• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 List of symbols
 Abstract
 Introduction
 Previous work
 Saltation model formulation
 Saltation model results
 Conclusion
 Appendix A. Formulation of force...
 Appendix B. Basset history force...
 Appendix C. Matlab programs used...
 Reference














Group Title: UFLCOEL-95012
Title: Evaluation and development of particle saltation models
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Permanent Link: http://ufdc.ufl.edu/UF00085020/00001
 Material Information
Title: Evaluation and development of particle saltation models
Series Title: UFLCOEL-95012
Physical Description: xiii, 93 leaves : ill. ; 29 cm.
Language: English
Creator: Krecic, Michael R., 1970-
University of Florida -- Coastal and Oceanographic Engineering Dept
Publication Date: 1995
 Subjects
Subject: Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (M.S.)--University of Florida, 1995.
Bibliography: Includes bibliographical references (leaves 90-92).
Statement of Responsibility: by Michael R. Krecic.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00085020
Volume ID: VID00001
Source Institution: University of Florida
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Resource Identifier: oclc - 33662099

Table of Contents
    Front Cover
        Front Cover
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Tables
        Page v
    List of Figures
        Page vi
        Page vii
        Page viii
    List of symbols
        Page ix
        Page x
        Page xi
    Abstract
        Page xii
        Page xiii
    Introduction
        Page 1
        Page 2
        Page 3
    Previous work
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
    Saltation model formulation
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
    Saltation model results
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
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        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
    Conclusion
        Page 68
        Page 69
        Page 70
    Appendix A. Formulation of force components
        Page 71
        Page 72
    Appendix B. Basset history force evaluation
        Page 73
        Page 74
    Appendix C. Matlab programs used for model development
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
    Reference
        Page 90
        Page 91
        Page 92
Full Text



UFL/COEL-95/012


EVALUATION AND DEVELOPMENT OF PARTICLE
SALTATION MODELS





by



Michael R. Krecic






Thesis


1995















EVALUATION AND DEVELOPMENT OF PARTICLE SALTATION MODELS


By

MICHAEL R. KRECIC













A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF
FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF
MASTER OF SCIENCE


UNIVERSITY OF FLORIDA


1995












ACKNOWLEDGMENTS


I would like to especially thank my advisor and supervisory committee chairman, Dr.

Daniel M. Hanes, for his support during my master's research here at the University of

Florida. I would also like to thank the members of my committee, Dr. Robert J. Thieke

and Dr. Ashish J. Mehta, who have helped me through many research and class problems.

Also my thanks go to Dr. Renwei Mei for his help in understanding some of the force

terms.

Thanks are also extended to the staff of the Coastal and Oceanographic Engineering

Department and the Coastal and Oceanographic Engineering Laboratory. Special

acknowledgment is given to Becky, Lucy, Helen, John, and Vernon.

A special thank you is given to my fellow young people, Santi and Leah, Mike,

Marshall, Al, Kenny and Kim, Annette, Craig, Justin, Paul, Mark and Allison, Susan, Chris

and Monica, Mark, Yigong, Bill, Rob and Cindy, Darwin, Eric, Wayne, and Jie.

Finally I would like to thank my parents, Russ and Trudy, for supporting me as I

meander through life. They have always encouraged me to do anything I wanted. That's

probably a good thing since I have yet to decide what I want to do with my life. Also I

would like to thank my two brothers, Matt and Dan, for not ribbing me too hard for

extending my professional student status.














TABLE OF CONTENTS

Page

ACKNOW LEDGM ENTS ........................................................................................... ii

LIST OF TABLES ..................................................................................................... v

LIST OF FIGURES....................................................................................................... vi

LIST OF SYM BOLS..................................................................................................... ix

CHAPTER 1: INTRODUCTION .................................................. ........................... 1

CHAPTER 2: PREVIOUS W ORK ...............................................................................

M echanism s of Grain Saltation .................................................... .......................... 4
Incipient M otion..........................................................................................................4
Particle Saltation .................................................................................................. 6
Force Analysis ......................................................................................................... 8
Gravity Force ....................................................................................................... 8
Added M ass Force ............................................................................................... 9
Drag Force........................................................................................................... ... 10
Shear Lift Force ................................................................................................. 10
Basset History Force .......................................................... .............................. 12
M agnus Lift Force...................................................... .......................................... 13
Applicability of Forces For Different Reynolds Number Ranges.............................. 14
Predictive M odels...................................................................................................... 15
M urphy and Hooshiari (1982) .................................................. ........................ 15
van Rijn (1984) .................................................................................................. 17
W iberg and Sm ith (1985) ..................................................... ........................... 19
Nino and Garcia (1992).......................................................................................... 23
Lee and Hsu (1994).............................................................................................. 24

CHAPTER 3: SALTATION MODEL FORMULATION ............................................ 27

Single Grain M odel................................................................................................ 27
Equation of M option ............................................................................................ 27
Boundary Conditions.................................................... .................................... 31
M ethod of Solution ............................................................................................ 32
Sensitivity Analysis................................................................................................. 32
Random Trajectory M odel......................................................................................... 38








Determination of Initial Angles............................................................................... 40
B oundary C onditions.............................................................................................. 42
Sensitivity Analysis........... ........................................................................ ......... 43

CHAPTER 4: SALTATION MODEL RESULTS ........................................................45

Single G rain M odel................................................................................................ 45
W iberg and Sm ith (1985) Shear Lift....................................................................... 45
Saffm an (1965) Shear Lift ...................................................................................... 49
Adding M agnus Effect..................................................... ................................. 53
R andom Trajectory M odel......................................................................................... 55
Com prison to Previous D ata Set.......................................................................... 55
Comparison of Grain Ejection Methods.............................................. .............. 56

CHAPTER 5: CONCLUSIONS ................................................................................ 68

APPENDIX A: FORMULATION OF FORCE COMPONENTS.............................. 71

APPENDIX B: BASSET HISTORY FORCE EVALUATION......................................73

APPENDIX C: MATLAB PROGRAMS USED FOR MODEL DEVELOPMENT........ 75

R EFER EN CES............................................................................................. .............. 90

BIOGRAPHICAL SKETCH ....................................................................................... 93














LIST OF TABLES


Table Page


3.1 Distribution of Initial Angles ......................................................... ...................41

4.1 Average Saltation Characteristics For Fernandez Luque And van Beek (1976).........56

4.2 Average Saltation Characteristics From Random Trajectory Model........................ 64

4.3 Saltation Characteristics With Grain Ejected With Shear Flow ............................... 67

5.1 Summary of Forces In M odels ................................................ ......................... 69














LIST OF FIGURES


Figure Page


2.1 Critical Shields' Parameter As A Function of Archimedes Number (Sleath, 1984) .....5

2.2 Shear Lift Results From Velocity Gradient Across Grain.......................................... 10

2.3 Magnus Force Results From Velocity Gradient Across Grain And Viscous Effects.. 13

2.4 Comparison of Typical Trajectory and Murphy, et al. (1982) Model..................... 17

2.5 Observed Trajectory and Model Comparison (van Rijn, 1984).............................. 19

2.6 Sign of Form Lift for Different 'P1 Angles of a Body Relative to Flow.................... 21

2.7 Comparison of Predicted and Observed Grain Saltation Trajectory..........................22

2.8 Comparison Between Predicted and Observed Trajectory (Nino et al., 1992) ...........24

2.9 Trajectories With Same Angle and Different Velocity (Lee and Hsu, 1994) .............25

3.1 Force Definition Sketch of Grain Saltation......................................... .............. 29

3.2 D definition of B ed Slope.......................................................................................... 29

3.3 The Saltating Grain Initial Position.................................................................. ..... 32

3.4 Particle Trajectories With A Particle Size Of 0.18 Centimeters................................. 34

3.5 Particle Trajectories With A Particle Size Of 3.1 Centimeters................................. 35

3.6 Particle Trajectories Sensitivity to Particle Rotation ............................................ 36

3.7 Particle Trajectory Sensitivity to Initial Trajectory Angle........................................ 37

3.8 Particle Trajectory Sensitivity to Initial Velocity....................................................... 37

3.9 Comparison Of Variable Bed Roughness Values .............................................. 39








3.10 Simulated Trajectories of Particles With Different Densities ............................... 39

3.11 Typical Random Trajectories With Initial Velocity 5ucr........................................42

3.12 Particle Trajectories Of Same Initial Velocity And Angles Symmetric About The
Vertical Plane ......................................................................................................................43

4.1 Model Comparison With Fernandez Luque and van Beek (1976) ...........................46

4.2 Reynolds Number and Drag Coefficient As Calculated By The Model...................46

4.3 Sim ulated Particle V elocities.................................................................................. 47

4.4 Sim ulated Particle Accelerations ...........................................................................47

4.5 Sim ulated Particle Forces....................................................................................... 48

4.6 Simulated Particle Basset History Force...................................................................48

4.7 Model Comparison With Fernandez Luque and van Beek (1976) ........................... 50

4.8 Reynolds Number And Drag Coefficient As Calculated By Model............................ 51

4.9 Sim ulated Particle Velocities.................................................................................... 51

4.10 Simulated Particle Accelerations ................................................................. 52

4.11 Sim ulated Particle Forces..................................................................... .............. 52

4.12 Simulated Particle Basset History Force............................................................... 53

4.13 Simulated Particle Trajectory With Magnus Effect ................................................. 54

4.14 Simulated Particle Combined Lift Force Effect..................................................... 55

4.15 Simulated Trajectories With Grain Ejected With And Against Shear Flow.............. 57

4.16 Simulated Trajectories With Grains Ejected With Shear Flow............................. 57

4.17 Simulated Trajectories With Initial Velocity, uc................................................. 58

4.18 Simulated Trajectories With Initial Velocity, 2ucr.............................. .............. 59

4.19 Simulated Trajectories With Initial Velocity, 3ucr.............................. .............. 59








4.20 Simulated Trajectories With Initial Velocity, 4ucr............................................... 60

4.21 Simulated Trajectories With Initial Velocity, 5ur................................ ........... .. 60

4.22 Simulated Trajectories With Initial Velocity, 6ucr................................................. 61

4.23 Simulated Trajectories With Initial Velocity, 7ucr............................................61

4.24 Simulated Trajectories With Initial Velocity, 8ucr.............................. .............. 62

4.25 Simulated Trajectories With Initial Velocity, 9ucr............................................ 62

4.26 Simulated Trajectories With Initial Velocity, 10ucr.................................................. 63

4.27 Simulated Trajectories With Initial Velocity, 100ucr......................................... 63

A.1 Particle Relative Velocity ................................................................................. 71












LIST OF SYMBOLS


A cross-sectional area of particle

Ar Archimedes number

b proportionality constant

ci proportionality constant

c2 proportionality constant

CD drag coefficient

CM added mass coefficient

CL lift coefficient

d grain diameter

Fa added mass force

Fb Basset history force

Fd drag force

Fg submerged weight

FiMagnus Magnus lift force

Flshear shear lift force

g acceleration due to gravity

ge effective gravity

H saltation trajectory maximum height

I moment of inertia

ks equivalent bed roughness

L saltation trajectory maximum length

Lb saltation length for random model

AL added saltation length









Re Reynolds number

Re* grain Reynolds number

s specific gravity of particle

S surface area of top half of sphere

Ts time particle travels to complete saltation

u saltating particle longitudinal velocity

ucr critical velocity for incipient motion

Uf fluid velocity

Ufrop fluid velocity at top of particle

UfBot fluid velocity at bottom of particle

UATop relative fluid velocity at top of particle

UABot relative fluid velocity at bottom of particle

uo initial longitudinal particle velocity

u* fluid shear velocity

V volume of saltating particle

Vr relative saltating particle velocity

w saltating particle vertical velocity

wo initial particle vertical velocity

x longitudinal coordinate

z vertical coordinate

zo zero fluid velocity level

P bed angle

0 horizontal plane angle

0c nondimensional Shields' parameter

v fluid kinematic viscosity

tL fluid dynamic viscosity

Q particle angular velocity









no initial particle angular velocity

TI flow angle of attack

T2 related to airfoil centerline curvature

p density of fluid

ps density of saltating particle

Tb bed shear stress

Tcr critical bed shear stress

0 azimuth angle

01 arbitrary surface angle

(2 arbitrary surface angle
AO difference between 01 and 02













Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

EVALUATION AND DEVELOPMENT OF PARTICLE
SALTATION MODELS
By

Michael R. Krecic

August 1995

Chairman: Daniel M. Hanes
Major Department: Coastal and Oceanographic Engineering



The sediment transport mode of saltation is very important in developing bedload

transport theories. If saltating particle trajectories are simulated, a bedload concentration

profile may be obtained. Theoretically, this can be done by determining the amount of

time particles spend in any arbitrary layer above the theoretical bed. Then the

concentration profile may yield a rate of sediment transport. Presently there are

theoretical models used to predict the trajectories of particles.

The theoretical models have been supported by experimental observations.

Laboratory observations of the saltation process in adjustable flumes have been reported.

Saltation characteristics such as trajectory height, length, and longitudinal particle velocity

are measured through the use of high speed photography and standard video imaging

techniques. The data presently available range from saltating sand grains to gravel.

These data are used to evaluate the predicative capabilities of five different theoretical

models. Comparisons are performed by plotting the measured particle trajectories and the

predicted trajectories. Comparisons are also made by investigating the proper application









of forces in the predictive models. The models are able to predict trajectories fairly well

except they usually include too many adjustable parameters.

When examining the predictive models it is found that the shear lift force has a greater

effect on large particles than on small particles whereas the opposite is true for the Basset

history force. The Magnus effect is found to have a large impact on particle saltation.

The initial value of angular velocity is not well known and needs further evaluation. The

particles that are ejected against the flow and with the flow as compared to particles

ejected with the flow only have greater average saltation lengths, heights, and velocities by

approximately ten percent.

