# Bedload transport of a bimodal sediment bed

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2 Table 1. Examples of Hiding Functions Commonly Found in the Literature a Hiding Function Reference Remark 2 ln Di=Dc 1 þ ð Þ Egiazaroff [1965] Theoretical modeling lnð19þ 2 ln Di=Dc 1 þ ð Þ for Di > 0:4 Ashida and Michiue [1973] Based on experimental data lnð19þ D c D 1 0:843 i D for i 0:4 D c D c D 0:98 i Parker et al. [1982] Based on field data D c D i D c D i D c 0:74 Powell et al. [2001, 2003] Based on field data g 0:69 with g ¼ 1 Wilcock and Crowe [2003] Based on experimental data 1 þ eð 1:5 Di=Dc Þ a Egiazaroff [1965] and Ashida and Michiue [1973] define D c as the arithmetic mean surface grain size whereas Parker et al. [1982], Parker [1990] and Wilcock and Crowe [2003] used the geometric mean surface grain size. where g is the gravitational acceleration, r is the fluid density, r s is the sediment density and R =(r s r)/r. The critical Shields number q c is the value of q below which sediment transport ceases [Shields, 1936; Meyer-Peter and Müller, 1948]. Equation (1) is practical because the only sediment data required are the representative size and the value of q c. It is however unable to predict changes of grain size distribution in the river. [4] A second approach consists of discretizing the grainsize distribution into a finite number N of populations of characteristic grain size D i and investigating how the transport rate of each population, q s,i, depends on the shear stress and the grain-size distribution at the bed surface [Parker, 2008]. It is usually observed that 1) q s,i increases with the proportion of grains belonging to the ith population within the surface layer, f i ; 2) far from the threshold of motion, q s;i q 3=2 i where q i = t/rrgd i is the Shields number calculated for the grain size D i [Liu et al., 2008]. Consequently, many authors hypothesized a relation of the following form [Wilcock, 1988; Parker, 2008]: q* s;i f i 3=2 ¼ F q i ; q c;i q i ; ð4þ p where q* s;i ¼ q s;i = ffiffiffiffiffiffiffiffiffiffiffi RgD 3 i is the dimensionless transport rate of the grains of size D i and F(q i,q c,i ) is a function capturing the complexity introduced by the grain-size distribution. Among other things, F(q i,q c,i ) depends on the critical Shields number of incipient motion associated with the grains of size D i, q c,i, which depends in turn on the grain-size distribution at the bed surface. Within the framework of this approach, authors have therefore focused on the determination of both F(q i,q c,i ) and q c,i from experimental data and theoretical modeling. [5] Parker et al. [1982], for example, proposed that h i F q i ; q c;i ¼ a q l p; i qm i b q m c;i ð5þ where a, b, l, m and p are adjustable coefficients depending on the configuration investigated [Parker et al., 1982; Parker, 1990; Wilcock and Kenworthy, 2002; Wilcock and Crowe, 2003; Powell et al., 2001, 2003]. As the Shields number approaches the value of incipient sediment motion, sediment transport becomes sensitive to local bed heterogeneities and pavement effects, which may not be captured by (5). This led to the formulation of several transport laws specifically designed to describe sediment transport in the vicinity of the threshold [Parker et al., 1982; Parker, 1990; Wilcock and Kenworthy, 2002; Wilcock and Crowe, 2003]. [6] The prediction of the critical Shields number of the grains of size D i, q c,i, rests on a qualitatively well observed phenomenon: the ability of a grain of size D i to move with respect to some representative grain size D c increases with the ratio D i /D c [Parker, 2008]. This is usually formulated by a relation of the form q c;i ¼ G q c;c D i D c ; ð6þ where q c,c = t c,c /rgd c is the critical Shields number of the representative grain size, t c,c is the value of the shear stress at which grains of size D c start moving and G is the so-called hiding function which incorporates relative size effects [Kirchner et al., 1990; Buffington et al., 1992]. [7] The frequently cited forms of the hiding function G(D i /D c ) are listed in Table 1. Most of them reduce to q c;i q c;c ¼ g ; ð7þ D i D c with 0 g 1. Equation (7) presents two interesting limiting cases. The first one corresponds to g = 0 which implies q c,i / q c,c = 1. In this case, a grain of given size D i within a mixture has exactly the same critical Shields number as it would have if the bed was composed entirely of grains of size D i. In such a scenario, the initiation of transport of sediment mixtures is highly selective, based on grain size. The second limiting case corresponds to g = 1 which implies q c,i /q c,c = D i /D c so that t c,i = t c,c where t c,i is the value of the shear stress at which grains of size D i start moving. In this limiting case, the effect of the mixture is to equalize the threshold of motion so that all grains are mobilized at the same absolute boundary shear stress, a configuration referred to as equal mobility. In practice, sediment mixtures appear to behave in between these two limiting cases. 2of13

