assumption can be
practically satisfied is shown in section 5.2.





NACA TM 1344


solutions increase linearly with x7. For this reason, we introduce
the dimensionless variable n = y/x and make the statements


4 = axm+l[i f(Ti


E =a2x2m() >


-u, = a22mt(T)


(5.5)


f(Tq), p(q), 0(D) are only functions of the variable 1. In
satisfy the prescribed boundary condition (eq. (5.4)) in case
y, f(n) must for large q tend asymptotically toward a
value that is, lim f'(r) = 0 must be time. Thus, there
Tj -4 m


follows from equation (5.1) for q --4 m


- p = a2,x2m-1
P x


(5.6)


results also directly from relation (5.4) and Bernoulli's equa-
After substitution of equations (5.5) and (5.6) into equa-
(5.1) to (5.3), one obtains after division by

-a2x2m-l a2x2m, a3x3m-1


respectively


2mf' mf2 (m + 1)(T f)f" =

(1 f')2m0 + (1 + m)(f I)s' =

-_f" 3/2 2 kq 3/2


cp = -KkTnIf"


-'p1


(5.7)


'The same results also from the momentum theorem (eq. (3.10)).


Therein
order to
of large
constant


which
tions.
tions






NACA TM 1344


In this system of ordinary differential equations, x
explicitly so that one actually has to expect systems
equations (5.5) where the boundary-layer thickness 5
ary ith x
arly with x


no longer appears
of solution of
increases line-


For unequivocal determination of a solution, five boundary condi-
tions must be prescribed. In order to satisfy the three boundary condi-
tions (relations (4.9)) at the inner edge, one has to put


lim f' = ;
S--4 0 KT


f(o) = o;


o(0)
.ko2


with 90 = c(0). Two conditions at the outer edge of the boundary layer
are added:


lim ;
n ---*


f' = 0;


S= 0


(5.9)


The first insures that the basic flow merges with the prescribed flow;
whereas, the second causes the turbulence intensity outside of the
boundary layer to die out to zero.


5.2 Properties of the Similar Solutions

Owing to the conditions at the inner edge, there appears, in addi-
tion to the parameter m occurring in equations (5.7), as a further
freely selectable quantity the value (0 which like the-local friction
coefficient cf' is according to relation (4.10):


cf TO
2 pU12


(5.10)


The solution of the system of equations (5.7) with the boundary condi-
tions (relations (5.8) and (5.9)) is, therefore, a two-parameter curve
family. The velocity profile of the outer part, most interesting in
these solutions, may be represented in the form


Uv U
v*


f'(Tl)

JSO


(5.11)


.5.8)


For instance, the displacement thickness 81 according to equa-
tion (3.8) is: 61 = xf(o).


9 Ul0





24 NACA TM 1344"


and is dependent on the two parameters m and v*/U1. Likewise, there
results, of course, for the pertaining "turbulence profile," that is
the plotting of the kinetic turbulence energy

E ( () (5.12)
V*2 CPO

over 1, also a two-parameter curve family dependent on m and v*/UI.

For turbulent flows in a pipe or between parallel walls, the
velocity profile corresponding to equation (5.11), plotted over the
wall distance y divided by the pipe diameter or, respectively, the
mutual distance of the walls (so-called "center law"), is independent
of the magnitude of the friction coefficient (compare, for instance,
ref. 8). It seems appropriate to point out this difference between
turbulent pipe and boundary-layer flow. Furthermore, attention should
be called to the difference compared to the laminar boundary layers
where the velocity profile is a function of only one parameter,
namely m.

The solution of equations (5.7), valid only for wall distances y w,
must be supplemented by the wall profile (relation (4.7)) in order to
obtain from it the complete velocity profiles. The condition for the
continuous junction of the outer part to relation (4.7) is obtained by
elimination of the quantity U/v*, with the aid of relation (4.8), from
the asymptotic form

U1- + K \ (5.13)
V* K MUl.

resulting for small n-values by integration of f" according to rela-
tions (5.8). In this manner, one obtains

U1 1 U1 v*\ 1 Ulx
+ in Km, v = n'--- + C (5.14)
v+K v* K =-

The constant K(m, v*/U1) in equations (5.13) and (5.14) may bE deter-
mined, in case of prescribed parameters m and v*/U1, from the system
of equations (5.7).

The solutions of the outer part herein discussed have real signifi-
Ulx
chance only when the Reynolds number Rex = and the wall roughness,
the effect of which is expressed in the quantity C, are such that equa-
tion (5.14) is identically satisfied for all x-values.






NAC A TM 1344 25


For extremely large Reynolds numbers, there exists a linear rela-
tion between C and the logarithm of the length characterizing the
roughness (for instance, of the grain size k)9, so that the right side
of equation (5.14) becomes independent of x when the grain size k is
proportional to x that is, when k/x = Const. For hydrodynamically
smooth walls and for constant roughness where C is a constant, the
condition (eq. (5.14)) cannot be rigorously satisfied for all x-values.
This wbuld be possible only in the case m = -1 which has, however, no
physical significance because then the flow separates from the wall.
However, since x in equation (5.14) appears in logarithmic form, it
will be permissible to regard, for sufficiently large x-values, the
expression on the right side of equation (5.14) as approximately con-
stant from x-interval to x-interval also for arbitrary m. Under this
assumption, one may regard the similar solutions with practically suf-
ficient accuracy as valid for the individual interval also for smooth
walls and for walls with constant roughness. It is, however, essential
that the value

5w 5wv* U1 V
x V v* Ulx


be so small that the function (U1 U)/v* at the point y/x = 5w/x
actually deviates only slightly from the asymptotic form (eq. (5.13)).
Otherwise, the method selected, joining the wall law (eq. (4.7)) to the
solutions obtained with neglect of the viscosity effect, does not lead
to useful results.

