PAUL S. HILL, ARTHUR R. M. NOWELL, AND PETER A. JUMARS .... vertical profiles of suspended sediment concentration [Rouse, ..... 4.95 cm s - ', zo = 0.45 cm, r2 = 0.71 ; (b) U, o = 40 crn s - ', uâ = 4.48 cm s - ', zo = 0.22 cm, r1 = 0.73 ; (c) U ...
JOURNAL OF GEOPHYSICAL RESEARCH, VOL . 93, NO . CIO, PAGES 12,499-12,509, OCTOBER 15, 1988
Flume Evaluation of the Relationship Between Suspended Sediment Concentration and Excess Boundary Shear Stress PAUL S. HILL, ARTHUR R . M . NOWELL, AND PETER A . JUMARS School of Oceanography, University of Washington, Seattle A flume study was undertaken to determine the source of the large variability in field estimates of y, the coefficient of proportionality between excess shear stress and concentration of suspended sediment at a reference height above the bed . Accuracy in predicting suspended sediment concentration and particle flux demands accurate knowledge of this coefficient . To avoid error due to incorrect assumptions concerning the eddy diffusion coefficient for mass, reference concentration was estimated from concentration profiles. A linear relationship was found adequate to describe the dependence of reference concentration on excess boundary shear stress . The value of y was determined to be 1 .3 x 10 -4 . Values for the coefficient range between only 0.8 x 10 -4 and 4.9 x 10 -4 at the 9594 confidence level . We show that the variability in previous field-derived estimates of y over 4 orders of magnitude is not attributable to unsound theory relating reference concentration to excess shear stress, but is most likely due to improper treatment of the eddy diffusion coefficient for mass and to measurement error .
INTRODUCTION
The utility of a general theory for predicting suspended sediment concentration and associated particle flux in open channels has been recognized for several decades . During transport as suspended load, sediment is carried along at a velocity near that of the fluid . This process lies in contrast to bedload transport, by which sediment grains move in a series of short hops, the series punctuated by periods during which the particle is at rest. This fundamental difference between the two primary modes of transport makes suspended transport the more effective means of dispersing sediment in the natural environment . As such, development of a sound theory to explain the mechanics of suspended sediment transport has attracted the attention of oceanographers, sedimentologists, sedimentary geologists, and hydraulic engineers . Success has been achieved in predicting the equilibrium distribution of suspended material in turbulent flows by balancing upward turbulent diffusion of sediment with its gravitational settling [cf. Vanoni, 1946] . Prediction of absolute sediment concentration, however, has been hampered by uncertainty regarding the proper means of setting the lower boundary condition on concentration. This lower boundary condition, termed the reference concentration, is coupled hypothetically to sediment concentration and distribution within the near-bed saltation layer [Einstein, 1950] . The lack of a routine method for measuring bedload concentration and distribution has forced reliance on indirect means for estimating reference concentration. Employing dimensional arguments, Yalin [1963] proposed that sediment concentration in the bedload layer is some function of excess shear stress S, defined as S = [(r e - rc )/r 1], where r y is boundary shear stress and r, is critical erosion shear stress . Yalin suggested that a simple linear relationship might adequately describe the functional dependence of bedload concentration on excess shear stress . Theory developed by Owen [1964] for grains saltating in air supports the hypothesis of a direct proportionality between excess shear and sediment concentration in the bedload layer . Copyright 1988 by the American Geophysical Union . Paper number 8C0414. 0148-0227/88/008C-0414$05 .00
The coefficient of proportionality has come to be known as y [Smith, 1977] . Setting reference concentration therefore requires a value for the excess shear stress and a value for y . Available estimates of y, all garnered from field studies, vary over 4 orders of magnitude (Table 1), apparently devaluating the theory . The sources of variability are unclear. Most of the field investigations have left relevant parameters undermeasured, unmeasured, or uncontrolled . For example, reference concentration often is estimated from concentration data taken at only one position in the vertical [McLean, 1976 ; Wiberg and Smith, 1983 ; Drake and Cacchione, 1988] . This strategy for setting reference concentration forces the imposition of assumptions concerning the distribution of mass in a turbulent boundary layer. Frequently, size and settling velocity distributions of suspended sediment are poorly constrained [Wiberg and Smith, 1983 ; Sternberg et al., 1986 ; Drake and Cacchione, 1988], owing to either lack of measurement or particle-particle interactions . Lastly, field estimates of critical erosion shear stress are affected to a varying degree by biological processes [Grant et al ., 1982] . Lack of comprehensive data sets in these studies disallows any attempt to distinguish between the role that experimental error plays in creating highly variable estimates of y and the role which a nonlinear dependence of reference concentration on excess shear stress takes in causing such variability. In the laboratory the degree of control necessary to elucidate the applicability of a linear relationship between reference concentration and excess shear stress can be achieved . Therefore a flume study was undertaken to examine the strength of the proposed relationship over a rippled bed of fine sand .