3 Figure 1. (a) Scheme of the experimental setup. (b) Sediment bed (top view) for an experimental run of the series 2 (f 1 = 0.71). On this image, the large grains are dyed in black whereas the small ones are white. [8] The transport laws discussed so far rely on the fit of empirical sediment transport rate curves. Consequently the physical meaning of the coefficients involved in equations (5) and (7) is unclear. An alternative approach to the problem of bedload transport is to consider that q s,i can be written q s;i ¼ dv i n i V i ; ð8þ where n i (dimensions [L] 2 ) is the surface density of moving particles of the ith population defined as the number of particles of size D i in motion per unit bed area, V i is their averaged velocity and dv i is the volume of an individual particle. A better insight in the problem of bedload transport can be gained from the separate measurements of V i and n i and the determination of their dependence on the control parameters. This approach has motivated several experimental investigations of bedload transport at the grain scale and the development of statistical transport models [Einstein, 1937, 1950; Francis, 1973; Fernandez-Luque and Van Beek, 1976; Abbott and Francis, 1977; Lee and Hsu, 1994; Ancey et al., 2008; Ganti et al., 2009; Ancey, 2010; Furbish et al., 2012a, 2012b, 2012c; Roseberry et al., 2012]. [9] Lajeunesse et al. [2010], in particular, reported the results of an experimental investigation of the motion of bedload particles, under steady and spatially uniform turbulent flow above a flat sediment bed of uniform grain size. Using a high-speed video imaging system to investigate the trajectories of the moving particles, they found that: (1) the surface density of moving particles increases linearly with (q q c ), n s ¼ aq ð q cþ; ð9þ where a = is a coefficient related to the erosion and deposition rate of bedload particles and s =1/D 2 is the number of particles at repose per unit surface of the bed; (2) the average particle velocity increases linearly with q 1=2 q 1=2 c, with a finite nonzero value at threshold, V V c ¼ b q 1=2 q 1=2 c ; ð10þ V s 3of13 p where V s ¼ ffiffiffiffiffiffiffiffiffi RgD is a characteristic sedimentation velocity and b = and V c /V s = are two fitting coefficients. [10] As far as we know, experiments similar to those of Lajeunesse et al. [2010] have been carried out with homogeneous sediment beds only. In this paper, our objective is therefore to extend the experimental approach of Lajeunesse et al. [2010] to the case of a mixture of sediments of different sizes. To this end, we investigate experimentally bedload transport above a bed composed of a bimodal mixture of small and large grains sheared by a turbulent flow in a small experimental flume. We focus on the measurement of the surface density of moving particles and of their average velocity for each size fraction. The experimental results allow us to characterize how sediment transport depends on the proportion of small and large grains on the bed. 2. Experimental Apparatus and Procedure 2.1. Experimental Setup [11] The experiments were conducted in a tilted laboratory flume of length 240 cm and width W = 9.6 cm (Figure 1a). The bottom of the flume was covered with a sediment bed composed of a mixture of small and large irregularly shaped quartz grains of density r s = 2650 kg m 3 (Figure 1b). The average size of the small grains determined from sieve analysis was D 1 = mm whereas the average size of the large grains estimated from image analysis was D 2 = mm. [12] The bed, typically 10 centimeters thick, was flattened by sweeping a rake whose orientation and distance from the bottom of the flume were constrained by two rails parallel to the channel. The bed slope S was measured with a digital inclinometer (accuracy 0.1 ). [13] Once the bed was ready, water was injected by a pump at the upstream flume inlet with a constant flow discharge Q w measured with a flowmeter (accuracy 0.01 L/minute). The discharge was always high enough for the flow to form across the full width of the flume. To prevent any disturbance of the bed, water was not injected as a point source but rather it overflowed smoothly onto the river bed via a small reservoir (see Figure 1). The reservoir extended across the full width of the channel and therefore guaranteed a flow injection that was uniform across the channel width.