Since the required conditions are rarely satisfied in actual cases,
the similar solutions will evoke chiefly theoretical interest. One has
here a type of solution of the boundary-layer equations which offers a
comparatively simple survey and is thus suitable for the study of theo-
retical problems. It could be shown that the solutions of the outer
boundary-layer part depend on v*/U1. On the problem regarding the
extent of this influence, which is one of the next-to-most-important
ones, one can, at the time, obtain information only from experiments,
as will be shown in section 6.

It is perhaps necessary to point out that the only assumption for
the derivation of these theoretical results was that the Reynolds number
of the turbulence should be sufficiently large (except in the thin sub-
layer 6w) so that the viscosity may be neglected in the boundary-layer
equations; aside from the customary boundary-layer simplifications no

For sand-rough walls, there applies, for instance, according to
the experiments by J. Nikuradse (ref. 8): C = 8.5 1 In *k.
K V





26 NACA TM 3134


restricting hypotheses were introduced. The findings thus have general
validity. However, if one wants to determine the solutions of equa-
tions (5.7) numerically, one cannot forego some hypotheses; that is,
one would have to introduce special formulations for c, k, and kq.
Thus, one would, for instance, insert constant values for the factors c,
k, kq. This we shall not do, however. Instead, we shall attempt to
obtain a survey of the solutions by empirical method by using the knowl-
edge attained from existing test results. Since nowadays measuring
series exist where the wall shearing stress was determined by a special
measurement refss. 9 and 10), a sorting of the experimental data can be
undertaken with greater success than was so far possible.


6. EMPIRICAL BOUNDARY-LAYER PROFILES

6.1 Velocity Profiles


Theoretically, for the boundary layer on the plate with constant
external pressure (the constant external pressure appears as special
case m = 0 in the system of eqs. (5.7)), a family dependent on the
local friction coefficient, thus a single-parameter family, would result.
Ul U ,
However, the plotting of 1 against y/5 according to F. Schultz-
v*
Grunow (ref. 11) shows that the profiles within the considered Reynolds
number range may be represented with practically sufficient accuracy by
a single curve. Since the boundary-layer thickness 6 used for the
plotting is a quantity which can hardly be exactly defined, the expres-
sion yv*/!(1Ul) instead of y/5 is introduced as reference value with
use oP the displacement thickness 51. Thereby the abscissa scale is
fixed so that the integral value becomes


SUl Ud i 1 (6.1)
O v4* 5U11

as one can see from a comparison with equation (3.8). In figure 1, the
UI U
values for the flat plate without pressure gradient were plotted
yv*
against log The test points of the smooth plate according to
51U1
reference 11 fall almost into a single curve; nevertheless, close.obser-
vation shows a small systematic influence of the value v*/U1. In con-
trast, the test points of the rough plate show according to reference 21
somewhat large deviations due to the greater variation of v*/U1. This






NACA TM 1344 27


investigation admits the conjecture that the magnitude of the local
friction coefficient is, in case of appropriate normalization of the
y-scale, of only moderate influence on the velocity profile.

The boundary-layer profiles measured for variable course of pres-
sure at the wall may be represented in the same manner. Figure 2, in
which several profiles of the quoted measurements by H. Ludwieg and
W. Tillmann (ref. 10) are represented, shows that the pressure vari-
ation exerts a considerably stronger influence on the profile shape
than v*/U1.

Performance of approximation calculations requires by no means
knowledge of the boundary-layer profiles to the last detail; it is,
on the contrary, quite sufficient to be oriented regarding the rela-
tions between the individual parameters (displacement thickness 51,
momentum-loss thickness 62, energy-loss thickness 53, and others).
Various authors refss. 10, 12, 13, 14, and 15) found empirically that,
for the profiles of turbulent boundary layers, for arbitrary pressure
increase, these relations are almost unequivocal that is, that the
boundary-layer profiles can be described approximately by only one
parameter. Further treatment of test material will be based on this
presupposition. The relation between the prescribed velocity distri-
bution Ul(x) and the profile parameter to be defined more closely is,
at first, not yet established. This relation is ascertained only by
application of the momentum theorem I.eq. (3.10)) and of the energy
theorem (eq. (3.15)) a method which, in principle, has been known for
a long time for the calculation of laminar and turbulent boundary layers
and has been very successfully applied in approximation methods. How-
ever, one should not forget that this type of single-parameter repre-
sentation is no more than an approximation as opposed td the fact that,
according to the theory, even in the simplest case of similar solutions
a two-parameter family is to be expected. The reason for the usefulness
of this approximation lies perhaps in the fact that (as was observed in
the case of the flat plate without pressure gradient) the influence of
one of the two parameters namely, of the local friction coefficient -
is generally probably little noticeable if the y-scale has been suitably
normalized.