THEORY
Schmidt [1925] first suggested that the vertical distribution of suspended sediment is defined by a balance between upward turbulent diffusion of sediment and its gravitational settling. This approach later received qualitative verification by Leighly [1934] . In a more rigorous treatment of the balance between upward turbulent diffusion and downward settling, O'Brien [1933] proposed that the distribution of sedi-
12,499
12, 5 00
HILL ET AL. : SEDIMENT CONCENTRATION VERSOS SHEAR STRESS
TABLE l . Estimates of y From Field Studies Source
7
1 .95 x 10 -3 2 .40 5 .00 1 .60 1 .50 2 .00
x x x x x
McLean [1976], Smith and McLean [1977a] Smith and McLean [ I 977b] Glenn [1983]* Wilbert, and Smith [1983] Kachel and Smith [1986] Drake and Cacchione [1988]
10 -3 10 -4 to 3 .00 x 10 - ' 10 -4 10 -2 10 -5 to 5 .00 x ] 0 -4
*This estimate comes in part from laboratory studies .
ment in a steady, horizontally uniform flow is defined by the equation w,C, + K,
Bz'
= 0
(1)
in which w, is the settling velocity of the suspended material, C, is the concentration of suspensate, and K, is the kinematic eddy diffusion coefficient for mass . Equation (1) was subsequently integrated by von Karman [1934a] in order to obtain an expression for relative concentration : In
-Pw,' J ,
K
(2)
where (C,) Z, represents the concentration of suspended sediment at a reference depth r,, p is the fluid density, z is depth in the flow, and K, represents the dynamic eddy viscosity . Von Karman's [1934a] analysis suggests that the relative distribution of suspended sediment may be predicted given adequate knowledge of the settling velocity of the suspended material and the appropriate eddy diffusion coefficient for mass . To evaluate absolute concentration at a given depth requires a value for concentration at some reference depth. The elegance of von Karma's [1934a] formulation spurred attempts to assess its validity in specific turbulent flow situations . For pipe and channel flows, von Karman [1934b] proposed that the eddy diffusion coefficient for momentum varies linearly with distance from the boundary . He also reasoned [von Karman, 1934a] that the eddy diffusion coefficient for mass should take the same form as that for momentum, noting that the magnitude of the two diffusion coefficients need not be the same . At the suggestion of von Karman, Ippen employed an eddy diffusion coefficient of the form K, = ku*z[l -(z/h)], in which k is von Karman's constant (= 0 .4), u * is the shear velocity, z is height above the bed, and h is boundary layer thickness or flow depth, in order to generate vertical profiles of suspended sediment concentration [Rouse, 1937] . The equation used by Ippen and published by Rouse [1937] is C, = z(h - z,)] (C,)„ [ z,(h - z)
(3)
The exponent p, is defined as p, = (w,/ku * ), where w, equals sediment settling velocity . The exponent is known as the Rouse parameter, and equation (3) is known as the Rouse equation . The Rouse equation for polydisperse suspension is [Smith and McLean, 1977b] Cm = [z(h- z,) -( z,(h - z) (Cm),4
(4)
where Cm is the concentration of size class m at a depth z, (C m ) :, is concentration of size class m at the reference depth z,, and w m is the settling velocity of size class m . The exponent pm defines the Rouse parameter for size class m . In these formulations the eddy diffusion coefficients for mass and momentum are considered identical . Early studies [Anderson, 1942 ; Vanoni, 1946] demonstrated that the form of the Rouse equation (3) was correct but that the exponent p, varied from its predicted value . Deviation of values for p, from theoretical values led to scrutiny of Rouse's assumptions . The assumption of identical eddy diffusion coefficients for mass and momentum has attracted the majority of attention . Dobbins [1944] proposed that rather than equating the two diffusion coefficients, a more general relation of the form K,
= BKm4
(5)
should be used . Deviations of the value of /i from unity could then be used to explain the difference between measured and predicted values of p, . The more general form of the Rouse parameter is thus p, = (w,/flku*), and p m = (w m //iku *) for a multicomponent system . Studies have examined the value of /1 in a variety of flow situations . To date, no consensus has been reached as to the value of /i or the relevant flow and sediment parameters upon which it depends . A large body of literature exists which suggests that /i must equal unity, except for the coarsest materials, for which /i may have a value of less than I [Rouse, 1938 ; Kalinske and Pien, 1943 ; Dobbins, 1944 ; Hunt, 1954] . In contrast, an equally large body of literature states that the value of /i exceeds unity [Ismail, 1952 ; Colby and Hembree, 1955 ; Laursen, 1958 ; Task Committee Report, 1963 ; Nordin and Dempster, 1963 ; Singamsetti, 1966 ; McTigue, 1981 ; Ikeda and Asaeda, 1983] . Theoretical investigations into possible controls on the value of /i suggest that vorticity dynamics play a dominant role in setting