A new predictive model is proposed to relate experimental observations and theory.

This model borrows from those previous models as well as adding some new features.

The particle's angular velocity is continuously updated and particles may be ejected into

the shear flow. Sensitivity analyses are performed to determine the relative importance of

the parameters involved. The model is also compared to experimental observations.

It is also deemed necessary to measure actual particle angular velocities to more

appropriately represent the particle's trajectory. In addition, an experiment involving

particles ejecting into a shear flow needs to be conducted. If this future experiment

validates particle ejection into the flow, the shear lift force may need to be reformulated.













CHAPTER 1
INTRODUCTION


Sediment transport may occur in several different modes. These modes depend on

the bed shear stress and the characteristics of the sediment. They include rolling, saltation,

suspension, and sheet flow. With increasing shear stress the transport progresses as stated

above. Rolling is where the grains maintain contact with the bed and are driven by grain-

to-grain forces. Suspension occurs when the bed shear velocity exceeds the particle fall

velocity. This causes the grains to remain supported in the fluid column by turbulence and

other fluid forces. Sheet flow may occur if the shear stress is very high and results in a

layer of grains, usually 5-10 grain diameters thick, moving above the bed in unison. All of

these modes of transport may occur simultaneously if the shear stress is sufficiently large.

The transport mode saltation is reviewed herein. The term saltation was first used by

Gilbert (1914) and comes from the Latin word "saltare" meaning to leap or dance.

Saltation occurs when the shear stress is great enough to allow a grain to overcome

gravity but not so great as to allow the grain to go into suspension. That is, saltation is

analogous to a ballistic trajectory.

Many have argued that sediment transport may be divided into two types: bedload

and suspended load. These two types of sediment transport are distinguished by the

dominant force mechanisms. Grain-to-grain forces are the dominant mechanisms in

bedload transport and fluid forces dominate suspended load transport; therefore, rolling

would be considered bedload transport and obviously suspension would be considered

suspended load transport. Saltation, on the other hand, does not fit neatly into either of

these categories. A grain is driven into saltation by another grain impacting it or by the

grain rolling and striking a neighbor grain in front of it, forcing the striking grain to move









into the fluid column. This scenario prescribes a grain-to-grain interaction; hence,

saltation may be considered bedload transport. Once the grain leaves the bed, however,

fluid drag, inertia, and other fluid forces act on the grain. Therefore, saltation could be

considered as suspended load transport. It is unclear how to categorize saltation in the

context of a bedload/suspended load dichotomy.

Perhaps more important than what category saltation fits, is the fact that it plays a

significant role in sediment transport. According to Bridge and Dominic (1984), saltation

is the dominant mode of bedload sediment transport. Rolling and sliding of grains does

not occur as often. This mode usually appears between individual saltation events (Bridge

and Dominic, 1984).

Saltation has been seen while conducting physical models. A flume, with an

adjustable bed slope, is used for these experiments. The bed roughness is set by

permanently placing particles on the base of the flume. Then bedload grains are allowed

to move freely in a shear flow. If the shear stress is great enough, saltation occurs.

Photography and video have been used to record grain flow characteristics. The initial

velocity and angle of a saltating grain is estimated through the use of photography (Abbott

and Francis, 1977) and video (Nino and Garcia, 1992). These quantities are only

estimates because it is difficult to see a grain leave the bed. The photographs and video

track a grain travel through its trajectory. Thus, a set of saltation trajectories are obtained

along with some initial condition estimates. Attempts are then made to match the paths

with numerical models.

The usual technique for developing a predictive model is to define all of the forces

acting on a grain, using numerical simulation, which includes the 'law of the wall' fluid

velocity profile and assuming ejection of grains from the bed with an initial velocity and

angle. The forces that act on a grain include gravity, drag, inertia, lift, Magnus lift, and

Basset history force. Some combination of these forces is used by most researchers

depending on the particle size and flow intensity. An equation of motion is developed and









solved numerically to yield a particle trajectory. From many trajectories, a concentration

profile may be obtained. Finally, a bedload transport model is developed from the

concentration and velocity profiles (Wiberg and Smith, 1989).

The focus of this study is to evaluate the existing models on their ability to predict the

trajectories of saltating grains. The next step then is to develop a new saltation model.

This model incorporates the best aspects of the earlier models and adds some new

features. The single grain model yields a particle trajectory as well as presenting figures

that demonstrate the relative importance of forces acting on the particle. In addition, a

random trajectory model is made as an extension of the single grain model where

numerous saltation paths are found. The grains are ejected against and with the shear

flow. This is the first time a particle saltation model will include grains ejecting into the

flow.













CHAPTER 2
PREVIOUS WORK



Mechanisms of Grain Saltation


Incipient Motion

Incipient motion occurs when the shear stress is great enough for the sediment to be

carried as bedload. That is, the grain begins to roll or slide along the bed surface. If the

shear stress increases further, the grain will gain more momentum and eject from the bed.

The grain will then travel in a ballistic manner and eventually strike the bed. It will either

rebound from the bed again or eject another grain through a momentum transfer. This

process continues until the shear stress lessens.

A nondimensional Shields' parameter is used to determine incipient motion. It is

defined as


0 = ( b equation 2.1
p (s 1)g d



where Tb is the critical shear stress, p is the density of the fluid, s is the specific gravity of

the grain, g is gravity, and d is the grain diameter. Shields (1936) plotted Oc values versus

grain Reynolds number (Re*),


u,d
Re, = equation 2.2


where u is the shear velocity of the fluid and v is the cinematic viscosity of the fluid.

where u. is the shear velocity of the fluid and v is the kinematic viscosity of the fluid.










For small grain Reynolds numbers, the Shields parameter varies linearly with the

Reynolds number. As the Reynolds number increases, the critical value goes through

some transitions with a minimum of approximately 0.03 at Re. equal to 10. For large

values of Re*, the critical value increases to 0.058 (Fredsoe and Deigaard, 1992).

Figure 2.1 shows the Shields' curve plotted against Archimedes number (Ar),




Ar = "2 d equation 2.3




where ps is the density of the particle.


100









_10

.-\








10-2

100 101 102 103
Archimedes Number

Figure 2.1 Critical Shields' Parameter As A Function of Archimedes Number (Sleath,
1984)









The Archimedes number can be a more convenient formulation to use if the shear

velocity is the desired quantity. That is, the original Shields' curve plotted versus the

grain Reynolds number requires iteration to obtain the shear velocity since it appears on

both axes. With Archimedes number replacing the grain Reynolds number, the shear

velocity is obtained directly.

Particle Saltation

When critical shear stress has been surpassed, sand grains will continue to build

momentum. As previously stated, the grain will roll along the bed and strike another

grain. This may cause the rolling grain to leave the bed and enter the shear flow. Once in

the fluid column, the grain will follow a ballistic trajectory unless the flow shear stress is

great enough to induce suspension. This situation can be attributed to drag forces. The

flow can also lead to lift forces due to the shearing across the grain surface that cause the

grain to leave the bed as well. The hydrodynamic forces, shear lift and Magnus lift can

account for this phenomenon (Nino, Garcia, and Ayala, 1992).

Einstein (1950) examined saltation in deriving his stochastic bedload formula. He

assumed that the particle's saltation length was proportional to grain size. It was later

found that the saltation length is inversely proportional to grain size.

Bagnold (1973) investigated saltation in water. He defined saltation as particles

transported by consecutive 'hops' over a bed driven by a fluid. He stated that the particles

'hop' by contacting other grains and the bed. The initial momentum comes from a grain

contacting the bed and rebounding higher into the flow.

Fernandez Luque and van Beek (1976) performed experiments in a flume with

different bed slopes. They were able to measure the mean critical bed shear stress, rate of

bedload transport, average particle velocity, and the average length of individual

saltations. This was accomplished through the use of high speed photography. Two

different sand types, gravel and magnetite were studied. When they compared their model









results, they concluded that a lift force was needed to explain the observed saltation

characteristics.

Abbott and Francis (1977) also conducted a series of experiments of particle saltation

in water. They too were able to obtain mean trajectory lengths and heights in addition to

the mean particle velocity with high speed photography. The particles that were studied

ranged in size from five to ten millimeters. Natural and artificial particles were both

utilized. The conclusion drawn again was that there were grounds for the existence of a

force that opposes gravity when a particle is in a vertical gradient of horizontal velocity.

Chepil (1958) examined the ratio of drag and lift forces on a fixed particle under the

influence of wind. At different wind intensities, the drag and lift forces on the particle

were calculated to determine a ratio. In addition to the lift force Chepil (1958) calculated,

a spinning lift force was also investigated. White and Schulz (1977), while inspecting

saltation in air, found that particles traveled higher and farther than if they were in a

vacuum. First they tried to explain the additional height and length with drag only. This

force proved insufficient. They concluded that the extra height and length was a result of

a Magnus or spinning lift force based on the work done previously by Rubinow and Keller

(1961). The latter derived the Magnus lift force and moment on a spinning sphere in a still

fluid.

Saltation may also occur when many layers of grains are in motion on the bed, in what

is commonly called sheet flow. Sheet flow conditions occur when the shear stress is

substantially greater than that required for grains to go into suspension. When sheet flow

is occurring, all modes of transport, rolling, saltation, and suspension may be taking place

as well. Sheet flow can cause grains to saltate through an event analogous to popping

popcorn being ejected from an air popper. The popped kernels are 'ejected' from the air

popper at different angles and speeds. As they strike the popcorn already in the receiving

bowl, they in turn eject popcorn at random angles and speeds. A similar effect can be









applied to sheet flow with sand grains striking each other. That is, the grains would be

ejected from the bed randomly. This can lead to grains saltating.

There is some debate as to what angles the saltating grains eject from the bed. Owen

(1964) examined grain saltation in air. When looking at the initial conditions for a grain

ejecting from the bed, he assumed that particles had nearly a vertical initial trajectory.

Since the height of a particle is related to the square of the vertical velocity, only those

particles that leave the bed nearly vertically with a certain initial speed will be able to

saltate (Owen, 1964). Others like White and Schulz (1977) believed that grains eject at

angles ranging from 30 to 69.5 degrees. This was also observed in air.

Other researchers have tried to derive a set of equations to describe the motion of

a particle in saltation from bed ejection to bed impact based on the fluid forces. Those

who developed models include Murphy and Hooshiari (1982), van Rijn (1984), Wiberg

and Smith (1985), Nino and Garcia (1992), and Lee and Hsu (1994). All of the models

explored by these researchers are from the Langrangian perspective. These models are

described in the section titled "Predictive Models."



Force Analysis


The search for forces has been motivated by the fact that computed trajectories based

on measured initial velocities and angle did not agree well with measured trajectories.

Models based on drag and gravity proved insufficient, so others were sought. Some of

these other forces are agreed to be significant while others can be ignored in certain

situations. Below is list of forces that researchers have used in saltation modeling.

Gravity Force

A force that all researchers can agree is the influence of gravity on the path of a

particle. Gravity always opposes the vertical ejection of a grain from the bed and usually









helps the horizontal ejection of a grain from the bed. The effect of gravity is usually

written as a submerged weight,


FG = (P p)gV equation 2.4



where V is the volume of the grain (van Rijn, 1984). Obviously, this force increases with

increasing grain size because of its dependence on the volume of a grain.

Added Mass Force

This force arises from the relative accelerations of the grain and the fluid. A

submerged body induces accelerations on a fluid if the body is moving with an acceleration

relative to the surrounding fluid. It can be thought of as a body having an 'added mass' of

fluid attached to its own mass when a submerged body induces an acceleration to some of

the surrounding fluid (Patel, 1989).

In general, the added mass force may be written

V fdu (duv dua\
FA du = p -du equation 2.5
Sdt dt dt)



where uf is the fluid velocity, u is the grain velocity, and CM is the added mass coefficient.

The added mass coefficient is defined as a ratio of the additional mass of fluid that is

accelerated with the particle to the mass of the displaced fluid by the particle (Dean and

Dalrymple, 1984). For a sphere, CM equals 0.5.

For a steady, shear flow, the added mass force may be reduced to


FA = dCV equation 2.6
dt









where the force acts opposite the grain's acceleration. The inertial effects are excluded.

Drag Force

A drag force is a fluid force exerted on a body. More specifically, it is a net force in

the direction of the fluid relative to the sphere due to pressure and shear forces on the

body. The drag force on a particle may be written as


F, = CDA pV,2 equation 2.7
2

where CD is the coefficient of drag, A is the cross sectional area of the grain normal to the

force, and Vr is the relative velocity (Fredsoe and Deigaard, 1992). The drag coefficient is

a strong function of Reynolds number and shape; thus, it is generally not constant. There

have been many empirical formulas such as those found by Graf (1984) and Morsi and

Alexander (1972). As to which formula is the best depends on the range of Reynolds

numbers that are encountered.

Shear Lift Force

This force is a direct result of a shear flow. The particle or sand grain develops a

pressure gradient across it and as a result causes a lift force. This phenomenon can be

attributed to the Bernoulli effect where the lift force acts up if a positive gradient is

present and acts down if a negative gradient is present. That is, the lift force acts up if

Ufrop is greater than Ufot and down if UfBot is greater than ufTop.



fr (top) ur (top)


uf (bot)


ur (bot)


Figure 2.2 Shear Lift Results From Velocity Gradient Across Grain









The shear lift force may be written as


F, (shear) = CLA p(u2To u ) equation 2.8



where CL is the lift coefficient and A is the cross-sectional area of the grain normal to the

force. The UATop and UABot are the relative velocities evaluated at the top and bottom of the

grain, respectively (Wiberg and Smith, 1985).