The next step is bringing the desired parameters into a form which
permits separate consideration of the influence of the viscosity and of
the wall roughness. For the momentum-loss thickness 52, there applies
according to equation (3.9)


2 f ( 1 dy Od=- dy fl1 U121 dy (6.2)





NACA TM 13 44'


which may also be written as


(6.3)


62 = 61( I i)


The value


l1 2


(6.4)


is, under the assumptions, made practically independent of the velocity
distribution for y < y5. Likewise there results for the energy-loss
thickness according to equation (3.13)


: dy = 2


f 1 dy


J0o(1


- U2 dy +
Ul)


3
-1 dy
Ul


(6.5)


3 = 5 2 3 U I +


I2 Uo L d (


also is practically independent of the profile form for

The profiles of the representation figures 1 and 2
to equation (5.13) for y --) 6y the form

U1 U 1 Iyv + K
v* K 61U1


(6.7)


y < 8w.


have according



(6.8)


53 = 0"


where


(6.6)


*ii


d 61
b 15


0 3 1






NACA TM 1344' 29


with the quantity K of a different amount for every profile form.
.Since, on the other hand, the velocity profiles have for small y-values
(y = y << 6) the form of expression (4.8), there results by substi-
tution of expression (4.8) into equation (6.8)

U1 1
1 In Rel + B (6.9)
K

if

UlI1
Re1 = (6.10)

is the Reynolds number formed with the displacement thickness and B is

B = C + K (6.11)

The quantities I1, 12, and K are pure form parameters which can be
immediately derived from the profile form and do not depend on the form
of the wall law (eq. (4.7)) if the condition 5w << 6 is sufficiently
satisfied. For a single-parameter profile family, there exists an
unequivocal relation between these quantities which can be determined
empirically from existing measurements. For the following consider-
ations, we shall regard I1 as characteristic form parameter and
express the others as function of Il.

In order to obtain some sort of numerical basis for this empirical
relation and thus to get away from the scatter of the test points, also
to facilitate the extrapolation in the region not comprehended in the
measurements, the velocity profile (y 5w) is represented by a simple
approximation expression which starts out from the wall law (eq. (4.8))


U = v* (zn y A) ] (6.12)


A is a freely selectable constant. The thickness 5 of the boundary
layer is defined for y = 6 by the condition U = Ul, so that the quali-
fying equation for 5 reads


U = v* In 6 + A) + (6.13)





30 NACA TM 1344


From expressions (6.12) and (6.13), there follows for the outer part of
the velocity profile

U1 A- y\ 1 yn (6.14)
v* K 8/ K &

For the displacement thickness, there results hence with equation (3.8)

1I U 1 + Al
= l -- U d () 2 (6.15)
6v* v* 5- K

Furthermore, the quadratures yield:

2 3 1 2
1 5U 1 U U ) U 2 + A + A





3 21 11 2 1 3
12 -1U1 J Y d 6 (6.17)
2 1 + A
2

Thereby the relation between II and 12 is given by a parameter
representation which is plotted in figure 3 for K = 0.4 and compared10
with the quoted measurements by H. Ludwieg and W. Tillmann (ref. 10) and
F. Schultz-Grunow (ref. 11).

For the quantity B in equation (6.9),

1 + A
B = C + A n 2 (6.18)
K K K

10The relatively large scatter of the test points for small I1-values
is without practical significance because the term in equation (6.6)
dependent on 12 contributes only a very small percentage to 53; the
scatter is explained by the fact that the experimental Il- and 12-values"
were not obtained directly by quadratures but were calculated backward
from the 51-, 52-, and 53-values determined by quadratures with use of
the experimentally ascertained v*/Ul from equations (6.3) and (6.6).






NACA TM 1344


would result from expressions (6.12) and (6.13). Although the expres-
sion (6.12) renders the velocity profile on the average quite satis-
factorily, deviations do appear in details, which take effect chiefly
in the quantity K according to equation (6.8). Therefore, the B-value
is not satisfactorily represented by equation (6.18); the modified form

A
S+A
B = C + 0.82 A 1 Zn 2 (6.19)
I K It K

is more appropriate, as shown by the comparison with measurements repre-
sented in figure 4 for C = 5.2 and a = 0.4.

Figures 3 and 4 may be regarded as a confirmation for the usefulness
of the representation dependent only on the form parameter I1 and of
the expression (6.12).


6.2. Turbulence Profiles

If one considers use of the energy theorem (3.15), one needs data
one cannot obtain from the velocity profile alone. In order to ascertain
the magnitude of the energy flow, one requires the turbulence profile
which in a dimensionless plotting corresponding to figures 1 and 2 is
represented as E/v*2 over (yv*)/(561U). Herein E/v*2 tends in the
neighborhood of the wall y -> y5 toward the universally valid value
given by expression (4.3) and decreases for y < bw very rapidly to
zero. Although it is fundamentally possible to determine, with known
hot-wire arrangements, the quadratic mean values of all three fluctuation
components experimentally and hence, according to relation (2.1), E
numerically, valuable measurements exist only for the component u
which, it is true, yields the most essential contribution to E. The

longitudinal oscillation profiles u/v* represented in figures 5
and 6 were measured by means of the turbulence-measuring device of
W. Tillmann refss. 16 and 21) developed by H. Schuh. The conjecture
following from the universal boundary-layer flow and 5 << 6 that the

u/v-* tends toward a universal value in the same manner as E/v*2 for
y -4 5w, is only insufficiently confirmed by these measurements. The
reason probably lies in inadequacies of measuring technique.

According to the considerations of section 5, two-parameter curve
families for E/v*2 would result for the similar solutions. If, how-
ever, the influence of the one parameter v*/U1 on the velocity profile
is small, which is probable according to the preceding section, one may





32 NACA TM 134 '-:


conclude with some certainty from consideration of the third part of
equation (5.7) that this parameter exerts only slight influence on the
turbulence profile as well. Figure 6 confirms the correctness of this
reasoning for the longitudinal-oscillation profiles of the plate flow
without pressure gradient for smooth and rough surfaces. Thus, the con-
jecture suggests itself that the quantities of interest in turbulence
proSiles may, like the parameters of the velocity profiles, approxi-
matively be described by an unequivocal relation to the form param-
eter II. This assumption is taken as the basis of the further
investigations.

