EARTH SURFACE PROCESSES, VOL. 5, 17-23 (1980)

TEST OF SCALE MODELLING OF SEDIMENT TRANSPORT IN STEADY UNIDIRECTIONAL FLOW

JOHN B. SOUTHARD

Department of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.

LAWRENCE A. BOGUCHWAL Shell Development Co., Houston, Texas 77025, U S A .

AND

RICHARD D. ROMEA

Department of Geological Sciences, University of Chicago, Chicago, Illinois 60637, U.S.A.

Received 3 November 1977 Revised 20 November 1978

SUMMARY

In two steady uniform flows at different physical scales in a small open channel, with variables characterizing flow, sediment, and fluid adjusted for dynamic similitude by means of four dimensionless modelling parameters (a Reynolds number, a Froude number, a density ratio, and a length ratio), measured frequency distributions of height, spacing, and migration rate of current ripples were almost identical when scaled, thus verifying that exact Reynolds-Froude modelling of loose-sediment transport is valid and workable. Modelling should be valid as well for a wide range of other transport conditions in the same kind of flow, because no additional kinds of forces or effects would be present in transport of loose grains in modes other than as ripples. In scaled-down modelling, a scale ratio of 2.5 is attainable without recourse to exotic fluids by use of water at 85°C to model natural flows at 10°C.

KEY WORDS Dynamic similitude Open channels Sediment transport

INTRODUCTION

Scale modelling is an important part of engineering practice, but true dynamic scale models have been little used in basic studies of sediment transport, seemingly for two reasons: first, as shown in a later section a model at a desirably large scale ratio necessitates use of a fluid with kinematic viscosity much lower than that of water; second, for a large scale ratio the grain size of the model sediment must be so fine that the resulting cohesion forces destroy the validity of the model. But models with more modest scale ratios would circumvent both of these difficulties and could be of great utility in experimental studies of grain movement and bed configuration in turbulent flows over loose sediment beds. Another motivation for testing a scale model of sediment transport is to have an independent check on the correctness of using a three- dimensional diagram involving dimensionless measures of mean flow depth, mean flow velocity, and sediment size to represent flume and field data on equilibrium bed configurations produced by steady uniform open-channel flows (Southard (197 1)); such plots involve dimensionless variables precisely equivalent to the modelling parameters tested here, and are simply a different way of viewing the same principle.

There seems to have been no definitive direct test of scale modelling of sediment transport, Yalin (1965) developed the possibilities for dynamic modelling of sediment transport and described a flow tunnel built to

0360-1269/80/0105-0017$01.00 @ 1980 by John Wiley & Sons, Ltd.

18 J. B. SOUTHARD, L. A. BOGUCHWAL AND R. D. ROMEA

test the applicability of his model parameters, but no results were reported in detail despite a suggestive photograph of similar bed configurations produced by two flows designed to be dynamically similar. A test involving two dynamically similar flume runs at different physical scales was therefore conducted in a laboratory open channel in order to verify that true dynamic modelling of sediment transport is possible.

MODELLING PARAMETERS

Assume that the following seven variables characterize completely the behaviour of steady uniform flow over a bed of loose sediment with given grain shape and sorting in an infinitely wide channel, in the sense that every set of values of these variables corresponds to one and only one state of the flow (Kennedy and Brooks (1963)): mean flow depth d, mean flow velocity U, mean sediment diameter D, fluid density p, fluid viscosity k, sediment density ps , and acceleration of gravity g. Proof of the correctness of this list lies in consistency of model studies or, equivalently, of dimensionless plots of fields or laboratory data.

The seven variables above can be combined into a set of four dimensionless variables that likewise characterize the flow, in the sense that specifying the values of these four variables fixes the dimensionless measure of any aspect of the behaviour of either fluid or sediment. One set consists of a Reynolds number, a Froude number, a density ratio, and a size ratio:

d E - U pud - CL ’ (gd)”” P ’ D

One equivalent set (among many others), in which the sedimentologically important variables d, U, and D are separated (Southard (1971)) is,

Mean velocity U is used here to characterize flow strength rather than boundary shear stress T ~ , because, for some combinations of variables, two distinct flow states correspond to a given combination of d and T,, owing to the resistance properties of the bed configurations (Brooks (1958)). If d and U are used to characterize the flow, then for every set of independent variables there is one and only one flow state. But since T~ is a function of d and U, even though partly double-valued, Sets 1 or 2 above could just as well involve T ~ . Use of a set of dimensionless variables equivalent or similar to Sets 1 or 2 is now common in work on sediment transport; for example, a variable of the form ro/psgD, commonly used as a dimensionless sediment-transport variable, could be one of the modelling parameters if T, is used instead of U (although this was not verified because we could not measure slopes with sufficient accuracy in our short channel).