the value of 3 for a particular flow [Taylor, 1932 ; Singamsetti, 1966 ; Jobson and Sayre, 1970 ; Tennekes and Lumley, 1972 ; Schlichting, 1979] . Consequently, bed geometry likely exerts an influence on the magnitude of mass diffusivity relative to momentum diffusivity in boundary layer flows [Jobson and Sayre, 1970] . In addition to the turbulent characteristics of the flow, particle inertial effects are also postulated to influence the value of Q [Singamsetti, 1966 ; Jobson and Sayre, 1970 ; Petkovic and Bouvard, 1983] . In light of the lack of a comprehensive theory for predicting the value of 9 in a given flow, the most prudent course for setting (C m)„ is extrapolation from measured profiles . Settling velocity of suspended sediment in low-concentration flows is a relatively well constrained parameter in many sediment transporting systems . It can be calculated quite accurately given grain diameter, shape, and roundness [Dietrich, 1982a] . Complications in settling velocity calculations arise when particle-particle interactions become important. Adequate theory for predicting the fall velocity of particle aggregates does not exist at present [Hawley, 1982 ; Gibbs, 1985 ; Kranck, 1986], thus limiting the accuracy of the Rouse equation for fine sediments [Sternberg et al ., 1986] . To predict suspended sediment concentration rather than simply vertical sediment distribution, all Rouse-type equations require the input of a reference concentration (C,)Z, . Accuracy in predicting total suspended load demands that this lower boundary condition on concentration be set at the lowest depth of applicability of the Rouse equation . Assuming that
4
HILL ET AL. : SEDIMENT CONCENTRATION VERSUS SHEAR STRESS
this depth is equal to the height of the top of the bedload layer and that concentration at this depth depends upon concentration and distribution of saltating grains within the bedload layer [Einstein, 1950], one can use Talin's [1963] dimensional arguments to hypothesize that reference concentration varies linearly with excess shear stress . Owen [1964] provided further theoretical basis for linear dependence of concentration on excess shear stress . Owen [1964] argued that stress in the bedload layer may be generated only by fluid acting on fluid, fluid acting on sediment, sediment acting on fluid, and sediment acting on sediment. To conserve momentum, the divergence of the fluid stress must balance the divergence of the momentum flux of the fluid . The same balance must hold for the divergence of sediment stress and sediment momentum flux. Assuming that grain-to-grain interactions are unimportant, a linear relationship between sediment concentration and the magnitude of the stress exerted on the fluid by the sediment emerges . Noting that the stress exerted on the fluid by the sediment must be equal and opposite to the stress exerted on the sediment by the fluid, the vertical flux of fluid momentum may be set equal to the sum of a fluid-fluid stress and a fluid-sediment stress which scales linearly with concentration . In horizontally uniform flow the vertical flux of fluid momentum may be equated with the total stress T,x in the fluid . Very near the bed, r„ is simply the boundary shear stress T, . Owen hypothesized that the fluid-fluid stress is maintained at the critical erosion shear stress due to the response of the sediment concentration field to changes in T,. If the hydrodynamic forces transmitted to the bed via the fluid-fluid stress exceed T,, the critical entrainment stress, grains saltate . The number of grains in the saltation layer increases until the momentum extracted from the flow by drag on sediment particles consumes a large enough portion of the total stress in the layer to force the fluid-fluid stress back down to its critical value . A concentration excess in the saltation layer decreases the erosive stress on the bed . As a result, concentration falls, channeling a greater portion of the total stress into fluid-fluid stress, until stress on the bed reaches its critical value . By this mechanism the equilibrium fluid-fluid stress near the bed is rendered insensitive to changes in the boundary shear stress . Solving for C, with Owen's [ 1964] momentum balance equations yields : C,=
Y(Tb
TJ
-YS
(6)
T,
In order to force the concentration to asymptote as S approaches infinity, Smith [1977] modified the linear dependence (6) to take the form
C,YS (C,)°°
(I + TS)
(7)
or im CbYS (1 + YS)
(8)
for a multicomponent system. In the above, C, represents the relative concentration of sediment in the bed (= 0.65 [cf. Blatt et al., 1980]), and im is the fraction which size class m composes of the bottom sediment . The final input required in the Rouse equation is a value of the boundary shear stress z, The total drag on a rippled bed consists of skin friction (equal to the integral of all shearing
12, 5 0 1