As could be determined so far, the variation of the turbulence
energy flow mostly does not make a very significant contribution for
two-dimensional boundary-layer flow, so that a somewhat liberal treat-
ment of this influence seems, as a rule, permissible. From a few older
measurements by H. Reichardt (ref. 17) in a rectangular channel, by
H. C. H. Townend (ref. 18) in a square pipe, and by A. Fage (ref. 19)
in a circular pipe, the order of magnitude of the v- and w-variation
components can be estimated. Near the wall, the v-component in particu-
lar is essentially smaller than the u-component; at larger distance from
the wall, the magnitudes of the v- and w-components approach that of the
u-component.

According to definition of the integral expressions practically
independent of the wall law (eq. (4.7))



l, E d and I w U1 U E d(yv* (6.20)
0 v2 1U12 0 v2


the turbulence-energy flow is determined to be


f UE dy = 51U12v* U I T (6.21)


For the presupposed single-parameter condition, IT and IT2 are only
functions of II. According to the existing data, the relation


f UE dy = 0.6551U12v* (6.22)


seems to be useful for the estimation independently of 11; it is, there-
fore, taken as the basis for further evaluations and calculations.





NACA TM 1344 33

6.3. Dissipation Function

For determination of the dissipation function D occurring in the
energy theorem (eq. (3.15)) according to expression (3.14), one will
again attempt by means of the results represented in section 4 to
express separately the influence of the viscosity and of the wall rough-
ness. For this purpose, one may determine S from equation (3.4) and
perform the quadrature for the part near the wall (0 < y < ) if one
puts T/p = v* = Const. and the left side of equation (3.41 equal to
zero which is admissible for small wall distances. One then obtains
for y <<

\ S dy' = v2U(y) Q(y) (6.23)
0

With U according to relation (4.8) and Q according to relation (4.4),
there results, hence, if the upper integration limit lies in the region
,w < y <<


SS dy = v n + C + K (6.23a)
0v k 3

For the outer part of the boundary layer y by one obtains with
the relation (3.18)

S S dy' =- / c d- 8 (6.24)
y yv*/l8U1 (y'v)/( ) /

Since, for 6by y << E/v-2 = l/k02 is valid according to expres-
sion (4.3) and cO = ko3 according to relation (4.6), there results
from equation (6,.24)

SS dy' = v J 1 In Y* (6.24a)


wherein the value of the integral expression

J1 E/v 2 3/2 y 1 y (.25)
J= c d(Y'V* + 1 Zn (y (6.25)
is = yv*/1U1 (y,) 1) \10l/ 511u
I 1U 1 1





NACA TM 1344


because of cO(E/v*23/2 = 1 is independent of the lower integration
limit y if it lies in the range 6wy y << 6. From equations (6.23a)
and (6.24a), one finally obtains for D the form


D = S dy = v*3(. In Rel + G (6.26)

with
k
G = C + + K (6.27)
ko3

We now again assume that for our single-parameter velocity and
turbulence profiles Js, and, therewith for equal wall properties, G
as well, is only a function of the parameter Il described by expres-
sion (6.4). Since nothing is known regarding the behavior of the func-
tion c, except for the region 5w y << 8, J, cannot be calculated
from expression (6.25), even if the turbulence profile is known. Thus,
there remains only the possibility of calculating the function D by
differentiation, by means of insertion of the experimentally determinable
quantities into the energy equation (3.15); this method suffers, however,
from serious uncertainties. The measuring series of F. Schultz-Grunow
(ref. 11) on the plate without pressure gradient could be evaluated quite
satisfactorily according to this method. The result represented in fig-
ure 7 is to be evaluated as a satisfactory confirmation of the correctness
of the relation (6.26).

If boundary-layer measurements with pressure increase are made in a
wind tunnel of rectangular cross section, the occurring secondary flows
represent a disturbance, as shown by a very careful investigation by
W. Tillmann (ref. 16). These secondary flows have the effect that the
flow is not two-dimensional (as had been assumed in the derivation of
eq. (3.15)) but at the location of the measurement usually convergent
with respect to the planes parallel to the wall. A. Kehl (ref. 13) has
shown how to consider in the momentum theorem (eq. (3.10)) a convergence
or divergence influence. In a similar manner, the energy theorem
(eq. (3.15)) for wedge-shaped flow may be ascertained; it is given herein
without derivation (compare fig. 8)


3UE dy
1 d( 38)1 U13 3=D (6.28)
+ f UE dy + (6.28)
2 d 2 X0 + x dxJ o X + X






NACA TM 1344


In the evaluation of the quoted measuring series by H. Ludwieg and
W. Tillmann (ref. 10), the following method was applied: The mean
measure of convergence 1 which is to express summarily the
X0 4 X
secondary-flow effects in equation (6.28), was estimated with the aid
of the momentum theorem given by A. Kehl since all quantities appearing
in it, with exception of the measure of convergence, were determined
experimentally. This measure of convergence then was introduced into
the energy theorem (eq. (6.28)) and, thus, the function D was deter-
mined. In this manner, it was possible to eliminate at least approxi-
mately the effect of the secondary flows. Aside from these measure-
ments, four further measuring series performed by W. Tillmann in the
same wind tunnel but not published were treated in the same manner11.

The result of this evaluation, which for the first time conveys an
indication for the magnitude of the dissipation function as a function
of the profile shape, is shown in figure 9. The scatter is sometimes
quite considerable; however, on the whole, the test points are grouped
fairly satisfactorily about a mean curve. Greater accuracy was hardly
12
to be expected in view of the circumstances described 2. For large
Il-values, the results may be approximatively rendered by

c = 7.5(I1 8.2) (6.29)


In order to make a more reliable determination of the dissipation
function (which is very important for the development of approximation
methods for the calculation of turbulent boundary layers), measurements
would be required for which by avoidance of secondary flows easily sur-
veyable flow conditions exist. Measurements in a rotationally symmet-
rical wind tunnel probably ensure clear conditions. These measurements
would have to include a very exact experimental determination of the
turbulence profiles, for instance, by hot-wire measurements. The reason
why turbulence measurements of boundary layers have been performed com-
paratively rarely can probably be found, amongst other reasons, in that
so far no immediate need for quantitative measurements of this kind
existed.