For two geometrically similar flows to be dynamically similar, each dimensionless variable in Set 1 must be the same in both the original flow and the model flow. Geometric similitude automatically fixes equality of d / D . Any p and ps can be chosen for the model provided that there is equality of ps/p. Since g is effectively invariable, choice of scale ratio dr (ratio of d in original to d in model) can be viewed as fixing model flow velocity by equality of Froude number U/(gd)’ in the two flows:

U, = (d,)’ (3)

Pr = pr(dr)’ (4)

Viscosity of the model fluid is then fixed by equality of Reynolds number p U d / p :

or, in terms of kinematic viscosity v = w / p ,

vr = (d,)t

Rearrangement of equation ( 5 ) shows that the scale ratio varies as the two-thirds power of the kinematic viscosity :

SEDIMENT TRANSPORT 19

There is nothing new about the set of modelling parameters developed above: it is merely an example of a Reynolds-Froude model. Although well known for many decades, Reynolds-Froude models have been little exploited up to now because of the severe constraint they place upon choice of possible model fluids through equation (6). The significance of the present work lies in a careful direct test of the applicability of this modelling law to transport of loose sediment by steady unidirectional currents.

MODELLING EXPERIMENTS

The modelling parameters of Set 1 or Set 2 above were tested by making two flume runs, at different physical scales, with flow variables and system variables in each run adjusted so that the two runs should be dynamically similar. Flow velocity was chosen to be in the range for which sediment transport is in the form of ripples; a convenient but sufficient criterion for correctness of modelling is that frequency distributions of ripple heights, spacings, and migration rates scale properly. It seems important that since migration of bed forms is intimately related to mode and rate of sediment transport, the test involves dynamics rather than merely geometry and kinematics. Total sediment transport rate should scale as well, but since it would have been extremely difficult to measure time-averaged total sediment transport rate accurately in our small flume, measurement of ripple behaviour was chosen in the interest of a more accurate and definitive test.

Water and quartz sand (p, = 2.65 g/cm’) were used for both runs; a scale ratio of 1.66 was attained by use of different water temperatures. Table I gives details of experimental conditions. The advantages of this combination of sediment and fluid are: a negligible difference in density ratio p,/p (about 3 per cent); a moderately large difference in kinematic viscosity, which by equation (6) can be viewed as fixing the scale ratio; and no complicating effects of differences in sediment wettability. Each sand was closely sieved to be of almost uniform size. The mean sizes were chosen to satisfy the scale ratio, and the standard deviations, although small, were also scaled. Grain shape was about the same in the two sands.

The runs were made in a tilting rectangular channel 8-Om long and 15.6cm wide arranged for recirculation of both water and sediment. In the cold-water (large-scale) run, the full channel width was used, and water temperature was controlled by diverting some of the return flow through a chiller; in the hot-water (small-scale) run, a false wall was used to narrow the channel to preserve the same width/depth ratio, and the water was heated by means of an immersion heater in the tailbox. Water temperature was controlled to within f 1 mm. Discharge was measured by means of a calibrated orifice meter, and is accurate to within 1 or 2 per cent. Accurate measurement of flow depth was more difficult. A run was started at approximately the desired depth; after ripples had become fully developed, a time-lapse motion picture was taken at a station well downstream of the inlet, and the mean sidewall depth was determined by taking from the motion-picture frames 250 depth measurements (a number found to be statistically sufficient) spaced widely enough in time to be uncorrelated. Water was then added or removed to establish the desired mean depth. Flow depths determined in this way, and therefore also flow velocities, are probably accurate to

Table I. Experimental conditions for flume runs, adjusted for dynamic similitude at a scale ratio of 1.66