The lift coefficient has been related to the drag coefficient by Chepil (1958). He

conducted experiments involving evenly-spaced hemispheres and allowed wind to flow

over them. He then proceeded to calculate drag, lift, and the ratio of lift to drag for

different wind speeds. It was determined that ratio was approximately equal to 0.85;

therefore, CL equals 0.85 times CD (Chepil, 1958).

Based on the above definitions, in a wall-bounded shear flow, the lift force will act

to lift the grain off of the bed when the local fluid velocity is greater than the grain

velocity. If the grain velocity is greater, the lift force will cause to the grain to accelerate

toward the bed as shown in figure 2.2.

Another definition of the shear force was developed by Saffman (1965). This

expression is for a sphere moving in a viscous flow. It is


S du 2
F,(shear)= CLpt2d2V 'z equation 2.9
Sdz )



where CL is a lift coefficient but different than Chepil's (1958). Saffman determines his lift

coefficient as a function of the grain Reynolds number, Re*; CL equals 1.6 when Re. is less

than 5 and equals 20 when Re* is greater than 70. CL varies linearly between those Re*

ranges.









The two shear lift forces are very close even though the lift coefficients are different

(Lee and Hsu, 1994). The Saffman expression does not have a negative lift associated

with it since the velocity gradient is always positive. That is, this lift force definition

always has positive lift in the plane normal to bed even if the grain is at a higher velocity

than the surrounding fluid (Lee and Hsu, 1994).

Basset History Force

A. B. Basset (1888) first acknowledged that a particle's history had a role in the

present particle path; hence, the force bears his name. Mei (1995), as told to the author,

described the force as

... derived from the diffusion of vorticity generated at the surface of the
particle at a rate proportional to the particle's relative acceleration. Since
the diffusion rate is finite, this means that the force is dependent on the
history of the particle motion. (Mei, Personal Communication, 1995)

Consider a sphere in a steady fluid and then instantaneously increasing the flow to

some higher value. It takes some 'time' for the boundary layer on the sphere to adjust to

the new flow intensity. This 'time' is accounted for through the Basset history force.

The Basset force is defined as


Sd 2 du duf
F f- dt dt dt equation 2.10
0 S



where pL is the dynamic viscosity, Ts is the total saltation time, and T is a dummy variable

(Mei, 1994). This force has both a vertical and horizontal component. For steady, shear

flow, the fluid acceleration is zero. So the term can be simplified to be a function of the

grain acceleration.








Magnus Lift Force

In addition to the shear lift force, the Magnus force results from the velocity gradient

across the grain, too. This force, named for Heinrich Magnus who first discovered the

phenomenon in 1853, is a pressure force due to the circulation around a spinning sphere

(Murphy and Hooshiari, 1982). As a result of viscous effects, a rotation is supplied to the

particle. This force accounts for the different types of pitches in baseball such as the

curveball and the slider (Munson, Young, and Okiishi, 1990). The shear flow induces a

rotation which causes the grain to saltate higher and further than without the inclusion of

this term. If the velocity gradient is positive, the grain will rotate clockwise (cw); hence,

an upward lift. If the velocity gradient is negative, the grain will rotate counterclockwise

(ccw); hence, a downward lift. See figure 2.3 below.


ur (top) ur (top)

grain ) FL grain FL

uf (bot) uf (bot)



Figure 2.3 Magnus Force Results From Velocity Gradient Across Grain And Viscous
Effects



For a moving sphere in a shear flow, the force is expressed as



F, (Magnus)= 8 d3pV Q-- equation 2.11
8 2 dz



with Q as the angular velocity of the grain with units in rad/s (White and Schulz, 1977).

This force was developed from the work of Rubinow and Keller (1961). Rubinow and

Keller (1961) looked at rotating a moving sphere in a still viscous fluid with low Reynolds









numbers only. Based on their analysis, the Magnus force was independent of viscosity.

From this they derived the form of the Magnus force and the moment acting on the

sphere. The force has been described previously. The moment has the form



Moment = I d= -_ 3 1 dUf equation 2.12
dt 2 dz



where I is the grain's moment of inertia. This moment equation is solved simultaneously

with the equations of motion to constantly adjust the grain rotation to that induced by only

the fluid. Note that the dynamic viscosity appears in equation 2.12. It acts to dampen the

effects of the initial particle rotation to that of the fluid. It must be said that the Magnus

force and moment equations are only valid for small Reynolds numbers. They are used in

this shear flow analysis only for the purposes of determining the effect the Magnus force

has on improving the correlation between experiments and theory (White and Schulz,

1977).

Applicability of Forces For Different Reynolds Number Ranges

The drag, added mass, shear lift, Magnus lift, and Basset history forces have been

formulated as previously published in the literature. The drag force, as defined here, is

valid for all Reynolds number flows. The form of the shear lift force was originally verified

for turbulent flow. Also, theoretically a solid sphere in an inviscid fluid has an added mass

coefficient value of 0.5; so, the added mass force defined herein is applicable to large

Reynolds number flows, too. From Rubinow and Keller (1961), the Magnus force was

derived for low Reynolds number flows only. The Basset history force is also not well

known for low Reynolds number flow (Nino and Garcia, 1992). For analysis purposes,

the forces are extended to the entire range of Reynolds numbers. This follows the

previous research procedures as done by Wiberg and Smith (1985) and Nino and Garcia

(1992) which have yielded satisfactory results.












Predictive Models


Murphy and Hooshiari (1982)

As mentioned earlier, Murphy and Hooshiari (1982) developed a model of particle

saltation. They obtained their own data set based on marble saltation with a diameter of

1.57 cm and a density of 2.49 g/cm3. The basis of their model was to neglect those forces

proportional to the relative velocity so that the result is a particle moving in a vacuum

under the influence of an equivalent gravity, ge,


2g(p, p)
ge = ( p) equation 2.13
2p, +p

The simple dynamic model, as it is called, yielded saltation characteristics such as height,

length, and duration of the trajectory. They are for height


2
H= equation 2.14
2g,

for length


L 2uo equation 2.15
ge

and for duration


2w
T 2wo equation 2.16
ge

where Uo is the initial horizontal velocity of the sphere and wo is the initial vertical velocity

of the sphere. They also included an additional length to the length of a trajectory by

including the effect of added mass due to fluid shear. This length is given by










equation 2.17


where


b=240p(p, p)
(2p, +p)2


equation 2.18


equation 2.19


with u* as the fluid shear velocity. The extra length to the trajectory comes from the

logarithmic fluid velocity profile given by


Uf = 2.5u.ln ..j


equation 2.20


where z is the vertical position of the center of the sphere and zo is the vertical position

where the fluid velocity is zero. The initial conditions, initial vertical and horizontal

velocity, were both based on the first observable, short, vertical displacement of the

marble from the bed.

The breakdown in the simple dynamic model is the fact the drag and lift both increase

as the initial vertical velocity of the sphere increases. According to this model, the

saltation height increases as the square of the initial vertical velocity. This means that the

height is not accurately predicted. Murphy and Hooshiari (1982) acknowledge this aspect.

They, however, maintain that the length of a trajectory is predicted accurately. They

needed to add the drag force back into the particle equation to more accurately match


wWubb' 1I l,
ALI -tah-


C1 gd 2i
C +W2











their obtained data set. After doing this, they believe the lengths of the trajectories are

matched fairly well.


2.5 I


2

E
E1.5
0o
I

1



0.5


Length (cm)

Figure 2.4 Comparison of Typical Trajectory and Murphy, et al. (1982) Model




From figure 2.4, the model appears to describe the path of a marble fairly well;

however, it may be too much of a simplification to exclude the lift forces entirely since

those forces may determine the height of a particle trajectory.

van Rijn (1984)

A model developed by van Rijn (1984) included added mass, gravity, drag, and a

shear lift force. The lift force used was the one developed by Saffman (1965) as described

previously in the section "Force Analysis." Recall that the expression was valid only for a

sphere moving in a viscous flow. The shear lift force has two components: one in the

plane normal to the bed and one in the plane horizontal to the bed. The Magnus force was


0 O






oo
00






0-

0 Experiment
m Model












neglected because Saffman (1965) showed that for a viscous flow, the Magnus force was

an order of magnitude smaller than the shear lift force (van Rijn, 1984).

The fluid velocity was given as the standard logarithmic profile with the velocity equal

to zero at the bed height, zo


z0 = 0.11 + 0.03k, equation 2.21




where ks is the equivalent bed roughness. The equivalent bed roughness is adjusted to

allow the model to match observed trajectories (van Rijn, 1984).

The initial conditions were obtained through others' observations. The initial

velocities of the grain used were gathered by Abbott and Francis (1977). They found that

the average initial velocity was approximately 2u*. White and Schulz (1977), looking at

saltation in air, found that the velocity varied in the range u* to 2u*. The take-off angles

from the bed were observed to vary from 300 to 700 (White and Schulz, 1977).

The theoretical bed is located at 0.25 times the grain diameter, d, below the layer of

grains that constitute the plane bed. The saltating grain is located 0.6 times the grain

diameter above the theoretical bed. Now the equation of motion can be solved

numerically.

A typical model trajectory is shown in figure 2.5 with an observed trajectory provided

by Fernandez Luque and van Beek (1976). The particle diameter, d, is 1.8 mm, s is 2.65,

u* is 0.04 m/s, the kinematic viscosity is 0.01 cm2/s, and the initial velocity is taken as 2u*.

The adjustable parameters in the model are the lift coefficient, bed roughness, and initial

velocity and angle. For the figure, the bed roughness, ks, is a grain diameter and the lift

coefficient, CL, is equal to 20. The initial angle is 450. The Fernandez Luque and van Beek

trajectory is probably wavy because of the upward fluid forces due to turbulence (van

Rijn, 1984).




















I
E2.5
a,
I
c= 2
0
U)
C,
1.5
0
z


0 5 10 15 20 25 30
Nondimensional Length, L/d

Figure 2.5 Observed Trajectory and Model Comparison (van Rijn, 1984)




Van Rijn (1984) was able to conclude from his model that small particles are

dominated by drag and therefore have long, flat trajectories. Large particles are

dominated by the shear lift force, particularly near the bed, and therefore have short, high

trajectories.

Wiberg and Smith (1985)


Wiberg and Smith (1985) derived an equation of motion that included gravity, added

mass, drag, shear lift force, Magnus lift force, and a form lift force. All but form lift have

been described previously under "Force Analysis." There is only a vertical component of

shear lift according to Wiberg and Smith (1985). The shear lift force coefficient that is

used is held constant at 0.2 as a representative value of the lift to drag ratio. In addition,

the Magnus force is defined here as










F (Magnus)= 27t pvQ equation 2.22



which is the theoretical form of an infinite, rotating cylinder in a two-dimensional flow

(Batchelor, 1967). They use this as an approximation of the force on a sphere. Wiberg

and Smith (1985) then proceeded to state that the Magnus force will cause the grain to

saltate too long and high when compared to observed results. Therefore a negative lift at

the top of the grain trajectory is needed to keep the hop length from increasing to such a

high value (Wiberg and Smith, 1985).

In response to the concerns expressed above, they defined a form lift force which acts

on non-spherical bodies. This lift force is the same as the one that acts on airplane wings.

Based on the general expression for form lift on an elliptic cylinder, the force is defined as


FL(form)= 4cpvc, sin(N1y + N2) equation 2.23



where c2 is proportional to the length of the centerline of the airfoil, YJ is the flow's angle

of attack relative to the leading edge, and T2 is a function of the curvature of the airfoil's

centerline. Here they take Y2 to be equal to zero and adjust the circulation so that the

trailing stagnation point is at the rear of the cylinder's long axis. As is shown in figure 2.6,

this force can be positive or negative based on the orientation of the particle (Batchelor,

1967).












FL>O FL







Figure 2.6 Sign of Form Lift for Different T' Angles of a Body Relative to Flow



Of course to use the form lift force, an elliptic grain was assumed. Wiberg and Smith

(1985) oriented the grain in a manner such that the form lift is a maximum at the beginning

of the trajectory and allows the grain to rotate so that the form lift is negative at the top of

the trajectory. For the downward portion of the trajectory, the grain orientation does not

change significantly (Wiberg and Smith, 1985).

The Magnus and lift forces are calculated together with one value of circulation that

accounts for the relative magnitude and signs of the two individual circulations. Once the

combined forces per unit length are calculated, the value is multiplied by the diameter of

the grain and by a coefficient. This coefficient is determined by comparing the model

result to the height of known trajectories, namely, Francis (1973). Figure 2.7 displays the

model result versus a Francis (1973) photographed trajectory with d = 0.7 cm, s = 1.3,

and u. = 6.6 cm/s. The velocity profile is the standard "law of the wall" profile and the

grain is initially positioned at one-half a grain diameter above the theoretical bed.




















:E
rn
.0
a)
a:


0 2 4 6 8 10 12
Length (cm)

Figure 2.7 Comparison of Predicted and Observed Grain Saltation Trajectory


The initial vertical and horizontal grain velocities are determined by decomposing the

forces on a grain into radial and circumferal components in polar coordinates. An

expression for the centrifugal force is obtained as a result of integrating the circumferal

force with respect to the angle the pivoting axis of the grain makes with the vertical. The

centrifugal force expression and the one for the radial force are balanced when the

pivoting angle is some critical value. The initial velocities are obtained from the resulting

equation. When this is done, the equation of motion is solved with a fourth order Runge

Kutta method (Wiberg and Smith, 1985).

From their model, Wiberg and Smith (1985) concluded that the saltation

characteristics, height and length, both increase with increasing particle density. This was

not supported by Abbott and Francis (1977). Without more data, Wiberg and Smith

(1985) state that it is very difficult to determine if the problem with the model is with the









force combination or the with the way the grain is oriented for the form lift force (Wiberg

and Smith, 1985).

