Proceedings of the U.S. Geological Survey (USGS) Sediment Workshop, February 4-7, 1997

RESISTANCE, SEDIMENT TRANSPORT, AND BEDFORM GEOMETRY RELATIONSHIPS IN SAND-BED CHANNELS

The prediction of equilibrium hydraulic conditions in sand-bed channels requires knowledge of the closely coupled phenomena of resistance to flow and bed material transport. Thus, given the unit water discharge, q, the water temperature, longitudinal bed slope, S, bed material of specified median size, d_50, and gradation, g , to predict the depth of flow, h, one must, in general, be able to predict the unit bed-material transport rate, q_t , and vice versa. Further, accurate representation of the physics of both the flow and transport processes requires an ability to predict the geometric characteristics of the range of bedforms common in sand-bed channels. Because resistance to flow is related to bedform size, prediction of equilibrium-state bedform geometry, and determination of the rate at which it evolves, are also essential for use in simulating unsteady flow in sand-bed channels. To predict bed-material transport when bedforms are present it is also necessary to be able to separate the resistance to flow due to the bedforms (form drag or form resistance) from that due to direct shear on the sand grains at the bed surface (skin friction). Bennett (1995) described the development of an algorithm that accurately relates flow depth to unit discharge and determines bed and suspended load transport for the entire range of bedforms found in sand-bed channels. In doing so, the algorithm also can be used to predict the equilibrium-state geometry of lower flow regime bedforms.

If one starts with an artificially flattened bed of sand finer than 0.6 mm, maintains a constant slope, and gradually increases the shear stress (discharge), a progression of bedforms appear as follows: ripples, dunes, transition (washed out dunes), plane or flat bed, antidunes, and chutes and pools. For sand coarser than approximately 0.6 mm, the ripple phase is absent. In this progression, the first three members are known as the lower flow regime, characterized with a relatively low sediment transport rate and Froude number much less than one. Plane bed is characterized by high suspended sediment transport rates and a Froude number in the range 0.8 to 0.95. The last two members of the progression are known as the upper flow regime with the Froude number above 1 but commonly less than 1.5. In the above progression, once ripples appear, the flow resistance gradually decreases until dunes form. Then it increases until the bedform amplitude begins to decrease in the transition range. The flow resistance achieves another minimum with plane bed and again increases into the upper regime. The size, shape, and spatial distribution of roughness elements on the channel bed determine the shape of the vertical profile of the longitudinal velocity and the overall resistance to flow. Accurate representation of this vertical variation of velocity and of the eddy diffusivity are necessary for computation of suspended load transport. Smith and McLean (1977) have presented a physically reasonable formulation for the longitudinally-averaged velocity profile over a wavy bed. The profile consists of a series of logarithmic segments. The number of segments required is determined by the number of different scales of superimposed bedforms that are present. The shear velocity and characteristic roughness height for each segment depends on the height, , to wavelength, , (steepness) ratio of that scale of bedform and on its drag coefficient. For ripple or dune beds, a two-segment version is appropriate, and has been adopted for use in the Bennett (1995) algorithm.

Use of the two-segment velocity profile requires partitioning the total boundary shear stress, , into a skin friction component, `, and a form drag component. Nelson and others (1993) have presented a drag coefficient closure which provides a physically realistic mechanism for determining `. The procedure, however, requires knowledge of the bedform dimensions and a value for the drag coefficient, which, fortunately, has been found to be relatively invariant. The skin shear stress is necessary not only for determining the lowermost section of the velocity profile, but also for predicting bedform type and dimensions, and for computing the bedload transport and the sediment concentration at the lower limit of the suspended load layer. In the Bennett (1995) algorithm, bedload transport is calculated using the Meyer-Peter formulation with the constant coefficient calibrated separately for each of three bedform types: ripples, dunes, and plane-upper- regime. Rather than couple directly to the bedload, Smith and McLean (1977) demonstrated greater accuracy for computing suspended transport using a separate relation for determining the concentration at the bottom boundary; their formulation was adopted by McLean (1992) and later by Bennett (1995).

McLean (1992) demonstrated that accurate computation of suspended sediment transport requires careful consideration of the effects of the size distribution of the sediment, the effects of bedforms, and often, the effects of stratification. The McLean (1992) procedure for computing suspended sediment transport (suspended load) which uses the Rouse suspended sediment concentration profile and numerical integration to compute transport was adopted essentially unmodified by Bennett (1995). Stratification occurs when suspended sediment concentrations are great enough to produce a density gradient and reduce the vertical turbulent fluxes of mass and momentum. For two flows of equal depth, the net effect of the reduced vertical momentum flux is for the two velocity profiles to be nearly identical near the bottom but for the stratified flow to have a greater near-surface velocity (see McLean, 1992, Fig. 1). Thus, for two flows with the same water discharge, a stratified flow will have a lesser depth. In addition, for equal bottom boundary concentrations, a stratified flow will have lower near-surface concentrations and the net effect is usually to reduce the suspended-load transport capacity with respect to an unstratified flow of equal flow rate. When stratification effects are significant, the flow and transport equations are coupled because of modification of the eddy viscosity, therefore an iterative solution technique must be used. The iterative solution procedure can be quite time consuming and, as a practical matter, is to be avoided whenever possible; guidelines are given by Bennett (1995).