Variable

Cold-water Hot-water run run

(large-scale) (small-scale)

Water temperature T (“C) Fluid viscosity p (poise) Fluid density p (g/cm’) Mean flow depth d (cm) Mean flow velocity U (cm/sec) Sediment size D (mm) Sediment density p , (g/cm’) Time-lapse filming rate (min)

13.5 1.19 X lo-’

0.999 9.3

29.2 0.38 2.65

15.0

49.9 0.55 x lo-’

0.988 5.6

22.7 0.23 2.65

11.6

20 J. B. SOUTHARD, L. A. BOGUCHWAL AND R. D. ROMEA

within 5 per cent. Great care in determining depth and mean velocity was thought to be important, because even casual flume observations show that ripple migration rate is strongly dependent on mean flow velocity.

Migration of ripples in a channel reach whose extent and position were properly scaled was photographed through the channel wall with a 16 mm time-lapse motion-picture camera, with a clock in the field of view. Height, spacing, and migration rate of ripples in this reach were measured by viewing the films frame by frame on a microfilm reader. Ripple height was taken to be the difference in elevation between crest and trough; ripple spacing was taken to be the horizontal distance between the crests of two successive ripples; ripple migration rate was characterized by the time between passages of successive ripples past a fixed station in the reach. It is unimportant which of the many possible definitions of height, spacing, and migration rate are used, so long as the methods of measurement are the same in both runs. Results for migration time are based on measurement of about 100 ripples, and, for height and spacing, about 200 ripples.

RESULTS AND DISCUSSION

Figures 1 to 3 show cumulative frequency distributions of height, spacing, and migration time of ripples in both runs; the means and standard deviations of the distributions are given in Table 11. When scaled, the curves for the hot-water run are nearly coincident with those for the cold-water run. Because perfect coincidence would mean perfect scaling, this indicates that Sets 1 and 2 are valid modelling parameters, and therefore that transport of loose sediment by steady unidirectional flows can be modelled exactly and without distortion by use of a Reynolds-Froude model.

Lack of perfect coincidence between the cold-water curve and the scaled hot-water curve in Figures 1 to 3 could have three causes: neglect of an unscaled variable; error in measuring scaled variables; or measurement of too few ripples to approximate the true distributions. Neglect of flume length and inlet geometry (which were not scaled in the experiments) indeed seemed to result in the presence of a few anomalous bed features, very long and very slow-moving, in both runs; these were readily separable and were not used in constructing the frequency curves. The only important scaled variable subject to major uncertainty is flow velocity. Error in flow velocity should affect migration time strongly but height and spacing only weakly, whereas agreement of the measured distributions is actually better for migration time than for height and spacing. The small discrepancies in Figure 1 seem most probably to be the result of using

100 I I 1 I I I

" 0 I 2 3 4 5 RIPPLE HEIGHT (CM)

Figure 1. Cumulative frequency distribution of ripple height

SEDIMENT TRANSPORT

80-

60

40

20

21

-

-

-

W

+ COLD-WATER R U N *HOT-WATER RUN -

"0 10 2 0 30

RIPPLE SPACING (CM)

Figure 2. Cumulative frequency distribution of ripple spacing

too few ripples to construct the cumulative curves: exclusion of randomly chosen subsets of 10 to 20 ripples shifts all the curves by a substantial fraction of these discrepancies.

It might be argued that use of only one set of flow and sediment conditions and a scale ratio not greatly different from unity does not constitute a sufficiently extensive test. With regard to the first point, there is no basis for assuming that if the scaling holds for the particular set of flow and sediment conditions used it would not hold also for a wide range of flow conditions, provided only that no different kinds of forces or physical effects are involved. A set of flow conditions resulting in ripples was chosen only for convenience, in order to avoid the necessity of using a much larger channel to generate larger bed forms. But there is nothing special about this set of conditions in terms of kinds of forces (i.e., viscous forces and gravity forces) involved in the flow system. To put this another way, the same equations of motion should hold for flows with different combinations of flow and sediment conditions so long as such effects as surface tension or

1001 I I I 1 I 1 1

a

a W J -J

5 ae -0-COLD-WATER RUN

-&HOT-WATER RUN -+-SCALED HOT-WATER

Figure 3.