stresses taken over the surface of the bed) and form drag (integral of normal forces) [Schlichting, 1979] . In sediment transport problems it is essential to be able to separate the skin friction from the spatially averaged boundary shear stress [Einstein, 1942 ; Einstein and Barbarossa, 1954] . The spatially averaged boundary shear stress includes form drag due to topographic features ; sediment grains during erosion, however, respond only to the skin friction . To date a sound procedure for extracting an estimate for skin friction in a ripple field from estimates of total drag on the boundary has not been proposed . Form drag on a rippled boundary is a complex function of bedform geometry and spacing [Nowell and Church, 1979 ; Grant and Madsen, 1982] . Wake interactions play a dominant role in the dissipation of turbulence at the spacings characteristic of rippled beds [Nowell and Church, 1979] . The application [Wiberg, 1987] of theories posed for flow over dunes [Smith and McLean, 1977a ; McLean and Smith, 1986 ; Nelson, 1988] to flow over ripples is unjustified [Paola, 1983], because studies which address flow over dunes either do not include or fail to adequately describe the stress field in the wake region . Until more information is available concerning flow over ripples, an empirical value of the skin friction based on bedload transport rate is deemed most accurate. While the precision of such an estimate is not good owing to the scaling between bedload flux and the boundary shear stress, the relationship between the measured T, and the "true" T, is least ambiguous . MATERIALS AND METHODS
Measurements were made in a large, recirculating, racetrack-shaped flume [Nowell et al., 1988] at the University of Washington's Friday Harbor Laboratories . The test section of the flume is 8 m long. Channel width is 70 cm, and flow depth was maintained at 12 cm . Flow speed is controlled by a paddle-type drive mechanism distributed over the return section which can provide free stream flow speeds up to 200 cm S
I .
All runs were performed with 3 .1 ± 0 .5% salinity (5 um) filtered seawater maintained at 17° ± 0 .5°C . During periods when no measurements were being made, the flume was supplied with fresh seawater. Outflow during these periods was through a standpipe . Moderately well sorted, subrounded to subangular, medium to fine, silica sand [Folk, 1980] (U.S . Mesh size 110, D5o = 168 pm) was elutriated in a 150-L cylindrical tank to remove the fine fraction . A grain size frequency distribution of the processed sand is shown in Figure 1 . Twenty 45 .5-kg bags of sand were required to cover the flume bed to an average depth of 5 cm . To remove adhesive effects associated with the presence of organic coatings or bacteria, the sand was washed in a 9% solution of hydrogen peroxide for 24 hours immediately prior to the running of the experiment . Following this treatment, three subsamples of the sediment were checked for bacteria under an epifluorescence microscope with the acridine orange technique [cf. Daly and Hobble, 1975] . There was no evidence of bacteria on any of the sand . At the end of the experiment (3 weeks later), the sediment was rechecked for buildup of organics and bacteria . Bacterial coatings had begun to form on some of the grains, but more than 90% of the grains had no bacteria at all . These findings suggest that adhesive effects were negligible. Measurements of velocity and concentration were made at
12, 5 0 2
HILL ET AL. : SEDIMENT CONCENTRATION VERSUS SHEAR STRESS
I
I
I
I
I
I
I
I
5
I
0 5163
6480
81100
101- 128- 161- 203- 256- 322127 160 202 255 321 406 D(,,,m)
Fig. 1 . Grain size histogram of bed sediment after sand was elutriated in order to drive off the fine fraction . Size analysis was carried out with a Coulter Counter model TA 11 equipped with a 1000-µm tube . three different flow speeds . Velocities at z = 10 cm (U,,) were 35 cm s - ', 40 cm s', and 45 cm s - ', respectively . The upper and lower bounds on workable flow velocities were set as follows. The lower limit was set by the need for noticeable suspension of sediment . The upper limit to flow velocity was set by the transmission capabilities of the velocity measuring system in a sediment-laden flow . At all flow settings, threedimensional ripples formed on the bed . Ripple migration rate, wavelength, and height were quantified at each flow velocity before making measurements of velocity and suspended sediment concentration . The vertical and streamwise components of mean and fluctuating velocity were measured with a TSI two-component, three-beam, laser Doppler velocimeter (LDV), model 9100-8, equipped with a Lexel model 85, 2 .5-W argon ion laser . The system was set up transverse to the direction of flow, and the receiving optics were positioned to measure backscattered light. The entire unit was mounted on the base of a Bridgeport number I milling machine to allow translation along the x, y, and z axes . The milling machine was fixed to a cart which rolled on a track bolted to the floor of the laboratory and afforded the mobility necessary to profile different parts of the test section .
