Diffusion of Sediment in the Lee of Dune-Like Bedforms. (Volumes I and II)
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LSU Historical Dissertations and Theses Graduate School
1976 Diffusion of Sediment in the Lee of Dune-Like Bedforms. (Volumes I and II). Chinmoy Chakrabarti Louisiana State University and Agricultural & Mechanical College
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Recommended Citation Chakrabarti, Chinmoy, "Diffusion of Sediment in the Lee of Dune-Like Bedforms. (Volumes I and II)." (1976). LSU Historical Dissertations and Theses. 3006. https://digitalcommons.lsu.edu/gradschool_disstheses/3006
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CHAKRABARTI, Chinmoy, 1943- DIFFUSION OF SEDIMENT IN THE LEE OF DUNE-LIKE BEDFORMS. (VOLUMES I AND II) The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1976 Geology
Xerox University Microfilms, Ann Arbor, Michigan 48106 DIFFUSION OF SEDIMENT IN THE LEE OF DUNE-LIKE BEDFORMS VOLUME I
A Dissertation
Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy
m
The Department of Geology
by Chinmoy Chakrabarti B.Sc.(Hons.), Jadavpur University, 1962 M.Sc., Jadavpur University, 1964 December, 1976 ACKNOWLED GEMENT
I am grateful to the members of my examining committee,
Drs. J. M. Coleman, P. H. Jones, D. R. Lowe, J. P. Morgan,
and R. H. Pilger for critically reading the manuscript.
Dr. Lowe went through several drafts of the dissertation and has been most helpful with constructive suggestions.
Dr. Morgan arranged for the experimental work and helped in many ways in carrying it out. His critical reading has improved the quality of presentation of this dissertation.
I am deeply indebted to the authorities of the Water ways Experimental Station, U. S. Army Corps of Engineers,
Vicksburg, Mississippi for facilitating the experimental work. Mr. James Glover, Chief, Waterways Branch, and Mr.
Tom Pokrefke, project engineer, WES, gave me invaluable assistance in the experimental design.
I am indebted to the Director General, Geological
Survey of India for granting me leave of absence for carrying out this work. My program was sponsored and initially financed jointly by the U. S. Educational
Foundation in India and the Institute of International
Education. I acknowledge with thanks the financial assist ance provided by the Department of Geology and the Student
Government Association of Louisiana State University, and the Sigma Xi to complete my study. Institute of Inter national Education also provided funds for finalization of the dissertation.
Finally, I am indebted to my wife, Champa, for moral support through many trying circumstances.
iii TABLE OP CONTENTS
Page
VOLUME I
ACKNOWLEDGEMENTS...... ii
LIST OF TABLES ...... v
LIST OP F I G U R E S ...... vi
LIST OF APPENDICES...... x
ABSTRACT...... xii
INTRODUCTION...... 1
DEPOSITION RATE AND SIZE DISTRIBUTION IN GRAIN SETTLED SEDIMENT...... 8
Theory ...... 8
Numerical Analysis ...... 23
Experiments ...... 65
SIZE DIFFERENTIATION IN AVALANCED SEDIMENT...... 116
T h e o r y ...... 116
Experiments ...... 125
CONCLUSIONS ...... 166
REFERENCES...... 169
VOLUME II
APPENDICES ...... 173
VITA ...... 401
iv LIST OP TABLES
Page
Table 1. Flow conditions for the experiments
17 to 2 7 ...... 82
Table 2. Calculated values of the coefficient,
c_,s in the deposition rate gradient regression equations ...... 99
Table 3. Calculated values of the constant, b,
in the deposition rate gradient
equation...... 99
Table 4. Values of the co-efficients a* and
n* from the mean size gradient
regression equations ...... 101
Table 5. Flow conditions for experiments 2
to 16 137
v LIST OF FIGURES
Determination of stream depth by the inverse path line method ...... 4
Principle elements of reattaching half-jet ...... 22
Flow chart outlining the main steps in the numerical simulation program 26
Friction-factor predictor for flat bed flows in alluvial channels - . . . 28
The threshold of sediment movement as a function of the boundary Reynolds Number ...... 30
Extrapolated curves for dimensionless measures of median path length and logarithmic standard deviation of dimensionless measures of starting height ...... 32
Effect of sorting of bed material on simulated mean size gradient of the leeside sediment ...... 38
Effect of sorting of bed material on simulated mean size gradient of the leeside sediment ...... 40
Effect of sorting of bed material on simulated mean size gradient of the leeside sediment ...... 43
Effect of sorting of bed material on simulated mean size gradient of the leeside sediment ...... 43
Effect of sorting of bed material on simulated mean size gradient of the leeside sediment ...... 46
vi Effect of sorting of bed material on simulated deposition rate gradient of the leeside sediment . . . . 52
Effect of sorting of bed material on simulated deposition rate gradient of the leeside sediment . . . . 54
Effect of sorting of bed material on simulated deposition rate gradient of the leeside deposit . . . . 56
Effect of the mean size of the bed material on the simulated deposition rate gradient of the leeside deposit . . 60
Effect of the mean size of the bed material on the simulated deposition rate gradient of the leeside deposit . . 62
Photograph of the Temperature Control Flume Facility of the U.S. Waterways Experiment Station, Vicksburg, Mississippi ...... 67
Schematic diagram of the flume used for experimentation ...... 69
End view of the sand feeder system of the flume looking downstream . . . . 72
Diagram of the bedform modelled on the flume floor and of the sediment collector ...... 75
Size frequency diagrams of sediment used as bed stock in the flume experiments ...... 80
Deposition rate gradient of grain settled sediment for experiments 20 and 21 using the bed material of sand 1 ...... 85
vii Page
Figure 23. Experimental deposition rate plotted against distance from the dune crest using bed material from sand 2 . . . 87
Figure 24 Experimental deposition rate plotted against distance from the dune crest using bed material from sand 3 . . . 89
Figure 25. Experimental deposition rate plotted against distance from the dune crest using bed material from sand 4 . . . 91
Figure 26. Graphs showing the variation of relative weight concentration as probability percentage with distance from the crest of the dune for experiments with sand 1 and sand 2 . 94
Figure 27 Diagram showing the variation of relative weight concentration as probability percentage with distance from the crest of the dune for experiments with sand 3 and sand 4 . 96
Figure 28, Variation of the mean size of the leeside sediment with increasing distance from the dune crest .... 103
Figure 29 Grain size gradient obtained numerically with different corrections introduced in the simulation program for experiment 17...... 112
Figure 30, Schematic diagram showing the mean size gradient predicted for various flow and sediment characteristics 124
Figure 31 Diagram of the dune-like bedform modelled on the flume floor . . 127
Figure 32 Measured velocity profiles of some of the experiments from the inlet end of the flume to a point beyond the reattachment zone ...... 132
viii Page
Figure 33. Size frequency diagram for the transported sediment of experiment 2 and experiment 5 ...... 135
Figure 34. Mean size gradient of the avalanched sand obtained in experiment 2 and experiment 6 ...... 141
Figure 35. Mean size gradient of the avalanched sediment obtained in experiment 3 and experiment 7 ...... 144
Figure 36. Mean size gradient of the avalanched sediment obtained in experiment 4 . . . 146
Figure 37. Mean size gradient of the avalanched sediment obtained in experiment 5 . . . 148
Figure 38. Mean size gradient of the avalanched sand obtained in experiments with bed material of sand 3 ...... 153
Figure 39. Mean size gradient of the avalanched sand obtained in experiments with bed material of sand 4 ...... 157
Figure 40. Mean size gradient of the avalanched sand obtained in experiments with bed material of sand 1 ...... 162 LIST OF APPENDICES
Page
Appendix 1. Listing of the computer simulation p r o g r a m ...... 173
Appendix 2. Input variables of the numerical simulation program ...... 176
Appendix 3. Results of numerical simulation .... 178
Appendix 4. Grain size gradients obtained numerically through the simulation p r o g r a m ...... 285
Figs. 1-6. Effect of sorting of bed material ...... 287 Figs. 7-11. Effect of flow velocity. . 309 Figs. 12-13. Effect of flow depth . . 319 Figs. 14-17. Effect of mean size of bed material ...... 323
Appendix 5. Deposition rate gradient obtained numerically through the simulation p r o g r a m ...... 330
Fig. 1. Effect of flow velocity . . . 332 Figs. 2-3. Effect of flow depth . . . 336
Appendix 6. Flow conditions for experiments 17 to 27 339
Appendix 7. Deposition rate in flume experiments . 340
Appendix 8. Regression equations of deposition rate g r adient...... 359
Appendix 9. Mean size gradient in flume experiments...... 360
Appendix 10. Regression equations of mean size g r a d i e n t ...... 370
x Page
Appendix 11. Results of numerical calculation for flume experiments ...... 371
Appendix 12. Bedform characteristics on different parts of the dune 394
Appendix 13. Size characteristics of avalanched sediment in experiments with sand 2 . . 395
Appendix 14. Size characteristics of avalanched sediment in experiments with sand 3 . . 396
Appendix 15. Size characteristics of avalanched sediment in experiments with sand 4 . . 397
Appendix 16. Size characteristics of avalanched sediment in experiments with sand 1 . . 398
Appendix 17. Size characteristics of sediment collected transverse to cross stratification ...... 399 ABSTRACT
An evaluation of theoretical and empirical studies of flow structure, sediment transport, and sediment size characteristics at the crest of dune-like bedforms indicates that it is possible to analyse quantitatively the diffusion processes and deposition of transported grains on the lee side of the bedforms. The numerical program based on these relationships simulates the deposition rate and grain size distribution on the leeside of the bedform for speci fied flow conditions and bed material. The results of systematic evaluation of each of the variables through the numerical simulation program have given considerable insight into the influence of each variable on the deposi tion rate and texture of leeside sediment before they are affected by avalanching. Sorting of the bed material is found to have great influence not only on the grain size gradient but also on the deposition rate gradient. The predicted and experimental grain size gradient become similar if certain corrections due to size sorting in smaller bedforms, which migrated on the dune stoss in the experiments, are incorporated in the simulation program.
The experimental deposition rate gradient is similar in form to the predicted trend. The absolute values are often
xii lower in magnitude than that predicted numerically because the potential transporting capacity of the flow was not fully satisfied in the experiments.
A semi-quantitative model of the texture of avalanched sediment on the dune lee indicates that the downslope mean size gradient is related to the mean size of the diffused sediment, nature of grain interactions during avalanching, flow speed and sediment transport rate. Experimental data collected broadly support the validity of the model. How ever, the experimental results also indicate that sorting of the transported sediment is as important as the other variables listed above in establishing a characteristic grain size gradient of avalanched sediment on the leeside of the dune-like bedform.
xiii INTRODUCTION
Cross-bedding or cross-lamination is one of the most common primary sedimentary structures. Generated by migrating ripples and dunes (Allen, 1963), cross-bedding forms within a turbulent flow system which exerts a strong sorting effect on the transported grains. It is to be expected, therefore, that theoretical analysis and numerical modelling of such a system will provide an understanding of the role that each independent variable plays in controlling the texture of cross-bedded deposits. Studies which have attempted semi-quantitative correlation between sand wave grain size distribution and flow properties include those of Jopling (1964a, 1964b, 1965a, 1965b, 1966a, and 1966b) and Allen (1965, 1968a, and 1968b). The present study aims in developing theoretical and empirical models of the deposition rate and grain size distribution on the leeside of a dune-like bedform and of subsequent avalanching which arises from the differential deposition rate.
Previous Work
Jopling (1966a) assumed that sediment accumulating in the bottomset zone of a delta-like sand body is in a state of suspension at the crest of the body. He estimated the
1 2 paleo-velocity of the stream depositing the sand from the competency relationship of threshold movement and suspension transport. Longitudinal velocity profiles were drawn from the velocity defect law (Sundborg, 1956) of open channel flow. Inverse path lines were sketched from the site of the bottomset samples in order to estimate from a knowledge of the settling velocities of the particles the projection height of the sampled particles above the stream bed. The paleo-depth was calculated by this so called inverse path line method in which the flow was assumed quasi-laminar
(Fig. 1). Serious objections were raised by Allen (1968b) about some of the simplifying assumptions made by Jopling, most critical of which was the assumption of quasi-laminar flow and exclusion of strong velocity fluctuations in the mixing zone of the leeside of the sand body. Allen (1968a, b) studied the diffusion of grains in the leeside of dunes before they are affected by avalanching. The settling rate at any point on the leeside was empirically determined to be a power function which was dependent on the total transport rate, grain size, and the mean flow velocity.
Allen (1968a) also argued that the mean size of the avalanched sand should have a downslope gradient which he related to the mean grain size, nature of grain interactions during avalanching, flow speed, and sand transport rate. 3
Fig. 1. Determination of stream depth by the inverse path
line method. The approximate path or locus of a
settling particle is retraced from the point of
particle lodgement in the bottomset to the level
of particle movement in the stream. The 'ceiling
height' of the reconstructed path lines defines
the depth of the stream (from Jopling, 1968a,
Fig. 2). Velocity Idealized path line Line of no diffusion Rough estimate of stream depth
Inverse path line of settling of zero velocii particle Locus
'Maximum travel of given size class of particle in downstream direction 5
Limited experimental data supported the predicted trend.
Statement of the Problem
No attempt has yet been made to analyze theoretically
the diffusion and deposition of sediment on the leeside of
dune-like bedforms. Based on theoretical and empirical
relationships, there exists no numerical model which re
lates the deposition rate and texture of settled sediment before avalanching to the flow and sediment properties.
It is worthwhile, therefore, to explore the possibility
of developing such a model.
A semi-quantitative model of downslope grain size
changes in avalanched sediment was proposed by Allen
(1968a). The model, if correct, is an important step
towards the semi-quantitative estimation of the relative
flow properties of depositional systems from the detailed spatial grain size variations of their cross-laminated deposits, even where the gross grain characteristics are identical. However, very little experimental evidence exists to confirm the predicted size gradient. A set of experiments is required to confirm the main features of the model and to clarify additional variables which might influence the spatial grain size distribution. 6
Objectives
The objectives of the present study are
1. to evaluate theoretical and empirical studies of flow
structure, sediment transport, and sediment size differen
tiation at the crest of dune-like bedforms and to analyze
diffusion processes and deposition of transported grains
on the leeside of the bedforms with the objective of
simulating numerically the deposition rate and grain size distribution on the leeside of such bedforms for specified
flow conditions and bed material?
2. to evaluate the influence of each variable on the proposed model?
3. to obtain new experimental data on deposition rate and texture of grain settled sediment for comparison with the simulation data, and to discuss the factors responsible for any differences between predicted and actual results?
4. to analyze the process of avalanching and to develop a semi-quantitative model of the texture of avalanched sediment? and
5. to perform a second series of experiments to obtain data on the downslope variation of texture of slipface deposits in a dune-like bedform under varying flow condi tions and with differing bed material in order to test the predicted grain size gradient and to discuss and explain 7 the correlation between the two.
Procedure of Analysis
For an analysis to the problem outlined, the processes affecting the grain size distribution before diffusion at the dune crest, during diffusion and sedimentation, and after deposition must be considered. These include:
1. deposition rate and size distribution in grain settled sediment including
a. hydraulic sorting or vertical diffusion of the
grains before they are convected on the leeside
of the dune; and
b. horizontal diffusion, convection, and settling of
the grains on the leeside of the dune, and
2. avalanching of the grains after they settle on the dune leeside. DEPOSITION RATE AND SIZE DISTRIBUTION IN
GRAIN SETTLED SEDIMENT
Theory
The theoretical analysis of the deposition rate and grain size distribution in settled sediment must be based on a knowledge of the following three characteristics:
1. height or flow levels from which the grains are
diffused,
2. grain velocity at each level, and
3. grain transport rate of each grain-size fraction
at each flow level.
These three properties must be determined from (a) the mean grain size, D, standard deviation, a , and the dis tribution function, f(D), of the material to be diffused at the dune crest, (b) the mean flow velocity, u, (c) the flow depth, d, and (d) the dune height, H. The flow is assumed to be turbulent and two dimensional and sediment is carried both as bed and suspended load. The flow at the dune crest may be visualized as composed of fluid layers of uniform thickness. Such division into discrete layers is advantageous for numerical calculation as each of the parcels is expected to have different characters
8 9
as outlined (1 to 3) above.
Flow Zones
Assuming that the flow is fully developed, turbulent, and rough, the flow field may be divided into two zones between which flow and sediment characteristics can be expected to differ: the bedload zone near the bed and the zone of suspended transport higher in the flow field.
Above the stationary bed, in which limiting static shear resistance is greater than the total applied shear stress, the bedload layer represents a distinct region. It is expected that its thickness will be complexly related to the fluid flow properties and grain size. The thickness of the zone was determined by Kalinske (1942) in the form of the height of rise of the saltating grains of a given diameter, (D), as r x D. The parameter r was defined by the relation
r = [1/ (2gD (pg - pf)/pf u2 + 0.75 Cd pf/ps)] (1) where = drag co-efficient of the grains,
pg = density of the grains, and
= density of the fluid.
Cd can be derived from the particle Reynolds Number
(Re = ~ ~ , where v = kinematic viscosity of the fluid) 10
from the relation
_ 24 , 0.687. (1 + 0.15 Re ) (2)
The approximation was given by Schiller et al. (1933) for
Re <_ 800. The rest of the flow field is occupied by the suspended-load zone.
Flow and Grain Velocity Profiles
The flow velocity profiles in the bed and suspended load zone are necessarily distinct and different. In the bedload layer, the fluid flow velocities are difficult to measure, but generally increase with increasing distance from the static bed. It is obvious that grains in the bedload zone would have a transport velocity substantially smaller than the fluid flow velocity. However, an indirect approach from some bedload discharge relationship may be sought. Bagnold (1973), from the approach of general physics, derived a bedload discharge equation which can be evaluated. In the equation given by Bagnold (1973) the group 0.37d . . UB = u - 5.75 u, log10 - Vq (3) corresponds to the grain velocity in the bedload layer, where VQ = fall velocity of the grain. The only undetermined quantity in equation (3) is *n' which was 11
empirically related by Bagnold (1973) as
n = 1.4 (u /u )°'6 (4) o
where u. = shear stress velocity.
u. can be obtained from the relation o
u* = V (0.756 C,)°*5 (5) *o o o d
where 9q = Shields parameter for initiation of erosion.
6q can be best obtained from the grain boundary Reynolds
Number (Re* = from the modified Shields diagram given by White (1970). The approach of Bagnold (1973), however,
gives only an average value for the grain velocity in the bedload zone which is useful in bedload discharge calcula
tion. If the bedload zone is several grains diameter
thick, as an approximation for the grain velocity profile, we may use the relationship developed by Lowe (1976) for
steady and uniform grain flows. The velocity profile in
such flows is given by
URB y = max [1 - (1 ~ y/y)3/2J (6) where u = grain velocity in the bedload zone at a y distance y from the static bed,
u = grain velocity at the top of the bedload surface, max 12 and
Y = thickness of the bedload zone.
The difference between grain flow, initiated and maintained by gravity acting on the grains directly, and bedload dis charge by fluid force is obvious and cannot be compared sensu strieto. In both cases, however, a thin layer of sediment is sheared against a static bed and a velocity gradient is established in the moving bed. Comparison between the velocity gradient curve given by Allen (1968b) for bedload transport and that given by Lowe (1976) for grain flow shows that they are similar in form. In the absence of a suitable function defining the grain velocity profile in the bedload zone, equation (6) developed for grain flow will be followed in this analysis. It is diffi cult to evaluate the magnitude of the error that will be introduced from this approximation.
The fluid flow velocity profile in the suspended load zone can be estimated conveniently using the procedure of
Brook (1963). Assuming the logarithmic velocity defect law,
u = u . + — lny/d (7) y max(at y = d) k and integrating for u equation (7) can be recast in the form 13
(8) where f = Darcy-Weisbach friction factor, and
k = von Karman's constant.
After substitution of u* by u “S/ f / Q and rearrangement equation (8) yields
u = u [1 + (1 + 2.3 log1Q y/d)] (9) Y
< has a value of about 0.4 for clear water, but determina tion of f is difficult. The approach of Lovera and Kennedy
(1969) can be used. They obtained the friction factor due to the grains alone from the flow Reynolds Number and the ratio d/D. Because we are considering that the bedload zone is defined by saltating grains (i.e., no other bedform migrates on the back of the dune-like bedform) no additional frictional component due to bedform is considered. If it is further assumed that the grain velocity of the suspended load zone at each flow level is very nearly equal to the fluid velocity at each corresponding level of the grain- free fluid, the fluid flow velocity profile given by equation (9) becomes the grain velocity profile in the suspended load zone. The accuracy of equation (9) in predicting the true velocity profile, however, depends on the assumptions that the fluid is not carrying any sediment and that it is moving past a fixed bed. Both these condi
tions are not satisfied in the present case and are sources
of error. However, experimental variation of k is very
small, and therefore the magnitude of error will be
minimum.
Sediment Transport Rates
The sediment concentration profile in the suspended
load zone is generally given in the form (Rouse, 1938)
V o /S k u * 2 L = fd-y . a_ (10) Ca y d-a where Cy = concentration at distance y from the bed,
Ca = concentration at a reference point located y a
above the bed, and
6 = ratio between sediment and momentum diffusion.
Equation (10) has several limitations which make it un suitable for use in the present study including the condi
tion that the reference concentration cannot be taken very near the bed. A more useful approximation is given by
Lane and Kalinske (1941) (see Raudkivi, 1967, p. 127;
Graf, 1971). 15 where t = VQ/u*.
The limitations and sources of error of equation (10) and
(11) have been discussed thoroughly in the literature (for example, Graf, 1971) and are not repeated here. Equation
(11) must, however, be applied to individual particle size fractions and can be applied iteratively, giving the total concentration at any level. Hie procedure of integration has been developed by Brush (1965). Assuming that 't* follows a normal distribution, then
where o standard deviation of V calculated from V,o o the particle size distribution, and y = mean Vq calculated from mean particle size.
The fraction of sediment in suspension of a discrete size class, 1, between two V limits can then be written as o
+ h. (vQ/u *) * f (V /u ) d (V /u ) (13) O * o *
Vo k - H (VQ/u*) u *
Also from equation (11) 16
(14) (Cx)a
Multiplying equation (13) by equation (14) and taking sizes i to j into consideration
where (CT)y = total concentration of sediment at any
level y, and
(CT)a = total concentration of sediment at a
reference level a.
For a completely deterministic model, the value of (cT)a for any chosen reference level must be considered and evaluated. Lane and Kalinske (1939) related the amount of the material in suspension at or very near the bottom
(a ~ 0) to the bottom composition and to the hydraulic factors tending to cause the material to be placed in suspension. For a given size fraction, i, the concentra tion at a * 0 is given approximately by
.1.61 (16) (C,) AF . X 5.55 Pi i a“0 cx 17
Fc^ = the percentage of the size class i present in
the total distribution, and
P± = (l/t±) f t e-t dt (17) t . i
The constant in equation (16) was, however, derived by
Lane and Kalinske (1939) from field data from large rivers
like the Mississippi. Assumptions were also made that the
turbulence follows the normal error law, and error function was introduced in evaluating the variable P .. Equation x (17) can be integrated in a closed form as
(18)
For the entire distribution, therefore, the total concentration at * a' is
3 1.61 ( O = y, Fc, X 5.55 P, (19) T a=0 4-1 , i i i=l
Substituting equation (19) into equation (15), an expression for total concentration can be obtained at any level as 18
3 (cT )y = Z (AFc±X 5.55 P.1 *61) i=l
X J f (t)dt X e
Equation (20) can be evaluated numerically to calculate the transport rate of each fraction of suspended sediment at each level of the flow for all flow levels.
The derivation of bedload transport rate function is one of the much disputed and discussed topics in the dynamics of solid-liquid systems. Many theoretical and empirical relationships exist, but most require the know ledge of parameters in addition to those which we have assumed to be known. A survey of the literature shows that the function best suited for our analysis is given by Bagnold (1973) as
= aruy tl _ 5.75_ u ^ h og10 (y/nD) + ur] (21) y
A number of approximations were made by Bagnold (1973) on the basis of strong empirical reasons as follows:
u* - u* a = ~ * o , and u, tan « = 0.63. 19
A further approximation for wide channels may be made for
t as follows (Schlichting, 1951, p. 356? Sundborg, 1956,
p. 154, especially footnote):
t = t (1 = y/d) * t o o
The total bedload transport rate can be divided among the
individual size fractions according to the probability
distribution of the bed material through the error function.
Unfortunately, measurement techniques of bedload discharge
have not reached a satisfactory stage and it is difficult,
therefore, to evaluate Bagnold9s equation in terms of the
magnitude of variation it can introduce under different
flow conditions.
Path Length
It is necessary now to review and examine critically
the flow characteristics due to presence of a dune on the
static bed. The flow past a dune has been linked to that past a negative step with a sharp edge. In real fluid
flow, separation at the crest of the step is the rule and
reattachment takes place some distance downstream of the
crest. The flow thus may be compared to a two dimensional plane half jet reattaching to a semi-rigid body at the downstream end of the negative step. A captive eddy or vortex termed the separation bubble is located behind the 20 attachment point in the corner. The flow characteristics past the step show important temporal and spatial fluc tuations, the understanding of which is essential for the study of sediment dispersion on the leeside of the dune.
Following Jopling (1965a) and Allen (1968a) a number of regions of contrasted flow structure may be identified
(Fig. 2). Above the boundary layer, the undisturbed stream shows low intensity turbulence. The free boundary layer shows steep velocity gradient. The mixing region resulting from the instability of the free shear layer shows strong velocity and pressure fluctuation. The other distinct region, the separation bubble, is characterized by rotating fluid with a horizontal axis perpendicular to the flow.
The turbulent intensity is low in this region with a sluggishly rotating mass of the fluid. The last region is the reattachment flow zone with large turbulence intensity.
As discussed by Allen (1968b), the settling of grains on the leeside of a dune cannot be treated simply as the resultant of the free fall velocity and the forward grain velocity in the mixing region. This is evidently due to the large temporal and spatial fluctuations of velocity in the mixing zone which makes any estimate of 'average' velocity meaningless. Grains having identical size and Fig. 2. Principal elements of reattaching half-jet
(from Allen, 1968a, Fig. 16.1). S and A
mark the separation and attachment points
respectively. 22
u Undisturbed stream
F ree b I.
Separation bubble 23 travelling at the same average velocity take a number of trajectories as they are diffused on the leeside of the dune. Allen (1968a) concluded from his experiments that the path lengths, i.e., the horizontal projection distance of the trajectories taken by the grains, follow a log normal distribution and so can be characterized by a median path length and logarithmic standard deviation. However, the experimental data were collected from the slipface region only. Also only a limited range of settling velocity (V ) to grain velocity (u ), and the height from which the grains were propelled (y) to the dune height (H) were investigated.
However, his extrapolation over a much larger field (Allen,
1968a, Fig. 18.11, p. 369) can be adopted for the calcula tion of the grain path length. It should be mentioned that Allen's experiment excluded aggregate and collective settling which are modifying factors influencing the grain path length.
Numerical Analysis
On the basis of the preceding discussion, it is now possible to simulate numerically the grain size and deposi tion rate gradient on the leeside of a dune knowing only five variables: particle mean size, sorting, depth of flow, flow velocity, and dune height. 24
Procedure
The numerical calculation is programmed for processing by computer. The program is written in Fortran language, a listing of which is given in Appendix 1. A flow chart des cribing the steps involved in the calculation is given in
Fig. 3. A few salient points of the program should be mentioned. From the five input variables, the Darcy-Weisbach friction factor is first obtained from the graph of Lovera and Kennedy (1969) (Fig. 4), and is incorporated as an input variable. Two regression equations are used to calculate
Shields parameter and the grain path length. These equations are derived from data points generated from the graph (Fig.
5) given by White (1970) and Allen (1968a) respectively
(Fig. 6). The free fall velocity of particles of a given size is calculated from the empirical relationship of Gibbs et a l (1971). The size range to be considered for integra tion is determined separately from the mean size and standard deviation so as to include 9 9 % of the size distribution curve. This is calculated easily assuming a log normal distribution in which 9 9 % of the size distribution curve lies between + 2.576 a of the mean. The dune is considered to be in equilibrium with the flow such that no smaller bedform migrates on its stoss side. It is assumed also that there is no size sorting in the bedload layer before the Fig. 3. Flow chart outlining the main steps involved
in the numerical simulation program. 26
INPUT VARIABLE: Mean velocity, particle h CALCULATE: Friction factor mean size, sorting, graphically. depth of flow and height of dune. INPUT: Friction factor. I INPUT: Constant: Density of solid and liquid, acceleration due to gravity, co efficient of dynamic viscosity. + CALCULATE: Particle Reynolds number, drag co-efficient, fall velocity. I CALCULATE: Thickness of bedload layer.
TO SUBROUTINE VEL I INPUT: Constant: von Karman's constant. I CALCULATE: Mean grain velocity in the bedload layer.------I DIVIDE: Suspended load zone into layers of 1 mm thickness. t CALCULATE: Flow velocity at each level. I CALCULATE: Concentration and transport rate at each level for each fraction of grains. I CALCULATE: The probability of each fraction from each level depositing in given segments of the slipface and bottom set zones.
CALCULATE: Bedload transport rate. ------t DIVIDE: Bedload layer into 1 mm thickness. I CALCULATE: The probability of each fraction from each layer depositing in given segments of the slipface and bottom set zones. I ADD: The two deposition rates. — — — ------I CALCULATE: Leeside deposition rate. I CALCULATE: Leeside size gradient. 27
Fig. 4. Friction-factor predictor for flat-bed flows in
alluvial channels (from Lovera and Kennedy, 1969,
Fig. 1). .04
.03
.02
.01
Re 29
Fig. 5. The threshold of sediment movement as a function
of the boundary Reynolds Number (from White,
1970, Fig. 2). 1
.7 ■
.4 •
e .2 ■ o .1
.07 ■
.04 ■
.02 >
• 0 1 — 11. i, ... — — * - - - i______■ ------j — — - ■ 01 .1 1 10 100 1000 Re*
u> o 31
Fig. 6. Extrapolated curves for dimensionless measures
of median path length (Xme(j/H) and logarithmic
standard deviation on dimensionless measures of
starting height (y/H) with u^/Vq taken as a
parameter (from Allen, 1968a, Fig. 18.11). 32
10*
10° mtd H
10
Experimental region
10r3 10°
/ H 33
grains are diffused at the crest. The flow zone is
divided into layers of 0.1 cm. thick for integration.
The probability of a given outcome, whether it be of the
fraction of solid present in the population or of a grain
of a given size taking a particular chosen path length
after diffusion, is calculated through the error function
as the total probability, Pr, up to that fraction or path
length. The probability of an individual outcome is cal culated by subtracting from Pr the total probability Pr^ where |Pr-Pr^| is the size interval or path length interval chosen. The concentration equation of suspended load gives finite values for all size classes under all flow conditions. It is obvious that for large grain sizes, although there is a finite calculated concentration, no grains are actually suspended. However, in the absence of any well defined way to eliminate such predicted values, it is left to the computer to eliminate and set equal to zero any concentration which is below a designated level *—68 (■^1 X 10 ). The size gradient and the deposition rate are calculated as if material is being deposited in
2 cm intervals starting from the crest of the dune up to
84 cm downstream from the crest. The output from the program is in the form of 3 sets of data each with 42 , 2 . values of size in phi units, deposition rate m gm/cm /sec, 34 and relative deposition rate compared with the total deposition over the 84 cm interval.
In order to evaluate the role of each variable in influencing the size and deposition rate gradient, a number of numerical experiments have been conducted. The input values of a particular variable are varied from one experi ment to another, keeping all other variables constant. The input data for the experiments are listed in Appendix 2.
Listed in Appendix 3 are the values of grain size variation in phi units, deposition rate, and relative deposition rate 2 in gm/cm /sec over the area under consideration. The results of these numerical experiments have given consider able insight into the role played by each variable in in fluencing the size and deposition rate gradient on the leeside of a dune.
Dimensional Analysis
Before attempting an analysis of the individual in fluence of each variable, it is necessary to make a dimensional analysis of the variables involved. The following eight variables are relevant:
u, d, p_, y , D, p , a , and g. f s
Setting aside for the moment the controversy over the 35
dimension of phi units (Blatt et al_., 1970) we may attach
a length dimension to both D and o . Dimensional analysis with the restriction that any two of the variables u, d, D,
and 0 cannot appear together in any grouping yields the
following dimensionless terms (Southard, 1971):
2 1/3 1/3 2 1/3 2 1/3 fS\ d, /Pf\ ujp fg\ D, and fp fg\ a
If we restrict ourselves to quartz sediment in water at constant temperature, the first dimensionless term and the co-efficients in the terms become constant. So the results of numerical analysis can be evaluated in terms of variation of only u, d, D, and cr . We may mention that another variable, H, the height of the dune, is treated here as constant at 15 cm. This is because the empirical data used for calculation of path length was collected by
Allen (1968a) for H = 15 cm only.
Size Gradient
A size gradient is defined here as positive when the mean size increases with increasing distance downstream from the dune crest. Similarly, a negative size gradient implies decreasing mean size with increasing distance down stream from the dune crest. The grain size gradients obtained by the numerical simulation program under varying 36
flow conditions with sediment of three different mean sizes are shown graphically in Appendix 4. The curves show a negative size gradient except for the first 2 cm of the leeside.
In all the sizes considered, very well sorted bed material ( cr = 0.35
90 cm/sec.
In the fine sand range (D = 2.5
36). For experiments with a low velocity and high water depth (Fig. 8), the leeside sediment formed from the poorly sorted bed material is coarser only within a few cm of the crest. Because of the very steep negative side gradient obtained with the poorly sorted bed material in these experi ments, more distal leeside sediment is actually finer grained than that with better sorted bed material. However, except in the two experiments with low velocity (u = 40 cm/sec) and 37
Fig. 7. Effect of sorting of bed material on simulated
mean size gradient of the leeside sediment. The
number of each curve corresponds to the experi
ment number.