Nino and Garcia (1992)

A gravel saltation model was developed by Nino and Garcia (1992) based on the data

they accumulated from Nino, Garcia, Ayala (1992). They include the forces of added

mass, gravity, drag, shear lift, Magnus lift, and Basset history. These forces appear as

previously described in section "Force Analysis". The shear lift force form is the same as

for Wiberg and Smith (1985) with CL equal to 0.2. The Magnus lift force is of the same

form as used by White and Schulz (1977) with the fluid velocity gradient as a part of the

force. The Basset history force is included for the first time of the models examined. This

force is not integrated by "brute force" means in the equation of motion, but it is

approximated using the analysis of Brush et al. (1964).

The fluid velocity profile is the turbulent velocity profile over a rough boundary given

as

uf=2.5 ln(30z) equation 2.24



where the roughness height is taken as equal to one grain diameter. The initial velocity of

the gravel was determined by examining the previously obtained video of gravel saltation

provided by Nino et al., (1992). The particle is initially positioned one-half grain diameter

above the theoretical bed. The angular velocity of the gravel was determined by choosing

the best value that allowed the model to match an observed gravel trajectory. A typical

trajectory is plotted in figure 2.8.















1.2

.2)1.1
c")
I 1


0 1 2 3 4 5 6 7
Nondimensional length, L/d

Figure 2.8 Comparison Between Predicted and Observed Trajectory (Nino et al., 1992)



Nino et al. (1992) were able to conclude the following: 1)The shear lift force does

indeed add to the agreement between theoretical and observed trajectories, 2)The Basset

history force had little effect on the predicted trajectory for gravel and 3)Particle rotation

does bring the predicted trajectory closer to the observed, however, only for some cases.

(Nino et al., 1992). The problem might be a result of the constant rotational value

assigned to the gravel instead of having the particle rotation adjust to the flow.

Lee and Hsu (1994)

Lee and Hsu (1994) closely followed the model developed by van Rijn (1984). The

main exception is that they included the Magnus force and compared the two shear lift

forces explored by Wiberg and Smith (1985) and Saffman (1965). The initial conditions

were very similar except that the initial position of the grain is at 0.5 times grain diameter

above the theoretical bed.










The two shear lift forces were calculated for a run and the values were comparable

even though the lift coefficients are very different. The Magnus force was explored by

trying different values to best match Fernandez Luque and van Beek (1976). Lee and Hsu

(1994) came to the conclusion that the trajectory has three distinct regions: rising, central,

and falling branches. For each region, there is an angular velocity associated with it. The

greatest rotation is in the rising branch and decreases through the other two branches.






- 2u* at 45 deg
2.5 ,'
-- 5u* at 45 deg
/ \





S2h S
o 2




E 1.5
0
z







0 2 4 6 8 10 12 14 16 18 20
Nondimensional Length, L/d

Figure 2.9 Trajectories With Same Angle and Different Velocity (Lee and Hsu, 1994)



When excluding the Magnus force altogether, an interesting result is obtained. As a

result of the shear lift force having two components (van Rijn, 1984), two particles fired at

the same angle with different velocities will exhibit a strange behavior. The particle with

the higher initial velocity will not go as high nor as far as the particle with the smaller

velocity. This is shown in figure 2.9. One particle has an initial velocity of two times the

shear velocity while the other has five times the shear velocity. Both were fired at the





26



same angle. The grain diameter is 1.36 mm with a shear velocity of 3.6 cm/s, CL equal to

20, and a ks of 2d. From this model it appears that maybe the shear lift force should act

only in the plane normal to the bed as done in Wiberg and Smith (1985) to achieve a more

physically realistic result.













CHAPTER 3
SALTATION MODEL FORMULATION



Single Grain Model


Equation of Motion

The objective of the single grain model is to yield a particle saltation trajectory. The

model developed by the writer contains many of the same forces as were used in the

previously discussed models. The forces that are considered important are added mass

(Fa), gravity (Fg), drag (Fd), shear lift (Flshear), Basset history (Fb), and Magnus or spin lift

force (F1Magnus). Recall that Saffman (1965) and Wiberg and Smith (1985) derived two

forms of the shear lift force. Both can be used in this model by simply exchanging one for

the other. This allows for an easy comparison of the two forms. The Basset history force

is used but in a different form than what is presented in Nino and Garcia (1992). They

borrowed from the simplifications of Brush et al. (1964) whereas in the writer's model the

integro-differential equation is handled directly. The form of the Magnus force is the same

as the form developed by White and Schulz (1976). Their form includes the constant

updating of the particle's angular velocity to that of the rotation induced by the fluid. All

the forces listed have a normal and a parallel component to the bed except for the shear lift

force. Since there exists a shear only in the plane parallel to the bed, the shear force has

only the component normal to the flow.

The forces are more readily seen in figure 3.1. The equation of motion for a saltating

grain may be divided into longitudinal and vertical components. The forces acting on a

saltating grain are described by the following equations:










tpV = -CDA
d 2 d
PSDAPu-(u)+w2(-u,)


-p VC du d
P du 6dt dt
dt 0r o -,-

S 2 p+-1 du, w
+-d22
8 2 dz (2uuy +W

+p(s )gV sin


and


dw 1 /\2
pVCdt = -CA p (u-u) +w(w)





8 2 dz (uf +w2
+2-CL~~TO ~0~- ps -1)gc u j3 ,


+ ACL(Urop -Uo )- p(s 1)gVcos P

where

where


equation 3.1


equation 3.2


2u = (U _-fop +W2
UATop ".- W2


equation 3.3


u2 Bot e-)2 +W2
UABot U UfBot-


equation 3.4









The variables, urrop and UfBot, are the fluid velocities evaluated at the top and bottom of the

grain, respectively. The quantities, u and w, are the particle's speed parallel and normal to

the bed, respectively. All these assume no w-component of fluid velocity.

Equation 3.1 is the bed-parallel component and equation 3.2 is the bed-normal

component of the equation of motion. The terms for equation 3.1 are, in order, mass

times acceleration, drag, added mass, Basset history, Magnus lift, and gravity. The terms

for equation 3.2 are mass times acceleration, drag, added mass, Basset history, Magnus

lift, shear lift, and gravity.




Fl(shear)
e Fl(Magnus)
Vr

/ Fd& '
wO ; Fb

uO 0.5d





Figure 3.1 Force Definition Sketch of Grain Saltation




Bed


Grains ---


Horizontal Plane



Figure 3.2 Definition of Bed Slope

The shear lift force currently in the model is the one derived by Wiberg and Smith

(1985). To compare the trajectories that result from the model with the shear lift of








Wiberg and Smith (1985) and those that result with the Saffman (1965) shear lift

expression, equation 3.2 needs to be modified. The shear lift term is replaced by


F = -C ,p0.5d2 U2+W2 C duf u-u,
r dz ) ( uf)2 +w2


equation 3.5


where this CL is related to the grain Reynolds number, Re* (Saffman, 1965). When using

the Wiberg and Smith (1985) shear lift force, the lift coefficient, CL, is taken to be 85% of

the drag coefficient, CD.

The added mass coefficient, CM, is taken to be 0.5 for a sphere. The coefficient of

drag, CD is calculated based on the empirical equations of Morsi and Alexander (1972).

Their empirical curves may be divided into the following regimes:


24
C =
Re


22.73 0.0903
C = + +3.69
D Re Re2


29.1667 3.8889
C +1.222
Re Re2


46.5 116.67
C = _+0.6167
D Re Re2


98.33 2778
C Re Re2 + 0.3644
S Re Re2


148.62 4.75e04
C Re 0.357
Re Re2


for Re<0.1


for 0.1


for 1


for 10


for 100

for 1000

equation 3.6


equation 3.7



equation 3.8



equation 3.9



equation 3.10


equation 3.11









-490.546 57.87e04
490.546 57.87e4 0.46 for 5000 S Re Re2


-1662.5 5.4167e06
CD 16. + +e6 0.5191 for 10000 Re Re2



The piecewise coefficient of drag formulation is chosen because it is very complete

and does not require much effort to use. It can, however, be replaced with a universal

equation valid for the entire range of Reynolds numbers.

The fluid velocity profile used in the model is the standard logarithmic "law of the

wall" profile with the velocity equal to zero at


zo = 0.11 + 0.03k, equation 3.14
U,



This profile is the same as van Rijn (1984) used. Note that the fluid velocity only has a

horizontal component. This fluid velocity can be easily modified in the model should it be

desired.

Boundary Conditions

The bed level is assumed to be 0.5 times grain diameter below the top of the grains as

is shown in figure 3.3. The initial position of the saltating grain is at one-half a grain

diameter above the theoretical bed. Although the grains aligned in the bed may saltate at

sufficiently high shear stresses, the model herein only yields the path of a grain at the given

position above the theoretical bed.

In order to solve the equations of motion given previously, the grain's initial vertical

and horizontal velocity components are needed. The values of these components come

from White and Schulz (1977). They found that the velocities varied from u* to 2u*. They

also found that lift-off angles varied from 30 to 69.5 degrees. In addition, when the









z


S .. Saltating
/ IGrain
0.2d zo






Figure 3.3 The Saltating Grain Initial Position



Magnus force is considered, an initial particle angular velocity is needed. This value is

adjusted to yield a best match to a known trajectory.

Method of Solution

The equations of motion along with the moment equation are defined as first order

ordinary differential equations. A fourth order Runge Kutta approach is used to yield a

vertical and horizontal grain velocity and an angular velocity. The vertical and horizontal

velocity components are numerically integrated with a simple Simpson's Rule approach to

obtain a particle trajectory.

The model includes some adjustable parameters. The bed roughness can be taken to

be any multiple of the grain diameter. The aforementioned initial velocities and angular

velocity also may be adjusted. Also, the initial grain ejection angle from the bed can range

from 0 to 180 degrees with the former meaning with the shear flow and the latter meaning

directly into the shear flow. This is the first saltation model to allow for the inclusion of

ejecting grains from the bed into the fluid flow.

Sensitivity Analysis

This section looks at the relative effect force combinations and initial conditions

have in determining the trajectory of a saltating grain. The parameters that are held

constant unless otherwise specified through the force sensitivity test are grain diameter,









specific gravity, initial particle velocity, initial angle, bed roughness, and the coefficient of

lift. They are 0.18 cm, 2.65, 2u*, 45 degrees, 2d, and 0.2, respectively. These are the

values that were either used or observed by Fernandez Luque and van Beek (1976). In

addition, it needs to be stated that the bed slope was taken to be zero in this case. The

critical velocity for incipient motion is found by calculating Archimedes number and using

a look-up table to find Shields' parameter. From this, the critical velocity is determined

with the relation,



Ucr -= equation 3.15



where Tcr is the critical shear stress.

The most basic form of the model contains the drag, added mass, and gravity forces.

The shear lift force developed by Wiberg and Smith (1985) and the history force from

Basset (1888) are successively added to the model. Figure 3.4 shows the results. For this

case, u. = 4 cm/s with uo = wo = 2u*.

The shear lift force increases the length of the trajectory by about one grain diameter

while the Basset history force appears to have a slightly lesser effect on the trajectory.

The results with the shear lift force included are consistent with our intuition. Only a small

velocity gradient develops across the grain because of its size. As a result, the shear lift

has a relatively small effect. The Basset history force is not as easy to predict. The height

is increased by less than one-tenth of a grain diameter while the length increased by

approximately one-half grain diameter when it is included. The increase in length is due to

the fact that the grain travels higher into the fluid column and thus attains a greater

velocity. This increase in length may occur because the vortices have a large impact on

small particles.





























0.5 I I X I \ I
0 0.5 1 1.5 2 2.5 3 3.5 4
Nondimensional Length, L/d

Figure 3.4 Particle Trajectories With A Particle Size Of 0.18 Centimeters



The next test was to run the model with a larger particle size. A diameter of 3.1 cm

was chosen, roughly in the gravel regime. The other parameters maintained their same

values with the exception of u* = 22.84 cm/s and P = 0.07 to match the experimental

conditions of Nino et al. (1992). Figure 3.5 shows that the shear lift force had more of an

effect than it did with the smaller grain. This result is expected because the velocity

gradient is large on a larger particle. The Basset force, however, did not posses the same

significance as it did in the other case. This force only slightly increased the particle

trajectory. The same result was obtained in Nino and Garcia (1992) with regard to the

Basset history force. Both forces do demonstrate that they act to increase saltation height

and length as a result of being included in the model.













1.4
No Shear Lift, No Basset


.. .. ..


0 2,. "4.6 \
8- I ."
/ -


08 \













Figure 3.5 Particle Trajectories With A Particle Size Of 3.1 Centimeters



The next force to consider adding to the formulation is the Magnus lift force. To

review, this force results from two effects. First, a grain may be transferred an angular

velocity from the shear flow. Since the fluid velocity is greater at the top of a grain than at

the bottom of a grain, it may acquire a rotation. Second, a grain on the bed may obtain an

angular velocity from another grain striking it. The magnitude of the rotation is dependent

on the speed and the placement of the blow that the grain striking the bed delivers.

Figure 3.6 displays the trajectories that result by varying the initial angular velocity.

The diameter of the particle is 0.18 cm with ks=2d, Cl=0.2, u*=4 cm/s, uo=2u*, and

wo=2u*. It is obvious and expected that the trajectories should both increase in length and

height with increasing initial angular velocity. The effect of the Magnus force on a

particle's path is quite significant. This is a discouraging result since little is known about

what angular velocities are appropriate for saltation.















/





I I

S
0.8 -



0.6

0 1 2 3 4 5 6 7 8 9 10
Nondimensional Length, L/dinitial angles are 30, 45, and 60

Figure 3.6 Particle Trajectories Sensitivity to Particle Rotation



As stated previously, the initial conditions of the model greatly effect the resulting
salvation trajectory. The initial angle and velocity of the grain is the next area of interest.