-0 40 80 120

MIGRATION TIME (MINI

Cumulative frequency distribution of ripple migration time

22 J . B. SOUTHARD. L. A. BOGUCHWAL AND R. D. ROMEA

Table 11. Means (M) and standard deviations (a) of frequency distributions of experimental characteristics

Ripple Ripple Ripple height spacing migration (cm) (cm) time (min)

~~~ ~~ ~

Cold-water run M 2.3 18.4 65.5 Cr 1.0 7.6 38.2

Hot-water run M 1.3 11.6 53.2 ff 0.6 4-5 31-0

Scaled hot-water run M 2.2 19.2 68.6 U 1.0 7.5 40.0

system rotation are absent or negligible. With regard to the second point, the measurement accuracy was purposely refined to the point that the validity of the modelling could be judged even though the scale ratio is not large.

IMPLICATIONS FOR RESEARCH ON SEDIMENT TRANSPORT

True and undistorted scale models involving even modest scale ratios would be useful in studying many aspects of sediment transport. Small-scale aspects of grain transport, like initiation, traction, and saltation, have been notably difficult to study experimentally, in part because of the small scales of observation needed. These phenomena could be viewed and measured much more easily in a scaled-up laboratory model. This would necessitate only ordinarily large channels, and, from Set 1, high viscosities readily attainable in water flows by means of solutes like glycerol or sucrose. For example, Taylor and Vanoni (1972) found by varying fluid viscosity as well as sediment size that bed-load transport rate in flat-bed low-transport flows scales in accordance with dimensionless parameters similar to those used in the present study.

Use of scaled-down models would be of obvious advantage in studying such large-scale aspects of sediment transport as bed configurations or fluvial channel pattern. Scaled-down models are limited through equation (6) by availability of suitable low-viscosity fluids. But even the difference between the viscosity of water at 10°C (reasonably representative of water temperatures in natural environments) and 85°C (feasible in properly designed laboratory channels, as shown by a pilot hot-water model study of large-scale bed configurations by Boguchwal(l977)) corresponds to a scale ratio of about 2.5. This scale ratio would extend the effective range of flow depths attainable in experimental channels no larger or more elaborate than existing large flumes to between two or three metres, and thereby make it possible to model flows with depths in the same range as those of small rivers and shallow tidal currents. In addition there is the considerable practical advantage of using everyday materials for fluid and sediment. The ratio of flow discharges attainable in such a model is more impressive than the scale ratio itself: since by equation (3) the velocity ratio varies as the one-half power of the length ratio, the discharge ratio varies as the five-halves power of the scale ratio, and would therefore be almost 10.

ACKNOWLEDGEMENT

Research supported by National Science Foundation, Grant No. 76-21979ENG. We thank Dr. Raymond Siever for use of the recirculating channel in the Science Center, Harvard University.

REFERENCES

Boguchwal, L. A. (1977). Dynamic Scale Modeling of Bed Configurarions, Massachusetts Institute of Technology, Department of

Brooks, N. H. (1958), ‘Mechanics of streams with movable beds of fine sand’, American Society of Ciuil Engineers; Transactions, 123, Earth and Planetary Sciences, Ph.D. Thesis, 147p.

526-594.

SEDIMENT TRANSPORT 23

Kennedy, J. F., and Brooks, N. H. (1963). Laboratory Study of an Alluvial Stream at Constant Discharge, U.S. Department of Agriculture, Agricultural Research Service, Misc. Publ. No. 970, Proceedings of the Federal Inter-Agency Sedimentation Conference, p. 320-330.

Southard, J. B. (1971). ‘Representation of bed configurations in depth-velocity-size diagrams’, Journal of Sedimentary Perrology, 41,

Taylor, B . D., and Vanoni, V. A. (1972). ‘Temperature effects in low-transport, flat-bed flows’, American Society of Civil Engineers,

Yalin, M. S . (1965). Similarity in Sediment Transporr by Currents, U.K. Ministry of Technology, Hydraulics Research Stations,

903-915.

Journal of the Hydraulics Division, 98, No. HY8, 1427-1445.

Hydraulics Research Paper No. 6, London, 24p.