u = 90 cm/secy D = 2 .5
Experiment 38 ( a = 1.5 ) ,
experiment 37 ( a = 0.71 (f> ), and
experiment 36 ( o = 0 . 3 5
31
36 2 .7 0 306 0 90 DISTANCE FROM DUNE CREST (cm) 39 Fig. 8. Effect of sorting of bed material on simulated mean size gradient of the leeside sediment. The numbers of each curve corresponds to the experi ment number. u = 40 cm/sec; D = 2.5 Experiment 73 ( 0 = 1.5 experiment 67 ( o = 0.71 experiment 40 ( o = 0.35 2 .3 67 2.6 40 73 s 290 30 60 90 DISTANCE FROM DUNE CREST (cm) 41 high water depth (d = 30 cm and 50 cm) there is a direct positive relationship between sorting and grain size in the diffused sediment within the 84 cm zone of the dune crest. This relation becomes more pronounced as the flow velocity is increased and/or flow depth is decreased. The same general relationship can be observed in the experiments with medium sand (D = 1.5 The distance over which the leeside sediment formed from poorly sorted bed material is also coarser increases with increasing flow strength until at very high flow strength (u = 90 cm/sec and d = 15 cm) (Fig. 10) the leeside sediment using poorly sorted bed material is coarser over the entire 84 cm zone than that deposited using well sorted sediment. In experiments with coarse sand (D = 0.50 40 cm of the dune lee than that from well sorted sediment 42 Fig. 9. Effect of sorting of bed material on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experi ment number. u = 40 cm/sec; D=1.5$;d=15cm. Experiment 42 (o=0.35 experiment 43 ( a = 0.71 experiment 44 ( o = 1.5 Fig. 10. Effect of sorting of bed material on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experi ment number. u = 90 cm/sec? D = 1.5 Experiment 48 ( a = 0.35^), experiment 49 ( o = 0.71 experiment 50 ( o = 1.5 ) . 43 FIG . 9 cr UJ »- UJ 2 42 < o z < cc (£) 43 z < UJ 2 44 2.4 0 30 60 90 DISTANCE FROM DUNE CREST (cm) 0 a : UJ »- 1.1 UJ 2 < o < on 1.4 o SO < 49 UJ 48 1.7 0 30 60 90 DISTANCE FROM DUNE CREST (cm) 44 (Fig. 11). Under most of the flow conditions, the leeside sediment deposited using poorly sorted bed material is coarser only within a few cm off the dune crest, beyond which it is actually finer. The above trend can be explained in terms of flow competence. When the bed material is poorly sorted, the flow is supplied with more coarse and fine particles than when the sediment has the same mean size but a lower sorting index. In the fine sand range, however, under most of the flow condition investigated, the flow is able to transport a greater amount of coarse sediment when poorly sorted sediment is used, and diffuse it over some distance on the dune lee. The increased finer grained material at the same time will be diffused to greater distances (beyond the 84 cm field used in the experiment). The leeside sediment in the case of the poorly sorted bed material is, therefore, coarser over some distance beyond the dune crest than that with the well sorted sediment. The distance over which the flow could diffuse the coarser grains, however, decreases with decrease in flow strength (Fig. 8) until at extreme low flow strength, it could diffuse them only to a short distance off the dune crest. Therefore beyond this dis tance the leeside sediment actually is fine grained. With medium and coarse sand bed material, the flow is 45 Fig. 11. Effect of sorting of bed material on simulated mean size gradient of the leeside sediment. Ihe number of each curve corresponds to the experi ment number. u = 90 cm/sec; D = 0.50 4> ? d = 15 cm. Experiment 60 ( o = 0.35 experiment 61 ( o = 0.71 experiment 62 ( ° = 1.5 supplied with coarser grains compared to the fine grained bed material. Comparisons between poorly sorted and well sorted medium- and coarse-grained sediment show that for the former, the flow will be supplied with some relatively coarser particles which can be diffused only a short dist ance beyond the dune crest, the increased finer grained material being carried much further downstream. As a result, the leeside sediment with the poorly sorted bed material is coarser than that with the well sorted bed material only over a small distance beyond the dune crest. This distance increases very slowly with increasing flow strength. Thus even at a very high flow strength (u = 90 cm/sec and d = 15 cm) the leeside formed from poorly sorted bed material is coarser only for a short distance. This distance decreases rapidly with increase in the mean size of the bed material. The evaluation of the influence of sorting on size gradient under varying flow conditions has already given indications about the effects of velocity on the size gradient of the leeside sediment. Variation of grain size gradient with changes in flow velocity holding all other variables constant is shown in Appendix 4. As expected, increasing the velocity not only makes the leeside sediment coarser but also decreases the size gradient. This is true 48 for all size classes with all degrees of sorting and for all water depths. The explanation lies in the fact that with increase in velocity, the competence of the flow is increased and the grain path lengths are also increased. The result is that more coarse particles are not only supplied to the dune lee but also they are diffused over a longer distance and larger area. It follows from the above discussion also that in creasing flow depth holding all other variables constant will have the same effect as decreasing the velocity. Figs. 12 and 13 in Appendix 4 show the results of plotting grain size with downstream distance for groups of data gen erated for constant velocity, particle size and sorting. As can be expected, the leeside sediment becomes coarser and the size gradient becomes flatter as water depth is decreased for flows with same velocity, mean size, and sorting. This can be explained in the same way as we have explained the effect of increasing velocity. The effect of changing particle mean size while holding flow conditions constant is straightforward. It is obvious that the mean size will increase, but it is also reasonable to expect that the gradient will be steeper. Figs. 14-17 in Appendix 4 show the variation of grain size gradient with increase in mean particle size of the bed material holding 49 all other variables constant. The explanation of the in crease in the grain size gradient with increasing size lies in the fact that, when grain size is increased, the flow is required to transport coarser materials, which it can diffuse only to a short distance, the overall capacity of the flow remaining the same. Therefore, the grain size in the upper part of the dune lee will receive more coarse particles com pared to that in the lower part, in case of a coarse bed stock. The result will be a steep grain size gradient in the dune lee with coarse bed material. Deposition Rate Gradient Except for the first 2 cm of the dune lee the data points of deposition rate obtained in the numerical experi ments fall in a straight line with a negative gradient on log-log paper. Increasing the sorting index of the bed material has a considerable influence on the deposition rate on the dune lee. Although the absolute gradient does not vary signi ficantly with change in the sorting of the bed material, the relative position (i.e., whether deposition rate de creases or increases over the first 84 cm of the dune lee) changes differently under different flow conditions and bed material size. 50 In the fine sand range (D = 2.5 In experiments with medium and coarse sand (D = 1.5 and 0.50 The effect of sorting on the deposition rate curves under varying flow conditions with different bed material can be explained as we have done for the mean size gradient. An increase in sorting index (decrease in sorting) implies that the flow will be supplied with relative greater amounts of both very coarse and very fine sediment. In the fine sand range and at low flow velocity (Fig. 12) the additional coarse material is transported in small quanti ties or may not be transported at all by the flow. The deposition rate decreases with increasing sorting index. However, at a high flow velocity (Fig. 12) the flow can transport the coarse fractions in much greater quantity and consequently the deposition rate on the 84 cm zone 51 Fig. 12. Effect of sorting of bed material on simulated deposition rate gradient of the leeside sediment. The number of each curve corresponds to experi ment number. D = 2.5 * . u = 90 cm/sec, d = 15 cm: o= 1.5$, experiment 38; 0 = 0.714>, experiment 37; o = 0.3541, experiment 36. u = 65 cm/sec, d = 15 cm: o=1.5 35; o = 0.71, experiment 34; ° = 0.35^, experi ment 33. u = 40 cm/sec, d = 50 cm; a =l.5 73; o = 0.71(f), experiment 67; o = 0. 35, experi ment 40. 52 E v E O) v o ac c o «n o a. ® a 10 Distance From Dune Crest (cm) 53 Fig. 13. Effect of sorting of the bed material on simulated deposition rate gradient of the leeside sediment. The number of each curve corresponds to experiment number. D = 1.5*. u = 40 cm/sec, d = 15 cm: experiment 42 ( a = 0.35*), experiment 43 ( 0 =0.71*), and experiment 44 ( o = 1.5*). u = 90 cm/sec, d = 15 cm: experiment 48 ( o = 0.35*), experiment 49 ( ° = 0.71*), and experiment 50 ( o = 1.5*). Deposition Rate (gm/cm2sec) 10 ' itne rm ue rs (cm) Crest Dune From Distance 50 43 42 55 Fig. 14. Effect of sorting of bed material on simulated deposition rate gradient of the leeside deposit. The number of each curve corresponds to experi ment number. D = 0.50 4>. u = 40 cm/sec, d = 15 cm: experiment 54 ( CT = 0.354*), experiment 55 ( 0 = 0.71$), and experiment 56 ( 0 = 1- 5 ^). u = 90 cm/sec, d = 15 cm: experiment 60 ( o = 0.3541), experiment 61 ( o = 0.7140, and experiment 63 ( a = 1.5 40 . 56 u 4) E \u 6 06 o oc c o vt o a a> Q 54 55 56 Distance From Dune Crest (cm) 57 increases with increase in sorting index. However, with medium and coarse sand bed material, even the highest flow stage (u = 90 cm/sec and d = 15 cm) is unable to entrain sufficient amount of additional coarse material when sorting index is increased. The result is that the deposition rate over the 84 cm zone actually decreases as the sorting index increases. The effect of velocity changes on the deposition rate gradient is direct. When velocity is increased, the capa city of the flow is increased. At high velocities, the flow is able to diffuse the particles at a greater rate, so the deposition rate increases over that under low velocity conditions. This is true for all flow conditions and sediment characteristics. Variations under some extreme flow conditions are shown graphically in Figs. 1A, B, and C of Appendix 5. Increasing flow depth has the effect of decreasing flow velocity. Increasing water depth, holding all other variables constant, decreases the deposition rate gradient over the 84 cm zone. This evidently implies that as the flow strength is reduced by increasing water depth, parti cles are transported at a lower rate, and the deposition rate is reduced. Plots of deposition rate against distance from the dune crest are shown in Figs. 2 and 3 of Appendix 5. 58 The effect on the deposition rate profile of changing the mean particle size of the bed material is somewhat complex, and is intimately connected with the flow velocity. In the low velocity flows (u = 40 cm/sec) the deposition rate over the 84 cm zone of the dune lee for coarse bed material (D = 0.50$) is lower than that for medium (D = 1.54>) and fine (5 = 2.5 ) bed material (Fig. 15). This is true for both poorly and well sorted sediment ( 0 = 0.35, 0.71, and 1.5$). However, the deposition rate with bed material of mean size 2.5 $ is lower than that with mean size 1.5$, except beyond 70 cm of the dune crest. When velocity is increased to 65 cm/sec, the deposition rate (Fig. 16) for 2.5 $ is much lower within the 84 cm zone than that for D = 1.5$ and 0. 5^ - Between the latter two in the same velocity range, the deposition rate for D = 1.5$ is lower up to about 15 cm from the dune crest beyond which it is higher. When velocity is increased to 90 cm/sec, the deposition rate (Fig. 16) for D = 2.5$ is lower than that with bed material with D = 1.5$ and 0.5$. Between the latter two, the deposition rate with D = 1.5$ is lower over the 84 cm zone, but tends to be higher beyond this zone. We may attempt to explain the above behaviour in more or less the same fashion as we have done before. At low flow velocity (u = 40 cm/sec), bed material with mean size 59 Fig. 15. Effect of mean size of the bed material on the simulated deposition rate gradient of the lee side deposit. Hie number of each curve corres ponds to experiment number, u = 40 cm/sec, d = 15 cm, and 0 = 0.35 ^. Experiment 54 (D = 0.5 1.5 u = 65 cm/sec, d = 15 cm, and cr = 0.71 Experiment 58 (D = 0.5 1.5 CM E a> e o oc c o o a 34 ® Q 30 54 Distance From Dune Crest (cm) 61 Fig. 16. Effect of mean size of the bed material on the simulated deposition rate gradient of the leeside deposit. Hie number of each curve corresponds to experiment number. u = 90 cm/sec, d = 15 cm, and o = 0.71 $. Experiment 61 (D = 0.50 1.5), and experiment 37 (D = 2.5$). u = 40 cm/sec, d = 15 cm, and 0 = 1.5$. Experiment 56 (D = 0.5$), experiment 44 (D = 1.5<{>), and experiment 32 (D = 2.5$). 62 a>i/» wE s E O) o QC c o w> o a « 37 Q c e F m 63 of 0.5 the dune crest: the deposition rate is low over the dune lee. When the mean size is decreased, more sediment is supplied to the dune lee and the deposition rate increases. However, for fine sand (D = 2.54>), the grains are convected to a greater distance compared to that for medium sand (D = 1.54>) . The result is that the deposition rate (Fig. 15) is lower for D = 2.5$ near the dune crest up to a distance of about 70 cm from the crest, and greater in the zone be yond this point compared to the deposition rate for D = 1. 5 D = 1.54> . When velocity is further increased to 90 cm/sec, the distance over which grains of the coarsest stock (D = 0.5 those with the finer stocks over the 84 cm zone of the dune lee. There is, however, a definite tendency for the deposition rate to be lower beyond this zone. It is necessary to say a few words on the grain size and deposition rate gradient obtained in the first 2 cm of the dune lee. Allen (1968a) has argued that the settling rate of a particular grade on the dune lee follows a proba bility density function which is very strongly skewed towards the larger values of the path length. Nonetheless, we have assumed that the grain path length is log normally distributed, ignoring the fact that the grain settling rate attains a maximum very close to the dune crest, the deposition at the very crest being zero. Nothing is known experimentally or theoretically about the distance of this settling maximum from the dune crest and the dependence of this distance on flow and sediment properties. It is likely, therefore, that the deposition rate and the size characteristics on the dune lee including this distance will show results which we might not be able to explain fully. Summarizing, the numerical experiments have shown that the size characteristics of bed material (mean size as well as sorting) play a significant role in determining the original size gradient of the leeside deposit of a dune. 65 Size distribution also has a major effect in controlling the deposition rate gradient on the dune lee. Since the deposition rate is intimately connected with the transport mechanism of open channel flow in general, it is essential that sorting of the bed material be considered as one of the variables influencing the transport rate of a stream. Experiments In order to verify the theoretically and numerically defined aspects of grain settled sediment on the leeside of a dune-like bedform, a series of flume experiments were devised. Experimental Arrangement The experiments were carried out at the 'Temperature Control Flume Facility' of the U. S. Army Corps of Engin eers, Waterways Experiment Station, Vicksburg, Mississippi. Although the temperature control devices were not employed, its other characteristics made the flume quite suitable for the designed experiments. The flume, which is nearly 30 meters long, 91.4 cm wide, and 30.5 cm deep, is constructed of plastic coated metal (Figs. 17 and 18). Mounted on the walls of the flume channel is a pair of rails which parallel the flume bed. Elevations of the bed and the water surface 6 6 Fig. 17. Photograph of the Temperature Control Flume Facility of the U. S. Waterways Experiment Station, Vicksburg, Mississippi. Flow of water is from right to left. 67 Fig. 18. Schematic diagram of the flume used for experimentation. 26.7 m. 21.3 m. I1.1 m .305 m. Tail bay Head bay Sum p Rails 11 .305 m . <*------.914 m. * Section A - A 70 were measured using a point gauge operated from a carriage mounted on these rails. The flume is also equipped with a push-button slope control by a system of jacks to obtain slopes up to 2.5 percent. Water is recirculated through the flume by means of one of the three electrically driven centrifugal pumps. The flow circuit is from a main reservoir tank, through the metal channel, and back into the reservoir. The channel is preceded in the water circuit by a stilling tank. Water discharged by the pump enters the tank from the top by a pipe, the outlet end of which remains submerged when the tank is half-full. The stilling tank is provided with wire netting to smooth the flow. Water that has passed down the channel is discharged over a continuously adjustable sharp- crested tailgate into the reservoir tank. The sediment feeder of the flume is a piston below a rectangular bucket. It is located beneath the bed at the upstream end of the flume between the stilling tank and the channel (Fig. 19). The rate of piston movement deter mines the sediment feed and can be controlled by the operating rate of the screw jacks which raise the piston. A metal plate completely covers the sediment feeder at its lower position. The difference between the sediment levels in the feeder before and after an experiment gives the 71 Fig. 19. End view of the sand feeder system of the flume looking downstream. Brace for guide rod alum, angle alum, angle slotted Jack o pass brace alum, wall extension (Guido rod] (open) .914 m t of jacks to mate Flume fl. line with g motor 1.6 m. guide rod Std. brass bush guide rod Jack bracing alum, angle alum, wall pipe 01?m Std. housing m i n . brass for lifting ciear. alum, floor bush. screw .064 m alum, channel (oxist.) Exist, sand elevator floor Flange to secure guido rod 73 volumetric sediment feed rate. Discharge was calculated by means of a mercury-water manometer mounted on the return pipe of the water circuit. Mean velocity was calculated from the measured discharge and measured mean depth, the width of the flume being fixed and known. Local point velocity was measured using a small Price-type current meter and an electronic counting unit for mean velocity over a thirty second period. Bed Modelling and Apparatus A wedge-shaped bedform was modelled on the flume floor near the inlet end. The bedform was modelled with a cement- sand mixture except the slip face. The first 12 cm of the stoss-side had a steep slope while the remaining part making up most of the stoss-side had a very gentle slope (less than a degree). The crestal platform was 35 cm long so that the height of the dune at the brink point was about 15 cm. A transversely partitioned sediment collector, of the type used by Allen (1968b), made in the shape of the leeward part of a dune-like bedform, was placed downstream of the wedge-shaped bed (Fig. 20). The collector fitted across the full width of the flume and was 84 cm long. It was divided into 42 compartments by thin but rigid metal walls spaced 2 cm apart along the flow line. Each Fig. 20. Diagram of the bedform modelled on the flume floor and the sediment collector. Flow. 32* "~T .121 m . .127 m . •03 m. i ItlLiiu iiiiimjiiijjiii.milii mi ■025 m .28 m. .415 m . Ol 76 compartment was thoroughly sealed with rubber cement. The first ten compartments represented the slipface and their tops sloped at about 32 degrees. The tops of the remaining 32 compartments were at an uniform height of 3 cm from the flume bed. These latter compartments represented the bottomset zones of the bedform. The apparatus was secured to the flume bed by rubber cement. Experimental Procedure and Initial Calculations The actual complete experiment involved the following steps: 1. The sediment feeder was lowered so that the cover plate was slightly below the level of the flume bed. The sediment level in the feeder in such a position was more than 15 cm below the flume bed. This insured that no sediment moved in the flume bed until the feeder was raised. 2. Water of pre-selected discharge was introduced into the flume channel. 3. By adjusting the tail gate and the slope of the flume, the desired flow depth was obtained. 4. The flow was maintained under these conditions for several hours during which the water surface profile was measured with the point gauge. When there was no change in the profile, current meter readings were taken along 77 and within the flow field. 5. A calculation was then made of the total sediment dis charge that may be appropriate for the flow condition established using the empirical relationship given by Colby (1964), and from this the sand feed rate was esti mated . 6. The sediment feeder was raised first at a faster rate than that calculated in (5) until the sediment level in the feeder attained a height equal to the gently sloping stoss side of the artificial dune in the flume bed. After wards it was operated at the calculated speed. The water surface profile was periodically measured during the experi ment . 7. The flow separated at the crest of the collector, and the grains were discharged into the separated region before finally settling in the collector and beyond the region occupied by the collector. 8. When any of the compartments became near full with sedi ment the flow was stopped, and the flume was drained very slowly. The sediment collector was then taken out of the flume bed. 9. Grains were collected from each compartment through a small hole near the base. These holes were kept completely closed during the experimentation. 78 10. Samples thus collected were dried and weighed. The rate of grain settling W, measured in dry weight/unit area/ unit time was calculated from the weight of each sample, the duration of the experiment, and the area from which the sample was collected. 11. If the weight of the sample was not more than 20 gms, it was sieved for 15 minutes at 0.25‘I* interval in rotap. In case of larger samples, they were split with a sample splitter into fractions of about 20 gms. 12. Size statistics of Folk (1968) were calculated by a computer program in which the critical percentile values were obtained by normal linear interpolation between data points plotted from sieve grain diameter and corresponding weight percents. All computing was done through the Louisiana State University IBM 360 computer facility. 13. Knowing the size distribution of each sample, the settling rate of a specified grade was calculated. The total sediment transport rate was calculated from the sum of the weights in the collectors and on the flume bed down stream of the collector. Bed Stocks The experiments were conducted using four quartz sands, each with a different mean size and sorting. Fig. 21 79 Fig. 21. Size frequency diagrams of sediment used as bed stock in the flume experiments. Frequency in Percent (wt.) 50r -2 en ri Diameter Grain Mean Sand2 Sand 4 ( illustrates the size distributions of the sediment. Sand 2 used in the experiments 25 to 27 was medium sand (M = Z 1.078$ ), moderately well sorted ( o = 0.604$), leptokurtic (K = 1.173), coarse skewed (S^ = =0.144). A somewhat finer grained sediment, sand 3, was employed in experiments 22 to 24. This sediment was unimodal (with M = 1.945 4, used in experiments 17 to 19, was also medium sand (M = 1.606$) and was also well sorted ( ° = 0.373$), mesokurtic (K = 0.946), and near symmetrical (S, = -0.033). The g k fourth bed stock, "sand 1", was actually pebble, with a mean size of -2.099$ and sorting of 0.372$ . However, it was strongly fine skewed (S^ = 0.492) and it did not have a log normal distribution. The grading of the bed stock was representative of the sediment that was put in the sediment feeder. However, the sediment feeding arrangement was such that a somewhat modified sediment whose size was appropriate for a particu lar flow condition was transported and entered the flume channel. As such, the size characteristics of the trans ported sand varied from experiment to experiment though the same stock was used for a set of experiments. 82 Entrainment and Transport Appendix 6 lists the flow and sediment properties, either measured or calculated from the measurements, a summary of which is given in Table 1. Table 1. Flow conditions for the experiments 17 to 27. Experiment Mean flow Flow Boundary Sediment Number velocity depth shear transport (u) (d> stress (t ) rate (i) (cm/sec) (cm) (gm/cm sec^) (gm/cm/sec) 17 63.33 13.594 60.785 0.0276 18 86.64 11.796 124.984 0.130 19 40.11 14.051 13.987 0.00154 20 86.64 11.796 21.869 0.00543 21 67.09 12.832 34.596 0.00138 22 64.78 13.289 36.804 0.0640 23 86.64 11.796 124.984 0.163 24 41.37 13.625 13.562 0.00976 25 40.11 14.051 13.987 0.00375 26 61.27 14.051 38.915 0.0188 27 87.32 11.704 124.015 0.0472 In all the experiments except those with sand 1, the finer sediment was entrained first from the sediment feeder, moved as isolated ripples up the stoss side of the artificial dune, and was diffused first. The main sediment front moved behind these isolated ripples at a slower speed, and never reached the brinkpoint of the artificial dune before the experiment was stopped. In experiments with sand 1 83 (experiments 20 and 21), individual rolling grains first separated out of the main sediment front and were diffused on the leeside. Subsequently, the main sediment front, a plane bed up to 3 grains diameter thick, travelled up to the brinkpoint of the dune and sediment was diffused from this front. Settling Rate of Mixed Sizes The grains of the bedload layer and of any accompanying suspended load, after entering the unstable free boundary layer, become widely dispersed in the lee of the dune. The rate at which grains arrive at any point on the bed down stream of separation can be defined most conveniently by the weight settling rate, W. Figs. 22-25 show, for the four bedstocks, the variation of W with the path length, X, the horizontal distance from the brink of the dune. The curves all have the same general form, although the measured settling rates vary widely from experiment to experiment. The settling rate is an inverse function of path length. The rate of change of W with X is large over the sloping part of the lee, and, for comparable values of t is greater in the case of the coarse grained stock (sand 2) than for finer grained (sand 3). The de cline of W over the bottomset for bed stock 2, 3, and 4 is Fig. 22. Deposition rate gradient of grain settled sediment for experiments 20 and 21 using bed material of sand 1. 85 u M E u E O) Experiments 86 Fig. 23. Experimental deposition rate plotted against distance from the dune crest using bed material of sand 2. Hie curves (solid lines) represent the results of numerical calculation with the same input values of flow and sediment variables. 87 Experiments u E \u 27 Eo> c to 1 X in cm 88 Fig. 24. Experimental deposition rate plotted against distance from the dune crest using bed material of sand 3. The curves (solid lines) represent the results of numerical calculations with the same input values of flow and sediment variables. 9°»/c m*, 9°»/c cm 90 Fig. 25. Experimental deposition rate plotted against distance from the dune crest using bed material of sand 4. The curves (solid lines) represent the results of numerical calculations with the same input values of flow and sediment variables. 91 Experiments 18 » 19 ® UE E o> c X i n cm 92 variable but gradual. However, for comparable values of tq, the rate of decline is highest for the coarse grained stock. There is, therefore, a fairly definite break in the slope of the deposition rate near the toe of the siipface. For the bed stock of pebbles (sand 1) there is no deposition on the bottomset zone. If the weight of grains that settled in a particular compartment is divided by the total weight of the grains deposited on the lee of the dune, a relative weight con centration of grains at the chosen path length is obtained. Multiplying by 100, we obtain the relative weight concen tration, C , in the form of a percentage frequency. Figs. 26 and 27 show the variation of C with log X. The curves w are of same general form. For smaller values of X the data are found to plot as roughly straight lines. At larger values of X a slight concavity is observed in all the experiments. Figs. 22-25 are log-log plots of W against X for the data listed in Appendix 7. With bed stock of sand 2 and larger values of To and i (experiments 26 and 27) the fit to straight lines is satisfactory though there appears to be two breaks in slope, one at X » 20 cm and the other at X ~ 40 cm. With the lowest values of t q and i (experiment 25), the fit to a single straight line is less satisfactory. 93 Fig. 26. Graphs showing the variation of relative weight concentration as probability percentage with distance from the crest of the dune for experi ments with bed material of sand 1 and sand 2. VO <0 «0 rol c/> Ca ro Relative W eight Concentration in Probabiity Percent ro in Q. ro 10 01 ro ro o X CQ in cm — t o 95 Fig. 27. Diagram showing the Variation of relative weight concentration as probability percentage with distance from the crest of the dune, for experi ments with bed material of sand 3 and sand 4. 4 <0 m Q. on t/> a. Q <0 CD ex Ul >» Relative Weight Concentration in Probability Percent 0) ro 09 Log10 X in cm 97 However, beyond X = 3 cm a single straight line could be fitted to the data points. With bed stocks of sand 3 and larger values of and i (experiments 22 and 23), the plots show identical results in detail as those of experi ments 26 and 27. With smaller values of t and i, the o results are identical as in experiment 25, though the fit to straight line for X greater than 3 cm is better. Ex periments with bed stock of sand 4 show identical trend of variation of W on path length. In experiments with sand 1 (pebble size particles), the data could be fitted to a straight line with a steep negative gradient as in the other experiments. However, the size distribution of the bed stock is strongly fine skewed compared to the near symmetrical log-normal distribution of common natural sand used in other experiments. Therefore, deposition rate and mean size gradients cannot be simulated numerically, neither will it be proper to compare the co-efficients of a general function relating these gradients with those obtained using the other three bedstocks. As such, the following discus sion excludes experiments with bed stock of sand 1. The above analysis of data suggests that the prob ability density function defining the variation of W with X for experimental data could be approximated by a power function over the combined length of the slipface and 98 bottomset zone. However, the degree of association improves if two regions are separately treated. Appendix 8 shows the results of regression analysis for each of the experiments with sand 2, sand 3, and sand 4. The power function can be represented by the following equation n W = aX (22) where a and n are co-efficients. If we restrict ourselves to slip face region only, which is of greater interest to us, we can relate the experimental transport rate to the variable a in the power function as (23) where c = co-efficient, of grain coarseness. The calcu- s lated values of c^ show that it generally decreases with decrease in mean size (Table 2). 