Again the grain size used is 0.18 cm. Figure 3.7 displays the results of varying the take-

off angles and maintaining a constant initial velocity. The initial angles are 30, 45, and 60

degrees. The initial longitudinal velocity and initial vertical velocity are both 2u*. Figure

3.8 shows what happens if the velocities are varied and the angles are held constant at 45

degrees. As the lift-off angles increased, the grains saltated farther. Also, the higher the

initial velocity, the longer the trajectory. These results are obviously from the fact that the

grain attains a higher velocity from the fluid with a larger angle and higher initial velocity.





















I"1




Ca

in
( 0.8
E
0
o
Z -,1


Nondimensional Length, L/d


Figure 3.7 Particle Trajectory Sensitivity to Initial Trajectory Angle


*a
I0.9
r
.0)
I<

cZ
0.8
.C

E
' a .
0.7
z


0.5 "
0 0.5 1 1.5 2
Nondimensional Length, L/d


Figure 3.8 Particle Trajectory Sensitivity to Initial Velocity









Another parameter that may be adjusted is the bed roughness. This value is usually

taken to be one or two times the grain diameter. The bed roughness affects the fluid

velocity intensity. Different bed roughness values were run with the model to determine

their effect on the particle's trajectory. The initial velocity used was 2u* for each

component with a particle size of 3.1 cm. The coefficient of lift, CL, equals 0.2, u. =

22.84 cm/s, and s = 2.65. These were chosen to match one of the cases of Nino and

Garcia (1992). Gravity, drag, shear lift, and Basset were the acting forces. The

trajectories are shown in figure 3.9. It shows that the higher the bed roughness, the

shorter the saltation length.

The particle density may also be adjusted; however, since the focus is on sand grains,

the density is held constant at 2.65 g/cm3. This model can also be used for particles other

than sand grains. As a way of showing the effect of particles of different densities under

the same fluid conditions, the model was run with s = 2.65 and s =1.3. As seen in figure

3.10, the less dense the particle, the farther it will saltate.

To summarize, the forces acting on the particle vary significantly with grain size. The

shear lift force and the Basset history force vary the greatest of the forces examined. The

Magnus effect increases saltation length and height greatly as initial angular particle

velocity increases. Finally, the larger the initial velocity and angle, the greater the saltation

length and height.



Random Trajectory Model



The random trajectory model is just an extension of the previous model. Instead of

ejecting only one grain, the random model ejects 42 grains over a specified range of initial

angles. Then for ten different initial velocities, a set of 42 trajectories would exist. The
























S1-










0.7 -


0.6


0.5
0 1 2 3 4 5 6 7 8
Nondimensional Length, L/d


Figure 3.9 Comparison Of Variable Bed Roughness Values


2.5


Ss=1.3

S- s =2.65

2-


F!
"-



0


01
c 1.5





z








0.5
0 5 10 15 20 25 30 35 40
Nondimensional Length, L/d


Figure 3.10 Simulated Trajectories of Particles With Different Densities









resulting trajectories could then be used to determine a concentration profile and finally a

bedload model.

Determination of Initial Angles

The angles were determined through the maintenance of equal increments of surface

area over the top half of a sphere. The objective was to have 21 increments of 0 and 21

increments of 0, using standard spherical coordinates. The angle, 0, is the angle between

the x and y axis and the angle, ), is the azimuth angle when a three-dimensional axes is

considered. Since this is a two-dimensional problem, however, only the 21 increments of

4 are considered. This angle ranges from 0 to 7t/2. The following equation is used for

surface area:



AS = sinqAA equation 3.16



where AS is the quasi-surface area because the angle, 0, is absent. This value should

remain constant while 4 changes. Or this equation can be written



sinodo = -d(coso) equation 3.17



which is also a valid expression. Since 0 ranges from 0 to 7c/2, coso ranges from 0 to 1.

Therefore for 21 increments of 0, coso is divided into 21 segments. Table 3.1 displays the

actual initial lift-off angles.

The angle phi can be found by integrating over the range 01 to 02. The following

shows how to obtain this angle.


2Sds equation 3.18
S= sin Odo = cos 1 -cos02 equation 3.18


Since S is a constant, it may be stated that










cos(, -cos2, =sin),(02 e,)


or solving for 0*


equation 3.20


ScosKd, -cos)2,
(. = arcsin -2 ----1
02 -1


where the angle 0* has no physical significance. It is only the angle that satisfies the

surface area equation.


Table 3.1 Distribution of Initial Angle
cos4


0
1/21
2/21
3/21
4/21
5/21
6/21
7/21
8/21
9/21
10/21
11/21
12/21
13/21
14/21
15/21
16/21
17/21
18/21
19/21
20/21
1


(deg)
90
87.27
84.53
81.78
79.02
76.23
73.40
70.53
67.61
64.62
61.56
58.41
55.15
51.75
48.19
44.41
40.37
35.95
31.00
25.21
17.75
0


(deg)
88.42
85.82
83.11
80.37
77.60
74.79
71.95
69.05
66.10
63.08
59.97
56.76
53.44
49.96
46.29
42.38
38.15
33.46
28.09
21.46
8.84


sin,*Ao


0.0476
0.0477
0.0476
0.0475
0.0476
0.0477
0.0476
0.0476
0.0476
0.0477
0.0476
0.0476
0.0476
0.0477
0.0476
0.0477
0.0475
0.0477
0.0476
0.0476
0.0476


----


I I


equation 3.19


;s









The surface area calculations are also shown in table 3.1. The values are nearly the

same and therefore validate this approach. All of these angles are found prior to running

the model.

Boundary Conditions

Once the angles have been determined, the initial velocities are needed. For this

study, the initial velocities were taken to be one to ten times the critical velocity for

incipient motion. That is, for case one, the initial velocity was ucr, for case two, 2ucr, and

so on. These numbers are purely arbitrary Statistically speaking, the grains are given an

equal chance to leave the bed at all of these angles and velocities, i.e., uniformly

distributed.

Besides ejecting with the flow, grains also may eject from the bed initially going

against the flow. This may occur during sheet flow described earlier in the section

"Particle Saltation." Of the models published, this is the first time that a model has


3.5-






S2.5



.


1.5






0.5
-2 0 2 4 6 8 10 12 14
Nondimensional Length, Lb/d

Figure 3.11 Typical Random Trajectories With Initial Velocity 5ucr
Figure 3.11 Typical Random Trajectories With Initial Velocity 5Ucr

















1 2.5
0)


Co
0)
E
-D
c 1.5
z


0.5'
-5 0 5 10 15
Nondimensional Length, L/d

Figure 3.12 Particle Trajectories Of Same Initial Velocity And Angles Symmetric About
The Vertical Plane


included grain ejection against the fluid. The angles previously found are translated to the

other plane by making the x-component of the initial velocity negative. This yields the

total of 42 different angles at ten different velocities.

Sensitivity Analysis

A typical result is plotted in figure 3.11. The relevant quantities are d = 0.1 cm, ks =

2d, CL = 0.85CD, 9o = 20 rev/s, and an initial velocity magnitude of 5ucr. Note that the

grains that are ejected into the shear flow at the same angle and velocity as ones that are

ejected with the fluid have greater saltation lengths and heights. This can be more readily

seen in figure 3.12. This figure shows grains ejected from the bed at the same angle with

respect to the bed. The difference exists because of the way the shear lift is defined by

Wiberg and Smith (1985). When the grain is ejected against the flow, it has a negative x-





44



component. This causes the magnitude of the shear force to be much greater than the

force acting on the grain ejected with the flow.













CHAPTER 4
SALTATION MODEL RESULTS



Single Grain Model


Wiberg and Smith (1985) Shear Lift

The Magnus Effect is neglected for the first comparison with a given data set to see

how well the shear lift force predicts the saltation trajectory. The trajectory data used for

the model comparison is provided by Fernandez Luque and van Beek (1976). The shear

lift force that was first tried in the model was Wiberg's and Smith's (1985) lift. From

Fernandez Luque and van Beek (1976), the grain diameter was 0.18 cm with a fluid shear

velocity of 4 cm/s. The initial conditions for the model were uo = 8 cm/s, wo = 8 cm/s, 3 =

0, CL = 0.85CD, and ks = 2d. The trajectory is shown in figure 4.1. It is easily seen that

the shear lift force alone does not adequately describe the motion of the particle as

observed by Fernandez Luque and van Beek (1976).

Figures 4.2 through 4.6 show some of the relevant velocities and forces as predicted

by the model. The fluid velocity is always in the rough, turbulent range as shown by the

Reynolds number. The drag coefficient was between 0.67 and 0.86. Those quantities are

graphically shown in figure 4.2 where time is nondimensionalized with the time it takes for

the grain to complete its trajectory, Ts. Figure 4.3 shows how the grain, fluid, and relative

velocities varied over the trajectory length of the grain saltation. The plot reveals that the

grain velocity is continuously increasing to equal that of the velocity of the fluid. So as a

result, the relative velocity is decreasing. This means that the drag and lift forces are both

decreasing over the grain trajectory. This is shown in figure 4.5. The negative lift results










b


4.5


4




-c
.
13.5





E
I 3-



"o
0



o 2-
z

1.5


1


0 5 10 15 20 25
Nondimensional Length, L/d

Figure 4.1 Model Comparison With Fernandez Luque and van Beek (1976)

3001 1 II1


E 250
z
-o
c200
cc


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Nondimensional Time

Figure 4.2 Reynolds Number and Drag Coefficient As Calculated By The Model


.. Fernandez Luque &
van Beek (1976) Range

- Model, d=0.18cm,ks=2d,
CI=0.85Cd, u*=4cm/s





~I










































0 1 2 3 4
Nondimensional Length, L/d


Figure 4.3 Simulated Particle Velocities


C


c)
0
0



. -2

E

Z-4


-6 k.


2 3 4
Nondimensional Length, L/d


Figure 4.4 Simulated Particle Accelerations


-- uf/u*
- ug/u*
..... v/U*


SHorizontal Acceleration

Vertical Acceleration













. . . . . . . . . . . .. ,. . ..






48





0.8- Lift Force, FI/Fg

0.7 Drag Force, Fd/Fg

0D
LL 0.6
LL
a 0.5

0.4

E 0.3 -
CO
.2 0.2
( -

E 0.1

0
z 0

-0.1

-0.2


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Nondimensional Length, L/d


Figure 4.5 Simulated Particle Forces


0.25





0.2

1.


0
L. 0.15-




C:
I- 0.1 -

E

o
z


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Nondimensional Length, L/d


Figure 4.6 Simulated Particle Basset History Force









from the fact that the difference between the fluid velocity at the top of the grain and the

grain velocity is less than the difference between the fluid velocity at the bottom of the

grain and the grain velocity. Also note that there is a "jump" in the lift force at the very

beginning of the curve as presented in the figure. This results from the way the lift force is

defined once again. At first the bottom of the particle has no fluid velocity. It then

immediately acquires a bottom velocity as the particle leaves the bed and causes the spike

in the curve.

The horizontal acceleration of the particle decreases as the particle velocity

approaches the fluid velocity as shown in figure 4.4. The vertical acceleration remains

fairly constant except near the bed. The accelerations are nondimensionalized by

multiplying the acceleration by TJu*. The particle decelerates significantly in the rising part

of the trajectory and continues to do so until the particle approaches the bed. The particle

accelerates again near the bed because of the aforementioned lift force. This shear lift

force is significant when compared to the drag force. The downward aspect of the lift

force is clearly displayed in figure 4.5. It is evident from this figure that the lift force is not

providing enough upward thrust for the particle to saltate as observed by Fernandez

Luque and van Beek (1976). The Basset force is smaller than both the lift force and the

drag force for much of the saltation as demonstrated in figure 4.6. Note that the Basset

force continually increased as the particle saltated.

Saffman (1965) Shear Lift

The lift force described by Wiberg and Smith (1985) was replaced by the Saffman

(1965) lift force to determine which allows the model to match the given trajectories more

closely. The same initial conditions were used as with the other shear lift force case. The

grain diameter was once again 0.18 cm, the grain roughness was two times the grain

diameter with the initial vertical and horizontal velocities both two times the shear

velocity. The shear velocity was 4 cm/s. The lift coefficient was constant at 20.










Figures 4.7 through 4.12 demonstrate the grain's properties with the inclusion of the

Saffman (1965) lift force. From figure 4.7, it is clear that this lift force has more of an

impact on the particle's trajectory than its predecessor. Its predicted trajectory exceeds

that of the range observed by Fernandez Luque and van Beek (1976). The height is

exceeded by a grain diameter while the length is by about five grain diameters. The

Reynolds number and therefore the drag coefficient both change dramatically as the

particle travels.

The grain's velocity approaches and eventually surpasses the fluid velocity near the

end of the trajectory. This occurs because the shear lift force increases near the bed and

causes an actual upward lift near the end of saltation unlike the lift defined by Wiberg and

Smith (1985). The Saffman (1965) lift keeps the grain higher in the fluid column and

allows it to maintain a higher velocity.



5.5- .. Fernandez Luque &
van Beek (1976) Range
5 Model, d=0.18cm,ks=2d,
CI=20, u*=4cm/s
4.5
2-






1.- 4
I(
S3.5

3
CO 3-
E
2.5 ." "

2 /







0 5 10 15 20 25 30 35
Nondimensional Length, L/d

Figure 4.7 Model Comparison With Fernandez Luque and van Beek (1976)











400


. 350
E
z
'300
0-

( 250
r:


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9



0.8

0.75

0.7

0.65

0.6

0.55
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Nondimensional Time


Figure 4.8 Reynolds Number And Drag Coefficient As Calculated By Model


14

uf/u*

12 ug/u*
.*. v/u*


10-






.8
0 4
6 \


4" / "" ,
0 4 / "'....