5 Figure 2. PDF of the streamwise velocity component of the large particles, u x,2, measured for q 2 = and f 1 = The dotted line corresponds to a fit of the data by equation (13). by computing this apparent velocity. This allowed us to define a cutoff velocity in the range 1 3cms 1 depending on the water flow rate, below which the velocity measurement was considered too noisy to be taken into account. [24] Measuring a large number (typically 7000 for each experimental run) of large particles instantaneous velocities allowed us to compute their experimental distributions. For the explored range of parameters, these distributions all exhibit a peak and are skewed to the large velocities (see Figure 2). They are well fitted by a chi-squared distribution: c 2 u k=2 1 x;2 e ux;2=2 u x;2 ; k ¼ 2 k=2 Gðk=2Þ ; ð13þ where G is the gamma function and k is a fitting coefficient, equal to the mean of the chi-squared distribution. The average velocity of the large particles, V 2, was therefore computed for each experimental run from a fit of the experimental distribution of u x,2 by equation (13). [25] For each experimental run, the transport rate of the large particles deduced from the average velocity and the surface density of moving particles, dv 2 n 2 V 2 (see equation (8)), was compared to the transport rate directly measured with the scale at the flume outlet. As illustrated by Figure 3, the two methods led to similar values thus confirming the consistency of the experimental data Measurement of the Velocity of the Small Grains [26] The noise introduced by the water surface oscillations did not prevent us from detecting whether a small grain is moving or not using the procedure described above. However, it did not allow us to measure the velocity of the small particles with a satisfying accuracy. As a result the automated technique used to measure the experimental distributions of the velocities of the large grains could not be employed with the small ones. The average velocity of the small particles, V 1, was therefore computed from the measurement of the transport rate q s,1 and the surface density of moving particles, n 1, using equation (8) 3. Experimental Results V 1 ¼ q s;1 dv 1 n 1 : ð14þ [27] We performed a total of 53 experiments with bed slopes ranging from to , water discharges ranging from 20 to 62 L min 1 and flow depth ranging from 1 to 3 cm. These experimental runs are organized in 4 series, labeled from 0 to 3. Each series correspond to a different bed composition, namely f 1 = 1 (uniform bed of small particles), 0.92, 0.71 and 0.54, respectively. We also used an additional set of data, labeled 4, extracted from Lajeunesse et al. [2010] and corresponding to measurements performed with a uniform bed of large particles i.e. f 1 = 0. Altogether, these 5 experimental series cover a relatively wide range of parameters summarized in Table Phenomenology [28] The images acquired with the fast camera reveal that the motion of the sediments entrained by the flow above a bimodal bed is qualitatively similar to the one reported for the case of a unimodal bed of sediment: 1) only a small fraction of the sediment bed particles is entrained by the flow; 2) these bedload particles exhibit intermittent behavior: periods of motion, called flights and characterized by a highly fluctuating velocity, alternating with periods of rest. [29] However the relative proportion of small and large particles at the bed surface influences quantitatively the 5of13

6 Figure 3. Transport rate of the large grains deduced from the average velocity and the surface density of moving particles, d 2 n 2 V 2 (see equation (8)), plotted as a function of the sediment transport rate directly measured with the scale at the flume outlet. Circles, squares and diamonds correspond to the experimental series 1, 2, and 4, respectively. The solid line corresponds to a linear relationship of slope 1. sediment transport rate. The total sediment flux measured by the scale at the flume outlet is plotted on Figure 4 as a function of the flow shear stress for two different experimental series. For a given shear stress, the total sediment transport rate varies with the surface fraction of fine sediments, f Surface Density of Moving Particles [30] By analogy with Lajeunesse et al. [2010], we chose to normalize the surface density of moving particles of a given size D i with respect to the number of static particles of the same size