99 Table 2. Calculated values of the co-efficient, V the deposition rate gradient equation. Experiment as Transport rate cs Number (gm/cm-vsec) (gm/cm/sec) (cm) 17 0.0117 0.0276 0.42 18 0.0397 0.1304 0.30 19 0.0003 0.0015 0.20 22 0.0062 0.0064 0.10 23 0.0114 0.1634 0.07 26 0.0051 0.0188 0.27 27 0.0142 0.0472 0.30 The exponent n decreases with increasing mean flow velocity and can be related to the ratio u/V . Introducing o a new co-efficient 'b' (Allen, 1968b) which is b = V /0.03nu (24) o it is found that b has a very small variation (Table 3) and Table 3. Calculated values of the constant, b, in the deposition rate gradient equation. Experiment Fall velocity n Flow velocity b Number cm/sec cm/sec 17 3.90 1.869 63.33 1.10 18 4.35 1.392 86.64 1.20 19 3.30 2.107 40.11 1.30 22 2.00 1.122 64.78 0.92 23 2.27 0.903 86.64 0.97 100 Table 3 continued Experiment Fall velocity n Flow velocity b Number cm/sec cm/sec 26 4.45 1.585 61.27 1.53 27 4.90 1.409 87.32 1.33 can be approximately taken to be constant in value. So the deposition rate on the slipface region can be recast (Allen, 1968b) as Vo/0.03bu (25) where the subscript L refers to some reference point. Equation (25) is of the same form as the suspended load concentration profile. So it seems that there is some physical similarity between vertical diffusion of sediment in open channel flow and horizontal diffusion and convection of sediment on the leeside of a dune. However, such physi cal similarity does not suggest that the processes operating are the same in the two cases. Mean Size and Grading of Settled Sand The preceding analysis has shown that, under any given experimental condition, the settling rates of grains decrease away from the dune crest. However, for coarser stocks, the rate of decline is much more pronounced, especially 101 over the slipface region, with a strongly negative mean size gradient over the slipface region and a gradual flattening of the gradient over the bottomset zone. The mean size of the settled grains will, therefore, fall with increasing distance from separation (Appendix 9). The plots (Figs. 28A-I) resemble those of W against log X, and a definite break in slope between the slipface and the bottomset zone can be recognized. Over the slipface region the variation of mean size with X can be approximated by a power function of the form (26) where a* and are two co-efficients. Appendix 10 lists the results of regression analysis on with X. The values of a* have a narrow range (Table 4) and increases Table 4. Values of the co-efficients a* and n^ from the mean size gradient regression equations n Experiment a* * Number 17 1.77 0.0457 18 1.573 0.0542 19 1.984 0.0356 22 2.308 0.039 23 2.073 0.0473 24 2.5175 0.0114 1 0 2 Fig. 23. Variation in the mean size of the leeside sediment with increasing distance from the dune crest. The solid lines show the grain size gradient obtained numerically through the simulation program. Discrepancies arc exp I a.inec'1 in the text. Mean Grain Diameter ( E X pt. 19 2.9 k . » I « Q c ko O c 2.2 o ® TS 4 b 63 Distance From Dune Crest (cm) C E X P t 23 £ 2.3 25 25 45 65 Distance From Dune Crest (cm) 105 2J E X p f. 22 2.5 a> 0) E 5 “ C E o L. O c D a> * 2J5 E X p t. 24 2.6 2.7 Distance From Dune Crest (cm) F 106 2.0 E o 2.2 Q c o k_ O 2.3 c o 0) 2* 2.4 2.5 85 Distance From Dune Crest (cm) 107 E X P f. 26 1.6 1.7 a> £ o a C o c o a> 2.2 23 7 5 4 5 Distance From Dune Crest (cm) Mean Grain Diameter (0) 2.05 1.85 1.75 1.95 1.65 145 1.55- Expt. itne rm ue rs (cm) Crest Dune From Distance 27 3 7 45 45 6 5 109 Table 4 continued Experiment a s* Number 25 1.946 0.0796 26 1.553 0.104 27 1.391 0.0969 with decreasing transport rate. On the other hand, the exponent nA generally increases with increase in u. How ever, the relationship of the co-efficient with the flow and sediment variables is expected to be complex. Sorting in particular plays a great role in determining the absolute or relative profile, and cannot be ignored. Therefore, it is not possible to evaluate the dependence of the co-effi cients a* and n# on these variables. Comparison between Experimental and Calculated Results Appendix 11 shows the calculated values using the numerical analysis program outlined earlier. The mean size gradient obtained numerically shows a very gentle decrease downstream in all the calculations. The absolute size values significantly differ from those of the experi ments in the case of the coarse stock of sand 2 . The deposition rate profiles are similar in form but differ in absolute values from those obtained experimentally. Although neither calculated nor experimental results give 110 impossible values, several factors have produced the observed differences between the two curves. Considering the size gradient first, in the numerical analysis it is assumed that the sediment placed in the flume and that which is transported by the flow is identical in size characteristics. Experimentally, however, when sediment was introduced into the flume, the finer fractions were entrained first from the sediment feeder and filled the sediment collectors. Therefore, the size characteristics of the transported sand were quite different from that placed in the sediment feeder. When a correction is introduced in the numerical program to take this into account, the overall size that is obtained numerically and that obtained experimentally be come nearly identical (Fig. 29). However, the form of the size gradient profile still is not similar or identical. Experimentally a major source of variation existed because the fixed bedform was always out of equilibrium with the flow and instead of a homogenous bed-load layer moving up the stoss-side and over the dune crest, the sediment moved mainly as individual small-scale ripples which successively encountered the breakpoint and spilled onto the slipface. A major effect of this condition is to increase the effective thickness of the bedload layer compared to that obtained numerically where the dune is I l l Fig. 29. Grain size gradient obtained numerically with different corrections introduced in the simulation program for experiment 17. Curve 1: size characteristics of the bed material changed to those of the transported sediment. The bedform is, however, plane bed on the dune stoss. Curve 2: size characteristics of the bed material are the same as those of the sediment in the sediment feeder. The bedform is rippled on the dune stoss. Curve 3: size characteristics of the rippled bedform are varied in the lower part to simulate a size sorting in the rippled bedload. The initial size characteristics, however, are those of the transported sediment. 112 25 45 6’ Distance From Dune Crest 113 assumed to be in such equilibrium with the flow that no other bedforms migrate on its stoss-side. However, an attempt was made to include this condition into the numerical program for experiment 17 from the empirical relationships of bedform size and flow conditions (Yalin, 1972). This increased slightly the similarity between the calculated and experimental profiles (Fig. 29). However, the smaller bedforms in the experiments were themselves size sorted with very coarse grained layers near the bottom. When such size sorted smaller bedforms migrated past the artificial dune crest, a strongly asymptotic size gradient was established on the dune slipface in the following way. Initially, when the upper layers of the small ripple were eroded at the dune crest, they supplied the dune lee with materials having size characteristics more or less identical to the average bed material. However, at the last stage, the very coarse basal layer of the ripple were eroded and diffused. These layers had lower starting heights which, along with the fact that they were diffused to shorter distance because of their larger mass and lower forward velocity, were deposited on the dune slipface with a very strong negative gradient. This mechanism apparently provided the strongly asymptotic negative 114 profile on the dune lee obtained experimentally. However, at the present state of our knowledge, there is no way to include and predict such size sorting changes in the smaller bedforms. In fact, it is one of the objectives of the present research. However, to test the validity of the explanation outlined above, the numerical program was modified for experiment 17 to take into account a very small arbitrary change in the mean size from 1 .8 1 .6 in the lower 0.3 cm of the smaller bedform. The size gradient curve obtained now shows a slight but defi nite steepening of the size profile (Fig. 29). The above discussion is also valid for the deposition rate gradient. Additionally, however, one important source of variation should be mentioned in predicting the absolute values of the deposition rate. Equations used in the numerical cal culations are developed on the assumption that the potential transporting capacity of the flow is always satisfied. This means that the size classes in the sediment are infinitely available for transport, limited only by the sorting or the standard deviation of the size distribution function. There is no certainty that such a condition was fulfilled in the experiments. In fact, it is likely, in the light of the fractionation of the sediment as it came out of the sedi ment feeder, that the potential transporting capacity of 115 flow was never satisfied. The experiments have, therefore, produced deposition rates which are in most cases substan tially lower than those predicted by the numerical calcu lations . SIZE DIFFERENTIATION IN AVALANCHED SEDIMENT Theory In the previous discussion, it has been shown that deposition rate and mean size of sediment accumulating on a dune slipface decrease downstream from its crest according to an approximate power function. The effect of continued deposition on the slipface is to steepen progressively its slope until a critical value, called the angle of initial yield ( is dependent on mean particle size and on the concentration of sediment accumulating on the slipface. An avalanche then begins near the brinkpoint and moves down the slip face. After avalanching the slope has a lower angle, known as the residual angle after shear ( very nearly constant for a given sediment. One major effect of avalanching is to change the size gradient on the leeslope established by grain sett ling. Bagnold (1954a, 1954b, 1956, and 1966) studied the role of grain to grain stresses within flows of granular particles. He suggested that the grains within such flows interact in two ways: inertial and viscous. In the inertial case, when grain collisions predominate, an intergranular dispersive pressure, P, exists which was 116 117 determined to be (27) where = proportionality factor, * = linear grain concentration at point y in the flow oc = dynamic internal angle of friction, and du = strain gradient in the flow, dy The velocity gradient in the avalanche was first given by Bagnold (1954a) as (28) l where c = grain concentration by volume above point y in the flow. From equations (27) and (28) it can be seen that the grain diameter is inversely proportional to the stress as given by the velocity gradient. Thus, the site of maxi mum shear should also be the plane on which the smaller particles tend to accumulate whereas the largest particles should tend to migrate to the zone of least shear at the free surface (Bagnold, 1954a). Bagnold (1954b) was able to demonstrate this by ingenious experiments using air and sand in which each grade had been dyed a distinctive color. 118 He observed that the coarser grains rose to the top of each avalanching mass while the finer particles settled to the bottom surface on which shearing was taking place. Several later workers have confirmed Bagnold1s finding by- measurement in flume-deposited cross-stratified deposit (Johansson, 1963, Allen, 1965, and Jopling, 1965a). Lowe (1976) integrated the equation for uniform and steady grain flow of inertial type and obtained the equa tion of the velocity profile as rj _ 2 rcg (P - Pf ) cos B i ^ 1_ TY3/2- (Y-y)3/2] (29) (y) ” 3 1 p a. cosl J *D L s i r where c = mean grain concentration by volume above point y in the flow, Y = flow thickness measured normal to the bed, B = bed slope, and y = distance above bed measured normal to the bed surface. Equation (29) shows that grain velocity increases progres sively away from the static bed. According equations (27) and (28), grain size is increasing simultaneously in the same direction due to grain dispersive pressure. During avalanching of a population of mixed grains, the larger ones, concentrated near the top of the flow, will move at 119 a faster speed towards the toe of the slipface and accumu late there at a faster rate than the smaller grains. There fore, the initial negative grain size gradient resulting from grain settling on the slipface should be changed to a positive gradient as a result of avalanching if, during avalanching, no additional grains settle on the leeslope. In fact, this should occur only if the avalanche moves with an infinitely large speed. However, equations (28) and (29) show that avalanches have a finite speed. The real time of descent of each avalanche, T, and the period between the end of one avalanche and the initiation of the next ava lanche, P, are given by L T and (30) ______(31) dw 1 cos 1 (41 + W = deposition rate per unit width on the slipface. The actual period between real avalanches is the sum of P and T, and an additional time which arises as a result of grain settling on the avalanche face during the time period, T (Allen, 1970). Equation (31) shows that the period of avalanching is 120 inversely related to the deposition rate on the slipface. The deposition rate in turn is directly proportional to the fluid flow velocity and sediment transport rate. When the avalanche period is great and flow speed low, the avalanches will be well defined and complete, and grain dispersive pres sure will be most effective in bringing about a positive downslope size gradient. At such low fluid flow velocities the intensity of deposition from grain settling alone will be minimal. At high fluid flow velocities, however, ava lanche period will be short, and grain settling will be intense resulting in a negative size gradient. Thus, the flow speed and sediment transport rate ultimately control whether grain settling or avalanching will be more effective in determining the characteristic size gradient in the slip face deposit (Allen, 1968a). Five cases may be predicted (Allen, 1968a); 1. When the grain transport rate and flow speed are low, the period of avalanching is large and sorting of grains by dispersive pressure is complete. At the same time, grain settling is low, and the effects of avalanching outweigh those of settling. A positive gradient of mean size is expected in the slipface deposits. 2. As the flow speed and transport rate are increased, the contribution of grain settling will also increase. At the 121 same time the period of avalanching will decrease. The combined effect will still produce a positive size gradient but it will be less pronounced than that in case 1 . 3. At still higher flow velocities differential settling just balances the effects of sorting resulting from ava lanching. The slipface deposits should then be of sub stantially the same mean size from crest to toe. 4. Over a range of still higher flow speeds and transport rates, when avalanching is continuous, one should expect the gradient of mean size to be negative. Grain size gradient imposed by settling will then have greater effect than that by avalanching. 5. If the flow speed and transport rate are increased sufficiently, settling will be so intense that avalanche will cease altogether, and slipface deposit will show the steepest negative size gradient. Back eddying in the separation bubble in high flow speed produces regressive ripples which can be identified and separated from the avalanched sediment. In flows dominated by viscous stresses, the grains interact without actually colliding (Bagnold, 1954a) and the normal grain stress for * > 2.5 was empirically determined to be The viscous and inertial state of flow are distinguished by a ratio, the Bagnold Number, defined as h 2 M _ V. P-S-D. du , . N = 5 — (33) Fully viscous flow is characterized by N < 40 whereas inertial flow by N > 450. Equation (32) is independent of grain diameter, D, so that no viscous size sorting occurs. In the viscous state of flow, therefore, the negative size gradient will be formed under all flow conditions except at high flow rates. Three cases may be outlined. 1. At low flow speed and transport rate, the mean size will decrease down the slipface at a moderate gradient. 2. At a higher flow speed and transport rate, when avalan ching is more frequent, the gradient of mean size will flatten out. 3. At a still higher speed and transport rate, when avalan ching is continuous, the mean size is expected to decrease only slightly from crest to the toe of the slipface. The two schemes of grain size variation in the slipface are summarized in Fig. 30. 123 Fig. 30. Schematic diagram showing the mean size gradient predicted for various flow and sediment charac teristics. Grain viscous and grain inertial states are differentiated by the Bagnold Number. Mean flow velocity, holding all other independent variables constant, increases for both the states from left to right. Each small diagram shows the variation of mean size from dune crest to toe. GRAIN INERTIAL STATE GRAIN VISCOUS STATE s N Si c « a 5 Si N a> c w (0 x Distance From Dune Crest u e n FlowMeanVelocity 124 125 Experiments In order to verify the size gradient predicted for different flow conditions a second series of experiments was designed. The same flume facility as for the previous series of experiments was utilized. Bed Modelling A dune-like bedform with the same stoss side dimensions as for the previous experiments was constructed at the up stream side of the flume with the exception that a shorter crestal platform, 17.7 cm long, was modelled. Additionally the slipface in the present case was an aluminum sheet coated with sand of mean size 1.08. The slipface had a slope of about 32 degrees (Fig. 31). The shape of the bedform modelled was in no way in tended to be representative of the bedform that may be found in equilibrium under flow conditions established during experimentation. In fact, our knowledge on the prediction of the bedform size is such that only empirical and semi-quantitative data are available, and there is no reliable function to relate the flow properties to the resultant bedform size. On the other hand, it was pre sumed that a dune-like bedform of higher order (higher length-height ratio) is one of the most common bed Fig. 31. Diagram of the dune-like bedform modelled on the flume floor, (not to scale), The inlet end of the flume is marked 0 . 12.1cm. 12Jcm. 12 cm. 28cm. 29 40cm 41.5 cm. cm. 127 128 features in an open channel flow on which smaller bedforms of appropriate size are superimposed. It is commonly the migration of smaller bedforms on the stoss-side and the diffusion of their sediment at the crest of larger dune-like features which produce large cross-stratification charac teristic of deposits of open channel flows (Allen, 1963). After each experiment, it was observed that sediment introduced filled the space behind the steep break in slope on the upstream end of the artificial bedform, and an appro- dune structure with superimposed smaller bedforms had been established. This insured that the experiments were a modest scaled down reproduction of an actual system that may be operative in an open channel flow under flow condi tions similar to that of the experiments. The only res triction was that smaller bedforms always migrated on the back of the larger bedform. Experimental Procedure The actual completed experiment involved the following steps in which the first 6 steps are the same as those previously described in the first series of experiments: 7. After a thick deposit was formed on the slipface, water and sediment discharge was stopped at a point when there was little or no avalanching over most of the slipface. At the same time the tail gate was raised quickly so that there was no rapid draining of water into the reservoir tank. 8 . The flume bed was then drained very slowly both from upstream and downstream ends. Extra care was taken when water depth became shallow at the dune crest. 9. The stoss-side was mapped with the point gauge after which all stoss sediment was removed. The bed profile was then again measured by the point gauge. 10. The slipface deposit was allowed to drain and dry slightly for about twelve hours. 11. Sediment in the 10 cm zone on either side near the walls was removed exposing a side profile of the slipface deposit. 12. Sediment samples about 1 cm wide and 2 cm deep were taken in 4 cm long strips down the slipface at a number of places across the slipface. Samples were also taken from exposed surface of the slipface in 4 cm long strips suc cessively downslope from the crest. 13. Sediment was dried in the oven and sieved with a rotap at 0.25 14. Size statistics were calculated as described earlier. 130 Velocity Fields The experiments now under discussion have at least one serious limitation? the velocity fields were not measured concurrently with the diffusion of the sediment. The velocity fields, shown in Fig. 32 for the values of u and d established in the experiments, were obtained in the absence of a grain load. Although it is known that the velocity field for a grainless fluid differs in detail from that of a fluid carrying grains, the two fields can be broadly similar. The velocity fields are given in terms of the mean horizontal component of u and were measured in the plane of the symmetry of the flow. Upstream from the step, velocity profiles are of the open channel type, and they show a progressive reduction of gradient towards the free surface. Downstream from the brinkpoint three zones could be identified to the reattachment point (Fig. 32). 1. A zone in which the velocity is downstream and is of the open channel type. 2. A zone in which velocity values are negative. 3. An intermediate or mixing zone of large fluctuations in velocity. This layer contains the isovel of zero forward velocity. Beyond the reattachment point which was located 2 to 4 131 Fig. 32. Measured velocity profiles of some of the experi ments from the inlet end of the flume to a point beyond the reattachment zone. The station number indicates the distance in meters from the inlet end to the point where velocity was measured. Station 0.9 and 3.4 represent the stoss-side of the dune, station 4.75 is on the dune brinkpoint, station 4.98 is on the dune troughpoint, and station 12.2 is downstream of the reattachment zone. Flow depth is marked with reference to an arbitrary datum point. FLOW DEPTH (cm) FLOW DEPTH (cm) 60 60 60 r 20 30 40 50 -si' 50 30 10 r 0 0 50 40 30 EOIY (cm/sec) VELOCITY aton .9 n tio ta S 9 * 73 t 20 10 VELOCITY 2 Station 40 0 0 0 O 60 SO 40 30 20 4.98 (cm/sec) 60 11 5 EOIY (cm/sec) VELOCITY 80 tto 3.4 Station 4 73 1 2 10 6 7 9 3 too 120 IS 30 EOIY (cm/sec) VELOCITY 20 tto 12.2 Station 0 0 60 50 40 30 0 0 0 0 10 2 10 140 130 120 110 100 90 00 70 EOIY cm/ ) e /u m (c VELOCITY tto 4.75 Station 132 133 dune-heights from the crest, the velocity profile is again of the open channel type. Bed Stocks Same bed stocks as in the earlier series of experi ments were used. However, the size characteristics of the transported sediment varied from experiment to experiment though the same bed stock was used for a set of experiments. This was particularly noticable in the case of experiments with sand 2. Although the mean size of the bed stock was 1.078 Fig. 33. Size frequency diagrams for the transported sediment of experiment 2 and experiment 5. Although the same bed stock, sand 2, was used in the experiments, size characteristics of the transported sand was different for the two flow conditions. Mean flow velocity was higher in experiment 2. For details of the flow conditions see Table 5. 1 3 5 2*r Expt. 5 • 16 Mean size in phi 136 situation duplicates many natural systems where a drainage basin has a stock of material of a certain mean size, but releases sediment of a different mean size appropriate to the steady state hydrologic regime established over certain intervals of time. Entrainment and Transport Table 5 lists the flow and sediment properties, either measured or calculated from the measurements. In the exper iments with sand 2, 3, and 4, when sediment was introduced into the flow, the finer sediment was entrained first from the sediment feeder, moved as isolated ripples up the stoss- side of the artificial dune, and avalanched down the slip- face reducing the artificial slope of 32 degrees by 5 to 8 degrees. The main sediment front moved behind these isolated ripples at a slower speed. As the stoss-side of the dune became covered with sediment, it was characteris tically covered also by small-scale bedforms, mostly ripples of various types and dimensions. Two to three successive ripples from this main sediment front were allowed to migrate past the brinkpoint of the artificial dune and form two or three corresponding sedimentation units on the dune slipface. Each of these units consisted of a basal zone of relatively fine-grained material and a Table 5. Flow conditions for experiments 2 to 16. Experiment Mean size discharge Flow depth Mean flow Flow Reynolds Froude Energy Sheer stress Boundary Sand feed Temperature Dynamic Number of the Q d velocity number number Slope velocity shear rate (°C) Viscosity transported (liters/sec) (cm) u Re F S u* stress (cc/cm/sec) (poise) sediment (cm/sec) (cm/cm) (cm/sec) (gm/cm.sec ) <*>) 2 0.958 100.81 12.101 91.109 117930 0.836 0.00755 9.466 89.347 0.777 22.8 0.00933 6 0.993 91.81 12.375 80.579 114280 0.731 0.00419 7.134 50.740 0.386 25.6 0.0088 3 83.25 13.686 66.525 99634 0.574 0.00333 6.684 44.541 0.382 23.9 0.00912 7 78.72 13.381 64.338 98663 0.562 0.00299 6.268 39.171 0.158 25.6 0.0088 4 70.23 14.112 54.424 84052 0.463 0.00209 5.379 28.849 0.014 23.9 0.00912 5 1.119 51.54 13.899 40.554 62445 0.347 0.00120 4.038 16.259 0.010 24.4 0.00900 10 1.693 93.45 12.192 83.824 114442 0.766 0.00499 7.726 59.518 0.416 25.0 0.008904 9 78.72 13.381 64.338 98663 0.562 0.00179 4.853 23.478 0.221 25.6 0.00880 11 51.54 14.204 39.683 62444 0.336 0.00120 4.082 16.615 0.013 24.4 0.00900 14 1.675 93.45 11.887 85.973 117058 0.796 0.00611 8.441 71.042 0.0324 26.1 0.00870 15 1.696 78.72 13.350 64.485 92000 0.564 0.00331 6.587 43.257 0.030 22.8 0.00933 16 51.54 13.350 42.220 60268 0.369 0.00184 4.903 23.967 0.061 23.3 0.0092 12 93.45 11.521 88.703 114442 0.834 0.00431 7.007 49.939 0.085 25 0.008904 13 78.72 9.510 90.527 97542 0.937 0.00331 5.559 30.813 0.097 25.6 0.0088 138 topmost very coarse layer one to two grain diameter in thickness. A single sedimentation unit was made up of several thin avalanche sheets. The flow attachment zone was wide, about 1.5 m, and there was appreciable back- eddying in the separation bubble in experiments with high discharge to produce low amplitude regressive ripples. These ripples never climbed up the slipface, however, but formed a thin, 2 to 5 mm high fine-grained pavement on which the slipface deposit prograded. Downstream from the reattachment, zone, small-scale ripples of different types and dimensions developed. The distribution of these bed- forms depended on the flow strength and the size of the sediment used. Appendix 12 summarizes the bedform charac teristics as observed in different parts of the dune during an experiment. In experiments using sand 1 (pebble-sized particles), characteristic bedform on the stoss-side of the dune was always plane bed. The pebbles were trans ported mainly by rolling and skipping. There was no re gressive ripple zone, neither were there any bottomset beds. Mean Size and Downslope Grading Experiments with sand 2 The sequence of experiments 2, 6 , 3, 7, 4, and 5 in 139 Table 5 represents decreasing order of water discharge with very little variation in the mean water depth. The flow Reynolds Number and Froude Number systematically decreased. Shear stress velocity, slope of the energy gradient, and boundary shear stress also decreased. Consequently, the sand transport rate was highest in experiment 2 , where appre ciable sediment was moved in suspension, and declined through the experiments to number 5, in which the sand was transported mainly as bedload with very little, if any, material in suspension. The variation in mean size of the slipface deposits with increasing distance from the dune crest followed a definite pattern as the flow speed and transport rate were decreased (Appendix 13). In the highest transport rate and flow speed (experiment 2 ), the mean size decreased from the crest to a position of about 16 cm downslope, but near the toe of the slipface there was a strong positive gradient in the mean size (Fig. 34). When the flow speed and transport rate were reduced slightly (experiment 6 ), the negative size gradient in the upper 16 cm of the slipface was modified to practically no change in mean grain size. However, for the lower 8 cm of the slipface, the positive size gradient be came slightly steeper (Fig. 34). At a still lower flow speed and transport rate (experiment 3), the upper 12 cm 140 Fig. 34A. Mean size gradient of the avalanched sand obtained in experiment 2 . B. Mean size gradient of the avalanched sediment obtained in experiment 6 . Mean size in phi Mean size in phi 1.2L 1.0 .6 .8 .6 2, ------Experiment Experiment Experiment e 1 1 2 24 20 16 12 8 4 * ------2 8 i X in cm. in X n m. cm in X 1 4 1 142 of the slipface attained a very gentle positive mean size gradient. The gradient became steeper from 12 to 16 cm down the slipface, and was the steepest in the lower 8 cm of the slipface. When the flow speed and transport were further reduced in experiment 7, the negative or very gently sloping mean size gradient in the upper two-third of the slipface obtained in experiments with high flow speed was no longer present. The whole size profile in stead could be approximated by a progressively downsiope steepening positive gradient (Fig. 35). With further de crease of flow speed and transport rate (experiment 4), the downslope grain size variation of the slipface followed a steep positive linear relationship (Fig. 36). In experi ments with the lowest flow speed and transport rate (experi ment 5), the grain size gradient was totally different from that of the earlier experiments with relatively high dis charge. Mean size gradient was positive and very gentle up to 20 cm down the slipface, but the grain size gradient was the steepest near the toe (Fig. 37). In the theoretical discussion, it was concluded that, at a very high flow speed and transport rate, the grain size gradient on the slipface will be negative. This stage was never realized in the experiments undertaken. Under the highest flow speed obtained (experiment 2), the Froude 143 Fig. 35A. Mean size gradient of the avalanched sediment obtained in experiment 3. Two sets of samples (A and B) along two transverse lines across the dune lee were analyzed. B. Mean size gradient of the avalanched sediment obtained in experiment 7. Three sets of samples (A, D, and E) along three transverse lines across the dune lee were analyzed. Mean Grain Diameter (0 ) 2 1 - • - • A -A Expt. itne rm oe rs (cm.) Crest Done From Distance Expt. 3 Expt. 7 24 4 4 1 145 Fig. 36. Mean size gradient of the avalanched sediment obtained in experiment 4. Two sets of samples (A and B) were analyzed. Samples of the sets were collected along two different lines trans verse to the dune lee. Mean Grain Diameter (0) 1.5 • - A Expt.4 Distance From Dune Dune From Distance rs (cm) Crest 146 147 Fig. 37. Mean size gradient of the avalanched sediment obtained in experiment 5. Two sets of samples (A and B) were analyzed. They were collected along two different lines across the dune lee. Mean Grain Diameter ( Number was 0.836. When attempts were made to obtain a still higher flow speeds by adjusting any or all of the flow variables, such as Q, d, and S ff the flow