11The magnitude of the wall shearing stress which had not been
experimentally determined in these measurements could be estimated by
means of the relations given by equations (6.16) and (6.19) and figure 4,
respectively.
12Particularly uncertain are the end points of the individual meas-
uring series which are denoted by "E" in figures 7 and 9 because the
variation of the curve to the differentiated is not exactly fixed at the
end of each measuring series.





36 NACA TM 1344


The energy equation in the form (3.15) will probably stimulate
carrying out of further turbulence measurements.

It would mean an essential progress in the determination of the
dissipation function if not only the wall shearing stress but also the
entire "shearing stress profile" could be determined experimentally.
Attempts to measure the mean value of the product uv, which is
according to expression (3.3) decisive for the shearing stress by means
of hot-wire probes, were made by H. Reichardt (ref. 17) and H. K.
Skramstadl3. Besides, H. Reichardt (ref. 20) has tried to measure
mechanically the mean value uv with an angle probe. Further develop-
ment of methods of this type will be of great advantage for the investi-
gation of turbulent boundary layers.


7. NUMERICAL ESTIMATION OF THE SIMILAR SOLUTIONS


The relations determined from the existing test material may serve
for developing approximation methods for calculation of turbulent bound-
ary layers with arbitrary pressure gradient. Here we shall use them for
quantitatively estimating the conditions for the similar solutions treated
in section 5 with the aid of the momentum equation (3.10) and the energy
equation (3.15).

In consequence of the results of section 5, according to which the
velocity distribution is prescribed in the form of a power law (rela-
tion (5.4)) and the boundary-layer thickness 6 increases linearly
with x, we make the statements


U, = ax"

52 = bx


therein a and b are quantities independent of x. If one takes
into consideration that the form parameters of the velocity profiles
H12 = 51/52 and H32 = 53/52 are, according to presupposition, also
independent of x, there results by substitution of equations (7.1) into
the momentum equation (3.10) and after division by the value (axm)2

2
(2m +- l)b + mbfH12 = ( (7.2)
VU-1


13National Bureau of Standards, Washington, D. C., USA.





NACA TM 1344 37


From the energy theorem (eq. (3.15)), one obtains in the same manner,
with use of relation (6.22)

8R32 v* 3
b- 0.65 -. H,1(3m + 1) = (7.3)
2 U1 Ul1 v*3

For the momentum loss thickness 62, there follows from equation (7.2)


52 (v*/U)2
-- = b = (7.4)
x (2m + 1+ mH12)

Since the calculation of the boundary layer for a prescribed velocity
distribution is troublesome, we choose a more convenient method and
determine for prescribed values of the boundary-layer profile the perti-
nent velocity variation along x, that is, the exponent m. For this
purpose, equation (7.3) is, after elimination of b and with the aid of
relation (7.4), solved with respect to m:


H32/2 (v*/Ul)(D/v*3 + 0.65H12)
m =(7.5)
3H32/2 (v*/U1)2 + H2)D/v*3 + 1.95H12

This equation is evaluated by calculation of the quantity v*/U1 for
assumed values of the Reynolds number Rel = U151/v and of the profile
parameter I1 with the aid of the relation (6.9) and figure 4. From
equation (6.3) then results H12 = 51/62- With equation (6.6) and fig-
ure 3, one may then proceed to calculate 63/61 and therewith
H32 = 63/52. Equation (6.26) and figure 9 make, furthermore, the deter-
mination of D/v*3 possible. With these quantities, it is finally
possible to determine from equation (7.5) the exponent m, from rela-
tion (7.4) the momentum-loss thickness 62/x referred to x, and from
relation (4.10) the friction coefficient cf' = 2(v*/U1)2. In figure 10,
the results of such a calculation are compiled for three different
Reynolds numbers, with the conditions of smooth walls taken as a basis
although they do not exactly satisfy the presuppositions of the similar
solutions.

It is an interesting result that a physically meaningful solution
does not exist for all m-values. This state of affairs is not immedi-
ately evident from the system of equations (5.7); it follows, however,





38 NACA TM 1344


at once from the momentum theorem. If 62 and cf' are to be positive,
one will according to equations (3.8) and (3.9), because of U/U1 1,
always have 51 > 62; thus, H12 > 1. Negative values of 62 and cf'
can occur only when reverse flow appears near the wall. However, in
this case, the boundary-layer theory loses its physical significance
since the flow separates from the wall. According to relation (7.4),
m must therefore be greater than -1/3. Figure 10 shows that the sepa-
ration is to be expected approximately in the range of m = -0.2.14
For comparison, it should be mentioned that, for the corresponding
similar solutions of the laminar boundary layers, the separation takes
place at m = -0.091. This confirms the well-known empirical fact that
turbulent boundary layers can overcome a larger pressure increase than
laminar ones.

Another noteworthy result is the dependence of the profile param-
eter H12 coordinated to a certain m-value on the Reynolds number.
The smaller the Reynolds number, the larger is H12. This dependence
comes about chiefly due to the fact that the wall law (eq. (4.7)) corre-
sponding to the respective Reynolds number is adapted, according to equa-
tion (6.9), to the single-parameter profile of the outer part (y _? 6w)
which is independent of Reynolds number. In this manner, the first-order
effect of the Reynolds number on the velocity profile is included so that
figure 10 actually is based on a two-parameter profile family. The
dependence of the outer profile parts on v*/U1, theoretically proved in
section 5, may be regarded as a Reynolds number effect of the second
order; this effect was not accurately expressed in the calculation for
figure 10. In an investigation by A. E. von Deonhoff and N. Tetervin
(ref. 15) who calculated similar solutions with the aid of the approxi-
mation method for calculation of turbulent boundary layers indicated by
them, a universal relation was found to exist between the exponent of the
the velocity law (eq. (7.1)) and the parameter H12; thus no dependence
on Re existed. However, as figure 10 shows, the influence of the
Reynolds number, the expression of which became possible only after one
had succeeded in the experimental determination of the wall shearing
stress, is rather important for the relation between HI2 and m.