available per unit surface of the bed, s i ¼ f i =D 2 i. We therefore introduce the dimensionless surface density of moving particles of size D i : n* i ¼ n i D 2 i f i with i ¼ 1; 2: ð15þ The dimensionless surface densities of large particles in motion, n* 2, measured for f 1 = 0.91 and f 1 = 0.54 (experimental series 1 and 3, respectively) are plotted as a function of the Shields number, q 2, on Figure 5. Two observations can be made: 1) n* 2 is 0 below a threshold value of the Shields number, which differs between the two series; 2) above this threshold, n* 2 increases linearly with q 2. These observations hold for all experimental series and both grain sizes. It seems therefore natural to define the critical Shields number of the grains of size D i, q c,i, from the threshold value below which n* i vanishes. A linear fit of n* i versus q i, i = 1,2, for each experimental series leads to the values of q c,i reported in Table 2. As commonly observed, the critical Shields number decreases when the proportion of fine sediments on the bed, f 1, increases. These variations will be discussed later. [31] Another interesting observation can be made from Figure 5: the slope of the relationship between the surface density of moving grains and the Shields number is identical for the two series of data plotted on this graph. This result can be generalized to both grain sizes (i = 1, 2) and all experimental series. Indeed, plotting the dimensionless surface densities measured for the 5 experimental series, n* i,as a function of (q i q c,i ) for both grain sizes reasonably merges all data on the same line (Figure 6). A fit of this line leads to the linear relationship n* ¼ n id 2 i ¼ ð4:2 0:2Þ q i q c;i : ð16þ f i Table 2. Range of Parameters Explored for Each Series of Experiments a Series f 1 S q 1 q 2 q c,1 q c,2 Re* 1 Re* 2 H/D 1 H/D a The series labeled 0 to 3 were performed by the authors, the data of the 4th series are extracted from Lajeunesse et al. [2010]. 6of13

7 Figure 4. Total sediment flux (g s 1 ) plotted as a function of the flow shear stress t (N m 2 ) for the experimental series 2 (f 1 = 0.71, solid circles) and 4 (f 1 = 0, open triangles). The solid lines correspond to the prediction of equation (23). This result is consistent with the observations of Lajeunesse et al. [2010] who reported a similar linear relationship above a uniform sediment bed (see equation (9)), with a coefficient equal to As a matter of fact, the measurements made by Lajeunesse et al. [2010] with uniform sediment beds of size 1.1 and 5.5 mm fall on the same trend (crosses on Figure 6) Average Velocities [32] As discussed in the experimental setup section, the average particle velocity of each size fraction, V i, was estimated for each experimental run, either from a fit of the velocity distribution for the large particles or from the ratio of the sediment flux to the surface density of moving particles for the small ones. By analogy with Lajeunesse et al. Figure 5. Dimensionless surface density of moving large particles, n* 2, as a function of the Shields number, q 2, for f 1 = 0.91 (experimental series 1, circles) and f 1 = 0.54 (experimental series 3, triangles). The two dotted lines correspond to linear fits of the data. 7of13

8 Figure 6. Dimensionless surface densities of moving grains n* i versus (q i q c,i ) for both grain sizes (i = 1, 2) and the 5 experimental series (see Table 2). Circles, squares and triangles correspond to the experimental series 1, 2 and 3, respectively, with solid symbols for large particles and open ones for small particles. Open and solid diamonds correspond to the series 0 (f 1 = 1) and 4 (f 1 = 0), respectively. The dotted line is a linear fit of our data. The data measured by Lajeunesse et al. [2010] with uniform sediment beds of size 1.1 and 5.5 mm are represented by crosses. [2010], we normalize the average particle velocities of each fraction p with respect to their characteristic settling velocity V s;i ¼ ffiffiffiffiffiffiffiffiffiffi RgD i, thus introducing the dimensionless average particle velocity V* i ¼ V i V s;i : ð17þ [33] As shown in Figure 7, the dimensionless particle velocities of both small and large grains converge onto a single line independent of the proportion of fine sediment, f 1, when plotted as a function of q 1=2 i q 1=2 c;i. Thus the particle velocity can be written V i V c;i ¼ b q 1=2 i q 1=2 p c;i ; V s;i ¼ ffiffiffiffiffiffiffiffiffiffi