became supercritical and a hydraulic jump occurred immediately downstream of the slipface. As this condition was supposed to be unnatural in the system under investigation, no experiments were conducted under supercritical conditions. In the sequence of experiments 2, 6 , 3, 7, and 4, the change in the grain size gradient from experiment to experiment was confined largely to the upper part of the slipface deposit. At the lowest flow speed and transport rate, experiment 4, the frequency of avalanching was low, little sediment settled on the slipface, and a steep linear positive gradient was obtained as predicted theoretically. As the flow speed was increased, avalanching continued to produce a steep positive gradient near the toe of the slipface, but the gradient decreased with the increase in flow speed. However, grains settled on the upper part of the slipface more rapidly than avalanching could remove and sort them. The result was a progressive flattening of the positive size gradient in the upper part of the slipface deposit until, at the highest flow speed and transport rate, the grain size gradient became negative as predicted theoretic ally. At the lowest flow speed, experiment 5, although the 150 same bed stock was used, the mean size of the transported sand was much finer, the coarse tail reduced (S^ changed from -0.195 to -0.136) and the sorting improved because of low velocity. These changes in grain size characteris tics of the transported sand makes comparison of grain size gradient obtained in experiment 5 with those in other ex periments difficult. As predicted theoretically, the positive grain size gradient was the steepest in this experiment, but it was confined only to the lower part of the slipface deposit. The implications of these results are important. Sequences of large scale cross-stratification are common in many stratigraphic sections. Two general conclusions about the flow strength can be made immediately from such sequences. First, the sediment must have been transported so that the critical shear stress necessary to erode the sediment was exceeded for the grain sizes present. It can also be safely assumed that the flow was in the ripple/dune field and as such it was subcritical (Froude Number less than 1). However, this range is quite wide and it does not help to discriminate between flow conditions in successive cross-stratified units which have essentially the same size characteristics but which may have been deposited under different flow conditions. Therefore, in a cross-stratified 151 sequence where mean size characteristics do not change appreciably, the size gradient in the slipface deposit may help discriminate beds laid down under different flow and transport rates. Experiments with sand 3 In the experiments 10, 9, and 11, water discharge de creased systematically, and sediment transport rate also decreased (Table 5). In experiment 10, the transport rate and discharge of water were high, and the grain settling was intense. Avalanching, although frequent, was not effective in sorting the sediment. The result was a low positive gradient with a slight increase in the grain size only at the toe of the slipface (Appendix 14, Fig. 38). When flow speed and transport rate were reduced in experi ment 9, avalanching was slightly more effective in sorting the sediment, and produced a rather steep positive gradient near the toe of the slipface. However, grain settling was high in the upper slipface and reduced the steep gradient produced by the avalanching. In experiment 11, where flow speed and sand transport rate were minimal, avalanching was less frequent but complete, and grain settling from suspen sion was appreciable. As a result, a very steep positive gradient in mean size was present near the toe of the 152 Fig. 38. Mean size gradient of the avalanched sand obtained in experiments with bed material of sand 3. Meah Grain Diameter ( A 153 154 slipface, whereas a gentle positive gradient was established in the upper 5/6 of the slipface. In the theoretical prediction for the grain size gra dient, predictions were made that in fully viscous region of avalanche flow, a negative size gradient will be formed under all flow conditions which will become progressively flatter with increasing flow speed. At a very high flow speed the gradient will be almost horizontal. In experi ments with sand 3, less than 1% of the sediment was above 0.25 Since it is expected (Bagnold, 1954a, p. 62; Allen, 1968a) that grains having diameter less than 0.2 mm will behave viscously during tangential shearing, a large number of grains in sand 3 probably behaved viscously during avalan ching. A fully viscous flow was, however, never realized in the experiments. Also, the sediment was well sorted and relatively few size classes were available to produce a marked difference in mean size between the upper and lower parts of the slipface. Finally, the bed stock always produced appreciable suspension in all the three experi ments and grain settling from sediment suspension was rather high even in experiments with low flow speed. Therefore, the grain size gradient was low as in the viscous stage, but a positive gradient as in the inertial 155 stage was established as opposed to a negative gradient of the viscous stage. Experiments with sand 4 In experiments with sand 4 , the sediment used was very well sorted and was intermediate in mean size between sand 2 and sand 3. Appreciable suspension was produced in the two experiments with high discharge (experiments 14 and 15). A thin basal layer of concentrated grains (heavy fluid layer of Jopling, 1965a) was found to extend down stream from the brinkpoint and grain settling was high on the slipface from this layer. In the experiment at a low flow speed (experiment 16) there was very little sediment suspension and avalanches with a low frequency were ob served. From experiment 14 through 16, the flow speed and transport rate decreased. This was reflected in progressive ly greater positive size gradient in the lower part of the slipface deposit (Appendix 15, Fig. 39). The grain size gradient on the upper part of the slipface, however, showed a progressive change opposite to the change obtained in experiments with sand 2. In experiments with sand 2, the upper part of the slipface gradually changed from a gently negative to progressively higher positive gradient with decrease in the flow speed. In experiments with sand 4, 156 Fig. 39. Mean size gradient of the avalanched sediment obtained in experiments with bed material of sand 4. 157 17 « o E Expt. 15 o Q c uo o c o ® * 1A Expt. 16 Distance From Dune Crest (cm) 158 the flow speed was never high enough to produce a negative size gradient, but the positive gradient in the upper part became flatter with decreasing flow speed. This apparent contradiction can be explained by the size characteristics of the two bed stocks. Sand 4 was well sorted sand and the mean size gradient of the grains settling during the period of avalanching changed within a very narrow limit. At a lower flow speed with low frequency avalanches most of the coarser fractions had sufficient time to travel at the toe of the slipface giving an uniform grain size in the upper part of the slipface. The above discussions suggest that sediment sorting and the percentage of the coarser fractions (skewness) are two of the important variables which have hitherto been neglected in studies of sediment transport. These factors are particularly important in comparing empirical results obtained with sediments having different size characteris tics . It is always the practice to include the mean size as the only sediment variable. This is undoubtedly the most important variable, but sorting and percentage of coarse fraction (skewness) play a significant role. In a series of experiments using the same sediment but with varying flow characteristics, any of the response elements, such $s the transport rate, show a consistent change in 159 value from experiment to experiment. However, when a diff erent sediment with different size characteristics is used, the change in value from one experiment to another under the same flow conditions cannot be explained fully. Sorting and skewness of the sediment introduce changes in detail which must be accommodated to explain the difference in values obtained in the two sediments under similar flow conditions. Experiments with sand 1 Two experiments were made with this bed stock which was in the range of pebbles. The grain size gradient in the slipface deposits is, therefore, likely to be modified because the theoretical gradient was developed for sand- sized particles. The primary objective of the experiments was to verify that a prediction can also be made in the case of very coarse grains similar to that made for sand-sized particles where transport rate was low and no suspended sediment was present even in very high flow speed. In both the experiments deposition on the slipface was very low and there was no well defined avalanching. When a number of grains accumulated on the upper part of the slipface, the reduction in the slipface gradient was brought about by sliding down of a number of grains down the slip face. They sometimes reached the toe of the slipface, in 160 other cases they stopped sliding in the middle of the slip face. In the next phase of sliding from the upper part of the slipface, the grains that were held temporarily in the middle of the slipface were entrained and reached the toe of the slipface. Since there was no grain settling from sediment suspen sion and since the transport rate was very low, there was no imposition of a negative size gradient under any flow conditions. At the same time, avalanching was always com plete . A positive size gradient on the slipface deposit was, therefore, expected. A reduction of gradient with increase in flow speed was also expected. In experiment 12, where the flow speed was slightly lower than that in experiment 13, the mean size in the slipface deposit had an uniform positive gradient. In experiment 13, the flow speed and the sediment transport rate were slightly higher? the fit to a straight line to the data points was less satisfactory, but still acceptable; and there was a slightly lower positive size gradient (Fig. 40). There was, therefore, satisfactory correlation between the predictions made and experimental results about the size gradient in the slipface deposit with a very coarse sediment. 161 Fig. 40. Mean size gradient of the avalanched sediment obtained in experiments with bed material of sand 1 . IWo sets of samples were analyzed for experiment 13. 162 "Mr Expt f 2'* 113 - A A A * A £ 1O - 2. 2|- r (9 c A S A 0 c 1 * 2 - 2.1 o -2 8 16 24 Distance From Dune Crest (cm.) 163 Mean Size and Grading Transverse to Slipface Stratification In the present experiments, the diffusion of sediment on the leeside of the dune-like bedform resulted from the migration of ripples past the dune brinkpoint. The ripples were themselves made up of small slipface deposit with con centration of larger particles at their toes. In a cross- section, the cross-stratification was clearly seen, but the grain size profile showed that a single ripple was made up of a floor of coarse grains which was succeeded by finer and finer 'layers' towards the top. When such a ripple reached the brinkpoint of the larger dune the initial sediment that built up the initial layers of the cross stratum came by erosion from the stoss-side of the ripple. The sediment thus supplied had about the same average size characteristics as that of the bed material. Gradually, as the top of the ripple is eroded, more of the lower layers of the ripple started supplying sediment to the dune slipface. The succeeding layers in the cross-stratum, therefore, be came coarse. At the last stages of a ripple migration past the brinkpoint of the dune, the floor of coarse material of the ripple base moved down the slipface. The sediment deposited on the leeside of the dune formed a cross-stratified unit or set as "a group of essentially conformable cross-strata" (McKee and Weir, 1953). Each 164 member of the group, forming a cross-stratum, was a single layer made up of a series of grain flow deposits. There is considerable controversy on the interpretation of 1 sedimen tation unit'. It was first defined by Otto (1938) as that part of a deposit that was laid down under essentially constant physical conditions. In the present experiments, a sedimentation unit in the cross-stratified set was iden tified by the laminae deposited by the complete migration and diffusion of the sediment of a ripple beyond the crest of the artificial dune-like bedform. Each sedimentation unit was, therefore, expected to be size differentiated transverse to the slipface stratification with relatively coarse layers near its top. Several samples from some of the experiments were taken to confirm the “transverse' sorting of sediment in a cross-stratum. The results of mechanical analysis are shown in Appendix 17. In each cross-stratum sampled, there was a progressive coarsening of the grain size transversely whether sampling was done at its upper, middle, or lower portion. The con trast was smaller in the upper and middle parts of the slipface than in the lower part. For example, in experi ments 4 to 7, where similar sediment was used (sand 2), the difference in mean size (M ) between the top and bottom CJ 165 layers in the upper and middle portions of the slipface was less than 0.25. However, towards the toe of the slipface the difference increased to more than 0.25 0.199$ ) . This transverse size differentiation in individual cross-strata provides a mechanism by which we can explain rhythmic layerings in large cross-strata which are charac teristic of deposits formed in open channel flows. Many of these large cross-strata are known to be associated with •bar-like' bedforms with a large wave length-height ratio, gently sloping stoss-side with superimposed ripples and a well defined slipface. When such a ripple migrates past the brinkpoint of the larger dune bedform, a cross-stratum is laid down on the slipface of the dune. The cross-stratum is size differentiated transverse to its stratification plane with coarser layers near its top. Through time, when several ripples migrate past the dune brink, a set of cross-stratum is deposited. Each member of the group is size sorted across the stratification plane and together they produce rhythmic layerings in the cross-set. CONCLUSIONS 1. Analysis of theoretical and empirical studies of flow structure, sediment transport at the crest of a dune, and subsequent sediment diffusion beyond the dune crest shows that it is possible to develop a numerical model which can simulate the grain size and deposition rate gradients of sediment deposited on leeside of a dune. 2. Results of numerical experiments have shown that, apart from the flow variables, the size characteristics of the bed material, especially its sorting, are important varia bles which control both mean size and deposition rate gradients on the dune lee. 3. Flume experiments with a fixed dune bed and a sediment collector, however, yield size gradient curves which are not of the same form as those reproduced numerically with the same flow and sediment characteristics. Discrepancies are explained by the movement of sediment up the dune stoss- side not as a homogeneous bedload layer but as discrete smaller bedforms with strongly size differentiated profiles. 4. Deposition rate gradients reproduced numerically have the same plotted form as those obtained through the flume experiments. However, the absolute rates in flume experi ments are lower than those predicted by numerical 166 167 calculations. This is because the potential transporting capacity of the flow could not physically be satisfied in flume experiments. 5. A schematic model of the texture of avalanched sediment on the dune slipface deposit has been developed in which the mean size gradient can be related to the mean size of the diffused sediment, nature of grain interactions during avalanching, flow speed, and sediment transport rate. 6 . A second series of flume experiments with a fixed dune bed generally supports the schematic model. Therefore, it is possible to discriminate between paleo-flow conditions in successive cross-stratified units which have essentially the same gross size characteristics but which may have been deposited under differing flow conditions. Again, sorting of the bed material was found to be an important variable in controlling the absolute grain size gradient. In summary, sorting of the bed material has a first order significance in controlling transport, deposition rate, and mean size gradient of sediment deposited before and after avalanching on the dune lee. Because such trans port and deposition rates are intimately associated with the flow characteristics of open channel flows, it is essential that we no longer ignore sorting of bed material 168 as one of the variables influencing fluvial transport and deposition rates. Although the numerical model has been developed for a dune bed only, it has a broader application. Deposition in a reservoir fed by a stream constructing a simple Gilbert-type delta and without underflow can be simulated by the present program. Such simulation will predict the deposition rate gradient in the reservoir from the debouch ing point of the stream into the reservoir, and hence its silting rate. REFERENCES Allen, J. R. L., 1963, Asymmetrical ripple marks and the origin of waterlaid cosets of cross-strata: Liverpool and Manchester Geol. Journ., v. 3, p. 187-236. ______, 1965, Sedimentation in the lee of small under water sand waves: an experimental study: Journ. Geol., v. 73, p. 95-116. ______, 1968a, Current Ripples: their relation to patterns of water and sediment motion: North Holland, Amsterdam, 433 pp. ______, 1968b, The diffusion of grains in the lee of ripples, dunes and sand deltas: Journ. Sed. Petrology, v. 38, p. 621-633. , 1970, The avalanching of granular solids on dune and similar slopes: Journ. Geol., v. 78, p. 326-351. Bagnold, R. A., 1954a, The physics of blown sand and desert dunes: Methuen, London, 265 pp. ______, 1954b, Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear: Royal Soc. London Proc., Ser. A, v. 225, p. 49-63. ______, 1956, The flow of cohesion less grains in fluids: Royal Soc. London Phil. Trans., Ser. A, v. 249, p. 235-297. , 1966, An approach to the sediment transport problem from general physics: U. S. Geol. Surv., Professional Papers, 422-1, 37 pp. ______, 1973, The nature of saltation and of 'bed-load' transport in water: Royal Soc. London Proc., Ser. A, p. 473-504. Blatt, H., G. Middleton, and R. Murray, 1972, Origin of sedimentary rocks: Prentice-Hall, New Jersey, 634 pp. 169 170 Brooks, N. H., 1963, Calculation of suspended load discharge from velocity and concentration parameters: Proc. Inter-agency Sedimentation Conference, U. S. Department Agri. Pub. 970, p. 229-237. Brush, L. M., 1965, Sediment sorting in alluvial channels: in Middleton, G. V. (ed.), Primary Sedimentary Structures and Their Hydrodynamic Interpreta tion : Soc. Econ. Paleontologists Mineralogists Spec. Pub. 12, p. 25-33. Colby, B. R., 1964, Discharge of sands and mean-velocity relationships in sand-bed streams: U. S. Geol. Surv., Professional Papers, 462-A. Folk, R. L., 1968, Petrology of sedimentary rocks: Hemphill1s, Austin, 170 pp. Gibbs, R. J., M. D . Mathews, and D. L. Link, 1971, The relationship between sphere size and settling velocity: Jour. Sed. Petrology, v. 41, p. 7-18, Graf, W. H., 1971, Hydraulics of sediment transport: McGraw- Hill, New York, 513 pp. Jopling, A. V., 1964a, Laboratory study of sorting processes related to flow separation: Jour. Geophysical Research, v. 69, p. 3403-3418. ______, 1964b, Interpreting the concept of sedimentation unit: Jour. Sed. Petrology, v. 34, p. 156-172. ______, 1965a, Laboratory study of the distribution of grain size in cross-bedded deposits: in Middleton, G. V. (ed.), Primary Sedimentary Structures and Their Hydrodynamic Interpreta tion: Soc. Econ. Paleontologists Mineralogists Spec. Pub. 12, p. 53-65. ______, 1965b, Hydraulic factors controlling the shape of laminae in laboratory deltas: Jour. Sed. Petrology, v. 35, p. 777-791. , 1966a, Some principles and techniques used in reconstructing the hydraulic parameters of a paleo-flow regime: Jour. Sed. Petrology, v. 36, p. 5-49. 171 , 1966b, Origin of cross-laminas in a laboratory experiments Jour. Geophysical Research, v. 71, p. 1123-1133. Johansson, C. E., 1963, Orientation of pebbles in running waters a laboratory study. Geografiska Annalar, v. 45, p. 85-112. Kalinske, A. A., 1942, Criteria for determining sand-trans port by surface creep and saltations Trans. Am. Geophysical Union, v. 23, p. 639-643. Lane, E . Vi., and A. A. Kalinske, 1939, The relation of suspended to bed material in riverss Trans. Am. Geophysical Union# v. 20, p. 637-641. , and ______, 1941, Engineering calculation of suspended sedimentss Trans. Am. Geophysical Union, v. 22, p. 603-607. Lovera, F. , and J. F. Kennedy, 1969, Friction-factors for flat-bed flows in sand channels s Jour. Hydraulics Div., Proc. Am. Society Civil Engineers, H. Y. 4, p. 1227-1234. Lowe, D . R ., 1976, Grain flow and grain flow depositss Jour. Sed. Petrology, v. 46, p. 188-199. Mckee, E. ;D., and G. W. Weir, 1953, Terminology for strati fication and cross-stratification in sedimentary rocks: Geol. Soc. America Bull., v. 64, p. 381-390. Otto, G. H ., 1938, The sedimentation unit and its use in field sampling: Journ. Geol., v. 46, p. 569- 582. Raudkivi, A. J., 1967, Loose boundary hydraulics: Pergamon Press, New York, 331 pp. Rouse, H., 1938, Fluid mechanics for hydraulic engineers: Dover, New York, 422 pp. Schiller, L., and A. Naumann, 1933, Uber die grundlegenden Berechnungen bei der Schwerkraftaufbereitung. Z. VDI, v. 77 (quoted by Graf, 1971). 172 Schlichting H., 1968, Boundary layer theory: McGraw-Hill, New York, 647 pp. Southard, J B., 1971, Representation of bed configuration in depth-velocity-size diagrams: Jour. Sed. Petrology, v. 41, p. 903-915. Sundborg, A , 1956, The River Klaralven: a study of fluvial processes: Geografiska Annalar, v. 38, p. 127-316. White, S. J , 1970, Plane bed thresholds of fine-grained sediments: Nature, London, v. 228, p. 152- 153. Yalin, M. S., 1972, Mechanics of sediment transport: Pergamon Press, Oxford, 290 pp. DIFFUSION OF SEDIMENT IN THE LEE OF DUNE-LIKE BEDFORMS VOLUME II A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy xn The Department of Geology by Chinmoy Chakrabarti B.Sc.(Hons.), Jadavpur University, 1962 M.Sc., Jadavpur University, 1964 December, 1976 173 Appendix 1. Listing of the computer simulation program. C LIST OF VARIABLES C us MEAN FLOW VELOCITT PHI* MEAN GRAIN DIAMETER OSs FLOW DEPTH C SIGMAs SORTING OF THE GRAINS BHs BEOFORM HEIGHT C F s DARCY-WEISBACH FRICTION FACTOR C LIST OF CONSTANTS C ROF * DENSITY OF FLUID ROS s DENSITY OF SOLID G = ACCELERATION DUE TO CGRAVITY MUs DYNAMIC VISCOSITY CO-EFFICIENT IMPLICIT REAL*8 (A-H.O-Z) REAL*© MU.NU DATA ROFoROS«G»MU/.9982300«2.65DQ.9 8 1 .0 0 0 ..01CU2D0/ I READ ISoIS eEND®100> IRUN,U.PHI.US.SIGMA.B h .F 10 FORMAT (5X.I5.F10.5.F10.3.F1 Q.5.F10.3.F5.0.F10.**) D=(2.**(-PHI)>/10. NUsMU/ROF V0=!DSQRTC9.*MU®*2.+G*( (D /2 . )**2.)*R 0F*(R 0S-R C F>*< .015*476 + . 198*41* 2 (0/2. > ) >-3.wMU)/(ROF*(.011607+(. 1*4881*(D/2. )) ) ) RE=D*V0/NU IF (R E .LE .30Q .) GO TO 30 20 CONTINUE WRITE (6«*40) *40 FORMAT ( *RE OUT OF RANGE*) 30 C0= (2*4./RE 8® (1 . + ( . 15 *R E **.6 6 7 )) R s l . / ( ( (2«*G *0*(R0S-R 0F) ) /(R 0 S * U * * 2 .) ) ♦ ( 0 «75*CC*( ROF/ROS) ) ) YSsR*0+D YSCsYS YSHCsYSC/BH 963 CALL VEL (IRUN.YS.OS.U.NU.CD.O*VO «ROS« ROF.PL.PHI.SIGMA.G.YSC. 2YSHC.BH.F) GO TO 1 100 STOP END SUBROUTINE VEL ( IRUN. YS.DS.U.NU.CO.D. VO. ROS »ROf-. MU. P H I«SIGMA.G. 2 YSC* YSHC »BH« F) IMPLICIT REAL*8 (A-H.O-.Z) DIMENSION P( IB ) «C0NS( 17) *RDEPR (*42 ) OIMENSION XX(17) DIMENSION SMT(17) *BI (17) .OEPR(*42) DIMENSION PP(<42) «FCONS( 17.*42) .PPP(*42) OIMENSION PAT(17) DIMENSION XML(1 7 ). DELX(17)«XM(17) DIMENSION AR1(17.<42>*AR2(17.*42> DIMENSION FC3n B(17.*42) . TFCONS (17.*42 ) » TxCON( *42 ) ,XFC0NS(H2) .SIZE (*42) DIMENSION AR9(17.<42> «AR5(17.*42> REAL«8 KtMU.NU DATA K /0 .*400/ TDEPRsO.DO 00 6 NX=ls*42 TXC0N(NX)=0.00 XFCONS(NX)=O.DO 6 CONTINUE DO 7 IIX=1.17 DO 7 IIY*1»*42 AR1 wro ra ig m m m o k o o z m >ONKN,nxoHr,ONXMxox^ "d t j w < x c w*noooo5«2*n ao^xwmjo-»ir*gNXHxnsr » s ox k co co c/j—*ii c o m m s i i o ~ h ~ iis u x r r ^ o h i i h \ #o«o*cowr'ONwx i i k k k m m k o h ^ c c o n-oo»nr^-»M^-*ff>*pM2Mw^MN* ii►•p x ^ v r -‘C — c -»nr*o— ©"nn-^ii it i i c c o c o m c o*iicoo> m 9 Z h k i i c o h '-'Kh -* rwMHNtnw^rwHxo ii-»r ^ o K r ^ r w o o ^ i im a m n a h i « i i i h ii ii c ^ iih k c o k oiicocor~« n^xr wr^* o * « \ ^ h o o o h ^ h p 9 ~ x • ^ h o o ^ ^ o c ^ m cocococ* a j o o ^ c r r ® x n m s * m i i s o • i ^ ii< ii> z \ w k m h u i » m i i i mmkcococokcooiicoo nnwLOMM «• x x p i «“osn oio oi{owT3r*coooww© «x m ox cfl«coe«sco* • o n o - “< r ~ f l w u n r m x h © • x m oor^>v>oo««M o • -*s© • x ♦ NO^ M# 174 M* M (3 <3 roro -S'S 9 ro\& OiS •3 Oi (3 SO Oil* Oi ro Oi rj H (3 13 roto o 130 9 9 sO o (30 9 9 H u o 9 SO at 9 «P(3 -4GD rv> H sO MH o Oi «M (MM e. s. t jo-ranowx-toon wo n »oo i»a>- Appendix 2. Input variables of the numerical simulation pro gram for mean size and deposition rate gradient. Friction factor, f ', due to grains is calculated separately from the other four variables. Experiment Flow Particle Flow Sorting Frictioi Number Velocity mean size Depth a factor u D d () f ' (cm/sec) (♦) (cm) 30 40 2.5 15 0.35 0.0170 31 40 2.5 15 0.71 0.0170 32 40 2.5 15 1.50 0.0170 33 65 2.5 15 0.35 0.0150 34 65 2.5 15 0.71 0.0150 35 65 2.5 15 1.50 0.0150 36 90 2.5 15 0.35 0.0140 37 90 2.5 15 0.71 0.0140 38 90 2.5 15 1.50 0.0140 39 40 2.5 30 0.35 0.0150 40 40 2.5 50 0.35 0.0140 41 65 2.5 30 0.35 0.0140 42 40 1.5 15 0.35 0.0235 43 40 1.5 15 0.71 0.0235 44 40 1.5 15 1.50 0.0235 45 65 1.5 15 0.35 0.0275 46 65 1.5 15 0.71 0.0275 47 65 1.5 15 1.50 0.0275 48 90 1.5 15 0.35 0.0325 49 90 1.5 15 0.71 0.0325 50 90 1.5 15 1.50 0.0325 51 65 2.5 50 0.35 0.0125 52 90 2.5 30 0.35 0.0150 53 90 2.5 50 0.35 0.0125 177 Appendix 2 continued. Experiment Flow Particle Flow Sorting Friction Number Velocity mean size Depth a factor u D d (*) f ' (cm/sec) <♦) (cm) 54 40 0.5 15 0.35 0.0325 55 40 0.5 15 0.71 0.0325 56 40 0.5 15 1.50 0.0325 57 65 0.5 15 0.35 0.040 58 65 0.5 15 0.71 0.040 59 75 0.5 15 1.50 0.040 60 90 0.5 15 0.35 0.045 61 90 0.5 15 0.71 0.045 62 90 0.5 15 1.50 0.045 * 66 40 2.5 30 0.71 0.015 67 40 2.5 50 0.71 0.014 6 8 65 2.5 30 0.71 0.014 69 65 2.5 50 0.71 0.0125 70 90 2.5 30 0.71 0.0150 71 90 2.5 50 0.71 0.0125 72 40 2.5 30 1.50 0.0150 73 40 2.5 50 1.50 0.0140 74 40 1.5 30 0.71 0.0175 75 40 1.5 50 0.71 0.0150 76 65 1.5 30 0.71 0 . 0 2 77 65 1.5 50 0.71 0.019 78 90 1.5 30 0.71 0.024 79 90 1.5 50 0.71 0.023 80 65 2.5 30 1.50 0.0140 81 65 2.5 50 1.50 0.0125 82 90 2.5 30 1.50 0.0125 83 90 2.5 50 1.50 0.0125 * No experiments were performed numbering 63-65. Appendix 3. Results of numerical simulation A. Mean size of sediment deposited on dune lee. X = distance in cm from dune crest. D = mean grain diameter in phi unit. 180 79 19 39 19 59 39 59 79 2.453 2.438 2.417 2.380 2.549 2.540 2.545 2.532 77 37 37 57 17 17 77 57 2.437 2.452 2.415 2.375 2.548 2.545 2.531 2.540 75 35 15 15 35 75 55 55 2.450 2.435 2.412 2.368 2.544 2.548 2.530 2.539 73 33 33 53 13 13 73 53 2.449 2.433 2.409 2.361 2.548 2.544 2.529 2.538 31 1 1 31 51 71 11 2.448 2.406 2.431 2.547 2.527 29 9 69 2.429 2.402 2.341 2.352 2.547 2.537 2.538 2.525 Experiment 31 Experiment 30 27 27 29 7 67 69 71 4767 49 7 9 47 49 51 2.427 2.445 2.446 2.399 2.327 2.547 2.542 2.543 2.543 2.523 2.536 65 5 25 25 65 45 5 45 2.443 2.425 2.309 2.395 2.546 2.542 2.520 2.535 3 23 63 23 63 83 3 83 43 43 2.455 2.442 2.422 2.390 2.549 2.280 2.546 2.541 2.534 2.515 21 1 61 21 61 1 81 41 81 41 2.454 2.440 2.386 2.420 2.522 2.546 2.549 2.541 2.533 2.599 X IQ X IQ X IQ X IQ XIQ XlQ XlQ XlQ X IQ X IQ Experiment 32 X 1 3 5 7 9 11 13 15 17 19 D 2.288 2 .009 2.098 2.149 2.185 2 . 2 1 2 2.234 2.252 2.268 2.281 X 21 23 25 27 29 31 33 35 37 39 D 2.293 2.304 2.314 2.323 2.331 2.339 2.346 2.352 2.358 2.364 X 41 43 45 47 49 51 53 55 57 59 D 2.370 2.375 2.380 2.384 2.389 2.393 2.397 2.401 2.405 2.408 X 61 63 65 67 69 71 73 75 77 79 D 2.412 2.415 2.418 2.422 2.425 2.427 2.430 2.433 2.436 2.438 X 81 83 D 2.441 2.443 Experiment 33 X 1 3 5 7 9 1 1 13 15 17 19 D 2.610 2.505 2.506 2.506 2.507 2.507 2.508 2.508 2.508 2.509 X 21 23 25 27 29 31 33 35 37 39 D 2.509 2.509 2.509 2.509 2.510 2.510 2.510 2.510 2.510 2.510 X 41 43 45 47 49 51 53 55 57 59 D 2.511 2.511 2.511 2.511 2.511 2.511 2.511 2.511 2.512 2.512 X 61 63 65 67 69 71 73 75 77 79 D 2.512 2.512 2.512 2.512 2.512 2.512 2.512 2.512 2.512 2.512 X 81 83 181 D 2.513 2.513 182 79 19 39 59 39 59 79 19 2.227 1.876 1.849 2.250 1.734 1.809 2.241 2.257 77 37 17 57 77 17 57 37 1.874 2.225 1.846 1.803 2.249 2.256 1.722 2.240 75 15 35 35 75 15 55 55 1.871 1.798 1.842 2 . 2 2 2 2.239 2.249 2.255 1.708 73 33 13 13 53 33 53 73 1.869 1.792 1.839 2.219 2.238 2.248 2.255 1.692 31 11 51 11 31 71 51 1.786 1 . 8 6 6 1.673 2.247 2.254 2.216 2.237 29 69 71 49 49 1.779 1.864 1.649 2.246 2.254 Experiment 35 Experiment 34 27 7 9 67 7 9 27 29 67 69 47 47 1.771 1.861 1.827 1.831 1.835 1.618 2.245 2.253 2.208 2.213 2.234 2.235 25 65 5 25 65 45 5 45 1.858 1.763 1.823 1.574 2.244 2.252 2.203 2.232 3 23 63 3 63 23 43 83 83 43 1.755 1.818 1.855 1.881 1.500 2.243 2.252 2.258 2.194 2.231 21 1 61 1 41 21 81 61 81 41 1.852 1.745 2.357 2.257 1.814 1.879 2.251 2.242 2.566 2.229 XlQ XlQ XlQ XlQ XlQ XlQ XlQ XlQ XlQ X l Q 183 79 39 59 39 19 19 79 59 2.207 2.204 2 . 2 0 0 2.506 2.194 2.505 2.506 2.506 6 ] 77 57 37 77 17 37 17 57 2.207 2 . 2 0 0 2.204 2.193 2.193 J 2.506 2.506 2.506 2.505 75 35 55 15 75 15 35 55 2.207 2.199 2.204 2.192 2.506 2.505 2.506 2.505 73 53 33 13 13 33 73 53 2.206 2.203 2.199 2.191 2.506 2.505 2.506 2.505 71 51 1 1 11 2.203 2.206 2.190 2.198 2.506 2.505 2.506 2.504 69 29 31 9 49 29 31 69 71 9 49 51 2.206 2.506 Experiment 37 Experiment 36 27 7 47 27 7 47 2.197 2.198 2 . 2 0 2 2 . 2 0 2 5 45 25 45 2.184 2.187 2.188 2.506 2.506 2.506 2.506 2.506 2.504 2.504 2.504 23 25 3 23 3 5 2.205 2.205 2.205 2.506 2 . 2 0 1 2 . 2 0 2 2.506 2.504 21 61 63 65 67 81 83 1 41 43 61 6381 83 65 67 1 2 1 41 43 2.207 2.208 2.205 2.195 2.196 2.196 2 . 2 0 1 2.588 2.181 2.506 2.506 2.506 2.505 2.505 2.5052.506 2.505 2.505 2.615 XlQ XlQ XlQ XlQ XlQ XlQ XlQ XlQ XlQ X l Q 184 79 59 39 19 39 79 59 19 2.580 2.574 2.567 2.555 1.513 1.571 1.547 1.449 77 57 37 17 17 77 37 57 2.579 2.574 2.554 2.566 1.569 1.439 1.508 1.545 75 55 15 35 75 15 55 35 2.579 2.573 2.552 2.565 1.567 1.428 1.503 1.542 73 53 33 13 73 53 13 33 2.578 2.573 2.564 2.549 1.565 1.495 1.414 1.538 11 11 2.572 2.563 2.547 1.563 1.535 69 71 29 31 9 29 31 69 71 9 2.562 1.532 1.379 1.398 Experiment 39 Experiment 38 67 27 7 47 49 51 27 7 67 47 49 51 2.570 2.571 2.577 2.577 2.578 2.540 2.544 2.561 1.558 1.560 1.481 1.4871.528 1.493 1.354 65 25 45 5 25 65 5 45 2.576 2.570 2.536 2.560 1.555 1.474 1.525 1.319 63 23 83 3 43 3 83 23 63 43 2.581 2.569 2.576 2.558 2.529 1.553 1.575 1.467 1.521 1.263 61 81 21 41 1 21 61 1 81 41 2.580 2.568 2.575 2.557 2.599 1.573 1.550 1.458 1.517 2.434 XlQ XlQ XlQ XlQ XlQ XlQ XlQ XlQ XlQ X l Q Experiment 40 X 1 3 5 7 9 11 13 15 17 19 D 2.601 2.547 2.557 2 .564 2.569 2.573 2.577 2.580 2.583 2.585 X 21 23 25 27 29 31 33 35 37 39 D 2.588 2.590 2.592 2.593 2.595 2.596 2.598 2.599 2.601 2.602 X 41 43 45 47 49 51 53 55 57 59 D 2.603 2.604 2.605 2.606 2.607 2.608 2 .609 2.610 2.611 2.612 X 61 63 65 67 69 71 73 75 77 79 D 2.612 2.613 2.614 2.615 2.615 2.616 2.617 2.617 2.618 2.619 X 81 83 D 2.619 2.620 Exper iment 41 X 1 3 5 7 9 11 13 15 17 19 D 2.607 2.513 2.515 2.516 2.516 2.517 2.518 2.518 2.519 2.519 X 21 23 25 27 29 31 33 35 37 39 D 2.520 2 .520 2.520 2.521 2.521 2.521 2.522 2.522 2.522 2.522 X 41 43 45 47 49 51 53 55 57 59 D 2.523 2.523 2.523 2.523 2.524 2.524 2.524 2.524 2.524 2.524 X 61 63 65 67 69 71 73 75 77 79 D 2.525 2.525 2.525 2.525 2.525 2.525 2.525 2.526 2.526 2.526 X 81 83 185 D 2.526 2.526 Experiment 42 X 1 3 5 7 9 11 13 15 17 19 D 1.589 1.590 1.618 1.637 1.651 1.662 1.672 1.680 1.687 1.693 X 21 23 25 27 29 31 33 35 37 39 D 1.699 1.704 1.709 1.714 1.718 1.721 1.725 1.728 1.731 1.734 X 41 43 45 47 49 51 53 55 57 59 D 1.737 1.740 1.743 1.745 1.747 1.750 1.752 1.754 1.756 1.758 X 61 63 65 67 69 71 73 75 77 79 D 1.760 1.762 1.763 1.765 1.767 1.768 1.770 1.771 1.773 1.774 X 81 83 D 1.776 1.777 Experiment 43 X 1 3 5 7 9 11 13 15 17 19 D 1.505 1.591 1.675 1.724 1.758 1.784 1.806 1.824 1.839 1.852 X 21 23 25 27 29 31 33 35 37 39 D 1.864 1.875 1.884 1.893 1.901 1.909 1.916 1.922 1.928 1.934 X 41 43 45 47 49 51 53 55 57 59 D 1.939 1.945 1.949 1.954 1.958 1.963 1.967 1.971 1.974 1.978 X 61 63 65 67 69 71 73 75 77 79 D 1.981 1.985 1.988 1.991 1.994 1.997 2 . 