0 5 10 15 20
Nondimensional Length, L/d


Figure 4.9 Simulated Particle Velocities











15


10


5


0



t--



E
< -5



0

-15
Z


-20-


-25-


-30
0 5 10 15 20
Nondimensional Length, L/d


Figure 4.10 Simulated Particle Accelerations


41 1 --1-1


L 3
-a
LL
0)
S2.5
LL


I, 2
o(
C
0
a 1.5
E

0 1
C


25 30 35


0 5 10 15 20 25 30 35
Nondimensional Length, L/d


Figure 4.11 Simulated Particle Forces


Horizontal Acceleration
Vertical Acceleration











. ... ........ ................................................ .


I I





























0 5 10 15 20 25 30 35
Nodimensional Length, L/d

Figure 4.12 Simulated Particle Basset History Force

Once again the horizontal component of the grain's acceleration changes for the

majority of the saltation as the grain is decelerating. The vertical acceleration varies

significantly near the bed. It remains constant for the central portion of the trajectory and

is decelerating and accelerating during the rising and falling portion of the trajectory,

respectively. The lift force is significantly larger than the drag force according to figure

4.11. The shear lift force is greatest near the bed since this is where the velocity gradient

across the grain is the largest. Also note that the curve for the lift force does not contain a

spike as it did in the previous case. This is because the Saffman (1965) lift is defined by

the velocity gradient directly instead of using relative difference. The Basset history force

has the same features as with the lift described by Wiberg and Smith (1985). The force

increases with increasing trajectory length.

Adding Magnus Effect

Since the Saffman (1965) shear lift force seems to overpredict the range from

Fernandez Luque and van Beek (1976), the Magnus force is only applied to the case






54




where the Wiberg and Smith (1985) lift force is concerned. Recall that the latter case did

not come close to matching the observed trajectory range. The Magnus force was added

to the equation of motion to help with the correlation. In addition, the moment equation

was also added to be solved simultaneously with the equation of motion. The initial

conditions were maintained with one exception. An initial angular velocity of 33 rev/s was

added. The model comparison with the data range is displayed in figure 4.13. The model

with the newly added Magnus force predicts the grain trajectory fairly well and certainly

much better than without it. The Magnus force significantly increases the overall lift effect

on the particle especially in the rising part according to figure 4.14. For this particular

case, the Magnus force increased the saltation height by 2.3 grain diameters and the length

by 21 grain diameters. With the Wiberg and Smith (1985) form of the lift force, the

Magnus force needs to be included. This is the same conclusion agreed upon by Wiberg

and Smith (1985).

4.5
0 Fernandez Luque &
4 o van Beek (1976)
- Model, 33 rev/s
Model, Shear lift
3.5 .- o
0 Only

3
0- / \o
0
I /
/ \
S2.5 /

E 2

Sl L ,
a) / /

/ \

/ \
/ \


o\

0 5 10 15 20 25 30
Nondimensional Length, L/d

Figure 4.13 Simulated Particle Trajectory With Magnus Effect
















1
LL


0.5
o 0 .5 ........................ .. ..........


o 0
z


-0.5



0 5 10 15 20 25 30
Nondimensional Length, L/d

Figure 4.14 Simulated Particle Combined Lift Force Effect



Random Trajectory Model


This model incorporated the use of the Wiberg and Smith (1985) shear lift force and

the Magnus lift force. Recall that this model ejected grains from the bed over a range of

42 angles: 21 against the shear flow and 21 with the shear flow. The initial velocities were

taken to be one to ten times the critical velocity for incipient motion.

Comparison to Previous Data Set

The random model was first run to try and predict the observed saltation range of

Fernandez Luque and van Beek (1976). From their experiment, the shear velocity was set

at 4 cm/s. The magnitude of the initial velocity was set equal to 11.3137 cm/s so that a

particle ejected at 45 degrees with respect to the bed will have a longitudinal and vertical

component of 2u* or 8 cm/s. This was done because White and Schulz (1977) observed

lift-off velocities of 2u. and this initial velocity was used by van Rijn (1984) when he









compared his model to this data. The grain size and bed roughness were 0.18 cm and

0.36 cm, respectively. An initial angular velocity of 20 rev/s was used in the model

simulation. This value was chosen because the trajectory had been matched satisfactorily

with 33 rev/s at 45 degrees. So 20 rev/s was chosen to compensate for using angles larger

than 45 degrees.

The resulting trajectories can be seen in figure 4.15. The saltation paths do fit into

the established experimental region. The trajectories of only the grains ejecting with the

flow were also examined. This allowed for the quantification of the effect of ejecting

grains against the flow. The results are displayed in table 4.1.


Table 4.1 Average Saltation Characteristics For Fernandez Luque And van Beek (1976)


How Grain Ejected Avg. Length (L/d) Avg. Height (H/d) Avg. Velocity (U/u*)


With And Against 20.94 3.75 6.74
Flow


With Flow 17.85 2.96 6.32



The average saltation length and height for Fernandez Luque and van Beek (1976)

were 24.5 and 2.8, respectively. The model run with grains ejected with the flow only was

closer to matching the height while the other combined effect was closer to the length.

The discrepancy may lie in the inaccuracy of the initial angular velocity.

Comparison of Grain Ejection Methods

The random trajectory model was also run by trying several different initial velocities.

As a simplification, the initial velocities were taken to be multiples of the critical velocity

for incipient motion. The objective of this was to generate a set of trajectories with grains




















S3.5
t-

I 3
07
C
2.5
E
C 2
o
z

1.5


1


0.5
-5 0 5 10 15 20 25
Nondimensional Length, Lb/d


Figure 4.15 Simulated Trajectories With Grain Ejected With And Against Shear Flow


0 5 10 15 20 25
Nondimensional Length, Lb/d


Figure 4.16 Simulated Trajectories With Grains Ejected With Shear Flow










ejected both with and against the flow and with grains ejected only with the flow to

compare their average saltation characteristics. That is, to determine the difference

between the two cases' average predicted length, height, and velocity over a range of

initial velocities. It is necessary to do this because a model has not been run before with

grains ejected into the fluid shear.

Figure 4.17 displays the results of the first 42 trajectories with an initial velocity of ucr.

Figures 4.18 through 4.26 show the trajectories that result from initial velocity values of

2ucr through O1ucr. In addition an initial velocity of 100ucr is considered to show an

extreme case. The grain diameter for all cases is 0.1 cm since this is the upper limit for

sediment to be considered as sand. The lift coefficient, CL, is equal to 0.85CD. This

means that the Wiberg and Smith (1985) lift force was incorporated into the model. The

equivalent bed roughness is equal to two times the grain diameter. The initial angular

velocity was chosen to be 20 rev/s.




1.1



"-o

"Z 0.9

Co
S0.8
E
0
zn7


-0.5 0 0.5 1 1.5 2 2.5
Nondimensional Length, Lb/d

Figure 4.17 Simulated Trajectories With Initial Velocity, ucr



















I 1.4




0
-Il




CO
t-

E 1

0
Z


-1 0 1 2 3 4
Nondimensional Length, Lb/d


Figure 4.18 Simulated Trajectories With Initial Velocity, 2ucr


-1 0 1 2 3 4 5
Nondimensional Length, Lb/d


Figure 4.19 Simulated Trajectories With Initial Velocity, 3ucr







































0.51 \1 I A '
-2 0 2 4 6
Nondimensional Length, Lb/d


Figure 4.20 Simulated Trajectories With Initial Velocity, 4ucr

r- I


0.5'
-2 0 2 4 6 8
Nondimensional Length, Lb/d


Figure 4.21 Simulated Trajectories With Initial Velocity, 5ucr


10 12 14


10


A- ------------------ ----- ----- -----











*1 I i I I I


3.5



3


o
T 2.5

"(

S 2
E
0
0-
OI


0.5
-2 0 2 4 6 8 10
Nondimensional Length, Lb/d


Figure 4.22 Simulated Trajectories With Initial Velocity, 6ucr


4.51 I


-o

3
-r,

2.5
C,


.E 2
*"
0
z
-o
c=
z


S 1\\\\\ 4 1I
12 14 16


-5 0 5 10
Nondimensional Length, Lb/d


Figure 4.23 Simulated Trajectories With Initial Velocity, 7ucr


20














4.5


4


S3.5




C
0
o
CO
S2.5

r
0
"Zi .


0.5' I I % '
-5 0 5 10 15
Nondimensional Length, Lb/d


Figure 4.24 Simulated Trajectories With Initial Velocity, 8ucr




5.5


5


4.5
-o
4-


3.5



z 2
O"5 3

E
5 2.5
O
2


1.5


1


0.5
-5 0 5 10 15
Nondimensional Length, Lb/d


Figure 4.25 Simulated Trajectories With Initial Velocity, 9ucr


20 25


25


I






63



7



6




I

0r
a)
14



O




2



1-


-5 0 5 10 15 20 25 30
Nondimensional Length, Lb/d


Figure 4.26 Simulated Trajectories With Initial Velocity, 10ucr

100

90


80



p 70
2-







20--
-5 0 5 10 15 20 25 3060
Nondimensional Length, Lb/d


Figure 4.26 Simulated Trajectories With Initial Velocity, lOucr


100C













C 50


E 40
r.
0
z 30


20

10


-200 0 200 400 600 800 1000
Nondimensional Length, Ud


Figure 4.27 Simulated Trajectories With Initial Velocity, 100ucr









The trajectories of the grains ejected against the flow consistently go higher and

farther than those grains that initially travel with the flow. This is a result of the higher

shear lift force magnitude for those grains shot into the flow. The average saltation

height, length, and particle velocity for each initial velocity are summarized in table 4.2.

An obvious result that may be seen from the table is that both the average saltation height

and length increase with increasing initial velocity.

The average particle velocity is the average longitudinal velocity of the saltating grain.

The velocity has been nondimensionalized by the fluid shear velocity which was 2.317

cm/s for all of the cases. This was equal to the critical velocity for incipient motion for

grain size of 0.1 cm. Like the other averaged quantities, the average particle velocity also

increased with increasing initial velocity. This is an expected result because the grain is


Table 4.2 Average Saltation Characteristics From Random Trajectory Model

Initial Vel. (n*Ucr) Avg. Length (Lb/d) Avg.Height (Hb/d) Avg.Velocity (u/u*)

1 3.18 0.98 3.34

2 4.70 1.35 4.03

3 6.24 1.73 4.61

4 7.85 2.12 5.14

5 9.55 2.53 5.63

6 11.29 2.96 6.09

7 13.22 3.39 6.52

8 15.14 3.83 6.93

9 17.14 4.28 7.32

10 19.20 4.73 7.69

100 526.67 61.59 30.95









allowed to travel higher into the fluid column and therefore acquire more of the fluid's

velocity. As a result of this momentum transfer, the grain travels farther.

These values that are chosen may or may not be physically realistic in grain saltation.

As stated previously, none of the models nor experiments have dealt with the ejecting of

grains into the flow. This makes it difficult to determine if this random trajectory model is

correct in ejecting grains into the fluid flow. If it is accurate to eject grains against the

flow, it may not be at the same velocities as those that are ejected with the fluid. All of

these points need to be looked at to either validate the above approach or modify it.

The saltation characteristic averages for the grains ejected both with and against the

flow are compared with the values obtained by ejecting the grains with the shear flow

only. The initial conditions were the same as those for the previous case. The grains were


Table 4.3 Saltation Characteristics With Grain Ejected With Shear Flow

Initial Vel. (n*Ucr) Avg. Length (L/d) Avg. Height (H/d) Avg. Velocity (u/u*)

1 2.86 0.87 3.20

2 4.35 1.16 3.92

3 5.86 1.46 4.55

4 7.44 1.77 5.15

5 9.17 2.10 5.72

6 10.92 2.44 6.23

7 12.88 2.79 6.82

8 14.87 3.15 7.34

9 16.97 3.52 7.86

10 19.14 3.90 8.36

100 615.50 58.19 49.72









































0 1 2 3 4 5 6 7 8 9 10
Initial Velocity, n*ucr


Figure 4.28 Average Length And Height Of Particle Saltation From Random Model



20 1
O Average Length O
18- + Average Height
Average Velocity 0
16
o 0
a)
> 14 -
6 o
.0
12-
.~ 0
T-
a10
i) o
a)
0

a) K
.-o o

E 6 0
C X
w -4.


5 6
Initial Velocity, n*ucr


Figure 4.29 Average Saltation Length And Height From Particles Ejected With Flow








Table 4.3 Saltation Characteristics With Grain Ejected With Shear Flow

Initial Vel. (n*Ucr) Avg. Length (L/d) Avg. Height (H/d) Avg. Velocity (u/u*)

1 2.86 0.87 3.20

2 4.35 1.16 3.92

3 5.86 1.46 4.55

4 7.44 1.77 5.15

5 9.17 2.10 5.72

6 10.92 2.44 6.23

7 12.88 2.79 6.82

8 14.87 3.15 7.34

9 16.97 3.52 7.86

10 19.14 3.90 8.36

100 615.50 58.19 49.72



ejected at velocities ranging from Ucr to 10ucr with the angles varying from 0 to 90 degrees

with respect to the bed. Like before a case for 100 ucr is also examined.

The average characteristics are summarized in table 4.3. It can be seen from the table

that the quantities are smaller by approximately ten percent compared to the values in

table 4.2. This would be expected because of the shear lift force effects on particles

ejected against the flow as described earlier. This is true for all but the 100Ucr case where

the average length and particle velocity are greater than in table 4.2. Still, the difference is

not that great to warrant abandoning this new approach. The process of ejecting grains

into the flow appears to have less of an effect than changing the initial angular velocity by

5 rev/s in the Magnus force as demonstrated in Chapter Three.