RgD i ; V s;i ð18þ where V c,i is the (non-zero) average particle velocity at the threshold. Fitting the data, the slope and the threshold velocity are found to be b ¼ 5:1 0:2; V c;i =V s;i ¼ 0:11 0:02 ð19þ [34] Again, we recover a result consistent with the measurements performed by Lajeunesse et al. [2010] who found a linear relationship (see equation (10)) above a uniform sediment bed. The slope coefficient reported by Lajeunesse et al. [2010], , is slightly smaller than the one measured in the present study. However, despite some dispersion, the velocity measurements of Lajeunesse et al. [2010] (crosses on Figure 7) are compatible with our data. It is therefore difficult to assess whether this difference of slope coefficient is significant or due to experimental uncertainties. [35] Finally, the above result suggests that the velocity is discontinuous at the threshold of sediment transport, particles moving there with a nonzero velocity V c. As pointed out by Lajeunesse et al. [2010], this result is supported by the experimental observation that, close to the threshold, a particle, once dislodged from the sediment bed, does not stop immediately but may travel over some distance. It is also consistent with the hysteretic nature of the threshold of motion first evidenced by Hjülstrom [1935] and with observations of the motion of a single particle on a fixed bed by Francis [1973] and Abbott and Francis [1977]. 4. Discussion 4.1. Summary of the Experimental Results [36] To summarize, we have investigated bedload transport of a bimodal sediment bed, composed of a mixture of two populations of quartz grains of size D 1 = 0.7 and D 2 = 2.2 mm, respectively. The particles of both populations are entrained by a steady and uniform turbulent flow above a flat topography. In this equilibrium regime, the erosion and deposition rates balance each other. The granulometric composition of the bed is characterized by the surface fraction of small grains, f 1, defined as the fraction of the bed surface covered with grains of the population 1. [37] Our experimental results show that: 8of13

9 Figure 7. Average dimensionless velocities versus q 1=2 i q 1=2 c;i for both grain sizes (i = 1, 2) and the 5 experimental series (see Table 2). Circles, squares and triangles correspond to the experimental series 1, 2 and 3, respectively, with solid symbols for large particles and open ones for small particles. Open and solid diamonds correspond to the series 0 (f 1 = 1) and 4 (f 1 = 0), respectively. The dotted line is the linear fit of all the experiments. The data measured by Lajeunesse et al. [2010] with uniform sediment beds of size 1.15 and 5.5 mm are represented by crosses. [38] 1. Particles begin to move for Shields number larger than a threshold q c,i, which was determined from the extrapolation to zero of the surface density of moving particles. This threshold differs from one population of grains to the other and decreases when the surface fraction of small grains increases (see Table 2). [39] 2. Above the threshold Shields number, the dimensionless surface density of moving particles, n* i, increases linearly with (q i q c,i ) following equation (16) independent of the fraction of small grains (see Figure 6). This linear dependence is consistent with the measurements performed by Lajeunesse et al. [2010] who found a similar linear relationship (see equation (9)) above a uniform sediment bed, with a coefficient equal to These experimental investigations demonstrate that the dimensionless number of particles in motion per unit surface is independent of both the grain size and the surface fraction of small particles, at least for the range of parameters explored. [40] 3. The dimensionless average particle velocity increases linearly with (q 1=2 i q 1=2 c;i ) following equation (18) independent of the fraction of small grains (see Figure 7). The slope of the linear law and velocity at threshold are given by b = and V c,i /V s,i = Again, we recover a result consistent with the measurements performed by Lajeunesse et al. [2010] and no significant dependence is noted with both the grain size and the surface fraction of small particles, at least for the range of parameters explored. This velocity law fully agrees with Lajeunesse et al. [2010, equation (13)] which arises from a simple force balance. The value of the coefficient b is close to that of previous investigations, which found b in the range for an