0 0 0 2.003 2.005 2.008 X 81 83 186 D 2 . 010 2 .013 Experiment 44 X 1 3 5 7 9 11 13 15 17 19 D 1.451 1.725 1.869 1.948 2 . 0 0 2 2.043 2.075 2 . 1 0 2 2.125 2.145 X 21 23 25 27 29 31 33 35 37 39 D 2.162 2.178 2.192 2.204 2.216 2.227 2.237 2.246 2.255 2.263 X 41 43 45 47 49 51 53 55 57 59 D 2.271 2.278 2.285 2.292 2.298 2.304 2.310 2.315 2.320 2.325 X 61 63 65 67 69 71 73 75 77 79 D 2.330 2.335 2.339 2.344 2.348 2.352 2.356 2.360 2.363 2.367 X 81 83 D 2.370 2.374 Experiment 45 X 1 3 5 7 9 11 13 15 17 19 D 1.600 1.547 1.555 1.560 1.564 1.568 1.571 1.573 1.576 1.578 X 21 23 25 27 29 31 33 35 37 39 D 1.580 1.582 1.583 1.585 1.586 1.588 1.589 1.590 1.591 1.593 X 41 43 45 47 49 51 53 55 57 59 D 1.594 1.595 1.596 1.597 1.598 1.598 1.599 1.600 1.601 1.602 X 61 63 65 67 69 71 73 75 77 79 D 1.602 1.603 1.604 1.605 1.605 1.606 1.607 1.607 1.608 1.608 X 81 83 187 D 1.609 1.609 Experiment 46 X 1 3 5 7 9 11 13 15 17 19 D 1.530 1.372 1.413 1.439 1.459 1.475 1.488 1.499 1.508 1.517 X 21 23 25 27 29 31 33 35 37 39 D 1.525 1.532 1.538 1.544 1.549 1.555 1.559 1.564 1.568 1.572 X 41 43 45 47 49 51 53 55 57 59 D 1.579 1.579 1.583 1.586 1.589 1.592 1.595 1.598 1.601 1.603 X 61 63 65 67 69 71 73 75 77 79 D 1.606 1.608 1.610 1.613 1.615 1.617 1.619 1.621 1.623 1.625 X 81 83 D 1.627 1.629 Experiment 47 X 1 3 5 7 9 11 13 15 17 19 D 1.288 1.145 1.283 1.360 1.414 1.454 1.487 1.513 1.536 1.556 X 21 23 25 27 29 31 33 35 37 39 D 1.574 1.590 1.604 1.617 1.629 1.640 1.650 1.660 1.669 1.677 X 41 43 45 47 49 51 53 55 57 59 D 1.685 1.692 1.699 1.706 1.713 1.719 1.725 1.730 1.736 1.741 X 61 63 65 67 69 71 73 75 77 79 D 1.746 1.750 1.755 1.760 1.764 1.768 1.772 1.776 1.780 1.783 X 81 83 188 D 1.787 1.791 Experiment 48 X 1 3 5 7 9 11 13 15 17 19 D 1.606 1.546 1.550 1.552 1.554 1.555 1.557 1.558 1.559 1.560 X 21 23 25 27 29 31 33 35 37 39 D 1.561 1.562 1.562 1.563 1.564 1.564 1.565 1.566 1.566 1.567 X 41 43 45 47 49 51 53 55 57 59 D 1.567 1.568 1.568 1.569 1.569 1.569 1.570 1.570 1.571 1.571 X 61 63 65 67 69 71 73 75 77 79 D 1.571 1.572 1.572 1.572 1.573 1.573 1.573 1.574 1.574 1.574 X 81 83 D 1.574 1.575 Experiment 49 X 1 3 5 7 9 1 1 13 15 17 19 D 1.554 1.343 1.361 1.374 1.384 1.393 1.400 1.406 1.412 1.417 X 21 23 25 27 29 31 33 35 37 39 D 1.421 1.425 1.429 1.432 1.436 1.439 1.442 1.444 1.447 1.450 X 41 43 45 47 49 51 53 55 57 59 D 1.452 1.454 1.456 1.458 1.460 1.462 1.464 1.466 1.468 1.469 X 61 63 65 67 69 71 73 75 77 79 D 1.471 1.472 1.474 1.475 1.477 1.478 1.480 1.481 1.482 1.483 X 81 83 189 D 1.485 1.486 190 79 39 59 19 39 79 19 59 2.538 2.536 2.533 2.529 1.230 1.344 1.405 1.447 77 37 57 17 37 77 57 17 2.536 2.538 2.533 2.528 1.336 1.443 1.400 1 . 2 1 2 75 55 35 15 75 35 55 15 2.537 2.535 2.532 2.527 1.440 1.328 1.395 1.190 73 33 13 53 73 33 53 13 2.535 2.537 2.527 2.532 1.436 1.319 1.390 1.165 71 51 11 71 11 51 2.535 2.537 2.532 2.526 1.432 1.135 29 31 9 49 29 31 9 2.525 1.299 1.309 1.099 1.378 1.384 Experiment 51 Experiment 50 27 67 69 7 47 27 67 69 7 47 49 2.537 2.537 2.531 2.531 2.534 2.535 2.523 1.424 1.428 1.287 1.372 1.050 65 25 5 65 45 25 45 5 2.536 2.534 2.530 2.522 1.419 1.275 1.366 0.982 63 83 23 3 43 23 83 3 63 43 2.538 2.536 2.534 2.530 2.519 1.454 1.415 1.863 1.262 1.359 61 21 81 1 41 81 21 61 1 41 2.538 2.536 2.533 2.529 2.605 1.450 1.410 1.247 1.352 1.306 XlQ X IQ XlP XlP X IQ X IQ X IQ X IQ X IQ X IQ Experiment 52 X 1 3 5 7 9 11 13 15 17 19 D 2.612 2.511 2.511 2.512 2.512 2.512 2.513 2.513 2.513 2.513 X 21 23 25 27 29 31 33 35 37 39 D 2.513 2.514 2.514 2.514 2.514 2.514 2.514 2.514 2.514 2.514 X 41 43 45 47 49 51 53 55 57 59 D 2.515 2.515 2.515 2.515 2 .515 2.515 2.515 2.515 2.515 2.515 X 61 63 65 67 69 71 73 75 77 79 D 2.515 2.515 2.515 2.515 2.516 2.516 2.516 2.516 2.516 2.516 X 81 83 D 2.516 2.516 Experiment 53 X 1 3 5 7 9 11 13 15 17 19 D 2.611 2.515 2.516 2.517 2.517 2.518 2.518 2.518 2.518 2.519 X 21 23 25 27 29 31 33 35 37 39 D 2.519 2.519 2.519 2.520 2.520 2.520 2.520 2.520 2.520 2.520 X 41 43 45 47 49 51 53 55 57 59 D 2.521 2.521 2.521 2.521 2.521 2.521 2.521 2.521 2.521 2.521 X 61 63 65 67 69 71 73 75 77 79 D 2.522 2.522 2 .522 2.522 2.522 2.522 2.522 2.522 2.522 2.522 X 81 83 191 D 2.522 2.522 Experiment 54 X 1 3 5 7 9 11 13 15 17 19 D 0.600 0.597 0.620 0.634 0.645 0.653 0.661 0.667 0.672 0.677 X 21 23 25 27 29 31 33 35 37 39 D 0.682 0 . 6 8 6 0.690 0.694 0.697 0.700 0.703 0.706 0.709 0.711 X 41 43 45 47 49 51 53 55 57 59 D 0.714 0.716 0.718 0.721 0.723 0.725 0.727 0.728 0.730 0.732 X 61 63 65 67 69 71 73 75 77 79 D 0.734 0.735 0.737 0.739 0.740 0.742 0.743 0.744 0.746 0.747 X 81 83 D 0.748 0.750 Experiment 55 X 1 3 5 7 9 11 13 15 17 19 D 0.537 0.528 0.608 0.659 0.696 0.726 0.750 0.770 0.788 0.804 X 21 23 25 27 29 31 33 35 37 39 D 0.818 0.831 0.842 0.853 0.862 0.871 0.880 0 . 8 8 8 0.895 0.902 X 41 43 45 47 49 51 53 55 57 59 D 0.909 0.915 0.921 0.927 0.932 0.937 0.942 0.947 0.952 0.956 X 61 63 65 67 69 71 73 75 77 79 D 0.960 0.964 0.968 0.972 0.976 0.979 0.983 0.986 0.990 0.993 X 81 83 192 D 0.996 0.999 Experiment 56 X 1 3 5 7 9 11 13 15 17 19 D 0.323 0.355 0.610 0.745 0.834 0.900 0.952 0.994 1.029 1.060 X 21 23 25 27 29 31 33 35 37 39 D 1.086 1 . 1 1 0 1.131 1.150 1.167 1.183 1.198 1 . 2 1 1 1.224 1.236 X 41 43 45 47 49 51 53 55 57 59 D 1.247 1.257 1.267 1.277 1.286 1.294 1.302 1.310 1.317 1.324 X 61 63 65 67 69 71 73 75 77 79 D 1.331 1.337 1.344 1.350 1.356 1.361 1.367 1.372 1.377 1.382 X 81 83 D 1.387 1.391 Experiment 57 X 1 3 5 7 9 11 13 15 17 19 D 0.600 0.574 0.590 0.600 0.607 0.613 0.617 0.622 0.625 0.628 X 21 23 25 27 29 31 33 35 37 39 D 0.631 0.634 0.636 0.638 0.640 0.642 0.644 0.646 0.647 0.649 X 41 43 45 47 49 51 53 55 57 59 D 0.650 0.652 0.653 0.654 0.655 0.657 0.658 0.659 0.660 0.661 X 61 63 65 67 69 71 73 75 77 79 D 0.662 0.663 0.664 0.665 0.665 0 . 6 6 6 0.667 0 . 6 6 8 0.669 0.669 X 81 83 193 D 0.670 0.671 Experiment 58 X 1 3 5 7 9 11 13 15 17 19 D 0.538 0.450 0.505 0.539 0.565 0.585 0.602 0.616 0.629 0.640 X 21 23 25 27 29 31 33 35 37 39 D 0.650 0.659 0.667 0.674 0.681 0 . 6 8 8 0.694 0.700 0.705 0.710 X 41 43 45 47 49 51 53 55 57 59 D 0.715 0.720 0.724 0.728 0.732 0.736 0.740 0.743 0.747 0.750 X 61 63 65 67 69 71 73 75 77 79 D 0.753 0.756 0.759 0.762 0.765 0.767 0.770 0.772 0.775 0.777 X 81 83 D 0.780 0.782 Experiment 59 X 1 3 5 7 9 11 13 15 17 19 D 0.326 0.094 0.299 0.414 0.493 0.552 0.599 0.637 0.670 0.698 X 21 23 25 27 29 31 33 35 37 39 D 0.723 0.745 0.764 0.782 0.799 0.814 0.828 0.840 0.852 0.864 X 41 43 45 47 49 51 53 55 57 59 D 0.874 0.884 0.894 0.903 0.911 0.920 0.927 0.935 0.942 0.949 X 61 63 65 67 69 71 73 75 77 79 D 0.955 0.961 0.967 0.97 3 0.979 0.984 0.990 0.995 1 . 0 0 0 1.004 X 81 83 194 D 1.009 1.014 Experiment 60 X 1 3 5 7 9 11 13 15 17 19 D 0 ..602 0.562 0.573 0.580 0.586 0.590 0.594 0.598 0.600 0.603 X 21 23 25 27 29 31 33 35 37 39 D 0 ..605 0.607 0.609 0.611 0.613 0.614 0.615 0.617 0.618 0.619 X 41 43 45 47 49 51 53 55 57 59 D 0 ,.620 0.621 0.622 0.623 0.624 0.625 0.626 0.627 0.628 0.629 X 61 63 65 67 69 71 73 75 77 79 D 0 ,.629 0.630 0.631 0.631 0.632 0.633 0.633 0.634 0.634 0.635 X 81 83 D 0 ,.636 0.636 Experiment 61 X 1 3 5 7 9 11 13 15 17 19 D 0 ,.543 0.406 0.448 0.475 0.495 0.511 0.524 0.536 0.545 0.554 X 21 23 25 27 29 31 33 35 37 39 D 0 ..562 0.569 0.575 0.581 0.587 0.592 0.597 0.601 0.606 0.610 X 41 43 45 47 49 51 53 55 57 59 D 0 ,.614 0.617 0.621 0.624 0.627 0.630 0.633 0.636 0.639 0.641 X 61 63 65 67 69 71 73 75 77 79 D 0 ..644 0.646 0.649 0.651 0.653 0.655 0.657 0.660 0.661 0.663 X 81 83 D 0 ,.665 0.667 196 39 79 19 59 19 39 79 59 2.428 2.492 2.469 2.508 0.471 0.625 0.705 0.758 57 77 17 37 77 37 57 17 2.421 2.490 2.506 2.466 0.699 0.445 0.614 0.754 75 15 35 55 75 35 55 15 2.463 2.488 2.505 0.692 2.414 0.749 0.415 0.603 73 53 13 33 73 13 33 53 2.460 2.504 2.486 2.406 0.685 0.744 0.381 0.591 66 Experiment Experiment 62 67 67 69 71 27 27 29 31 7 7 9 11 67 67 69 71 47 49 51 27 27 29 31 47 47 49 51 7 7 9 11 2.480 2.480 2.482 2.484 2.499 2.500 2.502 2.448 2.448 2.452 2.456 2.368 2.368 2.384 2.396 0.662 0.662 0.670 0.678 0.729 0.734 0.739 0.549 0.549 0.564 0.578 0.217 0.217 0.286 0.339 21 21 23 25 1 3 5 61 61 63 65 81 81 83 61 61 63 65 41 41 43 45 1 3 21 23 5 25 81 81 83 41 41 43 45 2.509 2.510 2.494 2.495 2.497 2.472 2.474 2.477 2.434 2.439 2.444 0.763 0.767 2.520 2.313 2.347 0.711 0.717 0.723 0.635 0.635 0.644 0.653 0.339 0.052 0.117 0.494 0.514 0.532 XlQ XlQ X IQ X IQ X IQ X IQ X IQ X IQ X IQ X IQ Experiment 67 X 1 3 5 7 9 11 13 15 17 19 D 2.524 2.348 2.386 2.410 2.427 2.440 2.451 2.460 2.468 2.475 X 21 23 25 27 29 31 33 35 37 39 D 2.481 2.487 2.492 2 .497 2.501 2.505 2.509 2.513 2.516 2.519 X 41 43 45 47 49 51 53 55 57 59 D 2.522 2.525 2.528 2.530 2.533 2.535 2.538 2.540 2.542 2.544 X 61 63 65 67 69 71 73 75 77 79 D 2.546 2.548 2.550 2.551 2.553 2.555 2.556 2.558 2.559 2.561 X 81 83 D 2.562 2.564 Experiment Ch 00 X 1 3 5 7 9 11 13 15 17 19 D 2.559 2.206 2.217 2.225 2.230 2.235 2.239 2.242 2.245 2.248 X 21 23 25 27 29 31 33 35 37 39 D 2.250 2.252 2.255 2.256 2.258 2.260 2.262 2.263 2.264 2.266 X 41 43 45 47 49 51 53 55 57 59 D 2.267 2.268 2.270 2.271 2.272 2.273 2.274 2.275 2.276 2.277 X 61 63 65 67 69 71 73 75 77 79 D 2.278 2.278 2.279 2.280 2.281 2.282 2.282 2.283 2.284 2.284 X 81 83 197 D 2.285 2.286 Experiment 69 X 1 3 5 7 9 11 13 15 17 19 D 2.557 2.211 2.223 2.231 2.238 2.243 2.247 2.251 2.254 2.257 X 21 23 25 27 29 31 33 35 37 39 D 2.260 2.263 2.265 2.267 2.269 2.271 2.273 2.274 2.276 2.277 X 41 43 45 47 49 51 53 55 57 59 D 2.279 2.280 2.281 2.283 2 .284 2.285 2.286 2.287 2.288 2.289 X 61 63 65 67 69 71 73 75 77 79 D 2.290 2.291 2.292 2.293 2.294 2.294 2.295 2.296 2.297 2.298 X 81 83 D 2.298 2.299 Experiment 70 X 1 3 5 7 9 11 13 15 17 19 D 2.580 2.197 2.202 2.205 2 .208 2.210 2.212 2.213 2.214 2.216 X 21 23 25 27 29 31 33 35 37 39 D 2.217 2.218 2.219 2.220 2.221 2.222 2.222 2.223 2.224 2.224 X 41 43 45 47 49 51 53 55 57 59 D 2.225 2.226 2.226 2.227 2.227 2.228 2.228 2.229 2.229 2.230 X 61 63 65 67 69 71 73 75 77 79 D 2.230 2.231 2.231 2.232 2.232 2.232 2.233 2.233 2.233 2.234 X 81 83 198 D 2.234 2.236 199 79 39 59 19 79 19 39 59 2.651 2.620 2.574 2.489 2.213 2.222 2.228 2.232 77 57 37 17 17 37 57 77 2.648 2.616 2.568 2.474 2.232 2.211 2.221 2.227 75 55 35 15 75 15 35 55 2.646 2.613 2.562 2.457 2.210 2.231 2.220 2.227 73 33 53 13 33 53 73 13 2.643 2.608 2.555 2.438 2.208 2.220 2.231 2.226 71 51 31 11 71 31 51 11 2.640 2 .604 2 2.548 2.416 2.219 2.206 2.230 2.226 69 29 49 9 69 29 9 49 .600 .218 2.637 2.540 2 2.387 2.225 2 .204 2 2 .230 2 2 Experiment 72 Experiment 71 67 27 7 47 27 67 47 7 2.634 2.595 2.531 2.350 2.230 2.201 2 .217 2 2.225 65 25 5 45 65 25 5 45 2.631 2 .590 2 2.522 2.297 2.198 2.224 2.229 2.216 63 83 23 3 43 63 3 23 83 43 2.656 2.627 2.585 2.512 2.203 2.229 2.233 2.193 2.215 2.223 61 81 21 1 41 61 21 1 81 41 .228 2.654 2 .624 2 2.580 2 .500 2 2.323 2 2 .232 2 2.214 2 .223 2 2 .579 2 XlP XlP XlP XlP XlP XlP XlP XlP XlP X l P 200 39 79 59 19 79 39 59 19 2.028 1.955 1.999 1.874 2.792 2.840 2.872 2.701 37 77 57 17 77 37 57 17 2.025 1.949 1.995 1.860 2.836 2.785 2.870 2.686 75 35 55 15 75 35 15 55 2.023 1.943 1.991 1.845 2.867 2.832 2.779 2.669 33 53 73 13 73 53 33 13 2.020 1.987 1.827 1.937 2.828 2.864 2.649 2.772 31 71 51 11 71 51 31 11 73 2.017 1.930 1.983 1.806 2.764 2.823 2.861 2.626 29 69 9 49 69 29 9 49 1.979 1.922 2.014 1.780 2 .819 2 2 .857 2 2 .596 2 2.756 Experiment 74 Experiment 67 7 27 47 67 27 47 7 1.975 2.011 1.745 1.914 2.854 2.557 2.747 .814 2 25 65 5 45 65 25 45 5 .008 1.970 2 1.906 1.696 2.501 2.851 2.809 2.737 23 63 3 43 83 83 63 23 43 3 2.033 1.965 1.896 2.005 1.610 2 .803 2 2.878 2.847 2.726 2 .402 2 21 61 1 81 41 61 21 81 1 41 1.960 .030 2 1.886 2.002 1.503 2.798 2.875 2.844 2.332 2.715 XlQ XlQ XlQ XlQ XlQ XlQ XlQ XlQ XlQ X l Q Experiment 75 X 1 3 5 7 9 11 13 15 17 19 D 1.505 1.640 1.728 1.779 1.814 1.841 1.862 1.880 1.896 1.909 X 21 23 25 27 29 31 33 35 37 39 D 1.921 1.932 1.942 1.950 1.958 1.966 1.973 1.979 1.986 1.991 X 41 43 45 47 49 51 53 55 57 59 D 1.997 2.002 2.007 2.011 2.016 2.020 2.024 2.028 2.032 2.035 X 61 63 65 67 69 71 73 75 77 79 D 2.039 2.042 2.045 2.048 2.051 2.054 2.057 2.059 2.062 2.065 X 81 83 D 2.067 2.070 Experiment. 76 X 1 3 5 7 9 11 13 15 17 19 D 1.524 1.372 1.418 1.447 1.468 1.485 1.499 1.511 1.521 1.530 X 21 23 25 27 29 31 33 35 37 39 D 1.538 1.546 1.552 1.559 1.564 1.570 1.575 1.580 1.584 1.588 X 41 43 45 47 49 51 53 55 57 59 D 1.592 1.596 1.599 1.603 1.606 1.609 1.612 1.615 1.618 1.621 X 61 63 65 67 69 71 73 75 77 79 D 1.623 1.626 1.628 1.630 1.633 1.635 1.637 1.639 1.641 1.643 X 81 83 201 D 1.645 1.647 Experiment 77 X 1 3 5 7 9 11 13 15 17 19 D 1.520 1.418 1.474 1.509 1.534 1.553 1.569 1.583 1.595 1.605 X 21 23 25 27 29 31 33 35 37 39 D 1.615 1.623 1.631 1.638 1.645 1.651 1.656 1.662 1.667 1.671 X 41 43 45 47 49 51 53 55 57 59 D 1.676 1.680 1.684 1.688 1.692 1.695 1.698 1.702 1.705 1.708 X 61 63 65 67 69 71 73 75 77 79 D 1.711 1.713 1.716 1.719 1.721 1.724 1.726 1.729 1.731 1.733 X 81 83 D 1.735 1.737 Experiment 00 X 1 3 5 7 9 11 13 15 17 19 D 1.548 1.336 1.360 1.375 1.388 1.398 1.406 1.413 1.420 1.426 X 21 23 25 27 29 31 33 35 37 39 D 1.431 1.436 1.440 1.444 1.448 1.452 1.455 1.458 1.461 1.464 X 41 43 45 47 49 51 53 55 57 59 D 1.467 1.469 1.472 1.474 1.476 1.479 1.481 1.483 1.485 1.487 X 61 63 65 67 69 71 73 75 77 79 D 1.488 1.490 1.492 1.494 1.495 1.497 1.498 1.500 1.501 1.503 X 81 83 202 D 1.504 1.505 Experiment 7 9 _V * 1 3 5 7 9 11 13 15 17 19 D 1.542 1.374 1.405 1.426 1.442 1.455 1.466 1.476 1.484 1.491 V 21 23 25 27 29 31 33 35 37 39 5 1.498 1.504 1.509 1.515 1.519 1.524 1.528 1.532 1.536 1.539 X 41 43 45 47 49 51 53 55 57 59 D 1.542 1.546 1.549 1.552 1.554 1.557 1.560 1.562 1.565 1.567 X 61 63 65 67 69 71 73 75 77 79 D 1.569 1.571 1.573 1.575 1.577 1.579 1.581 1.583 1.585 1.586 X 81 83 D 1.588 1.590 Experiment: 80 X 1 3 5 7 9 11 13 15 17 j_S D 2 . 323 1.673 1.755 1.802 1.836 1.862 1.882 1.900 1.915 1.928 X 21 23 25 27 29 31 33 35 37 39 D 1.939 1.950 1.959 1.968 1.976 1.983 1.990 1.996 2.002 2.008 >- 41 43 45 47 49 51 53 55 57 59 D 2.013 2.018 2.023 2.027 2.032 2 .036 2.040 2.044 2.048 2.051 X 61 63 65 67 6 . 71 73 75 77 79 D 2.054 2 .058 2.061 2.064 2 .06 7 2 .070 2.072 2.075 2 .078 2.080 X 81 83 203 D 2 .083 2.085 Experiment SI X 1 3 5 7 9 11 13 15 17 19 D 2.297 1.809 1.896 1.947 1.982 2.009 2.031 2.050 2.065 2.079 X 21 23 25 27 29 31 33 35 37 39 D 2.019 2.102 2.112 2.121 2.130 2.137 2.145 2.151 2.158 2.163 X 41 43 45 47 49 51 53 55 57 59 D 2.169 2.174 2.179 2.184 2.189 2.193 2.197 2.201 2.205 2.209 X 61 63 65 67 69 71 73 75 77 79 D 2.212 2.216 2.219 2 .222 2.225 2.228 2.231 2.234 2.237 2.239 X 81 83 D 2.242 2.244 Experiment 82 X 1 3 5 7 9 11 13 15 17 19 D 2.397 1.404 1.470 1.509 1.537 1.559 1.577 1.592 1.605 1.616 X 21 23 25 27 29 31 33 35 37 39 D 1.626 1.635 1.643 1.651 1.658 1.664 1.670 1.676 1.681 1.686 X 41 43 45 47 49 51 53 55 57 59 D 1.691 1.695 1.670 1.704 1.707 1.711 1.715 1.718 1.721 1.724 X 61 63 65 67 69 71 73 75 77 79 D 1.727 1.730 1.733 1.736 1.739 1.741 1.744 1.746 1.748 1.751 X 81 83 204 D 1.753 1.755 205 59 1.876 1.733 1.807 37 39 77 79 1.845 1.848 1.721 1.802 15 17 19 55 57 1.871 1.873 1.841 13 53 1.868 11 51 71 73 75 83 1.866 1.784 1.790 1.796 1.672 1.691 1.707 69 1.863 1.830 1.834 1.838 Experiment 7 9 47 49 1.860 1.618 1.648 5 45 1.857 1.822 1.826 1.762 1.770 1.777 63 65 67 3 23 25 27 29 31 33 35 83 43 1.880 1.854 1.753 1.504 1.576 1 21 61 81 41 1.878 1.851 1.743 1.812 1.817 2 .374 2 :iq xiq x ip Xiq x iq Sediment deposition rate on dune lee. X = distance in cm from dune crest. W = deposition rate on dune lee in gm/cm^/ s: x s: x s: x s: x s ix .00987 W x .0000223 .0000671 .0000132 .0000166 .0000337 43 57 71 29 15 1 ,0000128 ,0000160 ,0000213 ,0000315 i 0000589 000368 45 73 59 31 17 3 ,0000155 ,0000125 ,0000203 ,0000295 ,0000525 000212 61 47 75 5 33 19 xeiet 30 Experiment ,0000150 ,0000121 ,0000195 ,0000277 ,0000473 000149 63 49 77 7 35 21 .0000145 .0000118 .000114 .0000187 .0000261 .0000430 65 79 51 9 37 23 0000115 0000179 0000140 0000247 0000394 0000928 81 67 53 39 11 25 .0000112 .0000136 .0000173 .0000234 .0000779 .0000364 83 69 55 41 27 13 208 69 83 27 55 13 41 .0000100 .0000157 .0000753 .0000123 .0000340 .0000216 67 81 39 53 25 11 0000103 0000228 0000163 0000127 0000904 0000370 79 65 37 51 23 9 .0000106 .0000242 .0000170 .0000131 .000112 .0000405 77 35 63 21 7 49 0000135 ,0000109 ,000148 ,0000447 ,0000257 ,0000179 Experiment 31 75 61 33 19 47 5 ,0000112 ,000214 ,0000499 ,0000274 .0000186 .0000140 73 59 31 45 17 3 0000115 ,0000145 ,0000563 ,0000293 ,0000195 ,000382 71 57 29 15 43 1 0000151 0000315 ,0000119 ,0000205 ,0101 ,0000645 W X x is X & X £ x is x is Si X SIX six six six s ix 0024 0027 .0000193 .0000207 .0000224 0001 .0076 .00000762 .00000786 .00000811 .00000962 .00000999. .0000129 .0000104 .0000136 .0000143 0046 0042 .0000363 .000173 .0000412 .0000476 .000325 .0107 3 5 47 45 43 7 9 61 59 57 1 3 75 73 71 5 7 19 5 17 15 3 1 9 1 33 31 29 xeiet 32 Experiment .00000740 .00000928 .0000123 .0000181 .0000324 .000116 63 49 77 35 21 7 ,00000719 ,00000896 ,0000118 ,0000170 ,0000292 ,0000861 65 51 79 37 23 9 ,00000699 ,0000113 ,0000265 ,0000682 ,00000865 0000160 81 67 53 39 25 11 .0000151 ,00000680 .00000837 ,0000108 ,0000243 ,0000562 41 83 55 69 27 13 209 1 X SI X SIX Six SIX S IX .0619 W .0000883 .000131 .000256 .0000532 .0000664 43 57 71 29 15 0057 0053 .0000490 .0000503 .0000601 .0000517 .0000620 .0000774 .0000642 .0000807 .000109 .0000844 .000115 .000182 .000123 .000201 .000225 014 008 .000554 .000782 .00134 5 7 49 47 45 3 5 77 63 75 61 73 59 1 3 35 21 33 7 19 31 5 17 3 Experiment 33 Experiment .0000478 .0000582 .0000743 .000103 .000166 .000429 65 79 51 37 9 23 0000466 0000564 0000715 0000975 000153 000350 81 67 53 39 25 11 .0000454 .0000689 .0000548 .0000927 .000141 .000296 83 41 55 69 27 13 210 X S| X SIX six six s ix w .0629 .000142 .0000565 .000279 .0000708 .0000946 43 15 1 57 71 29 .0000903 .00150 .000132 .0000549 .0000683 ,000246 45 3 17 59 31 73 ,0000660 ,0000863 ,000870 ,000219 ,000124 ,0000533 47 5 61 19 33 75 Experiment 34 Experiment ,0000519 ,0000639 ,0000827 ,000613 ,000198 ,000117 7 63 49 21 35 77 0074 0073 .0000734 .0000994 .0000763 .0000794 .000105 .000110 .000473 0056 0043 .0000481 .0000493 .0000581 .0000506 .0000599 .0000618 .000153 .000165 .000180 9 5 7 69 55 67 41 53 27 65 39 51 25 37 23 9 1 83 81 79 .000384 11 .000324 13 211 SI X SIX Six SIX Six 2 X .0674 .000289 .0000908 .0000667 .0000524 ,000140 43 1 15 71 57 29 ,00180 ,0000864 ,000130 ,000252 ,0000509 ,0000642 45 3 17 31 73 59 ,000989 ,0000619 ,0000823 ,000121 ,000223 ,0000494 5 47 19 61 33 75 Experiment 35 Experiment .000199 .000675 .0000787 .000114 .0000480 ,0000598 49 7 63 21 77 35 .000510 .000180 .0000753 .000107 .0000467 .0000577 9 23 51 37 65 79 000164 000408 0000722 000101 0000454 0000559 11 81 25 53 67 39 .000151 .000339 .0000693 .0000956 .0000442 .0000541 41 55 83 69 27 13 212 Experiment 36 1 3 5 7 9 11 13 187 ,00276 ,00162 ,00115 .000889 000726 .000614 15 17 19 21 23 25 27 ,000531 ,000469 ,000419 ,000379 .000346 000318 .000295 29 31 33 35 37 39 41 ,000274 ,000256 ,000241 ,000227 .000215 000204 .000194 43 45 47 49 51 53 55 ,000185 ,000176 .000169 ,000162 .000156 000150 .000144 57 59 61 63 65 67 69 .000139 ,000134 ,000130 ,000126 .000122 000118 .000115 71 73 75 77 79 81 83 000112 ,000109 .000106 ,000103 .000100 0000978 .0000954 1 X SIX Six Six SIX SIX .189 W .000205 .000306 .000596 .000124 .000154 43 71 57 29 15 009 .018 .000180 .000253 .000188 .000269 .000196 .000286 002 .017 .000114 .000117 .000140 .000120 .000144 . .000149 .000424 .00129 .000469 .00183 .000525 .00313 5 7 49 47 45 3 5 77 75 73 9 1 63 61 59 1 3 35 21 33 19 31 17 5 7 5 3 xeiet 37 Experiment 003 .011 .000127 .000131 .000135 008 .036 .000329 .000356 .000387 001 .018 .000106 .000108 .000111 007 .016 .000160 .000216 .000166 .000227 .000173 .000239 .00100 5 7 69 67 65 9 1 83 81 79 1 3 55 53 51 9 7 9 41 39 37 3 5 27 25 23 .000816 11 .000689 13 214 X W Si X SIX Six SIX SIX ,000743 ,000178 ,200 ,000366 ,000141 000240 43 57 15 1 29 71 .000171 .000137 ,000341 ,000649 ,000229 00434 45 59 31 17 73 3 .000576 .000165 .000219 .00244 .000319 .000133 47 61 5 33 19 75 Experiment 38 Experiment ,000518 ,000160 ,000209 ,00169 ,000300 ,000129 63 49 35 21 7 77 .000200 .000468 .00129 .000155 .000282 .000126 65 51 37 23 79 9 000430 000150 000192 000104 000267 000122 81 67 53 39 25 11 .000185 .000253 .000396 .000119 .000145 .000866 41 83 55 69 27 13 s: x S! X SIX SIX S IX W X ,00835 ,0000119 ,0000204 ,0000311 ,0000631 0000151 43 71 57 15 1 29 .0000552 .0000115 .0000194 .0000290 .0000145 000359 45 73 31 17 3 59 ,0000490 ,0000271 ,000204 ,0000112 ,0000185 ,0000140 47 5 61 19 75 33 xeiet 39 Experiment .0000109 ,0000440 ,000142 ,0000135 ,0000177 ,0000255 49 63 7 77 21 35 .0000399 .000109 .0000106 .0000170 .0000240 .0000131 9 79 65 51 37 23 0000365 0000878 0000103 0000127 0000163 0000227 81 67 53 39 25 11 .0000735 .0000100 .0000123 .0000157 .0000215 .0000336 83 41 69 55 27 13 w X s: x s; x sx six s ix .00730 ,0000103 0000575 0000131 0000179 0000278 15 43 1 57 71 29 .000344 .0000126 .00000994 .0000171 .0000258 0000501 17 45 31 73 59 .000193 .0000443 ,0000121 ,00000965 ,0000162 0000241 5 47 19 61 33 75 Experiment 40 Experiment ,0000397 ,0000117 ,0000225 ,00000937 ,0000155 000133 7 63 49 21 35 77 .000101 ,0000359 .00000910 ,0000113 ,0000212 ,0000148 9 23 65 37 79 51 .0000809 .0000142 .0000109 ,0000327 ,00000885 ,0000200 81 11 67 39 25 53 .0000673 ,0000301 .00000861 ,0000106 ,0000136 ,0000189 41 83 55 27 69 13 Experiment 41 1 3 5 7 9 11 13 0578 00157 ,000915 ,000647 .000500 .000408 .000344 15 17 19 21 23 25 27 000297 000262 ,000234 ,000211 .000192 .000177 .000164 29 31 33 35 37 39 41 000152 000142 ,000133 ,000126 .000119 .000112 .000107 43 45 47 49 51 53 55 000102 0000972 ,0000930 ,0000891 .0000856 .0000823 .0000792 57 59 61 63 65 67 69 0000764 0000738 0000713 ,0000690 .0000668 .0000648 .0000629 71 73 75 77 79 81 83 0000611 ,0000594 ,0000577 ,0000562 .0000548 .0000534 .0000521 Experiment 42 1 3 5 9 11 13 . 0 1 1 1 000827 ,000420 .000270 .000194 000149 .000119 15 17 19 21 23 25 27 .0000987 0000836 ,0000721 .0000630 .0000558 0000499 .0000450 29 31 33 35 37 39 41 .0000408 ,0000373 .0000343 .0000317 .0000294 0000274 .0000256 43 45 47 49 51 53 55 .0000240 0000225 ,0000212 .0000201 .0000190 0000180 .0000171 57 59 61 63 65 67 69 .0000163 0000156 ,0000149 .0000142 .0000136 0000131 .0000126 71 73 75 77 79 81 83 .0000121 .0000116 .0000112 .0000108 .0000104 0000101 .00000976 Experiment 43 1 3 5 7 9 11 13 0112 .000704 .000344 .000219 .000156 .000120 .0000962 15 17 19 21 23 25 27 0000796 .0000676 .0000584 .0000512 ,0000455 ,0000408 ,0000369 29 31 33 35 37 39 41 0000336 .0000308 .0000284 .0000263 ,0000245 ,0000228 ,0000214 43 45 47 49 51 53 55 0000201 .0000190 .0000179 .0000170 ,0000161 .0000153 ,0000146 57 59 61 63 65 67 69 0000140 .0000133 .0000128 .0000123 ,0000118 .0000113 ,0000109 71 73 75 77 79 81 83 0000105 .0000101 .00000978 .00000945 ,00000914 .00000885 ,00000858 Experiment 44 1 3 5 7 9 11 13 0106 .000464 .000225 .000143 .000103 .0000797 .0000643 15 17 19 21 23 25 27 0000536 .0000458 .0000398 .0000351 ,0000313 ,0000282 ,0000256 29 31 33 35 37 39 41 0000235 .0000216 .0000200 .0000186 ,0000174 ,0000163 .0000153 43 45 47 49 51 53 55 0000144 .0000136 .0000129 .0000123 ,0000117 ,0000112 ,0000107 57 59 61 63 65 67 69 0000102 .00000979 .00000940 .00000904 ,00000870 ,00000839 ,00000809 71 73 75 77 79 81 83 00000782 .00000756 .00000731 .00000708 ,00000687 ,00000666 ,00000647 w X SI X Six SIX SIX SIX ,000228 .00101 .0968 ,000178 ,000485 000312 43 1 15 29 71 57 ,000219 ,000450 ,00597 000877 000172 000297 3 45 31 17 73 59 ,000283 ,000211 ,000420 .00336 ,000167 ,000775 5 47 19 61 33 75 Experiment 45 Experiment ,000162 ,000694 ,000203 .000393 ,000270 ,00232 7 49 21 63 35 77 .000628 .000158 .000196 .00176 .000258 .000370 9 23 51 37 79 65 000153 000190 000247 000349 000572 00142 11 81 53 39 25 67 .000149 .000184 .000237 .000330 .000525 .00118 41 83 55 13 69 27 222 Experiment 46 1 3 5 7 9 11 13 , 0984 00599 ,00328 ,00222 .00166 00132 .00109 15 17 19 21 23 25 27 000926 000802 ,000705 ,000629 .000566 000514 .000471 29 31 33 35 37 39 41 000433 000401 ,000373 ,000349 .000327 000308 .000291 43 45 47 49 51 53 55 , 000275 ,000261 ,000248 ,000236 .000226 000216 .000207 57 59 61 63 65 67 69 ,000198 ,000191 ,000183 ,000177 .000170 000165 .000159 71 73 75 77 79 81 83 ,000154 ,000149 ,000144 000140 .000136 000132 .000128 Experiment 47 1 3 5 7 9 11 13 103 ,00511 .00261 00171 .00125 .000981 .000801 15 17 19 21 23 25 27 000673 ,000579 .000506 ,000449 .000403 .000364 .000332 29 31 33 35 37 39 41 000305 ,000282 .000262 ,000244 .000228 .000215 .000202 43 45 47 49 51 53 55 000191 000181 .000172 ,000164 .000156 .000149 .000143 57 59 61 63 65 67 69 000137 000131 .000126 ,000122 .000117 .000113 .000109 71 73 75 77 79 81 83 000106 ,000102 .0000991 0000961 .0000932 .0000905 .0000880 Experiment 48 1 3 5 7 9 11 13 339 0168 .00968 ,00679 .00522 00423 .00355 15 17 19 21 23 25 27 ,00305 00268 ,00238 ,00215 .00195 00179 .00165 29 31 33 35 37 39 41 ,00153 00142 .00133 ,00125 .00118 00111 .00106 43 45 47 49 51 53 55 00101 000960 .000917 ,000878 .000841 000808 .000776 57 59 61 63 65 67 69 ,000748 000721 .000696 ,000672 .000650 000630 .000610 71 73 75 77 79 81 83 000592 ,000575 .000559 ,000543 .000528 000515 .000501 225 six six six six six six 00 025.00235 .00265 00303 04 018.00129 .00138 00148 344 000968 072.066.000661 .000686 000712 050.054.058.053.049.000485 .000499 .000513 .000528 .000544 000560 43 57 71 517 29 15 1 07 .00991 .0174 .000921 45 73 59 31 3 007 004 000 .000771 .000804 .000840 .000879 47 61 75 33 19 7 5 xeiet 49 Experiment .00688 1 2 1 0 0 . 1 1 2 0 0 . .000639 49 63 77 35 1 2 011.00174 .00423 .00191 .00525 .00114 001 .000597 .000617 65 51 79 37 23 9 .00108 81 53 67 39 25 1 1 .000741 2 0 1 0 0 . .00353 .00160 .000578 .000473 83 55 41 69 27 13 226 SIX six SIX SIX six s ix 003 .042 .000409 .000422 .000436 007 .075 .000699 .00105 .000735 .000774 .00113 .00122 006 .059 .000518 .000539 .000560 .00198 .00947 .00225 .0178 .00261 .357 3 5 47 45 43 1 3 75 73 71 7 9 61 59 57 5 7 19 5 17 15 3 1 9 1 33 31 29 Experiment 50 Experiment 009 .036 .000375 .000386 .000397 .000609 .000636 .000666 008 .090 .000866 .00144 .000920 .000980 .00159 .00374 .00177 .00473 .00635 000 .042 .000466 .000482 .000500 9 1 53 51 49 7 9 81 67 79 65 77 63 1 3 25 11 23 9 21 7 5 7 39 37 35 .000365 .000584 .000450 .000817 .00132 .00308 83 41 55 69 27 13 227 Si X Six SIX Six SIX S IX 2 0 1 0 0 859 00256 00504 02 .00123 00128 00171 345 43 57 931 29 15 71 .00163 .00239 .000992 .00443 .0269 17 59 73 006 .000939 .000965 00119 00156 00224 00395 0156 47 61 19 335 33 75 xeiet 51 Experiment .00115 .00149 1 1 2 0 0 . .00357 0 1 1 0 . 