Ripple geometry appears to depend primarily on the characteristics of the sand and on water temperature and is relatively independent of flow characteristics. Ripples achieve a length on the order of 1000d₅₀, and a maximum steepness of about 0.2. In contrast, dune bed geometry is primarily determined by the hydraulics of the flow. Without regard to their height, dune length is on the order of 6 to 7 times the depth of flow and maximum steepness is on the order of 0.06. Thus the maximum is approximately 0.4h. As shear stress and the suspended transport fraction increase, the dunes become more sinusoidal in shape and are often called transition or 'washed-out dunes'. For this bedform type the slope of the downstream face is much less than the angle of repose of the bed material and the strength of the eddy downstream of the crest is greatly reduced as compared to that for a dune bed. Thus transition bedforms have a lower flow resistance than do dunes. Field observations indicate that for deep flows, the transitional bedforms tend to maintain their height but become much longer; this phenomenon has not been reported in flume- scale flows. In either case, however, the result is a reduction in resistance to flow. Both ripples and dunes will form in closed-conduit flow so they are not free-surface phenomenon and do not require the Froude number in their classification scheme. Beyond plane bed, however, upper regime bedforms appear to be related to surface phenomena and their complete classification requires use of the Froude number. It has been observed that multiple scales of lower-regime bedforms often exist at equilibrium in flows that have depths in excess of 2m. Bennett (1995) does not deal with this situation nor did he find it necessary to attempt to distinguish between the various types of upper regime bedforms.

The above information should make clear that prediction of the type and geometric characteristics of the bedforms is a key to success in determining sand-channel hydraulic and sediment transport characteristics. A number of attempts have been made at doing this, with perhaps the most well known being due to van Rijn (1984). Van Rijn was primarily concerned with predicting the occurrence of the dune and transition regions, and he found that this could be done using a dimensionless particle-size parameter and the transport strength, which is defined as `/_cr-1 where _cr is the critical shear stress (that shear stress just necessary to initiate the transport of particles of the given size). Bennett (1995, Fig. 2) used a modified version of the van Rijn (1984) diagram for discriminating bedform types in the transport strength-size parameter plane. As mentioned above, if they occur, the geometric characteristics of ripples are essentially fixed, so their prediction is not difficult. However, dune height is variable and Bennett (1995) modified van Rijn's (1984) equilibrium-flow dune height predictor for this purpose; this relationship shows the ratio /h to be a strong function of transport strength and weakly related to d₅₀. Because they occur seldom in natural equilibrium situations and because the average hydraulic variables of upper regime differ little from those of upper-regime plane bed, Bennett (1995) concluded that it is appropriate to simulate the former using the better understood hydraulic and sediment transport characteristics of the latter.

The bedform classification scheme discussed above correctly identifies 73 percent of the 1192 mostly flume-scale calibration data sets used by Bennett (1995). For 194 sets of dune-transition calibration data, the bedform geometry predictor yields a geometric average predicted to observed height ratio of 1.00. After calibration using the complete data set, the Bennett (1995) algorithm produced both overall geometric averages of predicted to observed depth and of predicted to observed transport of 1.00. For a verification data set of 855 observations, mostly from rivers and canals, the overall geometric averages of predicted to observed depth and transport are 0.87 and 1.14.

The systematic under-prediction of depth and over-prediction of sediment transport noted above for flows with larger unit discharges occurs primarily because the Bennett (1995) algorithm sometimes mistakenly predicts upper-regime conditions where dune beds are actually present. In fact, Julien (1992) observed that for field-scale data (3< h< 30m , the ratio /h appears not to decrease once the transport strength reaches a value of 5, as has been observed in flumes and as is predicted by the Bennett (1995) and van Rijn (1984) relationships. Julien explained that this happens because for flows of this scale the Froude number does not become large enough to approach upper-regime conditions. Alternatively, the persistence may be due to the fact that such flows often develop smaller bedforms called sand waves on the backs of the dunes; these smaller bedforms interrupt the development of the boundary layer on the the dune and prevent the shift from saltation to suspension which also appears to accompany upper regime. Without regard for the reason, taking into account the persistence of dune bedforms at (apparently) elevated ` levels, the single-bedform drag-coefficient model still generally slightly under predicts observed depths where h is greater than 3m. Use of a reasonably parameterized drag- coefficient model incorporating a three-segment velocity profile and a second level of bedforms on the backs of the dunes removes the bias, but, because the second level of bedforms is not always apparent, only a moderate level of support can be found for such an approach in presently available field data. This and other possible mechanisms shape the current focus of our investigations into flow and sediment transport mechanics in deep sand channel streams.

REFERENCES

Julien, Pierre Y., 1992, Study of bedform geometry in large rivers, Colorado State U., Q 1386, 19 pp.

McLean, S. R., 1992, On the calculation of suspended load for noncohesive sediments, J. Geophys. Res., 97(C4), 5759-5770.

Nelson, Jonathan M., McLean, Stephen R., and Wolfe, Stephen R., 1993, Mean flow and turbulence fields over two-dimensional bedforms, Water Res. Res., 29(12), 3935-3954.

Smith, J. Dungan, and McLean, S. R., 1977, Spatially averaged flow over a wavy surface, J. Geophys. Res., 84(12), 1735-1746.

van Rijn, Leo C., 1984, Sediment transport, part III: bed forms and alluvial roughness, J. Hydr. Engrg., ASCE, 110(12), 1733-1754.