even larger range of grain sizes [Nino and Garcia, 1994] Erosion-Deposition Model for a Bimodal Bed [41] Our experimental observations suggest that the erosion-deposition model established by Charru [2006] and Lajeunesse et al. [2010] for a bed of uniform sediment can be generalized to the case of a bimodal one. These authors showed that the deposition rate scales as the surface density of moving particles divided by the time necessary for a particle to settle from a characteristic height equal to the grain size. Generalizing this argument to the case of a bimodal bed leads to _n d;i ¼ 1 n i p ; V s;i ¼ ffiffiffiffiffiffiffiffiffiffi RgD i ; ð20þ c d D i =V s;i where c d is a dimensionless coefficient. Similarly, the erosion rate is proportional to the number of particles at repose per unit surface of the bed _n e;i ¼ 1 c e f i D 2 i 1 t e;i ; ð21þ where c e is a dimensionless coefficient. The term t e,i is a typical hydrodynamic timescale which can be thought as the time needed for a particle at rest, submitted to the force exerted by the flow, to escape the small trough where it is trapped and reach some escape velocity of the order of the settling velocity [see Lajeunesse et al., 2010, equation (12)]. 9of13

10 Figure 8. Transport rate (g s 1 ) of all particles (solid circles) and of the small particles only (open circles) plotted as a function of the flow shear stress t (N m 2 ) for the experimental series 2 (f 1 = 0.71). Solid lines correspond to the predictions of equation (23). [42] Our experiments were performed above a steady state flat topography. In this equilibrium regime, the erosion and deposition rates balance each other so that n i D 2 i f i ¼ c e c d r r s q i q c;i : ð22þ Equation (22) is consistent with the experimental relation (16), with c e /c d r/r s = which leads to c e /c d = Note that this value of c e /c d is different from the one reported by Lajeunesse et al. [2010]. The contradiction is however only apparent. Indeed, Lajeunesse et al. [2010, equation 15] forgot the term r/r s. As a result, the term r/r s is included in their fitting coefficient c e. This being taken into account, the values of c e /c d are identical. [43] Finally, the sediment flux of each size fraction can be computed from (18) and (22) q s;i pffiffiffiffiffiffiffiffiffiffiffi RgD 3 i ¼ 56:6 r 1=2 f r i q i q c;i q i q 1=2 c;i þ 0:022 ; ð23þ s from which we deduce the total sediment flux per unit of width, q m = q s,1 + q s,2. The mass fluxes of small and large particles, and consequently the total sediment flux, calculated from (23) are consistent with the transport rate measurements performed with the scale as illustrated on Figures 4 and Variation of the Critical Shields Number With the Surface Fraction of Small Grains [44] Our experimental results demonstrate that, once in motion, the grains obey similar equations whether the bed is made of uniform sediment or of a bimodal mixture: in both cases the surface density of moving particles increases linearly with the Shields number and the average velocity increases linearly with the square root of the Shields number. The only difference evidenced by our experimental results concerns the value of the critical Shields number. Above a uniform sediment bed, the latter is uniquely determined from the particle Reynolds number through the Shields curve [Shields, 1936; Wiberg and Smith, 1987]. In the case of a bimodal bed, our experiments show that the critical Shields number for a given grain size depends also on the grain-size distribution at the bed surface as already observed in previous investigations [Wilcock, 1998]. This is illustrated in Figure 9 which displays the critical Shields numbers of both populations of grains, q c,1 and q c,2, as a function of the surface fraction of small grains, f 1. Two observations can be made: 1) the critical Shields number of the small grains is larger than that of the large grains, whatever the value of f 1 ; 2) the critical Shields numbers of both grain populations decrease linearly with f 1. [45] Despite a considerable amount of work partly summarized in the introduction, a mechanistic prediction of the critical Shields number of a sediment mixture remains an open problem which lies far beyond the scope of the present paper. Instead, we will now take advantage of the experimental observations to propose a simple statistical model that can describe the variation of the critical Shields numbers of the small and large grains composing a bimodal bed as a function of the surface fraction of small grains, f 1. [46] Let us consider a test grain of size D 2 at rest on a bimodal sediment bed characterized by its surface fraction of fine sediment, f 1. Our test grain has a probability f 2 =1 f 1 to repose on grains of size D 2. In that case, it is expected to start moving when the Shields number reaches a value q c,2/2 equal to the critical Shields number for a bed composed exclusively of grains of size D 2. The value of q c,2/2 is deduced from the experimental series 4 performed with a 10 of 13