49 63 1 2 77 001 .081 .000869 .000891 .000914 2 1 1 0 0 00851 00143 .00298 00199 00325 9 65 51 25 37 23 79 .00108 .00138 .00692 .00189 81 1 1 53 39 67 .00105 .00133 .00180 .00276 .00583 83 41 55 69 27 13 w X Si X SIX SIX SIX SIX .000139 .000343 .000667 .000174 .000231 182 43 57 71 29 15 1 ,000321 ,000588 000135 00348 000168 000221 45 73 31 17 59 3 .000132 .00203 .000162 .000301 .000525 .000211 47 5 61 19 75 33 xeiet 52 Experiment ,000128 ,000157 ,000284 ,000475 ,000202 ,00144 63 49 7 21 77 35 002 .012 .000119 .000122 .000125 .000242 .000255 .000269 005 .018 .000143 .000148 .000180 .000152 .000187 .000194 .00112 003 .039 .000369 .000399 .000433 5 7 69 67 65 9 9 1 83 81 55 41 79 53 27 39 51 25 37 23 .000911 11 .000770 13 229 Experiment 53 1 3 5 7 9 11 13 158 ,00349 ,00203 ,00144 .00112 .000910 .000769 15 17 19 21 23 25 27 000665 ,000586 ,000524 ,000474 .000432 .000397 .000367 29 31 33 35 37 39 41 000342 ,000320 ,000300 ,000283 .000267 .000254 .000241 43 45 47 49 51 53 55 000230 ,000219 ,000210 ,000201 .000193 .000186 .000179 57 59 61 63 65 67 69 000173 ,000167 ,000161 ,000156 .000151 .000147 .000142 71 73 75 77 79 81 83 000138 ,000135 ,000131 .000127 .000124 .000121 .000118 X w SIX SIX six six six ,00000576 .0000138 ,00000419 ,00000865 ,0000324 ,00406 1 43 15 29 57 71 ,00000810 ,0000126 ,00000548. ,0000277 ,000373 00000403 45 17 3 31 59 73 .0000240 ,00000761 ,0000126 ,00000522 ,000181 ,00000387 5 47 19 33 61 75 xeiet 54 Experiment ,00000373 ,00000717 ,0000116 ,00000499 ,0000211 ,000113 7 49 35 21 63 77 ,00000359 ,00000477 ,00000676 ,0000107 ,0000188 .0000792 37 23 51 65 79 .00000456 .00000640 .00000995 .0000168 .00000346 .0000473 11 81 53 39 25 67 ,00000334 ,00000437 ,00000607 ,00000926 ,0000152 .0000387 41 83 55 27 13 69 231 Experiment 55 X I 3 5 7 9 11 13 W .00405 .000350 .000165 .000102 .0000715 .0000538 .0000425 X 15 17 19 21 23 25 27 W .0000347 .0000291 .0000249 .0000216 .0000190 .0000169 .0000152 X 29 31 33 35 37 39 41 W .0000138 .0000125 .0000115 .0000106 .00000979 .00000909 .00000848 X 43 45 47 49 51 53 55 W .00000793 .00000745 .00000701 .00000661 .00000625 .00000593 .00000563 X 57 59 61 63 65 67 69 W .00000536 .00000511 .00000487 .00000466 .00000446 .00000428 .00000411 X 71 73 75 77 79 81 83 W .00000395 .00000380 .00000366 .00000353 .00000340 .00000329 .00000318 to to to 13 27 69 55 41 83 ,0000113 .00000649 ,00000441 ,00000328 ,00000258 .0000309 39 25 67 11 53 81 ,0000126 ,00000693 ,00000340 ,00000266 ,00000462 .0000391 79 37 23 51 65 9 ,00000743 ,0000141 ,00000486 ,00000354 ,00000275 .0000521 35 77 21 63 49 ,00000801 ,00000369 ,00000284 ,0000159 ,00000513 Experiment 56 33 75 19 61 47 5 ,000124 ,0000751 ,00000866 ,00000542 ,00000385 ,00000294 .0000183 31 17 59 73 3 45 ,00000574 ,00000942 ,00000402 ,00000305 ,000282 ,0000213 29 57 71 15 43 .00000316 .0000103 .00000609 .00000420 .0000253 w .00403 X X X 1 & X 3: X & XS X IS .0862 1 W X SI X SIX SIX SIX S I X .000984 .000429 .000138 .000182 .000260 43 15 71 57 29 004 .071 .000645 .00255 .000731 .000841 .00389 .00744 009 .034 .000338 .000364 .000394 003 .018 .000124 .000128 .000160 .000133 .000221 .000167 ..000174 .000233 .000246 5 7 49 47 45 5 7 5 3 9 1 63 61 35 21 59 33 19 31 17 3 5 77 75 73 Experiment 57 Experiment .000315 .000575 .00187 .000120 .000154 .000210 9 65 51 23 37 79 000517 00145 000148 000200 000295 000116 81 53 11 25 67 39 .000143 .000190 .000470 .00118 .000113 .000277 83 41 55 69 27 13 W X SI X SIX SIX SIX S I X , 0864 ,000894 ,000123 ,000163 ,000233 ,000386 43 71 57 1 15 29 ,00715 ,000762 ,000156 ,000220 ,000354 000119 45 73 59 3 31 17 ,000115 ,000149 ,000208 .00365 ,000661 ,000327 47 75 61 5 19 33 xeiet 58 Experiment ,000143 .000198 .000582 ,00237 ,000111 000303 49 63 77 35 7 21 .000518 .000138 .000107 .000188 .000283 .00172 79 65 51 37 23 9 000104 000133 000179 000264 000466 00133 81 67 53 39 25 11 .000101 .000128 .000170 .000248 .000422 .00107 83 41 55 69 27 13 s: x six SIX SIX six s ix .000280 .000662 .0873 .0000892 .000118 .000169 43 1 71 57 29 15 ,000561 .00625 ,000257 ,0000861 ,000113 ,000159 45 3 73 31 17 59 ,000485 .00296 ,0000832 ,000108 ,000150 ,000237 5 47 61 19 75 33 Experiment 59 Experiment .0000805 ,000425 ,00185 ,000104 ,000143 ,000219 49 7 63 77 35 21 011 010 .000801 .00100 .00131 0070 0076 .0000733 .0000756 .0000780 .000123 .000129 .000179 .000136 .000191 .000204 0097 0090 .0000925 .000Q960 .0000997 .000307 .000339 .000378 1 13 11 9 9 1 83 69 81 55 67 79 53 65 51 7 9 41 39 37 3 5 27 25 23 1 X • 365 W Si X Six SIX Six S i x .00428 .00196 .00122 .000874 .000672 15 43 29 57 71 039 033 .00287 .0105 .00323 .0155 .00369 .0286 016 010 .00105 .00110 .00157 .00116 .00168 .00181 003 .086 .000776 .000806 .000839 005 .069 .000610 .000629 .000650 5 7 5 3 7 9 21 19 17 5 7 49 47 45 1 3 35 33 31 9 1 63 61 59 3 5 77 75 73 xeiet 60 Experiment 028 024 .00213 .00234 .00258 .00779 017 018 .00130 .00138 .00147 009 .094 .000913 .000954 .000999 004 .070 .000695 .000720 .000747 009 .054 .000557 .000574 .000591 9 3 5 27 25 23 7 9 41 39 37 1 3 55 53 51 5 7 69 67 65 9 1 83 81 79 .00616 11 .00506 13 237 Experiment 61 1 3 5 7 9 11 13 ,367 ,0281 ,0149 ,00990 .00731 00574 .00469 15 17 19 21 23 25 27 ,00395 00340 ,00297 ,00263 .00236 00213 .00194 29 31 33 35 37 39 41 ,00178 ,00164 .00152 ,00142 .00133 00125 .00117 43 45 47 49 51 53 55 ,00111 ,00105 ,000994 ,000945 .000900 000859 .000822 57 59 61 63 65 67 69 ,000787 ,000755 ,000725 ,000697 .000671 000647 .000625 71 73 75 77 79 81 83 ,000603 ,000583 ,000565 ,000547 .000530 000514 .000499 X w s: x six SIX SIX s ix ,00306 ,000438 ,000574 ,000814 ,00133 374 43 15 1 57 29 71 ,00261 ,0260 ,000423 ,000550 ,000769 ,00122 45 3 17 73 59 31 ,0128 ,00227 ,000409 ,000729 ,00113 ,000528 5 47 19 61 33 75 Experiment 62 Experiment ,00821 .00105 ,00200 ,000396 ,000507 ,000692 7 63 49 35 1 2 77 .00178 .00591 .000384 .000488 .000658 .000981 23 9 65 51 37 79 00160 00456 000919 000372 000470 000628 81 1 1 53 25 67 39 .00146 .00368 .000361 .000453 .000863 .000600 83 41 69 55 27 13 Experiment 66 1 3 5 7 9 1 1 13 .00734 .000291 .000162 .000111 ,0000842 .0000674 .0000561 15 17 19 21 23 25 27 .0000479 .0000417 .0000369 .0000330 ,0000299 ,0000273 .0000250 29 31 33 35 37 39 41 .0000231 .0000215 .0000201 .0000188 ,0000177 .0000167 .0000158 43 45 47 49 51 53 55 .0000150 .0000143 .0000136 .0000130 ,0000124 .0000119 .0000114 57 59 61 63 65 67 69 .0000110 .0000106 .0000102 .00000984 ,00000951 .00000919 .00000890 71 73 75 77 79 81 83 .00000862 .00000836 .00000812 .00000789 ,00000767 .00000746 .00000726 240 SI X Six SIX SIX SIX S i x ,00000692 ,00000884 ,0000121 ,0000188 ,00598 ,0000392 43 57 71 29 1 15 ,00000850. ,0000115 ,00244 ,0000341 ,0000174 00000671 45 31 3 73 59 17 ,00000651 ,00000819 ,000135 ,0000110 ,0000301 ,0000163 47 5 61 75 33 19 xeiet 67 Experiment ,00000632 ,00000790 ,0000105 ,0000920 ,0000152 ,0000269 49 63 35 77 21 ,0000694 ,0000143 ,00000614 ,00000763 ,0000100 ,0000243 51 9 37 65 79 23 ,00000597 .00000958 .0000554 ,00000738 ,0000135 ,0000222 81 53 67 39 11 25 .00000920 ,00000581 ,00000714 ,0000460 ,0000128 ,0000203 41 83 55 69 13 27 241 X w SI X SIX Six SIX S I X ,000250 ,0513 ,000126 ,0000627 ,0000840 ,0000499 1 15 43 29 57 71 ,000220 ,000118 ,0000485 00136 0000801 0000605 3 17 45 31 59 73 .000196 ,000784 ,0000767 ,000110 ,0000584 ,0000471 5 19 47 33 61 75 Experiment 68 Experiment ,000176 ,000551 ,0000733 ,000104 ,0000459 ,0000565 7 21 49 35 63 77 .0000704 .0000981 .000161 .000424 .0000447 .0000547 9 23 37 65 51 79 000147 000345 0000676 0000929 0000530 0000435 11 25 53 39 81 67 .0000514 .0000882 .0000651 .000136 .000290 .0000424 41 55 83 27 13 69 242 Experiment 69 1 3 5 7 9 11 13 ,0422 .00116 ,000670 ,000471 .000362 .000294 .000247 15 17 19 21 23 25 27 ,000213 .000187 ,000166 ,000150 .000136 .000125 .000115 29 31 33 35 37 39 41 ,000107 .000100 .0000937 ,0000882 .0000832 .0000788 .0000748 43 45 47 49 51 53 55 ,0000712 .0000679 ,0000649 ,0000621 .0000596 .0000572 .0000551 57 59 61 63 65 67 69 ,0000531 .0000512 ,0000494 ,0000478 .0000463 .0000448 .0000435 71 73 75 77 79 81 83 ,0000422 .0000410 ,0000399 ,0000388 .0000378 .0000368 .0000359 243 1 X .171 W SI X SIX Six SIX S I X .000637 .000164 .000218 .000326 .000131 43 71 57 15 29 037 016 .00139 .00196 .00337 000 .026 .000269 .000286 .000452 .000304 .000501 .000561 002 .014 .000121 .000124 .000148 .000127 .000153 .000191 .000158 .000199 .000208 5 7 49 47 45 1 3 35 21 7 33 19 31 5 17 3 3 5 77 63 75 61 73 59 xeiet 70 Experiment 005 .021 .000229 .000241 .000254 .000737 .000873 .00107 008 .016 .000170 .000176 .000184 001 .039 .000350 .000379 .000412 001 .015 .000112 .000115 .000135 .000118 .000139 .000143 3 5 27 13 25 11 23 9 9 1 83 69 81 55 67 41 79 53 65 39 51 37 SI X SIX SIX SIX S I X 000106 000176 000132 000514 000263 137 43 71 57 1 15 29 .000103 .000127 .000168 .000245 .00272 .000452 45 73 59 3 17 31 .0000998 .000230 .000123 .00158 .000161 .000404 47 61 5 33 19 75 xeiet 71 Experiment .0000971 .000119 .000154 .000217 2 1 1 0 0 . .000365 63 49 7 77 35 1 2 .0000946 .000115 .000148 .000865 .000205 .000332 65 51 9 79 37 23 .0000922 2 1 1 0 0 0 . .000142 .000194 .000305 .000705 81 67 53 1 1 39 25 .0000900 .000109 .000137 .000185 .000594 .000282 83 41 55 69 27 13 245 s; x six SI X six six six .00868 .0000115 .0000181 .0000388 .00000649 .00000834 43 1 15 29 57 71 ,0000109 ,0000167 ,0000335 ,000271 ,00000629 ,00000802 3 45 17 31 59 73 ,0000104 ,0000156 ,0000295 ,000143 ,00000610 ,00000772 5 47 19 61 33 75 Experiment 72 Experiment ,0000951 ,0000262 ,00000592 ,00000744 ,0000146 00000991 7 49 21 63 35 77 .0000236 ,0000706 ,00000575 ,00000947 ,0000137 .00000718 23 51 37 65 79 .00000906 .0000129 .0000215 .0000557 ,00000559 ,00000693 11 53 25 81 39 67 ,00000868 ,0000121 ,00000544 ,0000196 ,0000458 ,00000671 41 55 83 13 27 69 246 Six SIX Six SIX SIX S ix .00712 .00000529 .00000681 .00000942 .0000148 .0000320 43 15 1 57 29 71 ,00000894 ,0000137 ,0000277 ,000229 ,00000513 ,00000654 45 17 3 31 59 73 .000119 .00000850 .0000128 .0000243 .00000630 ,00000497 5 47 19 33 61 75 xeiet 73 Experiment .00000482 .00000607 ,00000810 ,0000119 ,0000216 ,0000792 49 21 63 35 77 ,00000468 ,00000586 ,00000774 ,0000112 ,0000194 ,0000586 9 37 23 65 51 79 .00000455 .00000566 .00000740 .0000105 .0000176 ,0000462 11 81 53 25 67 39 .00000547 .00000709 .00000995 .0000161 .0000379 .00000443 41 83 55 27 69 13 247 SI X SIX Six Six Six Six .0000164 .00000513 .00000682 .00000983 .00574 .0000390 43 29 15 1 71 57 ,00000495 ,00000652 ,0000151 ,0000331 ,000350 00000926 45 3 31 17 59 73 ,00000478 ,00000624 ,00000875 ,0000139 ,0000286 ,000170 47 5 33 19 61 75 xeiet 74 Experiment ,00000462 ,00000599 .00000829 ,0000128 .0000251 ,000108 49 7 63 77 35 21 ,00000447 ,00000575 .00000787 ,0000119 ,0000222 ,0000769 51 37 65 23 79 .00000433 .00000553 .00000749 ,0000112 .0000199 .0000588 81 53 11 67 39 25 .00000714 .0000105 .0000471 .00000419 .00000532 .0000180 41 83 55 69 27 13 248 Si X SIX SIX six six SIX ,00354 ,0000228 ,00000954 ,00000394 ,00000296 ,00000569 1 43 15 57 29 71 ,0000193 ,000210 .00000536 ,00000874 ,00000376. ,00000285 3 45 17 31 59 73 .00000805 ,0000167 ,000101 ,00000507 ,00000275 ,00000360 5 47 33 19 61 75 xeiet 75 Experiment ,0000637 ,00000745 ,0000146 ,00000480 ,00000346 ,00000266 49 21 63 35 77 .0000130 .0000453 .00000693 .00000455 .00000332 .00000257 51 37 23 65 79 ,0000116 ,0000346 ,00000433 ,00000647 ,00000249 ,00000319 11 81 53 39 25 67 .00000606 ,0000105 ,0000276 ,00000413 ,00000307 ,00000241 41 83 55 27 69 13 249 Experiment 76 1 3 5 7 9 1 1 13 0633 ,00389 ,00210 ,00142 00106 .000839 .000691 15 17 19 21 23 25 27 000585 000506 ,000445 ,000396 000356 .000323 .000296 29 31 33 35 37 39 41 000272 ,000252 ,000234 ,000219 000205 .000193 .000182 43 45 47 49 51 53 55 000172 ,000163 ,000155 ,000148 000141 .000135 .000129 57 59 61 63 65 67 69 000124 ,000119 ,000114 ,000110 000106 .000103 .0000991 71 73 75 77 79 81 83 0000958 ,0000928 ,0000899 ,0000872 .0000846 .0000821 .0000798 s: x s: x s:x s:x six s ix 0077 0073 .0000690 .0000713 .0000737 .0000885 .0000921. .000121 .0000960 .000128 .000135 007 .040 .000359 .00178 .000410 .00336 .000476 .0521 001 .019 .000185 .000199 .000216 1 3 75 73 47 71 45 43 7 9 61 59 57 5 7 19 17 15 9 1 33 5 31 29 3 1 xeiet 77 Experiment 0081 0080 .0000790 .0000820 .0000851 001 .010 .000105 .000110 .000115 0069 0068 .0000629 .0000648 .0000669 001 .025 .000258 .000285 .000691 .000318 .000879 .00119 007 .011 .000151 .000161 .000172 3 5 67 65 63 9 1 53 51 49 7 9 81 79 77 5 7 39 25 37 11 23 35 9 21 7 .0000611 .0000763 .000100 .000235 .000143 .000565 83 55 41 69 27 13 251 SI X SIX SIX six six s ix .00206 .230 .00100 .000373 .000476 .000649 1 43 15 29 57 71 001 .058 .000562 .000588 .000617 .00143 .00473 .00159 .00684.00180 .0121 006 .031 .000341 .000351 .000362 .000814 .000868 .000930 005 . 004 .000426 .000441 ..000458 5 7 5 3 5 7 49 47 21 45 19 17 1 3 35 33 31 9 1 63 61 59 3 5 77 75 73 xeiet 78 Experiment 019 018 .00108 .00241 .00118 .00129 .00289 .00359 006 .073 .000684 .000723 .000766 003 .033 .000314 .000323 .000385 .000331 .000398 .000411 003 .055 .000495 .000515 .000538 1 13 11 9 7 9 41 27 39 25 37 23 1 3 55 53 51 5 7 69 67 65 9 1 83 81 79 252 1 X .187 W SI X SIX Six SIX S i x .000395 .000307 .000850 .000544 .00179 43 57 15 71 29 007 . 006 .000352 .000469 .000365 ..000379 .000491 .000687 .000516 .00122 .000734 .000788 .00421 .00137 .00615 .00155 .0111 009 .029 .000280 .000289 .000298 5 7 49 47 45 1 3 35 33 7 31 5 3 3 5 77 63 75 61 73 21 59 19 17 xeiet 79 Experiment 000340 000448 000645 0 1 1 0 0 00317 000272 9 65 51 37 79 23 .000328 .000429 .000607 .00253 .000264 0 0 1 0 0 . 81 53 67 39 1 1 25 .000257 .000411 .000574 .000317 .000921 0 1 2 0 0 . 83 41 55 69 27 13 Experiment 80 1 3 5 7 9 11 13 ,0626 .00184 , 0 0 1 0 0 ,000682 ,000513 ,000409 ,000339 15 17 19 21 23 25 27 ,000289 ,000251 ,000222 ,000198 ,000179 ,000163 ,000150 29 31 33 35 37 39 41 ,000138 ,000129 ,000120 ,000112 ,000106 .0000996 ,0000942 43 45 47 49 51 53 55 ,0000894 .0000850 ,0000810 ,0000773 ,0000740 ,0000709 ,0000680 57 59 61 63 65 67 69 ,0000654 ,0000630 ,0000607 0000586 ,0000566 ,0000547 .0000529 71 73 75 77 79 81 83 ,0000513 ,0000498 ,0000483 0000469 ,0000456 .0000444 ,0000432 Experiment 81 1 3 5 7 9 11 13 ,0534 ,00167 ,000902 ,000610 ,000457 ,000363 .000301 15 17 19 21 23 25 27 ,000256 ,000222 ,000196 ,000175 ,000158 ,000144 .000132 29 31 33 35 37 39 41 ,000122 ,000113 ,000105 0000988 ,0000928 ,0000875 .0000827 43 45 47 49 51 53 55 ,0000785 ,0000746 ,0000710 ,0000678 ,0000648 ,0000621 .0000596 57 59 61 63 65 67 69 ,0000573 0000551 ,0000531 ,0000512 ,0000495 ,0000478 .0000463 71 73 75 77 79 81 83 ,0000449 ,0000435 ,0000422 ,0000410 ,0000398 ,0000387 .0000377 3! X SIX SIX SIX SIX S I X 15 042 .00275 .00492 .195 002 .078 .000636 .000718 .000823 000 .034 .000350 .000374 .000402 006 .020 .000239 .000250 .000263 005 .018 .000144 .000180 .000148 .000153 .000187 .000194 5 3 1 3 5 47 45 43 5 7 19 17 15 7 9 61 59 57 9 1 33 31 29 1 3 75 73 71 xeiet 82 Experiment 019 014 .00115 .00144 .00189 007 .057 .000473 .000517 .000571 002 .039 .000292 .000309 .000328 002 .029 .000210 .000219 .000228 004 .016 .000133 .000136 .000163 .000168 .000140 .000174 9 11 9 7 3 5 67 53 65 51 25 63 49 23 35 21 7 9 81 79 77 37 37 39 .000435 .000961 .000277 .000158 .000129 .000201 41 83 55 69 27 13 256 1 X z z x s: x s: x six s ix .168 W .000757 .000176 .000240 .000368 .000139 43 43 15 57 29 71 006 .054 .000524 .000584 .00175 .000660 .00255 .00461 002 .027 .000208 .000217 .000228 003 .011 .000127 .000131 .000158 .000135 .000164 . .000170 .000300 .000320 .000342 5 7 49 47 7 45 5 3 1 3 35 21 33 19 31 17 9 1 63 61 59 3 5 77 75 73 xeiet 83 Experiment .00133 .000153 .000199 .000282 .000474 .000124 9 65 51 37 23 79 000148 000191 000266 000433 00106 000120 81 53 67 11 39 25 .000143 .000183 .000252 .000398 .000885 .000117 41 83 55 69 27 13 257 258 C. Deposition rate as fraction of the total sediment deposited 84 cm beyond the dune crest. X = distance in cm from dune crest. = deposition rate on dune lee as fraction of the total sediment. Experiment 30 X 1 3 5 7 9 11 13 15 17 19 00 wf • .031 .018 .013 .0097 .0079 0066 .0057 .0050 .0045 X 21 23 25 27 29 31 33 35 37 39 w£ .0040 .0037 .0034 .0031 .0029 .0027 0025 .0024 .0022 .0021 X 41 43 45 47 49 51 53 55 57 59 Wf .0020 .0019 .0018 .0017 .0017 .0016 0015 .0015 .0014 .0014 X 61 63 65 67 69 71 73 75 77 79 Wf .0013 .0013 .0012 .0012 .0012 .0011 0011 .0011 .0011 .0010 X 81 83 wf .00098 .00095 E xp e rim e n t 31 X 1 3 5 7 9 11 13 15 17 19 Wf .85 .032 .018 .012 .0094 .0076 0063 .0054 .0047 .0042 X 21 23 25 27 29 31 33 35 37 39 Wf .0038 .0034 .0031 .0029 .0026 .0025 0023 .0022 .0020 .0019 X 41 43 45 47 49 51 53 55 57 59 Wf .0018 .0017 .0016 .0016 .0015 .0014 0014 .0013 .0013 .0012 X 61 63 65 67 69 71 73 75 77 79 Wf .0012 .0011 .0011 .0011 .0010 .0010 00097 .00094 .00091 .00089 X 81 83 259 Wf .00086 .00084 Experiment 32 X 1 3 5 7 9 11 13 15 17 19 Wf .88 .027 .014 0095 .0071 0056 .0046 .0039 .0034 .0030 X 21 23 25 27 29 31 33 35 37 39 wf .0027 .0024 .0022 0020 .0018 0017 .0016 .0015 .0014 .0013 X 41 43 45 47 49 51 53 55 57 59 Wf .0012 .0012 .0011 0011 .0011 00097 .00093 .00089 .00086 .00082 X 61 63 65 67 69 71 73 75 77 79 wf .00079 .00077 .00074 00071 .00069 00067 .00065 .00063 .00061 .00059 X 81 83 wf .00058 .00056 Experiment 33 X 1 3 5 7 9 11 13 15 17 19 .90 .019 .0 1 1 0080 .0062 0051 .0043 .0037 .0033 .0029 wf X 21 23 25 27 29 31 33 35 37 39 Wf .0026 .0024 .0022 0020 .0019 0018 .0017 .0016 .0015 .0014 X 41 43 45 47 49 51 53 55 57 59 Wf .0013 .0013 .0012 0012 .0011 0011 .0010 .0010 .00096 .00093 X 61 63 65 67 69 71 73 75 77 79 Wf .00090 .00087 .00084 00082 .00079 00077 .00075 .00073 .00071 .00069 260 X 81 83 .00067 .00066 Experiment 34 X 1 3 5 7 9 11 13 15 17 19 wf .89 .021 .012 .0087 .0069 .0054 .0046 .0039 .0035 .0031 X 21 23 25 27 29 31 33 35 37 39 Wf .0028 .0025 .0023 .0022 .0020 .0019 .0018 .0017 .0016 .0015 X 41 43 45 47 49 51 53 55 57 59 wf .0014 .0013 .0013 .0012 .0012 .0011 .0011 .0010 .0010 .00097 X 61 63 65 67 69 71 73 75 77 79 Wf .00093 .00090 .00087 .00085 .00082 .00080 .00078 .00075 .00073 .00071 X 81 83 Wf .00070 .00068 Experiment 35 X 1 3 5 7 9 11 13 15 17 19 .89 .024 .013 .0089 .0067 .0054 .0045 .0038 .0033 .0029 wf X 21 23 25 27 29 31 33 35 37 39 Wf .0026 .0024 .0022 .0020 .0018 .0017 .0016 .0015 .0014 .0013 X 41 43 45 47 49 51 53 55 57 59 Wf .0013 .0012 .0011 .0011 .0010 .00099 .00095 .00092 .00088 .00085 X 61 63 65 67 69 71 73 75 77 79 w .00082 .00079 .00076 .00074 .00071 .00069 .00067 .00065 .00063 .00062 X 81 83 W .00060 .00058 261 Experiment 36 X 1 3 5 7 9 11 13 15 17 19 wf .93 .014 .0080 .0057 .0044 0036 .0030 .0026 .0023 .0021 X 21 23 25 27 29 31 33 35 37 39 w f .0019 .0017 .0016 .0015 .0014 0013 .0012 .0011 .0011 .0010 X 41 43 45 47 49 51 53 55 57 59 .00091 .00087 .00084 .00080 00077 .00074 .00071 .00069 w f .00096 .00067 X 61 63 65 67 69 71 73 75 77 79 .00064 .00062 .00060 .00059 .00057 00055 .00054 .00052 .00051 .00050 wf X 81 83 Wf .00048 .00047 Experimei 37 X 1 3 5 7 9 11 13 15 17 19 Wf .92 .015 .0089 .0063 .0049 0040 .0033 .0029 .0026 .0023 X 21 23 25 27 29 31 33 35 37 39 Wf .0021 .0019 .0017 .0016 .0015 0014 .0013 .0012 .0012 .0011 X 41 43 45 47 49 51 53 55 57 59 .0010 .00095 .00091 .00087 00084 .00081 .00078 .00075 .00072 w f .0010 X 61 63 65 67 69 71 73 75 77 79 Wf .00070 .00078 .00066 .00064 .00062 00060 .00058 .00057 .00055 .00054 X 81 83 Wf .00053 .00051 262 Experiment 38 X 1 3 5 7 9 11 13 15 17 19 Wf .90 .020 .011 .0077 .0058 .0047 .0039 .0034 .0029 .0026 X 21 23 25 27 29 31 33 35 37 39 Wf .0023 .0021 .0019 .0018 .0017 .0015 .0014 .0014 .0013 .0012 X 41 43 45 47 49 51 53 55 57 59 wf .0011 .0011 .0010 .00099 .00095 .00091 .00087 .00084 .00081 . 0007J X 61 63 65 67 69 71 73 75 77 79 .00075 .00072 .00070 .00068 .00066 .00064 .00062 wf .00060 .00058 .0005’ X 81 83 Wf .00055 .00054 Experiment 39 X 1 3 5 7 9 11 13 15 17 19 .82 .035 .020 .014 .011 .0087 .0073 .0062 .0054 Kf .0048 X 21 23 25 27 29 31 33 35 37 39 Wf .0043 .0039 .0036 .0033 .0031 .0029 .0027 .0025 .0024 .0022 X 41 43 45 47 49 51 53 55 57 59 Wf .0021 .0020 .0019 .0018 .0018 .0017 .0016 .0015 .0015 .0014 X 61 63 65 67 69 71 73 75 77 79 .0014 .0013 .0013 .0012 .0012 .0012 .0011 .0011 .0011 .0010 X 81 83 263 W .0010 .00099 Experiment 40 X 1 3 5 7 9 11 13 15 17 19 Wf .82 .039 .022 .015 .011 .0091 .0075 .0064 .0056 .0050 X 21 23 25 27 29 31 33 35 37 39 wf .0044 .0040 .0037 .0034 .0031 .0029 .0027 .0025 .0024 .0022 X 41 43 45 47 49 51 53 55 57 59 wf .0021 .0020 .0019 .0018 .0017 .0017 .0016 .0015 .0015 .0014 X 61 63 65 67 69 71 73 75 77 79 Wf .0014 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010 X 81 83 Wf .00099 .00096 Experiment 41 X 1 3 5 7 9 11 13 15 17 19 Wf .87 .024 .014 .0098 .0076 .0062 .0052 .0045 .0040 .0035 X 21 23 25 27 29 31 33 35 37 39 Wf .0032 .0029 .0027 .0025 .0023 .0021 .0020 .0019 .0018 .0017 X 41 43 45 47 49 51 53 55 57 59 Wf .0016 .0015 .0015 .0014 .0013 .0013 .0012 .0012 .0012 .0011 X 61 63 65 67 69 71 73 75 77 79 .0011 .0010 .0010 .00098 .00095 .00092 .00090 .00087 wf .00085 .00083 X 81 83 w f .00081 .00079 Experiment 42 X 1 3 5 7 9 11 13 15 17 19 w£ .79 .059 .030 .019 .014 .011 0085 .0070 .0059 .0051 X 21 23 25 27 29 31 33 35 37 39 Wf .0045 .0040 .0035 .0032 .0029 .0027 0024 .0023 .0021 .0019 X 41 43 45 47 49 51 53 55 57 59 Wf .0018 .0017 .0016 .0015 .0014 .0013 0013 .0012 .0012 .0011 X 61 63 65 67 69 71 73 75 77 79 W f .0011 .0010 .00097 .00093 .00089 .00086 00083 .00080 .00077 .00074 X 81 83 Wf .00072 .00069 Experiment 43 X 1 3 5 7 9 11 13 15 17 19 Wf .82 .051 .025 .016 .011 .0088 0070 .0058 .0049 .0043 X 21 23 25 27 29 31 33 35 37 39 Wf .0037 .0033 .0030 .0027 .0025 .0022 0021 .0019 .0018 .0017 X 41 43 45 47 49 51 53 55 57 59 Wf .0016 .0015 .0014 .0013 .0012 .0012 0011 .0011 .0010 .00097 X 61 63 65 67 69 71 73 75 77 79 Wf .00093 .00090 .00086 .00083 .00080 .00077 00074 .00071 .00069 .00067 X 81 83 265 .00065 .00063 wf W. X SIX SIX SIX SIX SIX SIX SIX SIX SIX .86 ,00054 ,0013 ,77 ,0026 ,0012 ,0017 41 81 1 0029 21 00077 61 41 1 0055 81 21 61 ,0012 , 038 ,00053 ,00074 ,048 ,0025 ,0050 ,0012 ,0016 43 0026 83 63 23 3 43 83 3 23 63 ,0011 ,00071 .0024 ,027 ,0016 ,0046 45 018 0023 5 65 25 45 5 25 65 ,0011 ,00069 ,019 .0023 47 012 7 0021 67 27 47 7 0042 0015 67 27 xeiet 44 Experiment Experiment 45 Experiment .014 .0039 .0015 .0022 .0010 ,0019 , 0084 ,00066 49 9 69 29 9 49 69 29 .011 .0021 .0036 .0014 ,0065 ,00064 ,00096 ,0018 51 11 71 31 11 51 31 71 ,00062 ,00091 ,0016 ,0053 ,0094 ,0014 ,0020 ,0034 53 33 13 73 13 53 33 73 ,00087 ,0015 ,0044 ,0080 ,00060 ,0019 ,0031 ,0013 55 15 75 35 55 15 35 75 ,00058 ,00083 ,0014 ,0037 ,0018 ,0030 ,0070 ,0013 57 77 17 37 57 17 37 77 ,00056 ,00080 ,0013 ,0033 ,0017 ,0028 .0062 ,0013 59 79 39 19 59 19 39 79 Experiment 46 X 1 3 5 7 9 11 13 15 17 19 .79 .048 .026 .018 .013 .011 0087 .0074 .0064 .0056 wf X 21 23 25 27 29 31 33 35 37 39 Wf .0050 .0045 .0041 .0038 .0035 .0032 0030 .0028 .0026 .0025 X 41 43 45 47 49 51 53 55 57 59 Wf .0023 .0022 .0021 .0020 .0019 .0018 0017 .0017 .0016 .0015 X 61 63 65 67 69 71 73 75 77 79 wf .0015 .0014 .0014 .0013 .0013 .0012 0012 .0012 .0011 .0011 X 81 83 Wf .0011 .0010 Experiment 47 X 1 3 5 7 9 11 13 15 17 19 Wf .84 .042 .021 .014 .010 .0080 0065 .0055 .0047 .0041 X 21 23 25 27 29 31 33 35 37 39 Wf .0037 .0033 .0030 .0027 .0025 .0023 0021 .0020 .0019 .0017 X 41 43 45 47 49 51 53 55 57 59 .0016 .0016 .0015 .0014 .0013 .0013 0012 .0012 .0011 .0011 w f X 61 63 65 67 69 71 73 75 77 79 .00092 00083 .00081 .00078 .00076 wf .0010 .00099 .00095 .00089 .00086 X 81 83 267 w. .00074 .00072 Experiment 48 X 1 3 5 7 9 11 13 15 17 19 Wf .80 .040 .023 .016 .012 .0099 0083 .0072 .0063 .0056 X 21 23 25 27 29 31 33 35 37 39 Wf .0050 .0046 .0042 .0039 .0036 .0033 0031 .0029 .0028 .0026 X 41 43 45 47 49 51 53 55 57 59 .0024 .0023 .0022 .0021 .0020 0019 .0018 .0018 .0017 wf .0025 X 61 63 65 67 69 71 73 75 77 79 Wf .0016 .0016 .0015 .0015 .0014 .0014 0014 .0013 .0013 .0012 X 81 83 Wf .0012 .0012 Experiment 49 X 1 3 5 7 9 11 13 15 17 19 o • 00 .041 .023 .016 .012 .0099 0082 .0071 .0062 .0055 wf X 21 23 25 27 29 31 33 35 37 39 .0049 .0045 .0041 .0037 .0035 .0032 0030 .0028 .0027 .0025 wf X 41 43 45 47 49 51 53 55 57 59 .0021 .0020 .0020 .0019 0018 .0017 .0017 .0016 wf .0024 .0023 X 61 63 65 67 69 71 73 75 77 79 wf .0015 .0015 .0014 .0014 .0013 .0013 0013 .0012 .0012 .0012 X 81 83 .0011 .0011 " f Experiment 50 X 1 3 5 7 9 11 13 15 17 19 wf .82 .041 .022 .015 .011 .0086 .0071 .0060 .0052 .0046 X 21 23 25 27 29 31 33 35 37 39 wf .0041 .0037 .0033 .0031 .0028 .0026 ,0024 0023 ,0021 .0020 X 41 43 45 47 49 51 53 55 57 59 W f .0019 .0018 .0017 .0016 .0015 .0015 ,0014 ,0013 ,0013 .0012 X 61 63 65 67 69 71 73 75 77 79 Wf .0012 .0012 .0011 .0011 .0010 .0010 ,00098 ,00095 ,00092 .00089 X 81 83 wf .00087 .00084 Experiment 51 X 1 3 5 7 9 11 13 15 17 19 .86 .027 .016 .011 .0085 .0069 ,0058 .0050 .0044 .0040 wf X 21 23 25 27 29 31 33 35 37 39 Wf .0036 .0032 .0030 .0028 .0026 .0024 ,0022 .0021 ,0020 .0019 X 41 43 45 47 49 51 53 55 57 59 Wf .0018 .0017 .0016 .0016 .0015 .0014 ,0014 ,0013 ,0013 .0012 X 61 63 65 67 69 71 73 75 77 79 wf .0012 .0012 .0011 .0011 .0011 .0010 ,00099 .00096 .00094 .00091 X 81 83 269 .00089 .00087 six six six SIX six six six six six six 009 .00067 .00069 41 81 61 21 1 91 41 00081 0012 0024 00061 21 89 1 1 83 81 61 0027 00092 0014 .0011 .017 .00078 .0022 .00059 .00089 .0025 .020 .0013 43 63 23 3 83 43 23 3 63 .0011 .00076 .010 .0020 .0012 .0023 .012 .00086 45 5 25 45 65 25 5 65 .0011 .0018 .00074 .0072 .00083 .0012 .0021 .0082 47 7 67 27 47 7 67 27 xeiet 52 Experiment xeiet53 Experiment .0056 .0010 .0017 .00071 .0011 .0019 .0063 .00081 9 49 29 69 49 9 29 69 .00097 .0045 .0016 .00069 .00079 .0011 .0018 .0052 11 31 71 51 31 11 51 71 .00093 .0038 .00067 .0015 .0017 .0011 .0044 .00076 13 33 53 73 53 33 13 73 .0033 .00090 .0014 .00066 .0016 .0038 .00074 .0010 55 35 15 75 35 15 55 75 .00087 .0029 .00064 .0013 .00098 .0015 .0033 .00072 17 57 37 77 37 17 57 77 .00062 .00084 .0026 .0013 .0014 .00095 .0030 .00071 59 19 39 79 59 19 39 79 270 Experiment 54 X 1 3 5 7 9 11 13 15 17 19 Wf .77 .071 .034 .021 .015 .011 0089 .0073 .0061 .0052 X 21 23 25 27 29 31 33 35 37 39 wf .0045 .0040 .0035 .0032 .0029 .0026 0024 .0022 .0020 .0019 X 41 43 45 47 49 51 53 55 57 59 Wf .0018 .0016 .0015 .0014 .0014 .0013 0012 .0011 .0011 .0010 X 61 63 65 67 69 71 73 75 77 79 wf .00099 .00094 .00090 .00086 .00083 .00079 00076 .00073 .00070 .00068 X 81 83 Wf .00065 .00063 Experiment 55 X 1 3 5 7 9 11 13 15 17 19 Wf .78 .068 .032 .020 .014 .011 0082 .0067 .0056 .0048 X 21 23 25 27 29 31 33 35 37 39 Wf .0042 .0037 .0033 .0029 .0027 .0024 0022 .0020 .0019 .0018 X 41 43 45 47 49 51 53 55 57 59 Wf .0016 .0015 .0014 .0014 .0013 .0012 0011 .0011 .0010 .00099 X 61 63 65 67 69 71 73 75 77 79 Wf .00094 .00090 .00086 .00083 .00079 .00076 00073 .00071 .00068 .00066 X 81 83 271 .00064 .00061 w f Experiment 56 X 1 3 5 7 9 11 13 15 17 19 W, ,82 ,058 ,025 015 .011 .0080 .0063 .0052 .0044 .0037 X 21 23 25 27 29 31 33 35 37 39 w. 0033 ,0029 ,0026 ,0023 .0021 .0019 ,0018 ,0016 ,0015 .0014 X 41 43 45 47 48 51 53 55 57 59 w. ,0013 ,0012 ,0012 ,0011 .0010 .00099 ,00095 ,00090 .00086 .00082 X 61 63 65 67 69 71 73 75 77 79 w. 00079 ,00075 00072 ,00070 .00067 .00065 ,00062 ,00060 ,00058 .00056 X 81 83 w. .00054 .00053 Experiment 57 X 1 3 5 7 9 11 13 15 17 19 75 ,064 ,034 022 .016 .013 ,010 ,0085 ,0073 ,0063 X 21 23 25 27 29 31 33 35 37 39 W, 0060 0050 0045 0041 .0037 .0034 ,0032 ,0029 ,0027 ,0026 X 41 43 45 47 49 51 53 55 57 59 w. ,0024 ,0023 ,0021 ,0020 .0019 .0018 ,0017 ,0017 ,0016 ,0015 X 61 63 65 67 69 71 73 75 77 79 w. ,0014 ,0014 ,0013 ,0013 .0012 .0012 0012 ,0011 ,0011 ,0010 X 81 83 272 W. ,0010 ,00098 Experiment 58 X 1 3 5 7 9 11 13 15 17 19 Wf .76 .063 .032 .021 .015 .012 0095 .0079 .0067 .0058 X 21 23 25 27 29 31 33 35 37 39 Wf .0051 .0046 .0041 .0037 .0034 .0031 0029 .0027 .0025 .0023 X 41 43 45 47 49 51 53 55 57 59 Wf .0022 .0021 .0019 .0018 .0017 .0017 0016 .0015 .0014 .0014 X 61 63 65 67 69 71 73 75 77 79 Wf .0013 .0013 .0012 .0012 .0011 .0011 0010 .0010 .00098 .00095 X 81 83 Wf .00092 .00089 E xp e rim e n t 59 X 1 3 5 7 9 11 13 15 17 19 Wf .80 .058 .027 .017 .012 .0092 .0074 .0061 .0052 .0045 X 21 23 25 27 29 31 33 35 37 39 Wf .0039 .0035 .0031 .0028 .0026 .0024 .0022 .0020 .0019 .0018 X 41 43 45 47 49 51 53 55 57 59 Wf .0017 .0016 .0015 .0014 .0013 .0013 .0012 .0011 .0011 .0010 X 61 63 65 67 69 71 73 75 77 79 Wf .0010 .00096 .00092 .00088 .00085 .0008! .00079 .00077 .00074 .00072 X 81 83 273 Wf .00070 .00068 Experiment 60 X 1 3 5 7 9 11 13 15 17 19 .75 .059 .032 .021 .016 .013 010 .0088 .0076 .0066 Kf X 21 23 25 27 29 31 33 35 37 39 Wf .0059 .0053 .0048 .0044 .0040 .0037 0034 .0032 .0030 .0028 X 41 43 45 47 49 51 53 55 57 59 Wf .0027 .0025 .0024 .0023 .0021 .0020 0020 .0019 .0018 .0017 X 61 63 65 67 69 71 73 75 77 79 Wf .0017 .0016 .0015 .0015 .0014 .0014 0013 .0013 .0012 .0012 X 81 83 Wf .0012 .0011 E xp e rim e n t 61 X 1 3 5 7 9 11 13 15 17 19 Wf .76 .058 .031 .021 .015 .012 0097 .0082 .0070 .0062 X 21 23 25 27 29 31 33 35 37 39 Wf .0055 .0049 .0044 .0040 .0037 .0034 0032 .0029 .0028 .0026 X 41 43 45 47 49 51 53 55 57 59 wf .0024 .0023 .0022 .0021 .0020 .0019 0018 .0017 .0016 .0016 X 61 63 65 67 69 71 73 75 77 79 Wf .0015 .0014 .0014 .0013 .0013 .0013 0012 .0012 .0011 .0011 X 81 83 274 v* .0011 .0010 Experiment 62 X 1 3 5 7 9 11 13 15 17 19 Wf .82 .055 .027 .018 .013 .0097 0078 .0065 .0056 .0048 X 21 23 25 27 29 31 33 35 37 39 Wf .0043 .0038 .0034 .0031 .0028 .0026 0024 .0022 .0021 .0020 X 41 43 45 47 49 51 53 55 57 59 Wf .0018 .0017 .0016 .0016 .0015 .0014 0013 .0013 .0012 .0012 X 61 63 65 67 69 71 73 75 77 79 Wf .0011 .0011 .0010 .0010 .00097 .00093 00090 .00087 .00084 .00082 X 81 83 Wf .00079 .00077 E xp e rim e n t 66 X 1 3 5 7 9 11 13 15 17 19 Wf 00 • .034 .019 .013 .0097 .0076 0064 .0055 .0048 .0042 X 21 23 25 27 29 31 33 35 37 39 .0038 .0034 .0031 .0029 .0027 .0025 0023 .0022 .0020 .0019 wf X 41 43 45 47 49 51 53 55 57 59 Wf .0018 .0017 .0016 .0016 .0015 .0014 0014 .0013 .0013 .0012 X 61 63 65 67 69 71 73 75 77 79 Wf .0012 .0 Oil .0011 .0011 .0010 .00099 00096 .00093 .00091 .00088 X 81 83 275 .00086 .00083 wf SIX SIX six six six six six six six 41 84 143 41 81 61 0018 0038 21 1 81 88 02.01.0011 .0011 0012 61 00084 0015 0030 123 21 1 00075 00.09 .00094 .00097 0010 01 .0016 .0017 .034 .0034 .00082 .00073 02 02 .0023 .0025 .0028 01 01 .0013 .0014 .0014 .023 83 43 3 83 6? 23 63 3 .019 .0031 03.0095 .013 547 45 5 547 45 25 65 65 5 25 03.0098 .013 01 01 .0014 .0015 .0015 .0010 .0026 .0029 001.08 .00086 .00088 .00091 9 7 67 27 67 7 27 xeiet 67 Experiment xeiet68 Experiment .0010 .0013 .0022 07 .0059 .0073 49 69 49 29 9 69 29 .0078 .0025 .00097 .0012 .0020 51 11 51 71 31 71 31 11 0065 13 0023 33 00094 0013 53 73 53 13 73 00083 0012 0019 33 0050 .0021 .0055 .0013 .00092 .0018 .0011 .0043 .00081 15 35 75 55 55 35 75 15 .0048 .00089 .0012 .0020 .0017 .0011 .00079 .0038 17 77 57 37 37 77 57 17 .0042 .00086 .0012 .0019 .0010 .0016 .00077 .0034 19 59 59 79 39 19 39 79 276 Experiment 69 X 1 3 5 7 9 11 13 15 17 19 w£ .88 .024 .014 .0098 .0075 0061 .0051 .0044 .0039 .0035 X 21 23 25 27 29 31 33 35 37 39 Wf .0031 .0028 .0026 .0024 .0022 0021 .0019 .0018 .0017 .0016 X 41 43 45 47 49 51 53 55 57 59 Wf .0016 .0015 .0014 .0013 .0013 0012 .0012 .0011 .0011 .0011 X 61 63 65 67 69 71 73 75 77 79 Wf .0010 .00099 .00096 .00093 .00090 00088 .00085 .00083 .00081 .00078 X 81 83 wf .00076 .00074 Exp e rime] 70 X 1 3 5 7 9 11 13 15 17 19 w .91 .018 .010 .0073 .0057 0046 .0039 .0034 .0030 .0026 f X 21 23 25 27 29 31 33 35 37 39 Wf .0024 .0022 .0020 .0019 .0017 0016 .0015 .0014 .0013 .0013 X 41 43 45 47 49 51 53 55 57 59 Wf .0012 .0012 .0011 .0011 .0010 00097 .00093 .00090 .00087 .00084 X 61 63 65 67 69 71 73 75 77 79 Wf .00081 .00078 .00076 .00073 .00071 00069 .00067 .00065 .0064 .00062 X 81 83 277 Wf .00061 .00059 Experiment 71 X 1 3 5 7 9 11 13 15 17 19 Wf .90 .018 .010 .0074 .0057 .0047 0039 .0034 .0030 .0027 X 21 23 25 27 29 31 33 35 37 39 Wf .0024 .0022 .0020 .0019 .0017 .0016 0015 .0014 .0014 .0013 X 41 43 45 47 49 51 53 55 57 59 Wf .0012 .0012 .0011 .0011 .0010 .00098 00094 .00091 .00087 .00084 X 61 63 65 67 69 71 73 75 77 79 Wf .00082 .00079 .00076 .00074 .00072 .00070 00068 .00066 .00064 .00063 X 81 83 wf .00061 .00060 Experiment 72 X 1 3 5 7 9 11 13 15 17 19 Wf .88 .028 .015 .0097 .0072 .0057 0047 .0039 .0034 .003 0 X 21 23 25 27 29 31 33 35 37 39 Wf .0027 .0024 .0022 .0020 .0018 .0017 0016 .0015 .0014 .0013 X 41 43 45 47 49 51 53 55 57 59 Wf .0012 .0012 .0011 .0011 .0010 .00096 00092 .00088 .00085 .00082 X 61 63 65 67 69 71 73 75 77 79 Wf .00079 .00076 .00073 .00071 .00068 .00066 00064 .00062 .00060 .00059 X 83 81 278 Wf .00057 .00055 Six six SIX six Six Six SIX Six SIX 88 41 0027 00078 61 0012 41 82 00056 81 1 81 21 0015 0036 00090 61 21 00062 1 .0024 .028 .00075 .0012 .050 .00055 .0014 .0032 .00060 .00086 43 3 43 83 83 63 23 3 23 63 .0022 .0011 .015 .00073 .024 .0013 .0029 .00083 45 45 5 25 25 5 65 65 .0020 .00070 .0011 .0098 .0013 .015 .0026 .00079 47 67 7 47 27 7 67 27 xeiet 73 Experiment Experiment .0018 .00068 .0010 .0073 .011 .0024 .0012 .00076 49 69 9 49 29 9 29 69 74 .0017 .0057 .00066 .00096 .0022 .0085 .0011 .00074 71 31 51 11 51 11 31 71 .0016 .0047 .00064 .00092 .0020 .0068 .00071 .0011 73 53 13 53 33 13 73 33 .00062 .0015 .00088 .0040 .0018 .0056 .0010 .00069 75 55 35 15 15 55 35 75 .00060 .0014 .0034 .00084 .0017 .0048 .00098 .00066 57 77 17 37 17 57 77 37 .00058 .0013 .00081 .0030 .0041 .0016 .00094 .00042 79 59 19 59 19 39 39 79 279 Experiment 75 X 1 3 5 7 9 11 13 15 17 19 wf .83 .049 .024 .015 .011 0081 .0065 .0054 .0045 .0039 X 21 23 25 27 29 31 33 35 37 39 Wf .0034 .0030 .0027 .0025 .0022 0020 .0019 .0017 .0016 .0015 X 41 43 45 47 49 51 53 55 57 59 wf .0014 .0013 .0013 .0012 .0011 0011 .0010 .00097 .00092 .00088 X 61 63 65 67 69 71 73 75 77 79 Wf .00084 .00081 .00078 .00075 .00072 00069 .00067 .00065 .00062 .00060 X 81 83 Wf .00058 .00057 Experimei 76 X 1 3 5 7 9 11 13 15 17 19 Wf .79 .048 .026 .018 .013 010 .0086 .0073 .0063 .0055 X 21 23 25 27 29 31 33 35 37 39 Wf .0049 .0044 .0040 .0037 .0034 0031 .0029 .0027 .0026 .0024 X 41 43 45 47 49 51 53 55 57 59 Wf .0023 .0021 .0020 .0019 .0018 0018 .0017 .0016 .0015 .0015 X 61 63 65 67 69 71 73 75 77 79 Wf .0014 .0014 .0013 .0013 .0012 0012 .0012 .0011 .0011 .0011 X 81 83 280 .0010 .0010 wf Experiment 1 3 5 7 9 11 13 15 17 19 .79 .051 .027 .018 .013 010 .0086 .0072 .0062 .0054 21 23 25 27 29 31 33 35 37 39 .0048 .0043 .0039 .0036 .0033 0030 .0028 .0026 .0024 .0023 41 43 45 47 49 51 53 55 57 59 .0022 .0020 .0019 .0018 .0017 0017 .0016 .0015 .0015 .0014 61 63 65 67 69 71 73 75 77 79 .0013 .0013 .0012 .0012 .0012 0011 .0011 .0010 .0010 . 00095 81 83 .00095 .00093 Experiment 8 1 3 5 7 9 11 13 15 17 19 80 .042 .024 .016 .012 ,010 .0084 .0072 .0062 .0055 21 23 25 27 29 31 33 35 37 39 0050 .0045 .0041 .0038 .0035 ,0032 .0030 .0028 .0027 .0025 41 43 45 47 49 51 53 55 57 59 0024 .0023 .0021 .0020 .0020 .0019 .0018 .0017 .0017 .0016 61 63 65 67 69 71 73 75 77 79 0015 .0015 .0014 .0014 .0013 .0013 .0013 .0012 .0012 .0012 81 83 0011 .0011 Experiment 79 X 1 3 5 7 9 11 13 15 17 19 Wf .79 .047 .026 .018 .013 .011 .0088 .0075 .0065 .0058 X 21 23 25 27 29 31 33 35 37 39 Wf .0051 .0046 .0042 .0039 .0036 .0033 .0031 .0029 .0027 .0026 X 41 43 45 47 49 51 53 55 57 59 Wf .0024 .0023 .0022 .0021 .0020 .0019 .0018 .0017 .0017 .0016 X 61 63 65 67 69 71 73 75 77 79 Wf .0015 .0015 .0014 .0014 .0013 .0013 .0013 .0012 .0012 .0011 X 81 83 Wf .0011 .0011 Experiment 80 X 1 3 5 7 9 11 13 15 17 19 00 00 Wf « .026 .014 .0096 .0072 .0058 .0048 .0041 .0035 .0031 X 21 23 25 27 29 31 33 35 37 39 Wf .0028 .0025 .0023 .0021 .0020 .0018 .0017 .0016 .0015 .0014 X 41 43 45 47 49 51 53 55 57 59 Wf .0013 .0013 .0012 .0011 .0011 .0010 .0010 .00096 .00092 .00089 X 61 63 65 67 69 71 73 75 77 79 w. .00086 .00083 .00080 .0007'/' .00075 .00072 .00070 .00068 .00066 .00064 X 81 83 W .00063 .00061 oo r N J Experiment 81 X 1 3 5 7 9 11 13 15 17 19 00 CO wf • .027 .015 .010 .0075 .0060 0049 .0042 .0037 .0032 X 21 23 25 27 29 31 33 35 37 39 wf .0029 .0026 .0024 .0022 .0020 .0019 0017 .0016 .0015 .0014 X 41 43 45 47 49 51 53 55 57 59 Wf .0014 .0013 .0012 .0012 .0011 .0011 0010 .00098 .00094 .00091 X 61 63 65 67 69 71 73 75 77 79 Wf .00087 .00084 .00081 .00079 .00076 .00074 00071 .00069 .00067 .00065 X 81 83 Wf .00064 .00062 Experiment 82 X 1 3 5 7 9 11 13 15 17 19 Wf .89 .023 .013 .0087 .0066 .0053 0044 .0038 .0033 .0029 X 21 23 25 27 29 31 33 35 37 39 w .0026 .0024 .0022 .0020 .0018 .0017 0016 .0015 .0014 .0013 f X 41 43 45 47 49 51 53 55 57 59 Wf .0013 .0012 .0011 .0011 .0010 .0010 00096 .00092 .00089 .00085 X 61 63 65 67 69 71 73 75 77 79 Wf .00082 .00080 .00077 .00074 .00072 .00070 ,00068 .00066 .00064 .00062 X 81 83 Wf .00061 .00059 Experiment 83 1 3 5 7 9 11 13 15 17 19 89 .024 .013 .0092 .0070 .0056 0047 .0040 0035 .0031 21 23 25 27 29 31 33 35 37 39 0028 .0025 .0023 .0021 .0019 .0018 0017 .0016 0015 .0014 41 43 45 47 49 51 53 55 57 59 0013 .0013 .0012 .0011 .0011 .0010 0010 .00097 00093 .00089 61 63 65 67 69 71 73 75 77 79 00086 .00083 .00080 .00078 .00075 .00073 00071 .00069 00067 .00065 81 83 .00063 .00062 284 285 Appendix 4. Grain size gradients obtained numerically through the simulation program. Figs. 1-6. Effect of sorting of bed material. Figs. 7-11. Effect of flow velocity. Figs. 12-13. Effect of flow depth. Figs. 14-17. Effect of mean size of bed material 286 Fig. 1. Effect of sorting of bed material on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experiment number, u = 90 cm/sec; D = 2.5 A. d = 15 cm; experiment 38 ( a = 1.5*), experi ment 37 ( o = 0.71 ( o = .33*) . B. d = 30 cm; experiment 82 ( o = 1.5*), experi ment 70 ( o = 0.71*), and experiment 52 ( a = 0.35*). C. d = 50 cm; experiment 83 ( o = 1.5, experi ment 71 ( ° = 0.71*), and experiment 53 ( a = 0.35*). 