It need not be explained further that corresponding calculations
may be carried out for flows at rough walls as well. For this purpose,
one has merely to perform a conversion of the values B and G intro-
duced in section 6 corresponding to the modified constant C. In
principle, however, these calculations would not offer anything new.

The test data at disposal is insufficient for exact determination
of the m-value corresponding to separation.






NACA TM 1344


8. SUMMARY


As equations of the turbulent boundary layer, this report indi-
cates the customary equation of motion, the continuity equation, and,
in addition, a balance for the kinetic turbulence energy from which one
may derive for approximation calculations besides the known momentum
theorem also an energy theorem for turbulent boundary layers.

Under the assumption (frequently confirmed by test observations)
that the influence of the kinematic viscosity and of the wall roughness
takes immediate effect only in a very thin layer bw at the wall, there
exists within the turbulent boundary layer a region (y, w y << 5) in
which a universal flow prevails which is determined by the magnitude of
the wall shearing stress but, for the rest, is not influenced either by
the wall conditions or the velocity distribution Ul(x) prescribed at
the outer edge. The presence of this universal boundary-layer flow
enables the division of the boundary layer into a part near the wall
(0 5 y 5 w) which is affected only by the viscosity and the wall prop-
erties and into an outer part (y 5w) independent of the viscosity in
which the flow is essentially determined by the velocity distribution
prescribed at the outer edge. The flows in these two parts show a
mutual influence only insofar as the asymptotic behavior of the inner
flow represents a boundary condition for the outer flow.

With the aid of the indicated boundary-layer equations, it can be
proved that, for a prescribed velocity distribution U1 = a xm and a
local friction coefficient which is almost independent of x, similar
solutions exist also for turbulent boundary layers; these solutions
depend on two parameters the exponent m and the local friction
coefficient. The boundary-layer thickness increases linearly with x.

With consideration of the findings obtained, the evaluation of
existing test data then yields empirically the relations between the
various quantities required for application of the momentum theorem
and the energy theorem. Finally, the established relations are used to
perform, with the friction laws valid for smooth walls taken as a basis,
approximation calculations for the similar solutions.


Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics




... :...

40 NACA TM 134 .'.


9. REFERENCES


1. Prandtl, L.: Ueber ein neues Formelsystem fur die ausgebildete
Turbulenz. Nachr.d.Akad.d.Wissensch. in GCttingen. Mathematisch-
physikalische Klasse aus dem Jahre 1945, S. 6.

2. Wieghardt, K.: Turbulente Reibungsschichten. Monographien Uber die
Fortschritte der deutschen Luftfahrtforschung seit 1939, Gattingen
1946. Teil B5 Oder: Ueber einen Energiesatz zur Berechnung lami-
narer Grenzschichten. Ing.-Arch. Bd. 16 (1948), S. 231.

3. Rotta, J.: Das Spektrum isotroper Turbulenz im statistischen
Cleichgewicht. Ing.-Arch. Bd. 18 (1950), S. 60.

4. v.Weizsacker, C. F.: Das Spektrum der Turbulenz bei grossen
Reynoldsschen Zahlen. Zeitschr.f.Physik, Bd. 124 (1948), S. 614.

5. Heisenberg, W.: Zur statistischen Theorie der Turbulenz. Zeitschr.f.
Physik, Bd. 124 (1948), S. 628.

6. Prandtl, L.: Fuhrer durch die Strbmungslehre, 3. Aufl. Vieweg u.Sohn,
Braunschweig 1949, S. 105 ff.

7. Rotta, J.: Das in Wandnahe gultige Geschwindigkeitsgesetz turbulenter
Stromungen. IErscheint im Ing.-Arch.)

8. Nikuradse, J.: Stromungsgesetze in rauhen Rohren. VDI-Forschungsheft
361 (1933). (Available as NACA TM 1292.)

9. Ludwieg, H.: Ein Gerat zur Messung der Wandschubspannung turbulenter
Reibungsschichten. Ing.-Arch. Bd. 17 (1949), S. 207. (Available
as NACA TM 1284.1

10. Ludwieg, H. and Tillmann, W.: Untersuchungen uber die Wandschub-
spannung in turbulenten Reibungsschichten. Ing.-Arch. Bd. 17
(1949), S. 288.

11. Schultz-Grunow, F.: Neues Reibungswiderstandsgesetz fur glatte
Platten. Luftf.-Forschg. Bd. 17 (1940), S. 239. (Available as
NACA TM 986.)

12. Gruschwitz, E.: Die turbulente Reibungsschicht in ebener Stromung
bei Druckabfall und Druckanstieg. Ing.-Arch. Bd. 2 (1931), S. 321.

13. Kehl, A.: Untersuchungen uber konvergente und divergente, turbulente
Reibungsschichten. Ing.-Arch. Bd. 13 (1943), S. 293. (Available
from CADO as ATI 38429.)






NACA TM 1344 41


14. Wieghardt, K. und Tillmann, W.: Zur turbulenten Reibungsschicht
bei Druckanstieg. Deutsche Luftfahrtforschung UM 6617 (1944).
(Available as NACA TM 1314.)