TABLE 2 .
Stationary time mean average velocity profiles were assembled via a stratified random sampling scheme . To obtain an accurate picture of mean velocity at a given vertical position over a nonuniform bed requires that the individual sample points which go into calculation of the mean must be distributed randomly with respect to structures on the bed . Further, to achieve independence, any two successive measurements at one vertical position should be decorrelated by setting an intersampling interval that is greater than the coherence time scale . In generating mean velocity profiles over a rippled bed, the above conditions may be met by separating successive measurements at one position by a random time interval long enough to ensure that the two measurements are not made over the same bedform . Thus intersampling intervals between consecutive measurements at a given vertical position were set by summing the maximum ripple migration period with the product of a random number (between 0 and 1) and that maximum ripple migration period . Velocity profiles were generated from a series of measurements taken at nine positions in the vertical over roughly 9-hour periods . The positions in z extended from 0 .1 cm to 10 cm in logarithmic increments . Velocity was measured at only five positions (at or above 1 cm) at the highest flow speed owing to problems with laser transmission through the nearbed suspended load . Accurate vertical positioning was achieved by first placing the LDV measuring volume on the sediment surface and then vertically translating the milling machine table by the required amount. Each individual velocity measurement represents the mean of 128 velocity readings . Sampling time for any one measurement ranged from 5 s to 60 s, which was short in comparison with ripple migration rate (Table 2). The brief sampling time prevented a situation in which vertical position of the LDV measuring volume relative to the bed changed significantly during sampling . Data reduction was carried out by an Apple He computer equipped with TSI software . Profiles were obtained before and after concentration measurements were made at each flow speed . This procedure provided a check on the steadiness of flow conditions during the course of the experiment . Profiles of average sediment concentration were taken in a manner analogous to velocity profiles . Sampling was carried out at five positions : 1 cm, 2 cm, 4 cm, 7 cm, and 10 cm above the bed. A series of samples was taken at each position, any two successive samples at one position being separated by a random time interval, calculated as for the LDV measurements . Sampling time was 5 s per sample. For a given flow speed all of the samples from a given position were introduced into a 1000-mL container spiked with 40 mL of formalin to inhibit subsequent bacterial growth . Five to eight samples
Estimates of Bed and Flow Parameters at Three Flow Speeds
l
H
Uia, cms - '
H,
A,
cm
cm
A
min
(11 .U ." cms '
zn. cm
("J", cm s - '
35 40 45
1 .4 1 .8 1 .2
12 .7 13 .9 13 .3
0 .11 0 .13 0 .09
19 .8 10 .2 7 .7
4 .95 x 0 .29 4 .48 ± 0 .25 5 .20 1 0 .26
0.45 0 .22 0 .29
2 .44 3 .31 3 .39
T,
1
0 .49 0 .74 0 .65
The variable His ripple height, A is ripple wavelength, H/A is ripple steepness, and T is ripple period . Other variables are defined in the text . Confidence intervals about are at the 95% level . No confidence intervals about (i1,) , are given because only one composite bedload sample was taken at each flow speed . The variable magnitudes of z„ and the ratios of (u,),„1 to reflect variable input of form drag to estimates of the stress in the outer layer .