287 QC LU h* LU 2 < o z < 24 36 2.7 30 60 DISTANCE FROM DUNE CREST (cm) A 2 8 8 1.2 1.3 t r UJ I- l.« < o < cr e> z < UJ 2.4 11, 0 30 60 90 D ISTA N C E FROM DUNE CREST (cm) B 2 8 9 O z < 0 1 O z < UJ S 2.4 2.7 DISTANCE FROM DUNE CREST (cm) c 290 Fig. 2. Effect of sorting of bed material on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experiment number, u = 40 cm/secy D = 2.5+ . A. d = 15 cm; experiment 32 ( cr = 1 .5+), experi ment 31 ( o = 0.71+), experiment 30 ( a = 0.35+). B. d = 30 cm; experiment 72 ( a = 1.5+), experi ment 66 ( a = 0.71+), and experiment 39 ( a = 0.35+) . C. d = 50 cm; experiment 73 (a =1.5+), experi ment 67 ( cr = 0.71+), and experiment 40 ( o = 0.35 + ) . 291 2.3 67 2.6 73 2.9 "S ' 2.3 a> 66 I “ Q c o c o 2.1 2.4 2.7 60 80 Distance From Dune Crest (cm) 292 Fig. 3. Effect of sorting of bed material on simulated mean size gradient of the leeside gradient. The number of each curve corresponds to the experiment number. u = 65 cm/sec; D = 2.5 A. d = 15 cm; experiment 35 ( a = 1.5(f)), experiment 34 ( a = 0.71 33 ( ° = 0.354>) . B. d = 30 cm; experiment 80 ( 0 = 1.5 experiment 68 ( a = 0.71(f)), and experiment 41 ( a = 0.35 C. d = 50 cm; experiment 81 ( o = 1.5 experiment 69 ( a = 0.71$), and experiment 51 ( a = 0.35(f)) . 293 1.5 cr 1.8 UJ t- UJ 2 < o z < a: o 34 z < UJ 2 2.4 33 2.7 30 60 9l DISTANCE FROM DUNE CREST (cm) A MEAN GRAIN DIAMETER (<*>) 1.5 2.7 1.8 2.1 2.4 ITNE RM UE RS (cm) CREST DUNE FROM DISTANCE B 0 0 9< 60 30 295 1.3 cr I > LU y— UJ S < Q 2.1 < cr CD < 69 UJ 2.4 2.7 0 30 60 90 DISTANCE FROM DUNE CREST (cm) 296 Fig. 4. Effect of sorting of bed material on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experiment number. u = 40 cm/secy D = 1.5*. A. d = 15 cm; experiment 42 ( a = 0.35*), experiment 43 ( a = 0.71 (o=1.5*). B. d = 30 cm; experiment 74 ( o = 0.71*). C. d = 50 cm; experiment 75 ( o = 0.71*). MEAN GRAIN DIAMETER (♦) 2.4 2.1 1.5 1.5 A ITNE RM UE RS (cm) CREST DUNE FROM DISTANCE 44 42 MEAN GRAIN DIAMETER («>) 2.1 2.4 1.3 i.a B ITNE RM UE RS (cm) CREST DUNE FROM DISTANCE 90 2 9 9 1.3 a: UJ H UJ 2 1.3 < Q < a: o a.i 2.4 o 3060 90 DISTANCE FROM DUNE CREST (cm) C 300 Fig. 5 Effect of sorting of bed material on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experiment number. A. u = 65 cm/sec; D = 1.5 experiment 45 ( G = 0.35 ( a = 0.71 B. u = 65 cm/sec; D = 1.5<}>; d = 30 cm; experiment 76 ( o = 0.714>). C. u = 65 cm/sec; D = 1.5 experiment 77 ( 0 = 0.7l4>). D. u = 90 cm/sec; D = 1.5 4>; d = 15 cm; experiment 48 ( ° = 0.35<)>)» experiment 49 (a = 0.714>), and experiment 50 ( a = 1.5 E. u = 90 cm/sec; D = 1.5 experiment 78 ( ° = O.?!^). F. u = 90 cm/sec; D = 1.5d = 50 cm; experiment 79 ( 0 = 0.71^). 301 (E UJ K- UJ 2 < a z < oc o z 50 < 49 UJ 2 48 1.7 K 13 UJ t- UJ 2 < Q 1.5 Z < 45 a: 46 e? z 1.7 < UJ 47 2 1.9 0 3U30 60 990 DIAMETER FROM DUNE CREST (cm) 302 B tr UJ h UJ 1.1 2 < Q < CL O 1.4 Z < -78 UJ 1.7 1.1 CL UJ 1.3 I- UJ 2 < Q < CL O 76 Z 1.7 < UJ 1.9 30 60 90 DIAMETER FROM DUNE CREST (cm) B 303 1.1 cr LU i- 1.3 LU < a 1.5 < cr & 1.7 < — 77 LU 1.9 0 30 60 90 DISTANCE FROM DUNE CREST (cm) C 304 Fig. 6. Effect of sorting of bed material on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experiment number. D = 0.50 A. u = 40 cm/sec? experiment 54 ( o = 0.35$), experiment 55 ( o = 0.71$), and experiment 56 ( o = 1.5 4») - B. u = 65 cm/sec; experiment 57 ( 0 = 0.35$), experiment 58 ( o = 0.71$), and experiment 59 ( o=l.5$). C. u = 90 cm/sec; experiment 60 ( o = 0.35), experiment 61 ( CT = 0.71$), and experiment 62 ( o=i.5$). 3 0 5 a: Ll I h- Ui S < Q 54 z < IT O z < 55 UJ 2 1.2 1.5 DISTANCE FROM DUNE CREST (cm) A MEAN GRAIN DIAMETER (0) 1.5 1.2 .6 .3 o B ITNE RM UE RS (cm) CREST DUNE FROM DISTANCE 307 o .3 QC UJ UJ S 6 < Q < (T i D 9 < UJ 12 1.3 0 30 60 90 DISTANCE FROM DUNE CREST (cm) c 308 Fig. 7. Effect of flow velocity on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experiment number. D = 2.5 A. o = 0.35 4> y experiment 36 (u = 90 cm/sec), experiment 33 (u = 65 cm/sec), and experiment 30 (u = 40 cm/sec). B. cr = 7i experiment 34 (u = 65 cm/sec), and experiment 31 (u = 40 cm/sec). C. a = 1.5 experiment 35 (u = 65 cm/sec), and experiment 32 (u = 40 cm/sec). 309 1.2 1.6 35 e E o c o k_ o 2.4 c o a) 37 34 B 2A 2.5 30 A 2.55 2.6 I 2.65]- 30 60 90 Distance From Dune Crest (cm) 310 Fig. 8. Effect of flow velocity on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experiment number. D = 1.5 (u = 90 cm/sec), experiment 47 (u = 65 cm/sec), and experiment 44 (u = 40 cm/sec). 3 1 1 1.2 1.8 47 2.1 44 2.4 30 60 Distance From Dune Crest (cm.) 312 Fig. 9. Effect of flow velocity on simulated mean size gradient of the leeside sediment. Ihe number of each curve corresponds to the experiment number. D = 1.5*: d = 15 cm. A. a = 0.35 experiment 45 (u = 65 cm/sec), and experiment 42 (u = 40 cm/sec). B. o = 0.71 experiment 46 (u = 65 cm/sec), and experiment 43 (u = 40 cm/sec). 313 1.3 49 1.5 46 V 1.7 V e a Q c 1.9 o O 43 ; 2 .i z 1.5 48 1.6 45 1.7 42 1.6 30 60 A Distance From Dune Crest (cm) 314 Fig. 10. Effect of flow velocity on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experiment number. D = 0.50 Experiment 62 (u = 90 cm/sec); experiment 59 (u = 65 cm/sec), and experiment 56 (u = 40 cm/sec). 315 1.2 56 1.4 30 60 Distance From Dune Crest (cm) 316 Fig. 11. Effect of flow velocity on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experiment number. D = 0.50$; d = 15 cm. A. a = 0.35$; experiment 60 (u = 90 cm/sec), experiment 57 (u = 65 cm/sec), and experi ment 54 (u = 40 cm/sec). B. o = 0.71$; experiment 61 (u = 90 cm/sec), experiment 58 (u = 65 cm/sec), and experi ment 55 (u = 40 cm/sec). 3 1 7 • L. ® © E 55 o B c k.o O c o ® % 57 54 30 60 Distance From Dune Crest (cm) A 318 Fig. 12. Effect of flow depth on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experiment n u m b e r . D = 1.54-; o = 0.714' . u = 90 cm/sec: d = 15 cm; experiment 49; d = 30 cm, experiment 78; d = 50 cm, experiment 79. u = 65 cm/sec: d = 15 cm; experiment 46; d = 30 cm, experiment 76; d = 50 cm, experiment 77. u = 40 cm/sec: d = 15 cm, experiment 43; d = 30 cm, experiment 74; d = 50 cm, experiment 75. 3^-9 320 Fig. 13. Effect of flow depth on simulated mean size gradient of the leeside sediment. The number of each curve corresponds to the experiment number. D = 2.5 A . o = 0 . 71 u = 90 cm/sec: d = 15 cm, experiment 37; d = 30 cm, experiment 71; d = 50 cm, experiment 70. u = 65 cm/sec: d = 15 cm, experiment 34; d = 30 cm, experiment 68; d = 50 cm, experiment 69. u = 40 cm/sec; d = 15 cm, experiment 31; d = 30 cm, experiment 66; d = 50 cm, experiment 67. B . 0 = 0.35 d = 15 cm, experiment 30; d = 30 cm, experiment 39; d = 50 cm, experiment 40. C. o = 1.54*; u = 65 cm/sec. d = 15 cm, experiment 35; d = 30 cm, experiment 80; d = 5 0 cm, experiment 81. 3 21 1.5 80 2.4 c 2.5 2.6 B 2.4 2S Distance From Dune Crest (cm) 322 Fig. 14. Effect of mean size of the bed material on the simulated mean size gradient of the leeside deposit. The number of each curve corresponds to the experiment number, u = 40 cm/sec, d = 15 cm, and a = 0 . 3 5 $ . D = 0.50$, experiment 54; D = l.S^, experiment 4 2 ; D = 2.5$, experiment 30. 3 23 54 « 1.7 2.3 2.6 90 Distance From Dune Crest (cm) 324 Fig. 15. Effect of mean size of the bed material on the simulated mean size gradient of the leeside deposit. The number of each curve corresponds to the experiment number, u = 40 cm/sec, d = 15 cm, and a = 1.54* . D = 0.50 4>, experiment 56; D = 1.5 4’, experiment 44; D = 2.54*, experiment 32. 325 1.3 56 O 1J6 2.2 2 .5 Distance From Dune Crest (cm) 3 26 Fig. 16. Effect of mean size of the bed material on the simulated mean size gradient of the leeside deposit. The number of each curve corresponds to the experiment number, u = 65 cm/sec, d = 15 cm, and o = 0.71 D = 0.50(f), experiment 58; D = 1.5$, experiment 4 6 ; D = 2.5 0.5 0.8 w e 46 c o O c o ® ^ 2.0 2.3 2J5 30 6 0 Distance From Dune Crest (cm) 328 Fig. 17. Effect of mean size of the bed material on the simulated mean size gradient of the leeside deposit. The number of each curve corresponds to the experiment number, u = 9 0 c m / s e c , d = 1 5 c m , a n d cr = 0.71 D = 0 . 5 0 experiment 61? D = 1.5 4 9 ? 5 = 2.5 A p p e n d i x 5 Deposition rate gradients obtained numerically through the simulation program. Fig. 1. Effect of flow velocity. Figs. 2-3. Effect of flow depth. 331 Fig. 1. Effect of flow velocity on simulated deposition rate gradient of the leeside sediment. The number of each curve corresponds to the experiment number. A. D = 1.5$, d = 15 cm, and o = 1.5^. Experiment 44 (u = 40 cm/sec), experiment 47 (u = 65 cm/sec ), and experiment 50 (u = 90 cm/sec). B. D = 2.5$, d = 15 c m , a n d o = 0 . 3 5 $ . Experiment 30 (u = 40 cm/sec), experiment 33 (u = 65 cm/sec), and experiment 36 (u = 90 cm/sec). C. D = 0.5$, d = 15 cm, and o = 0.71$. Experiment 55 (u = 40 cm/sec), experiment 58 (u = 65 cm/sec), and experiment 61 (u = 90 cm/sec). 332 ©10 47 10 - 44 Distance From Dune Crest (cm) 3 33 E o \ E o> 0) o oc c o o a 4 ) a Distance From Dune Crest (cm) Deposition Rate (g m /c m 2/sec) 10 itne rm ueCet (cm) Crest Dune From Distance 334 335 Fig. 2. Effect of flow depth on simulated deposition rate gradient of the leeside deposit. The number of each curve corresponds to the experiment number, u = 40 cm/sec, D = 2.5 Experiment 30 (d = 15 cm), experiment 39 (d = 30 cm), and experiment 40 (d = 50 cm). u = 65 cm/sec, D = 1.5 Experiment 46 (d = 15 cm), experiment 76 (d = 30 cm), and experiment 77 (d = 50 cm). u v» (M E V s E 0 1 o CK c o o a t> Q 46 77 30 10' 10 Distance From Dune Crest (cm) 337 Fig. 3. Effect of flow depth on simulated deposition rate gradient of the leeside deposit. The number of each curve corresponds to the experiment number, u = 9 0 c m / s e c , D = 2.5 Experiment 37 (d = 15 cm ), experiment 70 (d = 30 cm), and experiment 71 (d = 50 cm). 338 u -2 oc Disfance From Dune Crest (cm) Appendix 6. Flow conditions for experiments 17 to 27. Experiment Discharge (Q) Mean flow Flow depth (d) Flow Reynolds Froude Energy Shear stress Boundary sheer Total Sedim Number (liters/sec) velocity(u) (an) Number (Re) Number(F) slope (S) velocity (u») stress (t q ) transport r (cm/sec) (cm/cm) (cm/sec) (gm/cm. sec2) (gm/cm/sec) 17 78.72 63.33 13.594 96493 0.548 0.004567 7.804 60.785 0.0276 18 93.45 86.64 11.796 111835 0.805 0.010822 11.191 124.984 0.130 13 51.54 40.11 14.051 58965 0.342 0.001017 3.744 13.987 0.00154 20 93.45 86.64 11.796 109084 0.805 0.001894 4.681 21.869 0.00543 21 78.72 67.09 12.832 92384 0.598 0.002754 5.888 34.596 0.00138 22 78.72 64.78 13.289 93388 0.567 0.002829 6.073 36.804 0.0640 23 93.45 86.64 11.796 109084 0.805 0.010822 11.191 124.984 0.163 24 51.54 41.37 13.625 58965 0.358 0.001017 3.686 13.562 0.00976 25 51.54 40.11 14.051 58596 0.342 0.001017 3.744 13.987 0.00375 26 78.72 61.27 14.051 91043 0.522 0.002829 6.244 38.915 0.0188 27 93.45 87.32 11.704 109084 0.815 0.010822 11.147 124.015 0.0472 340 Appendix 7. Deposition rate on the dune lee in flume experiments. Experiment 17 Distance (cm) -2 -2 Deposition rate 0.969776 X 10 0.208057 X 10 (gm/cm2/sec) Distance (cm) - 3 Deposition rate 0.52508 X 10 0.33193 X 10 (gm/cm2/sec) Distance (cm) 11 - 3 - 3 Deposition rate 0.21272 X 10 0.12796 X 10 (gm/cm2/sec) Distance (cm) 1 3 1 5 - 4 - 4 Deposition rate 0.91821 X 10 0.66197 X 10 (gm/cm2/sec) Distance (cm) 1 7 1 9 - 4 - 4 Deposition rate 0.57832 X 10 0.65493 X 10 (gm/cm2/sec) Distance (cm) 21 2 3 - 4 - 4 Deposition rate 0.45396 X 10 0.41259 X 10 (gm/cm2/sec) Distance (cm) 2 5 2 7 - 4 - 4 Deposition rate 0.38180 X 10 0.30498 X 10 (gm/cm2/sec) Distance (cm) 2 9 3 1 - 4 - 4 Deposition rate 0.23746 X 10 0.23443 X 10 (gm/cmz/sec) Distance (cm) 3 3 3 5 - 4 Deposition rate 0.21533 X 10 0.16578 X 10' (gm/cm2/sec) Distance (cm) 3 7 3 9 -4 Deposition rate 0.22271 X 10' 0.19111 X 10 (gm/cm2/sec) Distance (cm) 4 1 4 4 3 4 Deposition rate 0.15185 X 10" 0.13086 X 10 (gm/cm /sec) Distance (cm) 45 4 47 4 Deposition rate 0.10323 X 10 0.12177 X 10 (gm/cm2/sec) Distance (cm) 49 -5 51 -5 Deposition rate 0.63868 X 10 0.93409 X 10 (gm/cm2/sec) Distance (cm) 53 4 55 4 Deposition rate 0.17845 X 10 0.13509 X 10~ (gm/cm2/sec) Distance (cm) 57 4 59 4 Deposition rate 0.15917 X 10" 0.11584 X 10~ (gm/cm2/sec) Distance (cm) 6 1 4 6 3 4 Deposition rate 0.11864 X 10 0.11647 X 10 (gm/cm2/sec) Distance (cm) 65 4 67 Deposition rate 0.10120 X 10 0.85979 X 10 (gm/cm /sec) Distance (cm) 69 -5 71 -5 Deposition rate 0.90669 X 10 0.92436 X 10 (g m / c m 2 / s e c ) Distance (cm) 73 -5 75 -5 Deposition rate 0.80657 X 10 0.86798 X 10 (gm/cm2/sec) Distance (cm) 7 7 5 7 9 Deposition rate 0.50674 X 10" 0.69066 X 10 (gm/cm2/sec) Distance (cm) 81 5 83 Deposition rate 0.55795 X 10" 0.10328 X 10 (gm/cm2/sec) 342 Experiment 18 Distance (cm) - 1 - 1 Deposition rate 0.328659 X 10 0.1103538 X 10 (gm/cm /sec) Distance (cm) -2 Deposition rate 0.47089 X 10 0.31167 X 10 (gm/cm /sec) Distance (cm) 11 -2 -2 Deposition rate 0.144466 X 10 0.136856 X 10 (gm/cm^/sec) Distance (cm) 1 3 1 5 -2 - 3 Deposition rate 0.115369 X 10 0.92457 X 10 (gm/cm2/sec) Distance (cm) 1 7 1 9 - 3 - 3 Deposition rate 0.71276 X 10 0.75546 X 10 (gm/cm^/sec) Distance (cm) 21 2 3 - 3 - 3 Deposition rate 0.64263 X 10 0.56008 X 10 (gm/cm2/sec) Distance (cm) 2 5 2 7 - 3 Deposition rate 0.53016 X 10 0.47522 X 10’ (gm/cm2/sec) Distance (cm) 2 9 3 1 - 3 Deposition rate 0.40976 X 10' 0.42288 X 10 (gm/cm2/sec) Distance (cm) 3 3 3 5 - 3 - 3 Deposition rate 0.39414 X 10 0.37165 X 10 (gm/cmVsec) Distance (cm) 3 7 3 9 - 3 - 3 Deposition rate 0.35012 X 10 0.32640 X 10 (gm/cm2/sec) Distance (cm) 4 1 4 3 Deposition rate 0.31397 X 10* 0.29862 X 10‘ (gm/cm2/sec) 343 Distance (cm) 45 -3 47 -3 Deposition rate 0.22840 X 10 0.24555 X 10 (gm/cm^/sec) Distance (cm) 49 -3 51 -3 Deposition rate 0.23137 X 10 0.21076 X 10 (gm/cm^/sec) Distance (cm) 53 -3 55 -3 Deposition rate 0.19878 X 10 0.18107 X 10 (gm/cm2/sec) Distance (cm) 57 -3 59 -3 Deposition rate 0.15567 X 10 0.16423 X 10 (gm/cm2/sec) Distance (cm) 61 -3 63 -3 Deposition rate 0.14270 X 10 0.13758 X 10 (gm/cm2/sec) Distance (cm) 65 -3 67 -3 Deposition rate 0.12056 X 10 0.11483 X 10 (gm/cm /sec) Distance (cm) 69 -3 71 -3 Deposition rate 0.10797 X 10 0.10677 X 10 (gm/cm2/sec) Distance (cm) 73 -4 75 -3 Deposition rate 0.93464 X 10 0.10599 X 10 (gm/cm2/sec) Distance (cm) 77 4 79 Deposition rate 0.77631 X 10" 0.62223 X 10 (gm/cm /sec) Distance (cm) 8 1 4 8 3 4 Deposition rate 0.57565 X 10" 0.87582 X 10 (gm/cm^/sec) 344 Experiment 19 Distance (cm) 1 - 3 3 - 4 Deposition rate 0.68346 X 10 0.14343 X 10 (gm/cm^/sec) Distance (cm) 5 - 5 7 - 5 Deposition rate 0.56059 X 10 0.39523 X 10 (gm/cm3/sec) Distance (cm) 9 5 1 1 5 Deposition rate 0.27318 X 10 0.23001 X 10 (gm/cm^/sec) Distance (cm) 13 -5 15 -5 Deposition rate 0.20268 X 10 0.17482 X 10 (gm/cm2/sec) Distance (cm) 1 7 5 1 9 Deposition rate 0.14669 X 10“ 0.19395 X 10 (gm/cm^/sec) Distance (cm) 21 -5 23 -5 Deposition rate 0.16523 X 10 0.15598 X 10 (gm/cm2/sec) Distance (cm) 25 -5 27 -5 Deposition rate 0.15523 X 10 0.15043 X 10 (gm/cm2/sec) Distance (cm) 2 9 5 3 1 5 Deposition rate 0.14445 X 10" 0.14971 X 10 (gm/cm3/sec) Distance (cm) 33 -S 35 -S Deposition rate 0.14253 X 10 0.13387 X 10 (gm/cmVsec) Distance (cm) 3 7 _ c 3 9 Deposition rate 0.13947 X 10 0.14342 X 10 (gm/cm3/sec) Distance (cm) 41 -5 43 -5 Deposition rate 0.13720 X 10 0.12683 X 10 (gm/cm3/sec) 345 Distance (cm) 45 47 -5 Deposition rate 0.11069 X 10 0.12358 X 10' (gm/cm2/sec) Distance (cm) 49 51 -5 -5 Deposition rate 0.11639 X 10 0.11169 X 10 (gm/cm2/sec) Distance (cm) 53 55 Deposition rate 0.11297 X 10 0.11537 X 10 5 (gm/cm2/s ec) Distance (cm) 5 7 -5 5 9 -5 Deposition rate 0.10179 X 10 0.10807 X 10 (gm/cm^/sec) Distance (cm) 61 /■ 63 v w /■ Deposition rate 0.96484 X 10 0.93052 X 10 (gm/cm2/sec) Distance (cm) 65 6 67 6 Deposition rate 0.93996 X 10 0.89141 X 10 (gm/cm2/sec) Distance (cm) 69 6 71 -fi Deposition rate 0.75389 X 10 0.83766 X 10 (gm/cm2/sec) Distance (cm) 73 6 75 fi Deposition rate 0.71658 X 10 0.73831 X 10 (gm/cm /sec) Distance (cm) 77 79 Deposition rate 0.76486 X 10 0.75942 X 10 6 (gm/cm2/sec) Distance (cm) 81 6 83 -ft Deposition rate 0.66470 X 10 0.86988 X 10 (gm/cm2/sec) 346 Experiment 20 Distance (cm) Deposition rate 0.64672 X 10“ 3 0.137978 X 10 2 (gm/cm^/sec) Distance (cm) 5 -3 Deposition rate 0.59451 X 10 0.20749 X 10“ 3 (gm/cm2/sec) Distance (cm) 9 -4 Deposition rate 0.4680 X 10 0.1421 X 10“ 4 (gm/cm2/sec) Distance (cm) 13 15 Deposition rate 0.918 X 10 0.700 X 10 (gm/cm^/sec) Distance (cm) 17 ^ Deposition rate 0.723 X 10 (gm/cm2/sec) Experiment 21 Distance (cm) 1 3 Deposition rate 0.1777 X 10 0.584 X 10 (gm/cm2/sec) Distance (cm) 5 7 Deposition rate 0.64 X 10 0.18 X 10 (gm/cm2/sec) Distance (cm) 9 11 Deposition rate 0.50 X 10 0.4 X 10 (gm/cm^/sec) Distance (cm) 13 15 Deposition rate 0.30 X 10 0.270 X 10 (gm/cm2/sec) Distance (cm) 17 Deposition rate 0.370 X 10 (gm/cm2/sec) 347 Experiment 22 Distance (cm) 1 -2 3 -2 Deposition rate 0.843058 X 10 0.163640 X 10 (gm/cm^/sec) Distance (cm) 5 -3 7 -3 Deposition rate 0.78915 X 10 0.61353 X 10 (gm/cm^/sec) Distance (cm) 9 -3 11 -3 Deposition rate 0.47843 X 10 0.35187 X 10 (gm/cm^/sec) Distance (cm) ^ 3 -3 -3 Deposition rate 0.33935 X 10 0.30340 X 10 (gm/cm^/sec) Distance (cm) 17 -3 1 9 -3 Deposition rate 0.26955 X 10 0.28874 X 10 (gm/cm^/sec) Distance (cm) 21 3 23 3 Deposition rate 0.26102 X 10 0.24588 X 10 (gm/cm2/sec) Distance (cm) 25 -3 27 -3 Deposition rate 0.24797 X 10 0.23848 X 10 (gm/cm^/sec) Distance (cm) 29 -3 31 -3 Deposition rate 0.22916 X 10 0.22171 X 10 (gm/cm^/sec) Distance (cm) 33 -3 35 -3 Deposition rate 0.21143 X 10 0.21120 X 10 (gm/cm^/sec Distance (cm) 37 -3 3 9 -3 Deposition rate 0.19870 X 10 0.19734 X 10 (gm/cm2/sec) Distance (cm) 4 1 3 4 3 3 Deposition rate 0.18833 X 10~ 0.18215 X 10 (gm/cm^/sec) 348 Distance (cm) 4 5 -3 4 7 -3 Deposition rate ,17149 X 10 0.17121 X 10 (gm/cm2/sec) Distance (cm) 4 9 -3 5 1 -3 Deposition rate 0.16202 X 10 0.15369 X 10 (gm/cm2 /sec) Distance (cm) 5 3 -3 55 - 3 Deposition rate 0.14333 X 10 0.13999 X 10 (gm/cm2/sec) Distance (cm) 5 7 -3 5 9 -3 Deposition rate 0.12857 X 10 0.13078 X 10 (gm/cm2/sec) Distance (cm) 6 1 -3 63 -3 Deposition rate 0.11414 X 10 0.11484 X 10 (gm/cm2/sec) Distance (cm) 6 5 -3 67 -3 Deposition rate 0.10271 X 10 0.10171 X 10 (gm/cm2/sec Distance (cm) 6 9 4 7 1 4 Deposition rate 0.93980 X 10" 0.93663 X 10~ (gm/cm2/sec) Distance (cm) 73 4 75 Deposition rate 0.89177 X 10" 0.91372 X 10 (gm/cm2/sec) Distance (cm) 77 4 79 Deposition rate 0.75610 X 10" 0.79411 X 10 (gm/cm2/sec) Distance (cm) 81 4 83 4 Deposition rate 0.66798 X 10" 0.78105 X lo” (gm/cm2/sec 349 Experiment 23 Distance (cm) 1 -1 3 -2 Deposition rate 0.1387783 X 10 0.431267 X 10 (gm/cm2/sec) Distance (cm) -2 -2 Deposition rate 0.215352 X 10 0.177109 X 10 (gm/cm^/sec) Distance (cm) 11 -2 -2 Deposition rate 0.154114 X 10 0.103991 X 10 (gm/cm2/sec) Distance (cm) 13 -2 15 -3 Deposition rate 0.103678 X 10 0.88298 X 10 (gm/cm2/sec) Distance (cm) 17 -3 1 9 -3 Deposition rate 0.90863 X 10 0.91637 X 10 (gm/cm^/sec) Distance (cm) 2 1 -3 23 -3 Deposition rate 0186223 X 10 0.80741 X 10 (gm/cm2/sec) Distance (cm) 25 -3 27 -3 Deposition rate 0.80055 X 10 0.77937 X 10 (gm/cm2/sec) Distance (cm) 2 9 -3 3 1 -3 Deposition rate 0.76776 X 10 0.72653 X 10 (gm/cm2/sec) Distance (cm) 33 -3 35 -3 Deposition rate 0.71224 X 10 0.70154 X 10 (gm/cm2/sec) Distance (cm) 37 -2 39 -3 Deposition rate 0.63502 X 10 0.66472 X 10 (gm/cm2/sec) Distance (cm) 4 1 -3 4 3 -3 Deposition rate 0.61188 X 10 0.57144 X 10 (gm/cm2/sec) 350 Distance (cm) 45 47 -3 -3 Deposition rate 0.54831 X 10 0.51629 X 10 (gm/cm2/sec) Distance (cm) 49 51 -3 -3 Deposition rate 0.47277 X 10 0.46186 X 10 (gm/cm2/sec) Distance (cm) 53 55 -3 -3 Deposition rate 0.45351 X 10 0.40577 X 10 (gm/cm2/sec) Distance (cm) 57 59 -3 Deposition rate 0.35390 X 10 0.37417 X 10‘ (gm/cm^/sec) Distance (cm) 61 63 -3 -3 Deposition rate 0.32602 X 10 0.33039 X 10 (gm/cm2/sec) Distance (cm) 65 67 -3 -3 Deposition rate 0.28772 X 10 0.28385 X 10 (gm/cm2/sec) Distance (cm) 69 71 -3 -3 Deposition rate 0.27130 X 10 0.25398 X 10 (gm/cm2/sec Distance (cm) 73 75 -3 Deposition rate 0.23695 X 10" 0.24888 X 10 (gm/cm2/sec) Distance (cm) 77 79 -3 Deposition rate 0.20623 X 10 0.18385 X 10 (gm/cm2/sec) Distance (cm) 81 83 -3 -3 Deposition rate 0.15172 X 10 0.21634 X 10 (gm/cm2/sec) 351 Experiment 24 Distance (cm) -2 -4 Deposition rate 0.120227 X 10 0.4435 X 10 (gm/cm2/sec) Distance (cm) -4 -4 Deposition rate 0.3160 X 10 0.3153 X 10 (gm/cm2 /sec) 11 Distance (cm) -4 -4 Deposition rate 0.2803 X 10 0.2512 X 10 (gm/cm2/sec) 15 Distance (cm) 13 -4 -4 Deposition rate 0.2409 X 10 0.234 X 10 (gm/cm2/sec) 17 19 Distance (cm) -4 Deposition rate 0.2293 X 10 0.2573 X 10' (gm/cm2/sec) 21 23 Distance (cm) -4 -4 Deposition rate 0.2205 X 10 0.2069 X 10 (gm/cm2/sec) 25 27 Distance (cm) -4 -4 Deposition rate 0.2230 X 10 0.2077 X 10 (gm/cm2/sec) Distance (cm) 29 31 -4 -4 Deposition rate 0.2159 X 10 0.1849 X 10 (gm/cm /sec) Distance (cm) 33 -4 35 Deposition rate 0.2122 X 10 0.2122 X 10' (gm/cm2 /sec) 37 39 Distance (cm) -4 -4 Deposition rate 0.2072 X 10 0.1991 X 10 (gm/cm2/sec) 41 43 Distance (cm) -4 -4 Deposition rate 0.1903 X 10 0.2021 X 10 (gm/cm2/sec) Distance (cm) 45 -4 47 Deposition rate 0.1882 X 10 0.1959 X 10' (gm/cm2/sec) Distance (cm) 49 51 -4 -4 Deposition rate 0.1835 X 10 0.1744 X 10 (gm/cm^/sec) Distance (cm) 53 55 -4 Deposition rate 0.1766 X 10' 0.1792 X 10 (gm/cm2 /sec) Distance (cm) 57 -4 59 -4 Deposition rate 0.1667 X 10 0.1766 X 10 (gm/cm^/sec) Distance (cm) 61 -4 63 -4 Deposition rate 0.1627 X 10 0.1672 X 10 (gm/cm^/sec) Distance (cm) 65 -4 67 -4 Deposition rate 0.1591 X 10 0.1598 X 10 (gm/cm^/sec) Distance (cm) 69 71 -4 -4 Deposition rate 0.1441 X 10 0.1551 X 10 (gm/cm2/sec) Distance (cm) 73 75 Deposition rate 0.1465 X 10' 0.1545 X 10' (gm/cm2 /sec) Distance (cm) 77 79 -4 -4 Deposition rate 0.1421 X 10 0.1442 X 10 (gm/cm^/sec) Distance (cm) 81 -4 83 Deposition rate 0.1369 X 10 0.1575 X 10' (gm/cm^/sec) 353 Experiment 25 Distance (cm) -2 -4 Deposition rate 0.164328 X 10 0.3807 X 10 (gm/cm^/sec) Distance (cm) — 4 Deposition rate 0.1554 X 10 0.1168 X 10 (g m / c m Vsec) Distance (cm) -5 11 -5 Deposition rate 0.898 X 10 0.693 X 10 (gm/cm^/sec) Distance (cm) 13 -5 15 -5 Deposition rate 0.677 X 10 0.618 X 10 (gm/cm2/sec) Distance (cm) 17 19 -5 -5 Deposition rate 0.543 X 10 0.296 X 10 (gm/cm^/sec) Distance (cm) 21 23 -5 Deposition rate 0.307 X 10 0.275 X 10" (gm/cm2/sec) Distance (cm) 25 -5 27 -5 Deposition rate 0.331 X 10 0.301 X 10 (gm/cm2/sec) Distance (cm) 29 31 -5 -5 Deposition rate 0.3 X 10 0.308 X 10 (gm/cm2/sec) Distance (cm) 33 35 Deposition rate 0.275 X 10' 0.262 X 10‘ (gm/cm2/sec) Distance (cm) 37 39 -5 -5 Deposition rate 0.307 X 10 0.247 X 10 (gm/cm2 /sec) Distance (cm) 41 43 Deposition rate 0.247 X 10‘ 0.267 X 10' (gm/cm2/sec) 354 Distance (cm) 45 47 Deposition rate 0.212 X 10 0.252 X 10~5 (gm/cm2/sec) Distance (cm) 49 51 Deposition rate 0.232 X 10 0.213 X 10~ (gm/cm2/sec Distance (cm) 53 55 Deposition rate 0.227 X 10 0.223 X 10- 5 (gm/cm2/sec Distance (cm) 57 59 Deposition rate 0.176 X 10~ 5 0.198 X 10~ 5 (gm/cm2/sec) Distance (cm) 61 63 Deposition rate 0.288 X 10 0.274 X 10~ (gm/cm2/sec) Distance (cm) 65 67 Deposition rate 0.330 X 10 0.294 X 10 (gm/cm^/sec) Distance (cm) 69 71 Deposition rate 0.262 X 10~ 0.278 X 10~ 5 (gm/cm^/sec) Distance (cm) 73 75 Deposition rate 0.276 X lo” 0.281 X 10_5 (gm/cm2/sec) Distance (cm) 77 79 Deposition rate 0.250 X 10~ 0.230 X 10 (gm/cm2/sec) Distance (cm) 81 83 Deposition rate 0.217 X 10-5 0.268 X 10~5 (gm/cm2/sec) 355 Experiment 26 Distance (cm) 1 -2 3 Deposition rate 0.702958 X 10 0.58949 X 10 3 (gm/cm2/sec) Distance (cm) 5 -3 7 Deposition rate 0.38983 X 10 0.23666 X io-3 (gm/cm2/sec) Distance (cm) 9 -3 11 Deposition rate 0.15557 X 10 0.955 X (gm/cm2/sec) Distance (cm) 13 4 15 -4 Deposition rate 0.8300 X 10" 0.66200 X 10 (gm/cm /sec) Distance (cm) 17 4 19 -4 Deposition rate 0.58533 X 10 0.57913 X 10 (gm/cm2/sec) Distance (cm) 21 23 -4 Deposition rate 0.49410 X 10“ 4 0.43088 X 10 (gm/cm2/sec) Distance (cm) 25 4 27 -4 Deposition rate 0.44989 X 10~ 0.42407 X 10 (gm/cm2/sec) Distance (cm) 29 31 -4 Deposition rate 0.40436 X 10" 4 0.38725 X 10 (gm/cm2/sec) Distance (cm) 3 3 4 3 5 -4 Deposition rate 0.36325 X 10 0.34835 X 10 (gm/cm2/sec) Distance (cm) 3 7 4 3 9 -4 Deposition rate 0.32083 X 10” 0.30451 X 10 (gm/cm2/sec) Distance (cm) 4 1 4 4 3 -4 Deposition rate 0.30198 X 10" 0.27080 X 10 (gm/cm2/sec) 356 Distance (cm) 45 47 Deposition rate 0.24962 X 10~ 4 0.24540 X 10~ 4 (gm/cm2/sec) Distance (cm) 4 9 -4 5 1 -4 Deposition rate 0.21239 X 10 0.20132 X 10 (gm/cm^/sec) Distance (cm) 5 3 - 4 5 5 -4 Deposition rate 0.19474 X 10 0.18203 X 10 (gm/cm^/sec) Distance (cm) 57 _ 4 59 _ 4 Deposition rate 0.14648 X 10 0.16259 X 10 (gm/cm^/sec) Distance (cm) 61 - 4 6 3 -4 Deposition rate 0.13421 X 10 0.12773 X 10 (gm/cm^/sec) Distance (cm) 65 -4 67 _ 4 Deposition rate 0.12330 X 10 0.11000 X 10 (gm/cm^/sec) Distance (cm) 6 9 -4 7 1 -4 Deposition rate 0.10992 X 10 0.11114 X 10 (gm/cm^/sec) Distance (cm) 7 3 -4 75 -4 Deposition rate 0.98269 X 10 0.11260 X 10 (gm/cm^/sec) Distance (cm) 77 79 Deposition rate 0.90822 X 10' 0.78870 X 10" (gm/cm^/sec) Distance (cm) 81 83 -5 Deposition rate 0.70103 X 10 0.12330 X 10* (gm/cm^/sec) 357 Experiment 27 Distance (cm) ^ -1 2 -2 Deposition rate 0.1231524 X 10 0.368189 X 10 (gm/cm2/sec) Distance (cm) 9 -2 7 _3 Deposition rate 0.161429 X 10 0.98793 X 10 (gm/cm2/sec) Distance (cm) 9 -3 11 -3 Deposition rate 0.66961 X 10 0.40739 X 10 (gm/cm2/ sec) Distance (cm) 13 3 15 3 Deposition rate 0.35084 X 10 0.27628 X 10 (gm/cm2 /sec) Distance (cm) 17 3 19 3 Deposition rate 0.24184 X 10" 0.23770 X 10~ (gm/cm2/sec) Distance (cm) 21 3 23 Deposition rate 0.20634 X lo” 0.18465 X 10 (gm/cm /sec) Distance (cm) 25 -3 27 -3 Deposition rate 0.17428 X 10 0.16253 X 10 (gm/cm2/sec) Distance (cm) 29 31 _ 3 Deposition rate 0.15801 X 10" 0.14658 X 10 (gm/cm2/sec) Distance (cm) 33 35 _ Deposition rate 0.13773 X 10" 0.13318 X 10" (gm/cm2/sec) Distance (cm) 37 -3 39 -3 Deposition rate 0.11131 X 10 0.11252 X 10 (gm/cm2/sec) Distance (cm) 4 1 -4 4 3 -4 Deposition rate 0.97909 X 10 0.90248 X 10 (gm/cm2/sec) 358 Distance (cm) 45 47 Deposition rate 0.81903 X 10" 4 0.77546 X 10~ 4 (gm/cm2/sec) Distance (cm) 4 9 -4 5 1 -4 Deposition rate 0.71074 X 10 0.59616 X 10 (gm/cm2/sec) Distance (cm) 5 3 -4 55 -4 Deposition rate 0.48759 X 10 0.46509 X 10 (gm/cm^/sec) Distance (cm) 5 7 -4 5 9 -4 Deposition rate 0.38022 X 10 0.41606 X 10 (gm/cm^/sec) Distance (cm) 6 1 -4 6 3 . 4 Deposition rate 0.36271 X 10 0.36370 X 10 (gm/cm^/sec) Distance (cm) 65 -4 67 -4 Deposition rate 0.30185 X 10 0.28186 X 10 (gm/cm2/sec) Distance (cm) 69 -4 71 -4 Deposition rate 0.25054 X 10 0.27008 X 10 (gm/cm^/sec) Distance (cm) 73 -4 75 -4 Deposition rate 0.24366 X 10 0.28648 X 10 (gm/cm^/sec) Distance (cm) 77 -4 7 9 -4 Deposition rate 0.19122 X 10 0.14530 X 10 (gm/cm^/sec) Distance (cm) 81 -5 83 -4 Deposition rate 0.97724 X 10 0.28630 X 10 (gm/cm^/sec) 359 Regression equations of deposition rate gradient for the leeside deposit from the experimental data. Distance from Equation for deposition Probability R Dune Crest rate gradient greater than cm gm/'cm^/sec F -1.869 17 1 - 17 Y (W) ss 0.0117 X 0.0001 0.99 17 17 - 83 Y (W) 2 = 0.0026 X 0.0001 0.87 18 1 - 17 Y (W) 3 2 0.0397 X 0.0001 0.98 -1.556 18 17 - 83 Y (W) = 0.0828 X 0.0001 0.95 v -2.107 19 1 - 15 Y(W) 2 = 0.000314 0.0001 0.92 19 15 - 83 Y(W) = 2 0.000009 0.0001 0.85 - 1.122 22 1 - 21 Y(W) =2 0.0062 X 0.0001 0.97 -0.985 22 21 - 83 Y (W) = 0.0065 X 0.0001 0.93 -0.903 23 1 - 23 Y (W) = 0.0114 X 0.0001 0.97 -1.260 23 23 - 83 Y(W) = 0.0568 X 0.0001 0.92 24 Non-significant correlation 25 1-17 Non-significant correlation 25 17 - 83 Y(W) = 0.0000176 X “0.528 0.0001 0.73 26 1-21 Y(W) = 0.0051 X _ 1 -585 0.0001 0.98 26 21 - 83 Y(W) = 0.0045 X “i-398 0.0001 0.94 27 1-23 Y(W) = 0.0142 X “ I-*09 0.0001 0.99 27 23 - 83 Y(W) = 0.174 X “2.062 0.0001 0.92 Appendix 9. Mean size gradient in flume experiments. Data are in phi unit. Experiment 17 Distance (cm) 1 3 5 7 9 11 13 15 17 MZ 1.785 1.851 1.891 1.924 1.955 1.964 1.989 2.026 2.023 °G 0.362 0.343 0.327 0.324 0.318 0.325 0.321 0.321 0.333 SK -0.085 -0.026 -0 . 0 1 1 -0.030 -0 . 0 0 2 0 . 0 2 0 0.034 0.079 0.107 1.134 1 . 0 2 0 1.051 k g 0.061 1.052 1.053 1.019 1.064 1.080 Distance (cm) 19 21 23 25 27 29 31 33 35 MZ 2.058 2.071 2.096 2.109 2.113 2.132 2 . 1 2 1 2.135 2.129 °G 0.325 0.325 0.324 0.329 0.331 0.334 0.337 0.336 0.332 SK 0.109 0.095 0.072 0.065 0.085 0.059 0.068 0.058 0.071 k g 1.040 1.057 1.043 1.037 1.036 1.037 1.040 1.033 1.054 Distance (cm) 37 39 41 43 45 47 49 51 53 M z 2.123 2.133 2.146 2.144 2.153 2.149 2.131 2.142 2.115 a G 0.334 0.336 0.330 0.332 0.327 0.325 0.341 0.326 0.325 SK 0.063 0.071 0.060 0.073 0.055 0.055 0.013 0.018 0.031 k g 1.036 1.069 1.019 1.046 1.030 1.014 1.069 1.023 1.023 Distance (cm) 55 57 59 61 63 65 67 69 71 Mz 2 . 1 1 1 2 . 1 1 0 2.125 2.108 2.114 2 . 1 1 2 2.104 2.116 2.093 *G 0.317 0.308 0.323 0.324 0.325 0.319 0.433 0.327 0.414 SK 0.042 0.036 0.029 0.047 0.015 0 . 0 1 0 -0.199 0.018 -0.144 k g 1.006 0.998 1.059 0.986 1.054 1.048 1.650 1.041 1.534 Distance (cm) 73 75 77 79 81 83 Mz 2.109 2 . 1 1 1 2.106 2.105 2.133 2.159 °G 0.378 0.339 0.356 0.321 0.333 0.325 SK -0.086 -0.004 -0.061 0.023 0 . 0 1 0 0.044 360 k g 1.316 1.076 1 . 2 0 0 1.024 1.058 1 . 0 1 0 Experiment 18 Distance (cm) 1 3 5 7 9 11 13 15 17 Mz 1.616 1.639 1.681 1.713 1.763 1.789 1.828 1.846 1.864 °G 0.378 0.375 0.362 0.364 0.351 0.357 0.345 0.339 0.343 SK -0 . 1 0 2 -0 . 1 1 0 -0.093 -0.079 -0.063 -0.048 0.005 0.019 0.007 k g 1.035 1 . 0 2 2 1.047 1.083 1.183 1.113 1.107 1.041 1.029 Distance (cm) 19 21 23 25 27 29 31 33 35 Mz 1.872 1 . 8 8 8 1.901 1.911 1.920 1.927 1»920 1.940 1.953 0.342 0.343 0.338 0.339 0.331 0.333 0.334 0.335 0.337 SK 0.006 -0 . 0 0 1 -0 . 0 0 2 0 . 0 0 2 0.016 -0.006 -O.OOx 0.009 0 . 0 1 2 k g 1.036 1.034 1.018 1.032 1 . 0 2 0 1.045 1.048 1.047 1.046 Distance (cm) 37 39 41 43 45 47 49 51 53 M 1.937 1.936 1.931 1.936 1.935 1.929 1.929 1.931 1.922 aZ G 0.324 0.335 0.326 0.338 0.332 0.327 0.326 0.320 0.323 SK 0.029 0.018 0.017 0.006 0.003 0 . 0 0 2 0.009 0.004 -0.003 k g 1.048 1.033 1.004 1.038 1 . 0 1 2 0.023 1.031 1.044 1.039 Distance (cm) 55 57 59 61 63 65 67 69 71 MZ 1.929 1.913 1.919 1.928 1.923 1.921 1.927 1.937 1.936 °G 0.320 0.316 0.321 0.309 0.306 0.311 0.317 0.304 0.306 SK -0.006 0.008 -0 . 0 1 1 0.028 0.034 0.031 0.016 0.066 0 . 0 2 0 k g 1.045 1 . 0 2 2 1.075 1.031 1.042 1 . 0 1 2 1.023 1.038 1.025 Distance (cm) 73 75 77 79 81 83 MZ 1.937 1.962 1.941 1.935 1.964 1.948 a G 0.302 0.310 0.302 0.311 0.308 0.301 SK 0.050 0.027 0.057 0.043 0.041 0.047 k g 0.994 1 . 0 1 2 0.992 1.029 0.995 0.993 361 Experiment 19 Distance (cm) 1 3 5 7 9 11 13 15 17 MZ 1.994 2.055 2.089 2 116 2.151 2.161 2.186 2.186 2.177 a G 0.298 0.302 0.329 0 301 0.310 0.310 0.309 0.320 0.325 SK 0.019 0.038 0.022 0 025 0.024 0.065 0.094 0 . 1 1 2 0.115 kg 0.959 1.034 1.013 0 950 0.993 0.973 1.028 1.059 1.058 Distance (cm) 19 21 23 25 27 29 31 33 35 Mz 2.206 2.231 2 .206 2 232 2 . 2 2 1 2.230 2.248 2.236 2.238 °G 0.345 0.331 0.340 0 366 0.346 0.328 0.355 0.326 0.323 SK 0.161 0.130 0.142 0 197 0.153 0.113 0.178 0 . 1 0 0 0 . 1 0 0 kg 1.161 1.119 1.137 1 278 1.134 1.081 1 . 