15. v.Doenhoff, A. E. and Tetervin, N.: Determination of General
Relations for the Behaviour of Turbulent Boundary Layers. NACA
Rep. No. 772 (1943).

16. Tillmann, W.: Ueber die Wandschubspannung turbulenter Reibungsschichten
bei Druckanstieg. Diplomarbeit, Gottingen 1947.

17. Reichardt, H.: Messungen turbulenter Schwankungen. Die
Naturvissenschaften, 26. Jg. (1938), S. 404.

18. Townend, H. C. H.: Statistical Measurements of Turbulence in the
Flow of Air Through a Pipe. Proc. Roy. Soc., A. Vol. 145 (1934),
S. 180.

19. Fage, A.: Turbulent Flow in a Circular Pipe. Phil. Magazine,
Ser. 7, vol. 21 (1936), p. 80.

20. Reichardt, H.: Zur-Frage der Schubspannungsmessung in turbulenter
Str6mung. ZAMM Bd. 29 (1949), S. 16.

21. Tillmann, W.: Untersuchungen uber Besonderheiten bei turbulenten
Reibungsschichten an Platten. Kaiser-Wilhelm-Institut fir
Stromungsforschung UM 6627, Gbttingen 1945.





NACA TM 1344


< K 5u UU
10 *U
----rv /< -u, t
*


o Smooth v /U =0.044
a Smooth v/UI= 0.040 4 6
Smooth vU, = 0.037 0 ^
Smooth v'UI = 0.036 4
x Rough vUi = 0.060 O
a- Rough v U J=0.054 -.~ 2
+ Rough vU,1 0-047

2.4 -2.2 -2.0 -1.8 -1.6 -1.4 -1.0 -0.8 -0.6 -.4
W" log ,


Figure 1.- Velocity profiles for constant pressure according to measure-
ments of F. Schultz-Grunow (reference 11) on smooth walls and of
W. Tillmann (reference 21) on rough walls.





NACA TM 1344


-c
log --
6,U,


uI-u
t


Figure 2.- Velocity profiles in case of pressure increase according to
measurements of H. Ludwieg and W. Tillmann (reference 10).





44 NACA .TM 1344.


700


600


,In





II
N


500


400


300


200


100



0


Sx Schultz Grunow, a p/d x = 0
-* '-I o Pressure increase
+ Pressure increase

5 10 15 20 25 31
0oo
F,=j\ v*/ \2,u,/ H,-2 v


Figure 3.- Relation between the profile parameters I1 and 12 according
to relations (6.16) and (6.17) and according to tests (references 10 and 11).
H12 = 1. 2'


K= Q4





*NACA TM 1344


S
0a
0



o.
-
+



t


K =0.4
G=5.2
25




PO


x Schultz Grunow,8p/d x 0
10 --
,++ + Pressure increase
So Pressure increase
5 0 Pressure decrease'

'"_____


H- I Ui
H12 vW


Figure 4.- Connection between the relation v*/U1, the Reynolds number
Rel, and the profile parameter 11 according to relations (6.16) and
(6.19) and according to tests (references 10 and 11) on smooth walls.




' V


NACA TM 1344


0 0.05 0.1 0.15 0.2


SIUI


Figure 5.-


Q3 0.35


Longitudinal-variation profiles for pressure increase according
to measurements by W. Tillmann (reference.16).





NACA TM 1344


Yr. p r 1


V*

f


o Smooth v /UI 0.036
3.5 ___ Rough v/U= 0.048 --
x Rough v'Ul= 0.052
I O


0 0.05


0.1 0.15


Figure 6.- Longitudinal-variation profiles for constant pressure according
to measurements by W. Tillmann (reference 21) on smooth and rough
walls.


S


y v
6 U,


0.35


~hLLLLL


~-~a~ah~J
\~k~~





48 NACA TM 1344, '.',


D



1


+.U 4.2


-- -log Re,

Figure 7.- Dissipation function for the boundary layer without pressure
gradient as a function of the Reynolds number according to measured
results by F. Schultz-Grunow (reference 11). E, end point of the
measuring series.


Divergent flow


* Flow
direction _
T 77~'


Convergent flow


Figure 8.- Designations regarding the energy theorem (equation (6.28)).


E




-0 7



D =v3 (-0.6+5.75 tog Re,)


.2 3.4 ..b 3.am


"






NACA TM 1344


-20


Figure 9.- Relation between the dissipation function according to
relation (6.26), the Reynolds number Rel, and the profile parameter
11 according to evaluation of measured results on smooth walls.
Each series of measurements is characterized by a special symbol.
E, end points of the individual measuring series.






NACA TM 1344 .':


8 1.0
pm


Figure 10.- Similar solutions of the equations for the turbulent boundary
layer on smooth walls.


NACA-L=gIe 2-12-53 -1000


-----4 m





C.;.
."I .>'. -

c*C- ( b

> 10
.s *o*C a5


Z- 0 .0 a
C2 K.' ^ 3hl :
e.. L. a~


lo i J i o0 ( k



00 cd -
B M -d C' D

mm I



'u Em
k Q0


iEi-lS ii"ili Zs u
B.dCaOW;,CU<




u 1 1 *o *s bDa r. 'd "o g ,3
?l~p^^- slti a~ll i S



E-4," G S
mo ci t v
9 ofS
tukagod r.
g 0 0 0 0 0
Zm .3 0;5 N .. VD mts
9-1 v Cd
:E -~I ~
U~ur31Jis I B~~c d t





r -ra 22 v
-.E-r 5-


CUo p ra 0o>QUW


SF.. 3
CUB. a> *- *
"g... '9 s.4 -

*g h-a S -*'
I4 34 -
e 5| | 4| | s
CU U W.^ LI as
'.4 || f E" 4WOU 3z1

a (S C D

i i OC ;


Llj Ft
g Z n ei
u 0 0 ?