12,50 3
HILL ET AL. : SEDIMENT CONCENTRATION VERSUS SHEAR STRESS
were taken at each position . Sampling was carried out with a peristaltic pump which allowed maintenance of a constant flow rate. To avoid biases stemming from sediment inertial effects, sampling was done isokinetically [cf. Hinds, 1982] . Prior to taking a sample, flow velocity at the appropriate position was measured . The sampling nozzle was removed from the flow while taking the velocity reading . Upon determination of mean velocity, the nozzle was reinserted into the flow and the LDV measuring volume was positioned immediately in front of the nozzle mouth . Flow rate on the pump was then adjusted until velocity at the nozzle mouth matched the mean velocity measured in the absence of the nozzle. Bedload transport rates were measured with Hele-Smith bedload samplers [Dietrich, 1982b] . Measurements were made concurrently with suspended sediment measurements with the identical method for setting the interval between successive measurements . Seven bedload samples were taken over 5-s intervals each to produce the composite bedload sample at each flow speed . Mass concentration of each composite suspended sediment sample was determined by filtering the sample through a preweighed 8-,um pore diameter Nuclepore filter . Filters were rinsed with distilled water, placed in preweighed plastic petri dishes, put in a desiccator for several days, and then weighed . Mass of bedload samples was determined by the same procedure except that larger filters were used on a porcelain funnel apparatus . Grain size analysis was done with a Coulter Counter model TA 11 equipped with a 1000-µm orifice tube, following procedures of Kranck and Milligan [1979] . Analysis was performed on from three to eight subsamples of each suspended sediment sample. Each subsample was sonicated for 7 min prior to analysis and counted for 180 s . Output of the machine, total number of particles per channel, was converted to sediment volume per channel by assuming the particles were spherical . Analysis was restricted to channels 7 through 14 because of the absence of sediment in finer and coarser size classes . Fifteen subsamples of bottom sediment were analyzed in an identical manner . Sampling order was randomized for all samples . Settling velocities for seven sieve-size classes were obtained by settling particles in a 1 .5-m-long settling tube with an internal diameter of 10 cm . The entire tube was jacketed by a flow-through water bath which minimized thermal convective effects on settling velocity . Twenty particles in each size class were timed individually as they fell a distance of 50 cm. Mean settling velocities for each size class fell on Dietrich's [1982a] curve for sand grains with a Powers index of 3 .5 and a Corey shape factor of 0.7 . This curve was used to estimate mean settling velocities of the eight Coulter Counter size classes . DATA ANALYSIS
In order to analyze velocity data the flow was divided into two layers . The inner region of the flow extended from the bed to roughly 1 .5- to 2-cm height and was characterized by high turbulence intensities and large scatter in mean velocity data . This region of the flow was termed the wake layer [cf. Arya, 1975] . At heights above 1 .5-2 cm, turbulence intensities were reduced and the mean velocity structure was more coherent . This region was termed the outer layer of the flow . Owing to the highly chaotic, nonuniform nature of fluid motion in the wake layer, attempts to model the flow in this region with an eddy diffusion type closure model were deemed
inappropriate [Richards, 1983] . While progress has been made recently in modeling flow over dunes [McLean and Smith, 1986 ; Nelson, 1988] by explicitly addressing the flow in the wake layer, the accuracy of the far-field wake solution degrades as one considers bed features of increasing aspect ratio . More involved theoretical treatment of the near-bed flow is beyond the scope of this study . Instead, attention was focused on gaining an estimate of stress in the outer layer . Velocity data were analyzed using an eddy diffusion coefficient of the form K m„ = ku,z[l - (z/h)] . Velocity is thus defined by the law of the wall :
U(Z)
= k In z
(9)
0
Estimates of the shear velocity u,, and the roughness parameter zv were obtained by regressing u(z) on In z . Because of inequality of variances at different depths, the regression equations were weighted [Kleinbaum and Kupper, 1978] . A suspended sediment stratification correction as described by Gel,fenbaum and Smith [1986] was applied to the eddy diffusion coefficient and was found to have an insignificant effect on the velocity profiles for the concentrations encountered in this study. Comparison was made of the slopes and intercepts of the best fits to the velocity data collected before and after concentration measurements were made [Kleinbaum and Kupper, 1978] . Slopes and intercepts were not significantly different (p > 0.05) for any of these pairs at any flow speed . Therefore data for each flow speed were pooled, and new estimates of the shear velocity and roughness parameter were made (Figure 2) . Confidence intervals about u, r (Table 2) were established as described by Gross and Nowell [1983] . In order to obtain values for the Rouse parameter pm and the outer flow reference concentration (C m) zw, the natural log of Cm was regressed on the natural log of {(z/z w )[(h - z_)/ (h - z)]} (Figure 2). The variable z w is the height of the top of the wake region (taken as 2 cm), and IQ_ is the concentration of size class m at this level . The slope of such a regression line is pm , and the intercept is In (C m) zw. The concentration at the top of