2 2 0 1.091 1.081 Distance (cm) 37 39 41 43 45 47 49 51 53 2.231 2.243 2.214 2 265 2.249 2.240 2.252 2.245 2.253 MZCT G 0.350 0.360 0.358 0 366 0.362 0.352 0.370 0.354 0.366 SK 0.165 0.169 0.164 0 179 0.187 0.159 0 . 2 0 0 0.188 0.155 k g 1.192 1.251 1.193 1 454 1.260 1.192 1.273 1.236 1.217 Distance (cm) 55 57 59 61 63 65 67 69 71 MZ 2.198 2.254 2.264 2 256 2.254 2.268 2.252 2.278 2.285 °G 0.358 0.331 0.382 0 345 0.354 0.363 0.364 0.370 0.377 SK 0.083 0.121 0.189 0 156 0.142 0.172 0.139 0.175 0.054 k g 1.155 1.146 1.472 1 177 1.192 1.389 1.233 1.402 1 . 2 2 2 Distance (cm) 73 75 77 79 81 83 Mz 2.289 2.292 2.273 2 253 2.270 2.281 °G 0.358 0.374 0.380 0 357 0.366 0.401 SK 0.145 0.190 0.183 0 177 0.189 0.188 362 Kr, 1.312 1.341 1.380 1 221 1.269 1.405 Experiment 20 Distance (cm) 1 3 5 7 9 11 13 15 17 --- M z -2.138 -2.222 -2.245 -2.268 ■-2.277 -2.291 -2.272 -1.188 °G 0.309 0.241 0.216 0.204 0.193 0.182 0.206 --- 0.043 --- SK 0.420 0.446 0.419 0.406 0.426 0.408 0.390 2.878 k g 0.930 1.023 0.996 1.093 1.223 1.344 1.297 -0.397 Experiment 21 Distance (cm) 1 3 5 7 9 11 13 15 17 MZ -2.137 -2.185 -2.185 -2.184 ■-1.970 -2.071 -1.973 -1.737 -2.244 a G 0.298 0.277 0.274 0.310 0.389 0.467 0.427 0.408 0.300 0.383 0.470 0.451 0.587 0.449 0.709 0.289 0.324 0.607 k g 0.919 1.006 1.110 1.158 1.205 0.910 1.039 0.959 1.940 Experiment 22 Distance (cm) 1 3 5 7 9 11 13 15 17 MZ 2.329 2.398 2.445 2.467 2.509 2.537 2.570 2.584 2.590 °G 0.374 0.349 0.355 0.354 0.359 0.368 0.361 0.361 0.356 SK -0.144 -0.135 -0.139 -0.169 -0.119 -0.089 -0.047 -0.052 -0.053 Kg 0.987 1.016 1.097 1.281 1.231 1.182 1.141 1.132 1.150 Distance (cm) 19 21 23 25 27 29 31 33 35 MZ 2.600 2.600 2.605 2.615 2.614 2.620 2.623 2.629 2.622 CG 0.358 0.355 0.356 0.350 0.351 0.357 0.346 0.345 0.351 SK -0.041 -0.050 -0.061 -0.034 -0.046 -0.047 -0.032 -0.030 -0.043 k g 1.120 1.135 1.150 1.101 1.125 1 . 1 1 1 1 . 1 0 0 1.106 1 . 1 1 2 363 Distance (cm) 37 39 41 43 45 47 49 51 53 MZ 2.632 2.629 2.635 2.632 2.625 2.642 2.644 2.638 2.640 QG 0.346 0.353 0.349 0.346 0.353 0.353 0.353 0.346 0.352 SK -0.032 -0.045 -0.032 -0.031 -0.039 -0.042 -0.036 -0.036 -0.035 k g 1.097 1.090 1.092 1.104 1.078 1.078 1.075 1.103 1.064 Distance (cm) 55 57 59 61 63 65 67 69 71 MZ 2.638 2.636 2.634 2.639 2.641 2.631 2.641 2.651 2.647 °G 0.346 0.342 0.348 0.350 0.350 0.352 0.348 0.352 0.347 SK -0.038 -0.039 -0.033 -0.036 -0.029 -0.038 -0.036 -0.031 -0.029 k g 1.103 1.125 1.086 1.092 1.071 1.085 1.090 1.088 1.099 Distance (cm) 73 75 77 79 81 83 Mz 2.633 2.649 2.624 2.641 2.644 2.623 aG 0.351 0.341 0.352 0.355 0.351 0.358 SK -0.057 -0 . 0 2 2 -0.047 -0.034 -.036 -0.038 k g 1.158 1.141 1.097 1.084 1.107 1.077 Experiment !3 Distance (cm) 1 3 5 7 9 1 1 13 15 17 MZ 2.106 2.175 2.203 2.246 2.283 2.311 2.346 2.370 2.385 °G 0.398 0.380 0.392 0.385 0.380 0.383 0.375 0.369 0.360 SK -0.094 -0.072 -0.055 -0.050 -0.044 -0.052 -0.077 -0.083 -0.077 0.962 k g 0.944 0.893 1.028 1.009 1.004 0.982 0.989 0.992 Distance (cm) 19 21 23 25 27 29 31 33 35 MZ 2.395 2.410 2.410 2.423 2.426 2.434 2.439 2.445 2.448 °G 0.357 0.347 0.351 0.343 0.345 0.342 0.349 0.340 0.345 -0 . 1 0 2 0 . 1 1 2 SK -0.096 -0.085 -0.104 - -0.117 -0.080 -0 . 1 2 2 -0.116 k g 1 . 0 1 1 1.037 1.013 1.053 1.052 1.067 1.074 1 . 1 0 0 1.077 364 Distance (cm) 37 39 41 43 45 47 49 51 53 MZ 2.444 2.458 2.455 2.445 2.453 2.467 2 459 2.455 2.470 °G 0.350 0.348 0.336 0.346 0.341 0.342 0 349 0.344 0.338 SK -0.096 -0.088 -0.132 -0.109 -0 . 1 2 1 -0.114 -0 103 -0.115 -0.111 1.089 k g 1 . 1 0 1 1.096 1.091 1.096 1.105 1 096 1.094 1.093 Distance (cm) 55 57 59 61 63 65 67 69 71 M 2.463 2.455 UZ 2.469 2.465 2.463 2.461 2 424 2.429 2.457 ° G 0.347 0.343 0.340 0.345 0.347 0.334 0 340 0.345 0.350 SK -0 . 1 1 1 -0.117 -0 . 1 2 2 -0.107 -0 . 1 0 1 -0 . 1 0 1 - 0 122 -0.093 -0.135 k g 1.085 1 . 1 1 2 1.354 1.092 1.105 1.086 1 049 1.034 1.055 Distance (cm) 73 75 77 79 81 83 MZ 2.464 2.469 2.468 2.465 2.478 2.461 a G 0.347 0.352 0.332 0.346 0.329 0.348 SK -0.123 -0.084 -0.118 -0 . 1 1 0 -0.152 -0.096 1.093 1.093 1.072 1.093 1.373 k g 1.074 Experiment >4 Distance (cm) 1 3 5 7 9 11 13 15 17 MZ 2.532 2.550 2.532 2.578 2.585 2.575 2 593 2.594 2.597 °G 0.390 0.419 0.424 0.414 0.411 0.420 0 411 0.397 0.412 SK -0.095 -0.149 -0.140 -0.104 -0.114 -0 . 1 1 2 -0 117 -0.101 -0.092 1.148 1.181 1.248 1.177 1 k g 1.183 1.173 207 1.208 1.145 Distance (cm) 19 21 23 25 27 29 31 33 35 M z 2.605 2.622 2.593 2.624 2.631 2.608 2 607 2.627 2.622 0.410 0.399 0 405 0.397 0.387 -0.055 -0 . 1 0 1 -0 082 -0.073 -0.059 1.147 1.188 1 149 1.158 1.171 w Cn Distance (cm) 37 39 41 43 45 47 49 51 53 2.638 2.607 2.637 2.640 2.620 2.627 2.637 2.627 2.636 aZ0 G 0.391 0.372 0.387 0.376 0.378 0.391 0.379 0.373 0.380 SK -0.060 -0.069 -0.041 -0.055 -0.032 -0.064 -0.048 -0.041 -0.052 k g 1.159 1 . 2 2 1 1.119 1.168 1.125 1.146 1.154 1.175 1.126 Distance (cm) 55 57 59 61 63 65 67 69 71 MZ 2.637 2.648 2.641 2.636 2.663 2.641 2.600 2.656 2.621 °G 0.366 0.362 0.371 0.367 0.373 0.366 0.383 0.362 0.400 -0.016 -0.027 -0.035 -0.019 -0.005 -0 . 0 2 2 -0.067 -0 . 0 1 2 -0.085 SKXX k g 1.137 1.119 1.153 1 . 1 2 1 1.098 1 . 1 0 1 1.153 1.142 1.172 Distance (cm) 73 75 77 79 81 83 2.634 2.658 2.644 2.638 2.674 2.630 0 G 0.400 0.360 0.368 0.405 0.395 0.378 SK -0.013 -0.008 -0.024 -0.035 -0.003 -0.052 k g 1.231 1.191 1.123 1.146 1.133 1.183 Experiment !5 Distance (cm) 1 3 5 7 9 11 13 15 17 1.930 2.136 2.230 2.279 2.334 2.342 2.407 2.385 2.428 MZc G 0.502 0.471 0.440 0.446 0.437 0.450 0.452 0.451 0.448 -0.003 -0.039 -0.017 -0.063 -0.077 -0.073 -0.042 -0.052 -0.033 SKX\ k g 0.994 0.985 0.941 0.974 1.030 1.049 1.103 1.054 1.079 Distance (cm) 19 21 23 25 27 29 31 33 35 MZ 2.401 2.401 2.439 2.438 2.445 2.457 2.432 2.461 2.481 °G 0.464 0.450 0.435 0.438 0.444 0.444 0.430 0,436 0.449 SK 0.029 -0.037 -0.047 0 . 0 1 2 -0.038 0.003 -0.006 -0.042 -0.007 k g 1.046 1.073 1 . 1 0 1 1.046 1.114 1.077 1.045 1.128 1.130 366 Distance (cm) 37 39 41 43 45 47 49 51 53 M z 2 .453 2 .491 2.473 2.472 2.473 2.477 2 .485 2.476 2.477 "g 0 .428 0.439 0.441 0.426 0.426 0.436 0.444 0.449 0.431 SK -0.053 -0.027 -0.009 -0.032 0.060 -0.038 -0.026 -0.022 -0.030 k g 1.096 1.161 1.155 1.135 1.308 1.239 1.218 1.287 1.297 Distance (cm) 55 57 59 61 63 65 67 69 71 MZ 2.466 2.486 2.498 2.481 2.492 2.504 2.509 2.529 2.496 CG 0.440 0.437 0.432 0.430 0.434 0.437 0.449 0.456 0.432 SK -0.018 -0.017 -0.025 -0.002 0.009 -0.046 0.019 0.030 -0.022 k g 1.095 1.113 1.178 1.109 1.192 1.210 1.138 1.215 1.187 Distance (cm) 73 75 77 79 81 83 MZ 2.510 2.522 2 .505 2.503 2.508 2.532 °G 0.449 0.446 0.435 0.434 0.465 0.438 SK 0.002 0.002 -0.016 -0.010 0.042 0.002 k g 1.215 1.188 1.285 1.234 1.134 1.278 Experiment 26 Distance (cm) 1 3 5 7 9 11 13 15 17 MZ 1.576 1.735 1.800 1.869 1.971 1.991 2.026 2.064 2.098 CG 0.428 0.439 0.425 0.407 0.416 0.417 0.415 0.415 0.415 SK -0.027 0.004 -0.011 0.063 0.055 0.080 0.040 0.025 -0.001 k g 1.050 1.100 1.045 1.027 0.981 0.969 0.970 0.927 0.913 Distance (cm) 19 21 23 25 27 29 31 33 35 M z 2.128 2.137 2.055 2.168 2.175 2.199 2.011 2.205 2.213 a G 0.413 0.399 0.357 0.399 0.395 0.406 0.362 0.400 0.401 SK 0.025 0.034 -0.072 -0.005 0.000 -0.016 0.095 -0.031 -0.020 k g 0.918 0.921 0.937 0.891 0.926 0.927 1.274 0.917 0.932 367 Distance (cm) 37 39 41 43 45 47 49 51 53 Mz 2.186 2.228 2.216 2.217 2.215 2.229 2.225 2.213 2.207 °G 0.397 0.394 0.404 0.398 0.396 0.399 0.393 0.396 0.391 SK -0 . 0 0 2 -0.053 -0 . 0 1 0 -0.035 0.018 -0 . 0 2 2 -0.040 -0.006 -0 . 0 2 1 k g 0.904 0.918 0.939 0.942 0.961 0.966 0.921 0.951 0.937 Distance (cm) 55 57 59 61 63 65 67 69 71 MZ 2 .207 2.215 2.239 2.194 2.218 2 . 2 2 2 2.243 2.245 2.243 °G 0.394 0.398 0.400 0.387 0.396 0.403 0.397 0.401 0.404 SK -0.025 -0 . 0 1 0 -0.038 -0.006 0 . 0 0 1 0.005 -0.034 -0.031 -0.017 k g 0.929 0.934 0.962 0.916 0.947 0.915 0.960 0.965 0.952 Distance (cm) 73 75 77 79 81 83 MZ 2.244 2.243 2.236 2.234 2.258 2.276 °G 0.398 0.403 0.400 0.411 0.401 0.400 SK -0.027 -0.005 -0 . 0 2 1 -0.008 0.031 -0.036 k g 0.959 0.969 0.962 0.951 0.960 0.983 Experiment 27 Distance (cm) 1 3 5 7 9 11 13 15 17 1.466 1.489 1.573 1.641 1.715 1.740 1.737 1.794 1.850 > G 0.436 0.433 0.419 0.413 0.416 0.425 0.423 0.460 0.427 SK -0.068 -0.076 -0.053 0.009 0.044 0.044 0.039 0.162 0.044 1 . 1 2 2 1.177 1 . 1 1 2 1 . 1 0 0 1.098 1 . 1 1 1 1.091 k g 1.373 1.036 Distance (cm) 19 21 23 25 27 29 31 33 35 Mz 1.879 1.910 1.921 1.941 1.957 1.954 1.964 1.976 1.979 °G 0.422 0.421 0.410 0.416 0.420 0.414 0.417 0.422 0.416 SK 0.064 0.059 0.087 0.063 0.056 0.058 0.066 0.054 0.051 1 . 0 2 0 1.023 0.984 1.018 1 . 0 0 1 0.990 1 . 0 0 2 0.995 1.005 k g 368 Distance (cm) 37 39 41 43 45 47 49 51 53 MZ 1.968 1.954 1.938 1 ,.945 1.960 1.967 1.959 1.945 1.975 °G 0.410 0.413 0.397 0 ,.408 0.407 0.402 0.405 0.400 0.399 SK 0.061 0.057 0.041 0 ,.085 0.061 0.067 0.071 0.077 0.081 k g 1 . 0 1 0 0.976 0.985 0 ,.978 0.994 0.987 1 . 0 1 0 1.027 1.017 Distance (cm) 55 57 59 61 63 65 67 69 71 MZ 1.954 1.949 1.966 1 ,.941 1.973 1.977 1.972 1.998 1.895 aG 0.395 0.406 0.408 0 ,.392 0.403 0.404 0.402 0.410 0.377 SK 0 . 1 0 2 0.094 0.084 0 ,.100 0.088 0.091 0.114 0.098 0.074 k g 1.007 1.005 0.995 1 ,.012 0.990 1.003 1 . 0 0 2 0.978 1.005 Distance (cm) 73 75 77 79 81 83 Mz 1.972 1.995 1.982 1 ..978 1.987 2.014 a G 0.400 0.404 0.416 0 ,.4 1 7 0.428 0.412 SK 0.098 0.089 0.089 0 ,.094 0.086 0.094 Kr, 0.997 0.998 0.994 0 ,.985 0.976 0.969 369 370 Appendix 10. Regression equations of mean size gradient in phi units for the leeside deposit from experimental data. Experiment Distance from Equation for mean size Probability Number dune crest gradient greater than cm 4> F 17 1 _ 17 Y SB 1.77 X °-0457 0.0001 0.97 17 17 - 83 Y m 2.019 X 0.0122 0.01 0.19 18 1 _ 17 Y as 1.573 X 0-0343 0.0002 0.89 18 17 - 83 Y =3 1.803 X 0.0174 0.0001 0.52 19 1 _ 15 Y as 1.984 X 0-0356 0.0001 0.98 19 15 - 83 Y as 2.068 X 0.0216 0.0001 0.69 _ as 22 1 21 Y 2.308 X °*03^A A A A A 0.0001 0.97 22 21 - 83 Y as 2.543 X 0-0089 0.0001 0.58 23 1 _ 23 Y = 2.073 X 0.0473 0.0001 0.96 23 23 - 83 Y st 2.335 X 0.0125 0.0001 0.48 24 1 - 83 Y “ 2.5175 X 0.0114 0.0001 0.81 25 1 17 Y = 0.0001 0.99 1,946 x n’cmn 25 17 - 83 Y = 2.222 X 0-0283 0.0001 0.85 26 1 _ 21 Y = 1.553 X 0-104 0.0001 0.99 26 21 - 83 Y as 1.858 X °-044 0.0001 0.51 27 1 23 Y = 1.391 X 0-0969 0.0001 0.92 27 23 - 83 Y = 1.8098 X 0.021 0.01 0.18 371 Appendix 11. Results of numerical simulation for flume experiments. A. Mean size of sediment deposited on dune lee. X = distance in cm from dune crest. D = mean grain diameter in phi unit. Experiment 17 X 1 3 5 7 9 11 13 15 17 19 D 1.926 1.853 1.858 1.861 1.863 1.865 1.867 1 . 8 6 8 1.870 1.871 X 21 23 25 27 29 31 33 35 37 39 5 1.872 1.873 1.874 1.875 1.876 1.877 1.878 1.878 1.879 1.880 X 41 43 45 47 49 51 53 55 57 59 D 1.880 1.881 1.881 1.882 1.883 1.883 1.884 1.884 1.884 1.885 X 61 63 65 67 69 71 73 75 77 79 D 1.885 1 . 8 8 6 1 . 8 8 6 1.887 1.887 1 . 8 8 8 1 . 8 8 8 1 . 8 8 8 1 . 8 8 8 1.889 X 81 83 D 1.889 1.890 Experiment 18 X 1 3 5 7 9 11 13 15 17 19 D 1.791 1.710 1.713 1.714 1.716 1.717 1.718 1.719 1.719 1.720 X 21 23 25 27 29 31 33 35 37 39 D 1.721 1.721 1.722 1.722 1.723 1.723 1.724 1.724 1.725 1.725 X 41 43 45 47 49 51 53 55 57 59 D 1.725 1.726 1.726 1.726 1.727 1.727 1.727 1.727 1.728 1.728 X 61 63 65 67 69 71 73 75 77 79 D 1.728 1.729 1.729 1.729 1.729 1.729 1.730 1.730 1.730 1.730 X 81 83 373 D 1.731 1.731 Experiment 19 X 1 3 5 7 9 11 13 15 17 19 D 2.113 2.068 2.075 2.080 2.084 2.087 2.090 2.092 2.095 2.096 X 21 23 25 27 29 31 33 35 37 39 D 2.098 2 . 1 0 0 2 .101 2.103 2.104 2.105 2.106 2.107 2.108 2.109 X 41 43 45 47 49 51 53 55 57 59 D 2 . 1 1 0 2 . 1 1 1 2 . 1 1 2 2.113 2.114 2.114 2.115 2.116 2.117 2.117 X 61 63 65 67 69 71 73 75 77 79 D 2.118 2.118 2.119 2 . 1 2 0 2 . 1 2 0 2 . 1 2 1 2.121 2.122 2.122 2.123 r~, X 81 00 D 2.123 2.124 Experiment; 22 X 1 3 5 7 9 1 1 13 15 17 19 D 2.626 2.514 2.515 2.516 2.516 2.517 2.517 2.517 2.518 2.518 X 21 23 25 27 29 31 33 35 37 39 D 2.518 2.518 2.519 2.519 2.519 2.519 2.519 2.519 2.520 2.520 X 41 43 45 47 49 51 53 55 57 59 D 2.520 2.520 2.520 2.520 2.520 2.520 2.521 2.521 2.521 2.521 X 61 63 65 67 69 71 73 75 77 79 D 2.521 2.521 2.521 2.521 2.521 2.521 2.522 2.522 2.522 2.522 X 81 83 374 D 2.522 2.522 375 79 59 79 19 19 39 59 39 2.280 2.285 2.621 2.625 2.283 2.286 2.609 2.617 77 17 57 37 37 77 17 57 2.624 2.280 2.285 2.286 2.608 2.621 2.283 2.616 75 55 75 15 35 15 35 55 2.621 2.624 2.285 2.286 2.607 2.279 2.282 2.615 73 73 13 33 53 53 13 33 2.624 2.284 2.286 2.620 2.279 2.282 2.605 2.615 51 71 31 1 1 31 51 71 11 2.623 2.604 2.284 2.620 2.278 2.282 2.286 2.615 29 69 9 69 49 29 9 49 .602 2.284 2.286 2.619 2.623 2 2.277 2.282 2.614 Experiment 24 Experiment 23 67 27 7 47 67 7 27 47 2.623 2.284 2.600 2.619 2.285 2.276 2.281 2.613 2.274 2.274 2.275 61 61 63 65 21 21 23 25 1 3 5 81 83 21 21 23 25 41 41 43 45 61 61 63 65 1 3 5 81 81 83 41 41 43 45 2.701 2.701 2.592 2.597 2.285 2.285 2.285 2.285 2.283 2.283 2.283 2.284 2.280 2.280 2.281 2.281 2.622 2.622 2.622 2.622 2.625 2.625 2.617 2.617 2.618 2.618 2.286 2.286 2.287 2.469 2.469 2.610 2.611 2.612 XlQ XlQ XlQ XlQ XlP X IQ X IQ XlQ X IQ X lQ 376 39 79 19 59 39 59 79 19 1.747 1.739 1.728 2.099 1.710 2.113 2.043 2.079 77 37 57 17 77 57 17 37 1.747 1.738 1.727 1.707 2 . 1 1 2 2.038 2.097 2.076 75 35 15 55 75 15 35 55 1.746 1.737 1.725 1.704 2.095 2 . 1 1 0 2.032 2.073 73 33 13 53 73 53 13 33 1.745 1.736 1.724 2.109 1.701 2.093 2.025 2.070 51 1 1 11 31 51 1.744 1.722 2.108 2.067 2.092 2.017 29 31 9 49 69 71 29 9 49 1.734 1.735 1.720 1.692 1.697 2.064 Experiment 26 Experiment 25 67 69 71 27 7 67 47 7 27 47 1.719 1.733 1.743 1.744 1.687 2.105 2.106 2.060 2.088 2.090 1.995 2.007 65 25 5 45 65 25 5 45 1.742 1.732 1.680 1.717 2.085 2.103 1.978 2.057 23 63 3 43 83 23 3 63 83 43 1.749 1.741 1.715 1.731 1.670 2 . 1 0 2 2.115 2.083 2.053 1.953 21 81 61 1 41 21 1 61 81 41 1.748 1.740 1.712 1.730 1.755 2.114 2.081 2 . 1 0 0 2.048 2.035 XlQ XlQ XlQ XlQ XlQ XlQ XlQ XlQ XlQ X lQ 377 59 79 19 39 1.595 1.580 1.600 1.590 57 77 17 37 1.595 1.589 1.579 1.599 75 55 15 1.594 1.577 1.588 1.599 53 73 33 35 13 1.587 1.594 1.576 1.599 11 51 1.586 1.593 1.574 49 1.593 1.572 Experiment 27 27 29 31 67 69 71 7 9 47 1.592 1.569 1.585 1.586 1.597 1.598 1.598 65 25 45 1.592 1.597 1.584 63 23 83 43 1.596 1.601 1.583 1.591 21 61 81 41 1.596 1.600 1.590 1.581 1.660 1.561 1.566 XlQ X | Q XlQ XlQ X lQ Sediment deposition rate on dune lee X = distance in cm from dune crest. W = deposition rate on dune lee in gm/cm^/sec. . 4 2 8 1 .0 W X Six Six SIX Six Six .000132 .000167 .000342 .000225 .000684 43 15 71 29 57 001 .028 .000280 .000298 .000318 002 .015 .000121 .000125 .000150 .000128 .000155 .000161 2 5 1 0 .0 8 1 2 0 .0 9 7 3 0 .0 001 .025 .000196 .000205 .000215 .000480 .000534 .000600 5 7 49 47 45 7 9 21 7 19 5 17 3 1 3 35 33 31 3 5 77 63 75 61 73 59 i 17 t n e rim e p x E 006 .020 .000237 .000250 .000264 001 .015 .000112 .000115 .000136 .000118 .000141 .000173 .000145 .000180 .000188 003 .040 .000368 .000400 .000436 7 1 1 0 .0 9 5 7 69 67 27 65 25 23 1 3 55 41 53 39 51 37 9 1 83 81 79 000948 4 9 0 0 .0 11 000795 9 7 0 0 .0 13 379 Si X SIX SIX Six Six SIX .000429 .000538 .000720 .00213 .295 ,00108 43 71 57 15 1 29 .000417 .000519 ,000687 00101 00187 0114 45 73 59 17 31 3 .000502 .000657 .000947 .00167 .00663 .000405 47 61 5 33 19 75 i 18 t n e rim e p x E ,000629 ,000891 ,00151 ,00468 ,000394 ,000485 49 63 21 7 77 35 .000384 .000604 .000841 .000470 .00137 .00361 65 79 51 37 23 9 00293 000374 000455 000580 000797 00126 81 67 53 25 11 39 .000365 .000442 .000558 .000757 .00247 .00116 83 41 55 69 27 13 380 X SIX SIX SIX six six w ,0102 ,0000927 0 1 5 016 0 ,00 ,0000211 ,0000448 ,0000289 43 15 71 57 1 29 ,000549 ,0000416 0000808 0000160 0000203 0000275 45 73 3 59 31 17 ,0000715 ,000309 ,0000155 ,0000196 ,0000262 ,0000388 47 61 5 19 75 33 i 19 t n e rim e p x E ,0000640 0000151 5 1 0 0 0 ,0 ,0000250 ,0000364 ,0000189 000214 63 49 77 21 7 35 0059 0058 .0000485 .0000528 .0000579 000162 . 000108 0 1 0 0 .0 0 3 1 0 0 .0 2 6 1 0 0 .0 0000147 4 1 .0000171 0 0 0 .0 .0000176 .0000220 .0000183 .0000229 .0000305 .0000239 .0000323 .0000342 5 7 69 55 67 79 53 27 65 13 51 25 11 23 9 7 9 41 39 37 0000143 4 1 0 0 0 .0 81 0000139 3 1 0 0 0 .0 83 1 X . 2 8 6 .0 W SI X Six SIX SIX S IX .0000943 .0000569 .0000710 .000140 .000273 43 57 15 71 29 00143 . 000591 9 5 0 0 .0 5 3 8 0 0 .0 3 4 1 0 .0 0053 0058 .0000524 .0000538 .0000553 .0000642 .0000663 .0000827 .0000686. .0000862 .000116 .0000901 .000123 .000194 .000131 .000215 .000241 5 7 49 47 7 45 5 3 7 9 21 19 17 3 5 77 63 75 61 35 73 59 33 31 i 22 t n e rim e p x E 000458 5 4 0 0 .0 .0000622 .0000794 .0000510 .000110 .000177 9 51 23 79 65 37 0000764 000104 000163 000374 0000498 0000603 11 81 53 25 67 39 .0000736 .0000486 .0000585 .000151 .000316 .0000990 83 41 55 27 13 69 382 Si X si X SIX SIX SIX S IX .000199 .000765 .204 000159 5 1 0 0 .0 .000264 .000393 43 57 15 71 29 1 .000192 .000252 .000674 .000155 .000368 ,00400 45 73 59 17 31 3 ,000345 ,000602 ,00234 ,000241 000151 5 1 0 0 ,0 ,000186 47 5 61 33 19 75 i 23 t n e rim e p x E ,000232 ,000180 ,000325 ,00166 0 147 0 ,00 000544 49 63 35 7 77 21 .000222 000143 4 1 0 0 .0 .000174 .000307 .000497 8 2 1 0 .0 65 79 51 37 9 23 000169 000214 000139 000292 000457 00105 81 67 53 39 25 11 000136 3 1 0 0 .0 .000164 .000206 .000277 .000423 4 8 8 0 0 .0 83 41 55 69 27 13 SI X six SIX SIX six 1 .0123 w X .0000381 .0000755 .0000151 .0000189 .0000254 43 15 57 29 71 ,0000242 ,0000356 ,0000663 ,000409 ,0000182 0000146 45 3 31 17 73 59 .0000231 ,0000334 ,0000591 ,000236 ,0000142 ,0000176 5 47 33 19 61 75 i 24 t n e rim e p x E .0000221 .0000314 .0000533 ,000166 0000138 3 1 0 0 0 ,0 ,0000171 49 7 63 35 21 77 .000128 .0000135 .0000212 .0000296 .0000485 .0000165 9 51 37 23 65 79 000104 0000204 0000281 0000445 0000131 0000160 11 53 39 25 81 67 0000875 7 8 0 0 0 .0 0000128 2 1 0 0 0 .0 .0000266 .0000411 .0000155 .0000196 41 83 55 69 27 13 SIX SIX SIX SIX SIX S IX .0000443 0000158 5 1 0 0 0 .0 .0000204 .0000282 .0000942 .0109 43 1 57 15 71 29 ,0000267 ,0000411 ,000602 ,0000196 0000153 0000817 45 3 31 17 73 59 0000148 4 1 0 0 0 .0 .0000254 .000331 ,0000382 ,0000719 ,0000188 47 33 19 61 75 i 25 t n e rim e p x E .0000243 0000144 4 1 0 0 0 .0 .0000357 .0000182 .0000641 .000225 49 63 77 35 21 .0000232 .0000335 .0000578 0000140 4 1 0 0 0 .0 .0000175 000169 6 1 0 0 .0 9 51 37 23 79 65 0000222 0000315 0000136 0000169 0000525 000134 81 11 53 67 39 25 .0000212 0000132 3 1 0 0 0 .0 .0000163 .0000298 .0000481 000111 1 1 0 0 .0 83 41 55 69 27 13 1 X s: x six six six s 4 5 7 i .0 x W .000710 000129 2 1 0 0 .0 .000164 .000224 .000346 43 71 57 15 29 002 .011 .000118 .000121 .000125 .000194 .000203 .000281 .000213 .000300 .000492 .000321 .000549 .000619 044 024 .00162 .00234 .00414 005 . 005 .000147 .000153 ..000158 5 7 49 47 45 3 5 77 63 75 61 35 73 59 33 31 7 9 21 7 19 5 17 3 i 26 t n e rim e p x E 008 .018 .000171 .000178 .000236 .000186 .000250 9 2 8 0 .000374 0 .0 .000265 .000407 4 9 9 0 0 .0 .000445 4 2 1 0 .0 001 .011 .000108 .000111 .000133 .000115 .000137 .000142 1 3 55 53 51 9 1 83 81 79 5 7 69 67 41 27 65 13 39 25 37 11 23 9 Six SIX SIX six six s i x . 336 .00278 .000545 .000687 .000924 .00140 15 1 43 57 71 29 ,0152 ,00244 ,000529 ,000662 ,000881 00130 3 45 17 31 73 59 .00217 ,00878 ,00122 ,000514 ,000639 ,000841 5 47 19 61 33 75 i 27 t n e rim e p x E .00196 ,00616 ,00115 ,000500 ,000618 ,000805 7 49 63 21 35 77 00474 7 4 0 .0 .00108 .00178 .000772 .000487 .000598 9 23 65 51 37 79 00384 00163 000741 00102 000474 000579 11 25 81 67 53 39 .00150 3 2 3 0 .0 .000561 .000713 .000971 000462 6 4 0 0 .0 41 83 55 27 13 69 3 8 8 C. Deposition rate as fraction of the total sediment deposited 84 cm beyond the dune crest. X = distance in cm from dune crest. = deposition rate on dune lee as fraction of the total sediment. Experiment 17 X 1 3 5 7 9 11 13 15 17 19 .81 .037 .021 .015 .012 .0093 ,0078 .0 0 6 7 ,0059 ,0053 wf X 21 23 25 27 29 31 33 35 37 39 wf .0047 .0043 .0039 .0036 .0034 .0031 ,0029 .0028 ,0026 ,0025 X 41 43 45 47 49 51 53 55 57 59 Wf .0023 .0022 .0021 .0020 .0019 .0019 0018 .0017 ,0016 ,0016 X 61 63 65 67 69 71 73 75 77 79 Wf .0015 .0015 .0014 .0014 .0013 .0013 ,0013 .0012 ,0012 ,0012 X 81 83 Wf . 0 0 1 1 . 0 0 1 1 Experiment 18 X 1 3 5 7 9 11 13 15 17 19 Wf .83 .032 .019 .013 .010 .0083 .0070 ,0060 ,0053 .0047 X 21 23 25 27 29 31 33 35 37 39 Wf .0043 .0039 .0036 .0033 .0031 .0028 0027 ,0025 0024 .0022 X 41 43 45 47 49 51 53 55 57 59 Wf . 0 0 2 1 . 0 0 2 0 .0019 .0019 .0018 .0017 ,0016 ,0016 ,0015 .0015 X 61 63 65 67 69 71 73 75 77 79 Wf .0014 .0014 .0013 .0013 . 0 0 1 2 . 0 0 1 2 0012 0011 ,0011 .0011 X 81 83 389 Wf . 0 0 1 1 . 0 0 1 0 Experiment 19 X 1 3 5 7 9 11 13 15 17 19 .80 .043 .024 .017 .013 0 1 0 wf 0084 .0072 .0063 .0056 X 21 23 25 27 29 31 33 35 37 39 .0050 .0045 .0041 .0038 .0035 0033 wf 0030 .0028 .0027 .0025 X 41 43 45 47 49 51 53 55 57 59 Wf .0024 .0023 . 0 0 2 1 . 0 0 2 0 . 0 0 2 0 0019 0018 .0017 .0017 .0016 X 61 63 65 67 69 71 73 75 77 79 Wf .0015 .0015 .0014 .0014 .0013 0013 0013 . 0 0 1 2 .0 01 2 .0 01 1 X 81 83 wf . 0 0 1 1 . 0 0 1 1 Experiment 22 X 1 3 5 7 9 11 13 15 17 19 Wf .90 .019 .011 .0078 .0060 0049 0042 .0036 .0032 .0028 X 21 23 25 27 29 31 33 35 37 39 Wf .0026 .0023 . 0 022 . 0 0 2 0 .0019 0017 0016 .0015 .0014 .0014 X 41 43 45 47 49 51 53 55 57 59 Wf .0013 . 0 012 . 0 0 12 . 0 0 1 1 . 0 0 1 1 0 0 1 0 0 0 1 0 .00097 .00094 .00090 X 61 63 65 67 69 71 73 75 77 79 .00087 .00085 .00082 .00080 .00077 00075 00073 .00071 .00069 .00067 wf X 81 83 390 Wf .00066 .00064 Experiment 2 3 X 1 3 5 7 9 1 1 13 15 17 19 .91 .018 . 0 1 0 .0073 .0057 .0046 0039 .0034 .0030 .0027 wf X 2 1 23 25 27 29 31 33 35 37 39 Wf .0024 . 0 0 2 2 . 0 0 2 0 .0019 .0017 .0016 0015 .0014 .0014 .0013 X 41 43 45 47 49 51 53 55 57 59 wf . 0 0 1 2 . 0 0 1 2 . 0 0 1 1 . 0 0 1 1 . 0 0 1 0 .00098 00095 .00091 .00088 .00085 X 61 63 65 67 69 71 73 75 77 79 Wf .00082 .00080 .00077 .00075 .00073 .00070 00069 .00067 .00065 .00063 X 81 83 Wf .00062 .00060 Experiment 24 X 1 3 5 7 9 1 1 13 15 17 19 Wf .85 .028 .016 . 0 1 2 .0089 .0072 0061 .0052 .0046 .0041 X 2 1 23 25 27 29 31 33 35 37 39 Wf .0037 .0034 .0031 .0028 .0026 .0025 0023 . 0 0 2 2 . 0 0 2 1 .0019 X 41 43 45 47 49 51 53 55 57 59 .0018 .0018 .0017 .0016 .0015 .0015 0014 .0014 .0013 .0013 wf X 61 63 65 67 69 71 73 75 77 79 Wf . 0 0 1 2 . 0 0 1 2 . 0 0 1 1 . 0 0 1 1 . 0 0 1 1 . 0 0 1 0 0 0 1 0 .00099 .00096 .00093 X 81 83 391 Wf .00091 .00087 Experiment 25 X 1 3 5 7 9 11 13 15 17 19 Wf .80 .044 .024 .016 .012 .0098 0081 .0069 .0060 .0053 X 21 23 25 27 29 31 33 35 37 39 Wf .0047 .0042 .0039 .0035 .0032 .0030 0028 .0026 .0025 .0023 X 41 43 45 47 49 51 53 55 57 59 Wf .0022 .0021 .0020 .0019 .0018 .0017 0016 .0016 .0015 .0014 X 61 63 65 67 69 71 73 75 77 79 Wf .0014 .0013 .0013 .0012 .0012 .0012 0011 .0011 .0011 .0010 X 81 83 Wf .0010 .00097 Experiment 26 X 1 3 5 7 9 11 13 15 17 19 Wf .79 .043 .025 .017 .013 .010 0087 .0074 .0065 .0058 X 21 23 25 27 29 31 33 35 37 39 Wf .0052 .0047 .0043 .0039 .0036 .0034 0031 .0029 .0028 .0026 X 41 43 45 47 49 51 53 55 57 59 Wf .0025 .0024 .0022 .0021 .0020 .0019 0019 .0018 .0017 .0017 X 61 63 65 67 69 71 73 75 77 79 Wf .0016 .0015 .0015 .0014 .0014 .0014 0013 .0013 .0012 .0012 X 81 83 392 Wf .0012 .0011 Experiment 27 1 3 5 7 9 11 13 15 17 19 81 .037 0 2 1 .015 Oil .0093 0078 .0067 0059 .0053 21 23 25 27 29 31 33 35 37 39 0047 .0043 0039 .0036 0034 .0031 0029 .0028 0026 .0025 41 43 45 47 49 51 53 55 57 59 0023 .0 022 0 0 2 1 . 0 0 2 0 0019 .0019 0018 .0017 0017 .0016 61 63 65 67 69 71 73 75 77 79 0015 .0015 0014 .0014 0014 .0013 0013 . 0 0 1 2 0 0 1 2 . 0 0 1 2 81 83 0 0 1 1 . 0 0 1 1 Appendix 12. Bedform characteristics in different parts of dune in experiments 2 to 1 6 . >erinent STOSS SIDE REGRESSIVE RIPPLE ZONE LEESIDE BEYOND REGRESSIVE RIPPLE ZONE Type of bedform progressively Height (cm) Length from the toe Type of bedform Height (mm) Type of bedstream progressively Height (mm) downstream of the slip face (cm) downstream 2 Transverse sinuous to linguoid 1.5 to 3.7 20 Linguoid ripples 5 Straight-crested, sinuous to 5 ripples isolated linguoid ripples 6 Transverse sinuous to linguoid 1.4 to 3.0 10 Linguoid ripples 4.5 Practically no bottomset ripples deposits 3 Straight-crested to transverse 1.2 to 2.5 10 Linguoid ripples 4 Practically no bottomset sinuous ripples deposits 7 Linguoid to transverse 1.3 to 2.0 10 Transverse sinuous 2.5 Practically no bottomset sinuous ripples to linguoid ripples deposits 4 Linguoid to transverse 1.0 to 1.6 4 Transverse sinuous 2 No bottomset deposits sinuous ripples to linguoid ripples 5 Transverse sinuous to linguoid 0.8 to 1.5 No regressive ripple No bottomset deposits ripples 10 Linguoid ripples 2.5 to 3.8 30 Linguoid to sinuous 10 Straight-crested, linguoid to 7.5, 3 to 2 ripples isolated linguoid ripples respectively 9 Linguoid ripples 2.0 to 3.2 25 Linguoid to sinuous 5 Straight-crested, isolated 6.5, 2.5 to 1 ripples small linguoid ripples respectively 11 Linguoid ripples 1.8 to 2.8 20 Linguoid to sinuous 3.5 Straight-crested to low ampli 3.2 to 0.8 ripples tude linguoid ripples 14 Isolated but closely spaced 4 15 Sinuous to linguoid 10 Linguoid to transverse 7.5 to 2 sinuous ripples ripples ripples 15 Isolated linguoid ripples 3.4 10 Sinuous to linguoid 7.5 Linguoid to transverse 5 to 2.5 ripples sinuous ripples Linguoid to transverse 16 Linguoid ripples 2.5 5 Sinuous to linguoid 5 4 to 1.5 ripples sinuous ripples 2 and 13 Plane bed No regressive ripple No bottomset deposit 394 Appendix 13. Size characteristics of avalanched sediment in experiments with sand 2. Data are in phi units. SLIPPAGE DEPOSIT TOESET DEPOSIT R2/10(8) R2/1KB) S ta p le No. R .V6( B) i’2/7(8) R2/8(5) R2/9(B) R2/1 R2/2 R2/3 R2/4 R2/5 " z 0.861 0.927 0.950 0.975 0.772 0.407 1.476 1 .52 9 1.556 1.865 2.099 Experiment 2 OC 0.55 5 0.55 6 0.53 7 0 .5 0 3 0.5 4 9 0.734 0.399 0.382 0.374 0.394 0.39 7 S|t - 0 . H 3 - C . 164 -0 .1 7 2 -0.143 -0.07 3 0.109 -0.063 -0.038 -0 .0 3 5 0.024 0.08 2 Ko 1 .0 )0 0.996 1.066 1.036 0.991 0.915 1.073 1.082 1.097 1.011 0.9 2 9 Sample So. R 3 /3 (6 ) R 3 /4 (6 ) R3/5(B) R3/6(B) R3/7(B) R3/8(B) R3/3(A) R3/4(A) R3/5(A) R3/6(A) R3/7(A) RJ/6(A) R3/1 E3/2 *2 1.228 1.232 1.199 1.106 0 .9 0 6 0.42 4 1.27 1 1.2 2 3 1.237 1 150 0.967 0.470 1.54 9 1.66 9 Experiment 3 cc 0.4 2 0 0 . 396 0.37 2 0.3 5 0 0.351 0.531 0.367 0.389 0.362 0 .3 5 7 0.37 5 0.5 8 6 0.427 0.34 1 H -0.090 -0.104 -0.082 -0.028 -0 .0 4 8 -0 .0 2 0 0.02 3 -0.047 •0.042 -0.009 -0 .0 7 4 -0 .0 5 8 -0 .1 4 7 -0 .0 5 2 1.002 1.031 1.015 0.983 0.992 1.004 1.006 1.067 1.052 0.9 9 3 1.10 6 1.06 0 1.202 1.167 Sesple So. 84/2(4) R 6/3(A ) P.4/MA) R 4/5 (A ) ' R 4/6 (A ) R 4/7 (A ) R6/2(B) B4/3(B) R4/4(B) R6/S(B) **/6 Staple Vo. R5/KA) K5/2(A> F5/37A) R5/4(A) R5/S(A) R5/6A+7A RS/1(B) A 5 /2 (B ) R5/3(B) R5/4(B) R5/5(B) R5/6(B) H 1.421 1.387 1.273 1.273 1.181 0.653 1.310 1.252 1.147 1.07 3 1 .0 7 0 0.5 7 8 Experiment 5 K 0.360 0.35 6 0 .35 4 0.27 2 0 .2 9 4 0.407 0.39 5 0 .3 5 3 0 .31 7 0.29 8 0 .3 7 6 0 .4 6 8 H -0.111 -0.110 -0.173 -0.016 -0.037 -0.053 -0 .0 5 9 -0 .0 1 4 0 .0 0 7 0.064 0.058 -0.121 Kc 0.980 0.966 0.997 1.209 1.021 0.98 4 0 .9 7 6 1.017 1.033 1.108 0.94 1 1.036 Sanple No. R 6/3(A ) R6/4(A) R6/5(A) R6/6 Sample No. R7/2CA) R7/3CA) R 7 /4 { A) R 7/5 (A ) R 7/6(A ) P7/7A+6A R7/2(D) R7/3CD) R 7/4(D) R 7/5(D) R7/6(D) R7/7(D) R7/2 SLIPFACE DEPOSIT TOESET DEPOSIT Sample No. 1(D) 2(D) 3(D) 4(D) 5(D) 6(D) 1 2 3 4 5 6 7 Mz 2.204 2.145 2.131 2.066 2.028 1.874 2.391 2.372 2.379 2.493 2.621 2.661 2.662 Experiment 9 °G 0.326 0.330 0.329 0.316 0.307 0.324 0.325 0.314 0.317 0.333 0.368 0.394 0.481 SK -0.065 -0.014 -0.027 0.012 0.025 0.004 -0.080 -0.083 -0.077 -0.169 -0.078 -0.104 •0.142 Kg 0.959 0.943 0.949 0.968 0.997 1.071 1.022 0.989 1.017 1.320 1.170 1.257 1.326 Sample No. 1(D) 2(D) 3(D) 4(D) 5(D) 6(D) 1 2 3 4 5 6 7 8 1.978 1.930 1.886 1.918 1.930 1.860 2.236 2.248 2.298 2.296 2.320 2.425 2.488 2.513 Experiment 10 °G 0.431 0.436 0.449 0.442 0.471 0.605 0.370 0.354 0.336 0.386 0.400 0.391 0.433 0.444 SK -0.058 -0.040 -0.057 -0.020 0.034 -0.132 ■0.118 -0.114 -0.060 -0.104 -0.156 -0.162 -0.127 -0.136 Kg 1.059 1.070 1.055 1.088 0.972 0.880 0.940 0.955 0.926 1.001 0.015 1.066 1.174 1.166 Sample No. 1(E) 2(E) 3(E) 4(E) 5(E) 6(D) 1 2 3 4 5 6 7 8 2.564 MZ 2.355 2.342 2.305 2.252 2.202 1.998 2.461 2.568 2.568 2.621 2.571 2.582 2.605 Experiment 11 °G 0.344 0.325 0.304 0.330 0.347 0.486 0.324 0.325 0.351 0.378 0.481 0.545 0.599 0.643 SK -0.086 -0.075 -0.086 -0.100 -0.093 -0.025 -0.185 0.023 -0.056 -0.088 -0.130 -0.203 -0.169 ■0.125 Kg 0.964 0.951 0.921 0.956 0.976 0.940 1.903 1.190 1.203 1.241 1.203 1.137 1.007 0.953 396 Appendix 15. Size characteristics of avalanched sediment in experiments with sand 4. Data are in phi units. SLIPFACE DEPOSIT TOESET DEPOSIT Sample No. 14/KC) 14/2(0) 14/3(0 14/4(C) 14/5(C) 14/6(067(0 I II III IV V VI VII VIII «Z 1.602 1.558 1.521 1.481 1.497 1.414 1.883 1.911 1.924 1.924 1.987 2.069 2.126 2.158 Experiment 14 CG 0.335 0.342 0.358 0.364 0.366 0.361 0.318 0.311 0.308 0.291 0.286 0.300 0.307 0.308 SK -0.073 -0.070 -0.093 -0.010 -0.113 -0.083 0.011 0.007 0.012 0.066 0.083 0.012 0.028 0.003 k g 1.035 1.070 1.034 1.023 1.035 0.944 1.033 0.996 1.016 0.969 0.994 0.954 0.961 0.994 Sample No. 15/HD) 15/2(D) 15/3(D) 15/4(D) 15/5(D) 15/6(D)S7(D) I II III IV V Mz 1.860 1.825 1.809 1.776 1.673 1.610 2.077 2.105 2.134 2.203 2.215 Experiment 15 °G 0.339 0.333 0.371 0.372 0.301 0.295 0.294 0.294 0.290 0.280 0.283 SK -0.014 0.049 -0.041 -0.096 -0.033 -0.028 -0.009 -0.029 -0.004 -0.009 -0.007 k g 1.024 1.053 1.042 1.111 1.134 1.124 0.923 0.938 0.940 0.971 0.968 Sample No. 16/HD) 16/2(D) 16/3(D) 16/4(D) 16/5(D) 16/6(D) I II III IV V VI MZ 1.688 1.687 1.690 1.642 1.511 1.403 1.908 1.949 1.964 2.083 2.203 2.194 Experiment 16 °G 0.333 0.320 0.327 0.327 0.315 0.352 0.330 0.299 0.297 0.302 0.292 0.283 SK -0.074 -0.058 -0.066 -0.040 -0.041 -0.030 -0.001 0.048 0.043 0.002 -0.018 0.000 Kg 1.130 1.097 1.126 1.102 1.106 -0.954 1.035 0.957 0.954 0.960 0.951 0.958 397 Appendix 16. Size characteristics of avalanched sediment in experiments with Sand 1. Data are in phi units. Sample No. 1(B) 2(B) 3(B) 4(B) 5(B) 6(B) -2.025 -2.059 -2.092 -2.183 -2.258 -2.288 0.371 0.335 0.310 0.248 0.197 0.182 Experiment 12 S„ 0.297 0.251 0.271 0.360 0.376 0.415 0.864 0.885 0.831 0.914 0.997 1.303 kg Sample No. 1(B) 2(B) 3(B) 4(B) 5(B) 6(B) 1(C) 2(C) 3(C) 4(C) 5(C) 6(C) -2.054 -2.153 -2.240 -2.254 -2.284 -2.301 -2.120 -2.107 -2.169 -2.244 -2.259 -2.273 0.182 0.310 0.301 0.256 0.223 0.201 G 0.356 0.284 0.214 0.206 0.177 0.191 Experiment 13 Sjr 0.311 0.382 0.385 0.408 0.402 0.388 0.362 0.273 0.312 0.434 0.377 0.404 0.904 0.960 0.987 1.010 1.193 1.390 0.892 0.882 0.999 1.076 0.981 1.108 398 399 Appendix 17. Size characteristics of sediment collected transverse to the cross-stratification in experiments 4, 5, 6 , 7, 11, and 15. Experiment Downslope distance Position in Mean Number from brinkpoint Cross-stratum Size cm $ 4 4 Upper 1.186 4 4 Middle 1.242 4 4 Lower 1.325 4 20 Upper 1.032 4 20 Middle 1.152 5 10 Upper 1.217 5 10 Lower 1.389 5 20 Upper 0.745 5 20 Lower 1.100 6 12 Upper 1.087 6 12 Lower 1.227 7 12 Upper 1.126 7 12 Lower 1.243 11 6 Upper 2.415 11 6 Lower 2.450 11 6 Upper 2.406 11 6 Lower 2.441 15 4 Upper 1.830 15 4 Middle 1.883 15 4 Lower 1.901 4 0 0 Appendix 17 continued. Experiment Downslope distance Position in Mean Number from brinkpoint Cross-stratum Size c m 15 4 Upper 1.873 15 20 Upper 1.548 15 20 Middle 1.690 15 20 Lower 1.747 VITA Chinmoy Chakrabarti was born on January 19, 1943 in Calcutta, India. He attended elementary and high schools in Calcutta and entered Jadavpur University, Calcutta in 1958. He received B.Sc.(hons.) and M.Sc. degree in Geology in July, 1962 and September, 1964 respectively. Subsequent ly he was employed by the Geological Survey of India. He entered Louisiana State University in 1973 for a Ph.D. degree in Geology. He is married to former Champa Upadhyay of Calcutta. 4 01 EXAMINATION AND THESIS REPORT Candidate: Chinmoy Chakrabarti Major Field: G e o lo g y Title of Thesis: D iffusion of Sediment in the Lee of dune-like bedforms. Approved: Major Professor and 'Chairman Dean of the Graduate SSchool MINING COMMITTEE: Co-Chairma: Date of Examination: November 23/ 1976