CHAPTER 5
CONCLUSIONS



The objective of this study was to evaluate existing saltation models and develop a

new model that would hopefully bridge the gap between theory and observation. The new

model incorporates the best aspects of the previous models. In addition a model was

developed to eject grains from the bed with varying angles and initial velocities under the

assumption of grains having an equal likelihood of leaving the bed at all of those angles.

Finally, the model was compared to experimental observations.

Many of the models examined were quite similar. Most contained the same basic

forces and combined them in different manners. This can be seen in table 5.1. All of the

models include gravity, added mass, and drag forces. Meanwhile, Wiberg and Smith

(1985) developed the only model with a form lift in addition to the shear and Magnus lift

forces. Also, Nino and Garcia (1992) had the only model of those reviewed that included

the Basset history force. The model that was developed in this study included all of the

forces listed in table 5.1 except for the form lift force. It was very flexible in that many

parameters could be changed to adapt to many different situations.

A more complete summary is given below:

1. Five particle saltation models were examined in this study. They were: the

Murphy and Hooshiari (1982), the van Rijn (1984), the Wiberg and Smith (1985), the

Nino and Garcia (1992), and the Lee and Hsu (1994) theoretical models. The models

were compared with data sets either obtained by the researchers themselves or with

trajectories provided by Fernandez Luque and van Beek (1976) or Abbott and Francis

(1977).








Table 5.1 Summary of Forces In Models

Forces Murphy van Wiberg Nino and Lee and Current
and Rijn and Garcia Hsu Model
Hooshiari (1984) Smith (1992) (1994) (1995)
(1982) (1985)

gravity X X X X X X

added mass X X X X X X

drag X X X X X X

shear lift X X X X X

Magnus lift X X X X

form lift X

Basset X X
history____



2. The models did show some ability to predict the observed particle trajectories;

however, each had some discrepancies. The Murphy and Hooshiari (1982) model was

unable to accurately predict saltation height because they ignored forces related to the

relative velocity except for the drag force. The van Rijn (1984) model had both

longitudinal and vertical components of the shear lift even though the shearing was only

in the plane normal to the bed. The Wiberg and Smith (1985) model included a form lift

where the particle needs to rotate a certain number of times for the desired effect. In

addition, their model had the particle's rotation remain constant throughout the saltation.

The Nino and Garcia (1992) model also had the rotation remain constant. The model

developed by Lee and Hsu (1994) had the same problem as van Rijn (1984) and assigned a

different angular velocity for three distinct regions of the particle saltation trajectory.

3. The model developed in this study included gravity, added mass, drag, shear lift,

Magnus lift, and Basset history forces. The shear lift force is the same as defined by









Wiberg and Smith (1985) but can be exchanged for the Saffman (1965) lift. When the

Magnus lift force is applied, the moment equation from White and Schulz (1977) is used

to continually update the particle's rotation.

4. The forces have been assumed to apply for the entire range of Reynolds numbers

in this study.

5. The major problem is obtaining a realistic initial angular velocity of a saltating

particle, which is currently considered an adjustable parameter. The Magnus force

expression is very sensitive to the value of the initial angular velocity

6. The random model assumes that the lift-off angles are uniformly distributed. Also,

it allows the grains to be ejected into the shear flow.

7. The average saltation characteristics for grains ejected against and with the flow

differed from those ejected with the flow by approximately ten percent.

8. An experiment needs to be run with grains being ejected into the oncoming flow,

since this phenomena has not been investigated before. How to do this without disturbing

the flow is still not as yet resolved.














APPENDIX A
FORMULATION OF FORCE COMPONENTS


U -Uf


Figure A. 1 Particle Relative Velocity



The formulation of longitudinal and vertical force components is given here. This

section describes, for example, how the components of the drag force were developed

for the grain saltation model.



Drag Force Component Derivation:



The relative velocity is


Vr = (u-u +w2


equation A.1


From figure A. 1


cos =
u(u-uf) +w2


equation A.2









W
sinO =
(U- f)2 +W2


equation A.3


The drag force is defined as


Fd =CDA pV,2
2


equation A.4


The longitudinal component is


Fd = -CDA plVIVcos
2


1 p(u 2 w(u- r +w2 u -Mu
=-CDA'p (-U) +fw2 (U) +2 W-2 U
2 (Juu-u) +w2


=-CDA 2 (ju-u)2 +w2 (u-u )


equation A.5


The vertical component is

1
F = -CDA plVIVsinO
2

1 p2u- u, 2 22 2 W
=-CDAP (u-uf) +w2 [(u-uf+w2 W
2 V(u-uf) +w


equation A.6


=-CDA 2 pl uufY)2+ W (W)













APPENDIX B
BASSET HISTORY FORCE EVALUATION


The Basset history force integral expression is given here as it appears in the model, a. k.

a., the "brute force" approach.


The Basset history force is defined as


diu
Fb = const dt dt
0 !t-


equation A.7


Let al, a2, a3, etc. represent the particle's acceleration over two time steps with al as the

initial acceleration. For the first time step, At, the Basset history integral may be written




( da = a+ a2 =
00 2 0

= -(a + a2X At- At


= (a, + a2 PA


equation A.8


For the next time step, the Basset history integral may be written









a1 + a2
JAt dt+
If2At-T


a2 +a3DJ
2At 2At-'
.4 --t -,


= a2 2A t_ t -(a 2+ 2At -t 2
l2 )2f2 2At -r 2 2 At
i a,+ a,,,, :~)liZ~~At


=[(a, + a2 X )+(a2 + a)]At


Therefore in summation notation, the term can be expressed as







(du\
dt

Sa + aj+1 (2At) + j + n-j)+ a +a (2At
j=1 2 2


equation A.9


equation A. 10
















APPENDIX C
MATLAB PROGRAMS USED FOR MODEL DEVELOPMENT



The following programs were used for the analysis of the saltation models, and for the

development of a new predictive model using the previous available literature.






"arch.m"


%%Determines taucr from Archimedes # (from van Rijn)
%%for traject.m
global s rho g nu d taucr

medes=((s-l)*rho*g/(nu^2))^(1/3)*d;
if medes<4
taunon=0.24/medes;
elseif medes<=10
taunon=0.14*medes^(-0.64);
elseif medes<=20
taunon=0.04*medes^(-0.1);
elseif medes<=150
taunon=0.013*medes^(0.29);
elseif medes>150
taunon=0.058;
end
taucr=(s-l)*gam*d*taunon;



"coeffd.m"


%%Calculation of drag coefficient for traject.m

global Re Cd

if Re<0.1
Cd=24/Re;
elseif Re Cd=22.73/Re + 0.0903/(Re^2) + 3.69;
elseif Re<10
Cd=29.1667/Re 3.8889/(Re"2) + 1.222;
elseif Re<100
Cd=46.5/Re -116.67/(Re^2) + 0.6167;









elseif Re<1000
Cd=98.33/Re 2778/(Re^2) + 0.3644;
elseif Re<5000
Cd=148.62/Re 4.75e04/(Re^2) + 0.357;
elseif Re<10000
Cd=-490.546/Re + 57.87e04/(Re^2) + 0.46;
elseif Re<50000
Cd=-1662.5/Re + 5.4167e06/(ReA2) + 0.5191;
end





"fluidvel.m"


%%Calculation of velocity profile for traject.m
global d uf z z0 flow fltop flbot Ks

%flow=2.5*uf*log(29.7*((z+0.5*d)/Ks) +1);
flow=2.5*uf*log(z/z0);

%fltop=2.5*uf*log(29.7*((z+d)/Ks) +1);
fltop=2.5*uf*log((z+0.5*d)/z0);

%flbot=2.5*uf*log(29.7*(z/Ks) +1);
if z>0.5*d
flbot=2.5*uf*log((z-0.5*d)/z0);
else
flbot=0;
end
if flbot<0
flbot=0;
end


"motion.m"


%%Details the equation of motion of a saltating grain
%%The forces included are gravity, drag, added mass, shear lift,
%%Basset history, and Magnus lift force.
%%This is the m-file called in traject.m and solved with a
%%fourth or fifth order Runge Kutta.


function veldot = motion(t,vel);
global dudt dwdt s g tf d flow fltop flbot Cl Cd bassetx bassetz dragx
dragz..
admass btu btw mu V Ar nu Cm Re coeff dudz gravx gravz beta vrel
liftx...
liftz magx magz rho utop ubot history drag shear spin saff

vrel=sqrt((vel(1)-flow)^2+vel(2)^2);
Re=vrel*d/nu;

%%Call coeffd.m to determine drag coefficient based on Morsi and
Alexander
coeffd
%Cd=(24/Re)+7.3/(l+sqrt(Re))+0.25;

if saff=='N'









Cl=0.85*Cd; %%or remark this out if C1=0.2
end

gravx=rho*(s-l)*g*V*sin(beta);
gravz=-rho*(s-l)*g*V*cos(beta);

if history=='N'
bassetx=0;
bassetz=0;
end

if drag=='Y'
dragx=-Cd*Ar*0.5*rho*vrel*(vel(1)-flow);
dragz=-Cd*Ar*0.5*rho*vrel*vel(2);
else
dragx=0;
dragz=0;
end

if shear=='Y'
if saff=='Y'
liftx=0;
liftz=-Cl*rho*sqrt(nu)*(d^2)*vrel*sqrt(dudz)*((vel(1)-flow)/vrel);
else
utop = sqrt((vel(l)-fltop)^2 + vel(2)^2);
ubot = sqrt((vel(l)-flbot)^2 + vel(2)^2);
liftx=0;
liftz=(rho/2)*Ar*Cl*(utop^2 ubot^2);
end
end
if shear=='N'
liftx=0;
liftz=0;
end

admass=l+Cm/s;

if spin=='Y'
magx=(pi/8)*(d^3)*rho*vrel*(vel(3)-0.5*dudz)*(vel(2)/vrel);
magz=-(pi/8)*(d^3)*rho*vrel*(vel(3)-0.5*dudz)*((vel(1)-flow)/vrel);
else
magx=0;
magz=0;
end

if spin=='Y'
dudt=(bassetx + dragx + gravx + liftx +magx)/(rho*s*V*admass);

dwdt=(bassetz +gravz + dragz + liftz +magz)/(rho*s*V*admass);
%%bassetz acts in same direction as grave

domegadt=-coeff*(vel(3)-0.5*dudz);

veldot=[dudt;dwdt;domegadt];
else
dudt=(bassetx + dragx + gravx + liftx +magx)/(rho*s*V*admass);

dwdt=(bassetz +gravz + dragz + liftz +magz)/(rho*s*V*admass);
%%bassetz acts in same direction as grav

veldot=[dudt;dwdt];
end
















"random.m"


clear
global dudt dwdt tf btu btw s d g flow fltop flbot Cl Cd bassetx bassetz
gravx..
gravz beta dragx dragz admass mu V Ar nu Cm Re coeff dudz vrel liftx
liftz...
z z0 rho magx magz Ks utop ubot uf history drag shear spin taucr t


ang=[17.75 7.46 5.79
2.92...
2.87 2.83 2.79 2.76
ang=(pi/180)*ang;


4.95 4.42 4.04 3.78 3.56 3.4 3.26 3.15 3.06 2.99

2.75 2.74 2.73];


history='Y';
drag='Y';
shear='Y';
spin='Y';


beta=0;
deltat=1/500;
s=2.65;
d=0.136;
g=981;
nu=0.009796;
Cm=0.5;
rho=0.9985;
mu=nu*rho;
Ar=(pi/4)*d^2;
V=(pi/6)*d^3;
const=(6*pi*mu*
results=[];
yous=[];


((d/2)^2))/sqrt(pi*nu);


%Calculation of velocity profile
gam=rho*g; %%(g/(cm^2-s^2)
%taucr=0.058*gam*(s-l)*d; %%(g/cm-s^2) --From Shield's diagram (0.058)

%%Call arch.m to determine taucr from Archimedes #
arch

ucr=sqrt(taucr/rho); %(cm/s)
uf=ucr; %%Make friction velocity a multiple
%%of the critical velocity

%%Scaling factors
vscale=sqrt((10/3)*(s-l)*g*d);
tscale=(s+Cm)*sqrt((10/3)*d/(s-l)/g);
lscale=(s+Cm)*(10/3)*d;

ufnon=uf/vscale; %%Nondimensional shear velocity


Ks=2*d;
z0=0.11*(nu/uf) + 0.033*Ks;









%%Initial grain rotation
inert=(2/5)*(rho*s)*V*(d/2)^2;
coeff=pi*mu*(d^3)/inert;

for n=1:10 %%This loop is for 10 cases
initnon=n*ufnon;
init=initnon*vscale; %%Dimensional initial grain velocity

loop=l; %%This loop is for firing grains into and with flow
for loop=l:2
for m=l:21 %%This loop is for 21 angle increments (0-90)
t0=0;
tf=deltat; %%Step size
if loop==l
if m==l
u0=sqrt((init^2)/(1+(tan(ang(m)))^2));
w0=abs(u0)*tan(ang(m));
else
direct=sum(ang(l:m));
u0=sqrt((init^2)/ (+(tan(direct))^2));
w0=abs(u0)*tan(direct);
end
elseif loop==2
if m==l
u0=-sqrt((init^2)/(1+(tan(ang(m)))^2));
w0=abs(u0)*tan(ang(m));
else
direct=sum(ang(l:m));
u0=-sqrt((init^2)/(1+(tan(direct))^2));
w0=abs(u0)*tan(direct);


end


velO = [uO wO 2*pi*20]'; % initial conditions
uvel(l)=vel0(1);

z=0.5*d;

%%Call fluidvel.m to determine fluid velocity
fluidvel
fluid(l)=flow;

A=[];
i=2;

jones=0;

%%Determination of relative velocity
vrel=sqrt((vel0(1)-flow)^2+vel0(2)^2);

%%Determination of Reynolds number
Re=vrel*d/nu;

%%Call coeffd.m to determine drag coefficient
%%based on Morsi and Alexander(1972)
coeffd

%%Set lift coefficient
Cl=0.85*Cd;

%%Initial gravity force/buoyancy force
gravx=rho*(s-1)*g*V*sin(beta);
gravz=-rho*(s-1)*g*V*cos(beta);


uO = ? wO = ?