the bedload layer was obtained by assuming concentration was constant through the wake region. This assumption is analogous to that proposed by Glenn [1983] for suspended sediment distribution in wave boundary layers . The high turbulence intensities and the concentration measurements made within the wake layer suggest that this assumption is reasonable (Figure 2) . Determination of the Rouse parameter and reference concentration for each size class required that the total volume concentration at each depth be divided into contributions from each size class. A heterogeneity chi square test [Sokal and Rohlf, 1981] was run to determine if grain size distribution data from subsamples of each suspended sediment sample could be considered descriptors of the same parent population, and in no case could they at the 95% confidence level . Therefore the median value of the volume fraction data was used to estimate volume of each size class at each depth . The 95-% confidence intervals on reference concentrations were established according to procedures given by Kleinbaum and Kupper [1978] for determining confidence intervals about an estimate of an intercept obtained from a linear regression . Boundary shear stress at each flow speed was determined by converting mass bedload transport rate to volume bedload transport rate and then backsolving Yalin's [1963] bedload
12,504
HILL ET AL . : SEDIMENT CONCENTRATION VERSUS SHEAR STRESS
10 1
E 100 N
b 10
20 30 40
10 20 30 40
U(cm s')
U(cm s'')
10
20 30 40
U(cm C')
10 1
10 -' 10-5 10 -4 10-5 io- 5 Cm Cm Cm Fig. 2 . Plots of velocity and concentration versus depth at three different flow speeds . Best fits to the data are shown . (a-c) Each velocity datum represents the mean of 18 independent measurements of U . Each independent measurement is the average of 128 velocity readings taken over 5- to 60-s periods. Error bars indicate one standard deviation . Owing to inequality of variances at different depths, the contribution of each point in determining the best fit was made inversely proportional to the variance in U at that depth . A simple linear eddy diffusion coefficient of the form K m, = ku I z(1 - z/h) was considered adequate for the purposes of this study. Results of the best fits are as follows : (a) U,0 = 35 cm s - ', = 4.95 cm s - ', zo = 0.45 cm, r2 = 0.71 ; (b) U, o = 40 crn s - ', u„ = 4 .48 cm s - ', zo = 0 .22 cm, r1 = 0.73 ; (c) U, o = 45 cm s - ', u w = 5.20 cm s - ', z, = 0.29 cm, rz = 0 .78. (df) Concentration data and best fits for three of seven size classes (squares, 64-80 pm ; plus signs, 101-127 pin ; circles 161-202 pm) at (d) U, o = 35 cm s - ', (e) U 10 = 40 cm s - ', and (f) U,o = 45 cm s"' . Regression was carried out with K, = Qku * z(l - z/h) as the eddy diffusion coefficient for mass . Height of the wake layer, z,,, is indicated by a dashed line . Concentration at the top of the wake layer is assumed equal to the reference concentration . 10-6
equation for (u s ),f , where the subscript sf denotes skin friction (Table 2) . Yalin's equation reads q, = 0 .635p,D, o u,S[1 (1/as) In (1 + as)], where a = 2 .45(p/p,) o 'IT, AP,- P)gDso] o .s In this expression, q„ is volume transport, p, is sediment density (2.65 g cm -3 ), g is acceleration due to gravity (980 cm s -z), and D, o is median grain diameter (168 pro) . Confidence limits could not be placed on these estimates of u s because only one average bedload sample was collected at each flow speed . Critical erosion shear stress was calculated for each size class using White's [ 1970] version of the Shields curve . Excess shear stress for each size class at each flow speed was then calculated (Table 3) . Because the variances of the estimated values of reference concentration are neither equal nor distributed normally about the mean, Mood's nonparametric linear regression model [Tate and Clelland, 1957] was used to elucidate the strength of the linear relationship between excess shear stress and reference concentration (Figure 3) . Mood's model relies on a graphical approach to determining the line which best fits the data. Confidence intervals on an estimate of slope are established by iteratively changing the slope until a line results which may not be considered a realistic descriptor of the data at the preset confidence level .
RESULTS AND DISCUSSION
Values for total shear velocity, skin friction shear velocity, the ratios of the two, and values for the roughness parameters z o are given in Table 2 . The value of the total shear velocity (u,) o,,, is greatest for the high flow speed and least for the medium flow speed . The trend in values of (u„),f differs from the trend in (u„)O0L . The lowest value for (u .),f is at the low flow speed, and the highest value is at the high flow speed . The trend in (u„),f is borne out by the trend in reference concentration (Table 3). The observation that the trends in (u*),,, and (u„),f differ as one goes from low to high flow speeds indicates that form drag on the ripples does not account for equal proportions of the total stress at the various flow speeds . Unequal contribution of form drag to the total stress in the outer region of the flow is most clearly demonstrated by comparing the ratios of (u * ),f to (u .),,,, for each case. Form drag has the greatest impact on the total stress at the low flow speed and contributes the least to the total stress at the medium flow speed . The roughness parameters zo for the outer flow regions also reflect this variable input to the total stress by form drag . The greatest value of zo exists at low flow speed, and the lowest value for z o is at the medium flow speed .
t
t
HILL ET AL. : SEDIMENT CONCENTRATION VERSUS SHEAR STRESS
TABLE 3 .