Emw B,"w
EE
ou
.2 ar CU0C~




00
(JO C



4E- tm
S E 2 a

uo si"'a'



r-
0> iZ ,C 2
5.S|Ue
P^PII
<~Sz< s S
zo U-Sa-a


d" ca M2
>S a
0 S
B .UCUB!* CUi~


01s .2E g
- B.5 ais

C a a cc -
Ci 0.-
w w ca a 0 CL
W, w 0 5 ~404
CU C



a 0-J*.- ,, a Ci D



cc 2 2,

0 S o t '0 0 00
H P =a- I r- Gh ^ 5.
a- .5 oa
a~ ~ ~ J '

Wii^*O %
L ia m iiil
a 0 r C a O
E- a Cm CO.2


a,'-. C- p4 ;z
Ih SIs



c aa CU 1

- 0~- &


I i cq c. B



B; Cd ff~ t. e a ;i a
I' &- SR : ~
00~

Cjr

4'0 0)U 4;
I o,1' tJsllg |^ i
42 V I |. U ;' C
Cd 0 %. Cd*"

) 0



aCU0CUCC~. -Oo
sl as :|; cd w a




0 O Wo ~ 4,
lli-j Zllw i-C 0
C41 ?; UQ a ^ C W a n **;

Q^^s l. ^ B .!^
I^ 3 ^stinl
(D lil blllla M


oC|&|t !|^: ii I
L) 0 M C bD C r cd z
1 gm 1f ~Lo c 0 r o
E. Iz a) WV. w 7 w
k M 7 '' VbOA=-

45 m co h
Cd B =
o d p ko -e or aow ww
z 0 '1P |0|:1
cd Z .4 0 0-0 >





























84
















am)
NO
CD

0


0



cs




0.- .


(CU
a)



0 ,P
'Wa)


~i












(U

I-r
.0
r
















e
o
c






.n
a
b

u
<
2


o




0.
0
Us


*1




as
a)








S
Cb






- I -


DWI. r
.01 0%
0 a .
05
Cu
rt0 i
L .C! >

La
I ii


$5m c

GO d
- 9.0


z S
crz~QInO


- N m 3-4


CL

br,


a b .
0 Cu "v


.0
E 2, o

^B5 25
E



S. mtw w fl-
02W '~ .2
0 Hd L 2



uOu
~t
0
c
zzolo6u
A a m

Wr
ai 1
i Z F.
- B e z .
zzs 0
^1^U,^S


U Cu 0 C

6Cu06 i Z ^ S e
E02%U~u 0601

E F


Go MEh0 o CU
g c. 2 .6


'0 > 4c c a, U N *
2 ig A 12
CL 6 .)



N'- 0
w cc '0 3




*g .s a j r.0 aj.-

-J Q. 0
0 6 g SL E -






Cu 4,
--3 ; 19 Q0 ^ V-'
o.'CIC Cu g .,








I s WE: l s a^,
il-ii^5 N






C. u V m u r 0.
c~ S b' iMS-



<1-*3C 0 D o -a
CCu






2 I .C-a a .S


rI 04 02 ---


b.
0






0rzZ75 02
f2 w 6 gd4

p in a,P~ r
ti 0 0










E- o
CJ IC


g .40 Z a. .M -
ZZO: O'a P~
rr > 1<* .

5lllSi
^g-|5 |
^Iliiul


0 '
cd axB cd V>S t
$4> I^ 'g c uc i s

w 0P
35 >>, -S III

*t> Z z c 0 a
lagls^Scdsg
CE o ia C ia ru a
S m 0$4i -1,
0t a UN




v01a Na N.C
!,I a= 0 :



s i S^ 0 c, ay





-~~~c $4 Cu
*0
oe cg 0o

.0 C d $40 .a
Cuu DC 0 t- C
C, 0 *r0 Cu 0- 0'
t<3 'W .2 u ,


.a '| 'adl i -- 14

D _l l -to! G, k, -0
Co C d 0 d
QC~ -& 54m~
0. U...


- I I
1 (Lu :.,-
,5 "
=41
01 .a


ol
c c~



I1 I
.L II
P9E _
o C u
Ce


o a
*| -


0, d
.r

2 x 0
s 2 :5 z
CJa iC


- 04 V -4 H


><


0 5
2!~c < guI



m d
.a C
d:C$ k-
il 0l


5? .0^|'

t% S j..4o1
q I r.0 Cu
uou->slg
boaa


E02



.23 E-*t-
o o o a k

0 InI
PP
zzO-iu-ooE


0- $4
c$ t- '- '0


*I 0 d1 M.2
02"' fl'~
S 0 -
$ -.V
: C~
S D'a 0
g >. 3: .a a
Er ps $i^ ^^s
hi C< $4ra
0 '0
> 0<


(D- 0 V >
> a S4 a

ahi- o h a m B
o F>,g 1| 0

> U N
2vr F4 .Bi:= i2Z

S O C

.0 .0~ C .
Cu- a -C 1 3

aX ," 'U m F

C 0 al$ N

io....s sC*u'aNCs'su

m s5
$4C e ~ o au u$


o = -' 4 0 g
ka a SC

*% '0 0 5.
a.= ~ 0 ti u
a-i U..
Bh r ea-sS^ .a8


I -




























4u
4)
5
0


Cd
ci



4)
ga
0
U






00
hfl

i*
co o


1 0
ci -
- $4




41


01


0



o














0 k
io
OC
is








.00."







UNIVERSITY OF FLORIDA
31 I III 11111105 82 8
3 1262 08105 829 8