%%Initial drag force
dragx=-Cd*Ar*0.5*rho*vrel*(velO(1)-flow);
dragz=-Cd*Ar*0.5*rho*vrel*vel0(2);

%%Initial shear lift force
utop=sqrt((vel0(1)-fltop)^2 + velO(2)^2);
ubot=sqrt((vel0(1)-flbot)^2 + vel0(2)^2);
liftx=0;
liftz=(rho/2)*Ar*C1*(utop^2 ubot^2);

%%Initial Magnus force
magx=(pi/8)*(d^3)*rho*vrel*(velO(3)-0.5*dudz)*(vel0(2)/vrel);
magz=-(pi/8)*(d^3)*rho*vrel*(vel0(3)-0.5*dudz)*((vel0(1)-...
flow)/vrel);

%%Initial added mass
admass=l+(Cm/s);

%%Initial Basset force is zero
bassetx=0;
bassetz=0;

%%Initial grain accelerations
dudts(1)=(bassetx+dragx+gravx+liftx +magx)/(rho*s*V*admass);
dwdts(l)=(bassetz +gravz + dragz + liftz...
+magz)/(rho*s*V*admass);

%%Initial grain position
dep(l)=0.5*d;
x=0;
len(l)=0;
time(l)=0;

%Calculation of velocities until grain hits bed
while jones==0
[t,vel] = ode45('motion',t0,tf,vel0);
B=[t vel];
z = 0.5*deltat*(vel0(2)+vel(length(vel),2))+z;
x = 0.5*deltat*(vel0(1)+vel(length(vel),l))+x;
len(i)=x;
dep(i)=z;
time(i)=(i-l)*deltat;
if z<0.5*d
jones=l;
end
%%Keep accelerations tabbed
dudts(i)=dudt;
dwdts(i)=dwdt;

%%Update of Basset history force
btu=0; %% Need these conditions
btw=0; %% Need these conditions
if i==2
btu=(dudts(i-l)+dudts(i))*sqrt(deltat);
btw=(dwdts(i-l)+dwdts(i))*sqrt(deltat);
else
for k=l:(i-2)
btu=(dudts(k)+dudts(k+l))*sqrt(deltat)*...
(sqrt(i-k)-sqrt(i-l-k))+btu;
btw=(dwdts(k)+dwdts(k+l))*sqrt(deltat)*...
(sqrt(i-k)-sqrt(i-l-k))+btw;
end
btu=(dudts(i-l)+dudts(i))*sqrt(deltat)+btu;
btw=(dwdts(i-l)+dwdts(i))*sqrt(deltat)+btw;









end

%%Call fluidvel.m to determine fluid velocity
fluidvel
fluid(i)=flow;

%%Update initial conditions
vel0=[vel(length(vel),1) vel(length(vel),2)...
vel(length(vel),3)]';
t0=tf;tf=tf+deltat;
A=[A;B];
uvel(i)=vel0(1);

%%Update Basset history force
bassetx=-const*btu;
bassetz=-const*btw;

i=i+l;
end

%%Plot of particle trajectory
figure(n)
len=len./d; scalel;
dep=len./d; scalel;
plot(len,dep)
hold on


time=time./max(t); %tscale;
vg=zeros(length(len),l)+n;
hey=[vg uvel' fluid'];
yous=[yous;hey];
interim=[vg time' len' dep'];
results=[results;interim];
nines3=[9 9 9];
nines4=[9 9 9 9];
results=[results;nines4];
yous=[yous;nines3];

clear len dep vel B
end
m=l;
end
hold off
xlabel('Nondimensional Length, L/d')
ylabel('Nondimensional Height, H/d')
end
save results
save yous



"traject.m"


%%This m-file determines the trajectory and forces
%%acting on a single particle in saltation


clear
global dudt dwdt tf btu btw s d g flow fltop flbot Cl Cd bassetx bassetz
gravx...
gravz beta dragx dragz admass mu V Ar nu Cm Re coeff dudz vrel liftx
liftz...









z z0 rho magx magz Ks utop ubot uf history drag shear spin taucr saff
t0=0;
deltat=1/500;
tf=deltat; %%Step size

%%Grain size
d=input('Input the grain size, (cm) ');

%%Initial velocities
init=input('Input the initial velocity, (no units) ');
ang=input('Input the initial take-off angle, (0-180 deg) ');

jones=0;
s=2.65;
g=981; %cm/s^2
nu =0.01; %%0.01; %cm^2/s
Cm=0.5;
rho=l; %1; %g/cm^3
mu=nu*rho; %g/(cm-s)
Ar = (pi/4)*d^2;
V = (pi/6)*d^3;
const=(6*pi*mu*((d/2)^2))/sqrt(pi*nu);

%Calculation of shear velocity
gam=rho*g; %%(g/(cm^2-s^2)

%%Call arch.m to determine taucr
arch

%taucr=0.058*gam*(s-l)*d; %11.1 %%(g/cm-s^2) --From Shield's diagram
(0.058)
ucr=sqrt(taucr/rho); %(cm/s)
uf=ucr; %Make friction velocity a multiple
%of the critical velocity

%%Nondimensional scales (Jenkins)
vscale=sqrt((10/3)*(s-l)*g*d);
tscale=(s+Cm)*sqrt((10/3)*d/(s-l)/g);
lscale=(s+Cm)*(10/3)*d;

%%Transform nondimensional velocity to cgs units
ucrp=ucr/vscale;
init=init*ucrp;
init=init*vscale;

%%Determine velocity components
if ang<=90
ang=pi/180*ang;
xdot=sqrt((init^2)/(l+(tan(ang))^2));
else
ang=180-ang;
ang=pi/180*ang;
xdot=-sqrt((init^2)/(l+(taan(ang))^2))
end
zdot=abs(xdot)*tan(ang);

%%Bed slope
beta=input('Input the bed slope, (rise/run) ');

%%Forces to include in saltation model
history=input('Include Basset history force? Y/N, [Y] ','s');
if isempty(history)
history='Y';









drag=input('Include drag force? Y/N, [Y] ','s');
if isempty(drag)
drag='Y';
end
shear=input('Include shear lift force? Y/N, [Y] ','s');
if isempty(shear)
shear='Y';
end
if shear=='Y'
saff=input('Saffman shear lift force? Y/N, [Y] ','s');
if isempty(saff)
saff='Y';
end
end
spin=input('Include Magnus lift force? Y/N, [Y] ','s');
if isempty(spin)
spin='Y';
end
if spin=='Y'
rev=input('Input initial spin rate of grain, (rev/s) ');
vel0=[xdot zdot 2*pi*rev]';
else
rev=0;
vel0=[xdot zdot];
end
rough=input('Input bed roughness as multiple of grain diameter ');
Ks=rough*d;
z=0.5*d;
z0=0.11*(nu/uf) + 0.033*Ks;

%%Call fluidvel.m to determine fluid velocity
fluidvel

%%Initial fluid velcity w/respect to center of grain
fluid(1)=flow;

%%Initial relative velocity at top of grain
top(1)=fltop;

%%Initial relative velocity at bottom of grain
bot(1l)=flbot;

%%Calculation of velocity gradient across grain
dudzs(1)=dudz;

%%Determination of relative velocity
vrel=sqrt((velO(1)-flow)^2+vel0(2)^2);
Vr(l)=vrel;

%%Determination of Reynolds number
Re=vrel*d/nu;
%Cd=(24/Re)+7.3/(l+sqrt(Re))+0.25;

%%Call coeffd.m to determine drag coefficient
%based on Morsi and Alexander (1972)
coeffd
Rno(1)=Re;
Cdrag(l)=Cd;

%%Set lift coefficient
if saff=='Y'
Restar=uf*d/nu;
if Restar<=5
C1=1.6;









elseif Restar>=70
C1=20;
else
Cl=interpl([5 70],[1.6 20],Restar);
end
end
if saff=='N'
%C1=0.2;
C1=0.85*Cd;
end
Clift(1)=C1;

%%Initial grain rotation
inert=(2/5)*(rho*s)*V*(d/2)^2;
coeff=pi*mu*(d^3)/inert;
if spin=='Y'
larsson(l)=vel0(3);
end

A=[];
i=2;

%%Initial Basset force
bassetx=0;
basx(1)=bassetx;
bassetz=0;
basz(1)=bassetz;

%%Initial gravity force/buoyancy force
gravx=rho*(s-l)*g*V*sin(beta);
gravz=-rho*(s-1)*g*V*cos(beta);

%%Initial drag force
if drag=='Y'
dragx=-Cd*Ar*0.5*rho*vrel*(velO(1)-flow);
dragz=-Cd*Ar*0.5*rho*vrel*velO(2);
end
if drag=='N'
dragx=0;
dragz=0;
end
%%Initial shear lift force
if shear=='Y'
if saff=='Y'
liftx=0;
liftz=-Cl*rho*sqrt(nu)*(d^2)*vrel*sqrt(dudz)*((vel0(1)-
flow)/vrel);
else
utop=sqrt((vel0(l)-fltop)^2 + velO(2)^2);
ubot=sqrt((vel0(1)-flbot)^2 + velO(2)^2);
utops(1)=utop;
ubots(1)=ubot;
liftx=0;
liftz=(rho/2)*Ar*Cl*(utop^2 ubot^2);
end
end
if shear=='N'
liftx=0;
liftz=0;
end
%%Initial Magnus force
if spin=='Y'
magx=(pi/8)*(d^3)*rho*vrel*(velO(3)-0.5*dudz)*(velO(2)/vrel);
magz=-(pi/8)*(d^3)*rho*vrel*(velO(3)-0.5*dudz)*((velO(1)-flow)/vrel);









end
if spin=='N'
magx=0;
magz=0;
end
magsx(1)=magx;
magsz(1)=magz;

%%Initial added mass
admass=l+(Cm/s);

%%Initial grain accelerations
dudts(1)=(bassetx+dragx+gravx+liftx +magx)/(rho*s*V*admass);
dwdts(1)=(bassetz +gravz + dragz + liftz +magz)/(rho*s*V*admass);

%%Initial grain position
dep(1)=z; % z=0 from velocity profile calculation
x=0;
len(l)=x;
time(1)=0;

%Calculation of velocities until grain hits bed
while jones==0
[t,vel] = ode45('motion',t,0,tf,vel);
B=[t vel];

%%Calculation of grain position
z = 0.5*deltat*(vel0(2)+vel(length(vel),2))+z;
x = 0.5*deltat*(velO(1)+vel(length(vel),l))+x;
len(i)=x;
dep(i)=z;
time(i)=(i-l)*deltat;

%%Check if grain reached bed
if z<0.5*d
jones=l;
end
%%Keep accelerations tabbed
dudts(i)=dudt;
dwdts(i)=dwdt;

%%Update of Basset history force
btu=0; %% Need these conditions
btw=0; %% Need these conditions
if i==2
btu=(dudts(i-l)+dudts(i))*sqrt(deltat);
btw=(dwdts(i-l)+dwdts(i))*sqrt(deltat);
else
for k=l:(i-2)
btu=(dudts(k)+dudts(k+l))*sqrt(deltat)*...
(sqrt(i-k)-sqrt(i-l-k))+btu;
btw=(dwdts(k)+dwdts(k+l))*sqrt(deltat)*...
(sqrt(i-k)-sqrt(i-l-k))+btw;
end
btu=(dudts(i-l)+dudts(i))*sqrt(deltat)+btu;
btw=(dwdts(i-l)+dwdts(i))*sqrt(deltat)+btw;
end

%%Call fluidvel.m to determine fluid velocity
fluidvel


%%Fluid velocity at middle of sphere
fluid(i)=flow;









%%Relative velocity at top of grain
top(i)=fltop;

%%Relative velocity at bottom of grain
bot(i)=flbot;

dudzs(i)=dudz;

%%Update initial conditions
if spin=='Y'
vel0=[vel(length(vel),1) vel(length(vel),2)...
vel(length(vel),3)] ';
else
vel0=[vel(length(vel),1) vel(length(vel),2)]';
end
t0=tf;tf=tf+deltat;
A=[A;B];

%%Keep record of contributing forces
bassetx=-const*btu;
bassetz=-const*btw;
basx(i)=bassetx;
basz(i)=bassetz;

Vr(i)=vrel;
Rno(i)=Re;
Cdrag(i)=Cd;
Clift(i)=C1;

if shear=='Y'
utops(i)=utop;
ubots(i)=ubot;
end
if spin=='Y'
larsson(i)=vel0(3);
end
magsx(i)=magx;
magsz(i)=magz;
i=i+l;
end

%%Plot of particle trajectory
p=l;
for q=l:18:length(A)
ts(p)=A(q,l);
us(p)=A(q,2)
ws(p)=A(q,3);
p=p+1;
end
ts(p)=A(length(A),1);
ws(p)=A(length(A),3);
u=us/uf;

figure(1)
plot(len./d,dep./d)
xlabel('Nondimensional Length, L/d')
ylabel('Nondimensional Height, H/d')

figure(2)
subplot(2,1,1)
plot(time./max(ts),Rno)
ylabel('Reynolds Number')


subplot(2,1,2)




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Last updated October 10, 2010 - - mvs