12,505
Estimates of Excess Shear Stress and Suspended Sediment Concentration and Distribution by Size Class
VI0'
cm s - '
D, µm
S
(C,,,); ., x 10-64
(17,,,),'t
(Pm)„
0
35
51-63
6 .3
0 .14
0 .40
0 .35
64-80
5 .3
0 .20
0 .49
0 .41
81-100
4 .6
0 .29
0 .50
0 .58
101-127
4 .0
0 .45
0 .55
0 .82
128-160
3 .4
0 .62
0 .64
0 .97
161-202
3 .0
0 .95
0 .84
1 .13
203-255
2 .5
1 .26
1 .05
1 .20
256-321
2 .1
1 .75
0 .96
1 .82
51-63
12 .4
0 .16
0 .06
2 .67
64-80
10 .7
0 .22
0 .10
2 .20
81-100
9 .2
0 .32
0 .19
1 .68
101-127
8 .1
0 .49
0 .37
1 .32
128-160
7 .1
0 .70
0 .54
1 .30
161-202
6 .4
.06 1
0 .77
1 .38
203-255
5 .4
0 .5 (+1 .4) (-0.4) 1 .9 (+3 .9) (-1 .3) 6 .4 (+6 .6) (-3 .3) 11 .8 (+9.6) (-5 .3) 18 .3 (+18 .1) (-9.1) 16 .0 (+11 .5) (-6 .6) 6 .1 (+9.8) (-3 .8) 1 .3 (3 .3) (-0.9) 0 .7 (+0.8) (-0 .3) 3 .1 (+8 .2) (-2 .2) 13 .4 (+15 .6) (-7 .2) 42 .8 (+34 .3) (-19 .1) 73 .4 (+74 .4) (-37 .0) 56 .0 (+87 .4) (-34 .1) 17 .2 (+17 .8) (-8 .7)
1 .38
0 .97
1 .42
256-321 51-63 .
4 .7 13 .0
0.14
0 .26
0 .54
64-80
11 .2
0.19
0 .19
1 .00
81-100
9 .7
0.28
0 .31
0 .90
101-127
8 .6
0.42
0.40
1 .05
128-160
7 .4
0 .60
0.54
.11 1
161-202
6 .7
0 .91
0 .65
1 .40
203-255
5 .7
1 .20
0 .76
1 .58
256-321
5 .0
2 .4 (+4 .5) (-1 .6) 11 .2 (+29 .1) (-8 .1) 46 .3 (+28 .5) (-17 .1) 107 .3 (+39 .0) (-28 .6) 163 .3 (+22 .7) (-19 .1) 97 .1 (+43 .3) (-30.1) 22 .5 (+17 .8) (-9.9) 2 .9 (+2 .8) (-1 .4)
1 .65
0 .55
3 .02
40
45
*Numbers in parentheses define 95% confidence intervals . tTheoretical Rouse parameter is calculated assuming identical eddy diffusion coefficients for mass and momentum . The estimated value of the coefficient y in this study is 1 .3 x 10 -4 (Figure 3) . The 95% confidence intervals on this estimate range from 0.8 x 10 -4 to 4.9 x 10 -4 . Confidence intervals were obtained nonparametrically [Tate and Clelland, 1957] and are conservative in relation to intervals obtained with a parametric regression model. The magnitude of the confidence intervals is small in relation to the enormous variability in published estimates of y (Table 1) . This finding suggests that a linear relationship adequately describes the functional dependence of reference concentration on excess shear stress in nonuniform flows at the excess shear stresses measured . Several assumptions were made in arriving at the above result . First, Yalin's [1963] model for bedload transport was selected to derive boundary shear stresses from bedload transport rates . Other commonly used bedload models are those of
and Muller [1948], Einstein [1950], and Fernandez-Luque and van Beek [1976] . The Yalin [1963] Meyer-Peter
model was deemed the best choice because it includes explicit size dependence lacking in the models of Meyer-Peter and Muller [1948] and Einstein [1950], which were both fit to data from bed material coarser than that considered in this study . Additionally, Wiberg [1987] has shown that the Yalin equation provides a better overall fit to available data . Nonetheless, estimates of the boundary shear stress were derived with the other models to check whether significantly different values of y resulted when employing the various estimates of r6 to calculate excess shear stress . The estimates of boundary shear stress yielded by the Yalin [1963] equation exceed the estimates produced by the Einstein [1950] and Meyer-Peter and Muller [1948] equations by roughly 50% . Boundary shear stress estimates from the
12,506
HILL ET AL. : SEDIMENT CONCENTRATION VERSUS SHEAR STRESS
0.006
0.005
0.004 (C n )z o i,a C b
0.003 0
1
a 0.002
0 C
a
0.001
a13
0
0
2
~~
4
1313 . 19
a I
8
I
10
C I
1
1
14 S Fig . 3 A plot of excess shear stress S versus normalized reference concentration (C_),o/imC1 . The normalization factor defines the relative concentration of size class m in the bed . The slope of the best fit line represents y . A nonparametric regression yielded the result y = 1 .3 x 10 -4. The 95% confidence intervals about this estimate are 0 .8 x l0 - ` < y