11 Figure 9. Critical Shields number of the small (q c,1, open circles) and the large (q c,2, solid circles) grains as a function of the surface fraction of small grains f 1 on the bed. These critical Shields values and the corresponding error bars were obtained from a linear fit of the surface density of moving grains as discussed in section 3.2. Crosses and triangles correspond to measurements of the critical Shields number of a single large grain protruding above or partly buried in a bed of small grains, respectively. Stars correspond to measurements of the critical Shields number of a few small grains deposited on a bed of large grains. Dotted lines represent the equations (24) and (25). bed of uniform sediment of size D 2 leading to q c,2/2 = [47] Our test grain has also a probability f 1 to repose on grains of size D 1. In that case, it is expected to start moving when the Shields number reaches a value q c,2/1 similar to the critical Shields number of a grain of size D 2 which would be at repose on bed composed exclusively of grains of size D 1. Consequently, on average, we expect that the critical Shields number of the large grains is q c;2 ¼ f 1 q c;2=1 þ ð1 f 1 Þq c;2=2 ¼ q c;2=1 q c;2=2 f1 þ q c;2=2 : ð24þ [48] Equation (24) predicts a linear relationship between q c,2 and f 1 which is consistent with the data plotted on Figure 9. To further test this equation, we need to estimate the value of q c,2/1. This was achieved by a series of specific experimental runs in which we prepared a sediment bed exclusively composed of small grains of size D 1 onto which we added a single large grain of size D 2. The critical Shields number of the latter was determined by increasing slowly the flow discharge until it was carried away. We performed several runs using different large grains to account for the slight variations of shape and size. Some of the experimental runs were conducted with the large grain half-buried in the bed (triangles on Figure 9). In other experimental runs, the large grain was posed above the sediment bed so that it fully protruded (crosses on Figure 9). Both protocols led to quasiidentical values (see Figure 9) so that, on average, q c,2/1 = As shown on Figure 9, the critical Shields number of the large grains calculated from equation (24) using the experimental measurements of q c,2/1 and q c,2/2,isconsistent with our experimental data. [49] A similar reasoning leads to the following expression for the critical Shields number of the small grains: q c;1 ¼ q c;1=1 q c;1=2 f1 þ q c;1=2 : ð25þ where q c,1/1 is the critical Shields number of a bed of uniform grain size D 1 and q c,1/2 is the critical Shields number for a single small grain of size D 1 at repose on bed composed exclusively of large grains of size D 2. The experimental series 0 performed with a bed of uniform sediment of size D 1 gives q c,1/1 = As for q c,1/2, it was estimated from specific experimental runs in which we deposited a few small grains of size D 1 onto a bed exclusively composed of large grains of size D 2. The critical Shields number of these latter was determined by increasing slowly the flow discharge until they were carried away, leading to q c,1/2 = (stars on Figure 9). The critical Shields number of the small grains calculated from equation (25) using the experimental measurements of q c,1/1 and q c,1/2 is plotted as a function of f 1 on Figure 9. Again, the prediction of this very simple model is in good agreement with the experimental data. [50] According to the model developed in this section, the critical Shields number of each of the two populations of grains forming a bimodal sediment bed is a linear combination of two asymptotic configurations: a first one where the grains have all the same size and a second one where a grain 11 of 13

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