Theoretical development on the effects of changing flow hydraulics on incipient bed load motion



[1] Several decades of flume and field measurements have indicated that in rough turbulent flows the critical Shields stress increases with increasing slope and associated decreasing relative depth. This result contradicts the usual consideration of a decreased critical Shields value on very steep slopes because of increased gravitational effects. However, recent studies have demonstrated that these experimental results could be reproduced with a force balance model if the classical logarithmic velocity profile was replaced with a velocity profile that was more compatible with available velocity measurements over gravel beds. These measurements indicate the existence of a roughness layer that is a zone of almost constant velocity close to the bed, whose properties (mean velocity and turbulence) depended on the flow's relative depth. Unfortunately, velocity profile measurements for low relative depth associated with steep slopes are scarce, and it is still difficult to include such flow properties in a force balance model. Flow resistance data (on the basis of depth average velocity measurements) are very common and cover a wide range of slopes and relative depths. In this paper these data are used to fit a velocity profile including a roughness layer. When used in a force balance model for incipient motion, it adequately reproduced a data set composed of 270 critical Shields values measured in a flume with near-uniform sediments. The relevance of this research to field problems is discussed using a data set composed of 92 critical Shields stresses obtained from field measurements. Finally, a model is proposed for field applications taking into account the slope effect.

1. Introduction

[2] Bed load prediction is of primary importance for river engineering, fluvial geomorphology, eco-hydrology, environmental surveys and management, and hazard prediction. Using similarity principles, Shields [1936] established a framework for bed load prediction that is still in use today. He considered bed load a threshold phenomenon and established a diagram relating the dimensionless critical shear stress θc = τc/[(ρsρ)gD] (where τc = ρgHS is the critical shear stress, ρs is the sediment density, ρ is the water density, g is the acceleration of gravity, D is the grain diameter, H is the water depth and S is the energy slope) to the roughness Reynolds number Re* = u*D/ν (where u* = (τ/ρ)1/2 is the shear velocity). Although this curve is not easy to use (because both θc and Re* depend on the shear velocity u*, which implies an iterative approach), one interesting practical issue is that θc was hypothesized by Shields to be constant when Re* > 1000, which is the case for most natural flow conditions (rough and turbulent flows). Thus, knowing the value of this constant, the calculation of threshold flow conditions (characterized by a flow depth H) for a given sediment (characterized by its grain-size distribution curve) and a given energy slope S should be straightforward.

[3] However, whereas Shields [1936] proposed an asymptotic value of 0.06 for θc, the appropriate value for this constant has been widely and continuously discussed since that time. For instance, the well-known bed load transport equation proposed by Meyer-Peter and Muller [1948] considered θc = 0.047. Values as low as 0.01 were also proposed [Fenton and Abbott, 1977; Carling, 1983; Mueller et al., 2005] as well as values higher than 0.1 [Mizuyama, 1977; Church, 1978; Reid et al., 1985; Mueller et al., 2005]. More generally, values were proposed in the range 0.03 [Parker et al., 2003] to 0.07 (an exhaustive review was provided by Buffington and Montgomery [1997]), with a mean value at approximately 0.045 [Gessler, 1971; Miller et al., 1977; Yalin and Karahan, 1979; Saad, 1989]. The importance attached to this question can easily be understood when considering that in most natural gravel bed rivers the Shields number θ barely exceeds 20% of the critical value θc [Parker, 1978; Andrews, 1983; Mueller et al., 2005; Ryan et al., 2002; Parker et al., 2007] and that for these flow conditions, transport rates increase by several orders of magnitude for very small changes in shear stress, which can lead to very large errors in bed load prediction if θc is not correct.

[4] Numerous explanations can be given for the uncertainty on θc [Buffington and Montgomery, 1997], including the definition of incipient motion itself, the shear stress definition (mean or instantaneous) and calculation (from the energy slope, the velocity profile or the Reynolds stress profile), sediment specificities (near-uniform or nonuniform) and the general protocol used (measurement techniques, duration, and sidewall correction method). A subject that has received less attention is the natural dependence of θc on flow parameters. In particular, the shear stress τ is usually calculated from the energy slope S and the flow hydraulic radius R. Calculating it thus, the Shields parameter (equation (1)) indicates dependence with the relative depth R/D (the ratio between the hydraulic radius and the grain diameter) and the slope S (considered here in the streamwise direction):

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where s = ρs/ρ is the sediment's relative density. Shields himself first recognized these dependences [Shields, 1936] and observed increasing critical Shields stress with increasing slopes. This result has been confirmed since that time by several researchers, on the basis of both flume and field experiments [Tabata and Ichinose, 1971; Aksoy, 1973; Bathurst et al., 1982; Bettess, 1984; Bathurst, 1987; Graf and Suszka, 1987; Tsujimoto, 1991; Shvidchenko and Pender, 2000; Shvidchenko et al., 2001; Mueller et al., 2005; Armanini and Gregoretti, 2005; Vollmer and Kleinhans, 2007; Lamb et al., 2008] (see also J. Bogardi, Sediment transportation in alluvial streams, lecture notes from an international postgraduate course on hydrological methods for developing water resources management, Research Institute for Water Research Development, UNESCO, Budapest, Hungary, 1980). Other researchers proposed a θc(R/D) equation [Mizuyama, 1977; Torri and Poesen, 1988; Suszka, 1991; Lenzi et al., 2006] instead of a θc(S) equation, but both types of equations give similar results. This is illustrated in Figure 1, where Mizuyama [1977] and Suszka [1991] flume experiment results are compared to other flume results after being rearranged in a θc (S) equation using equation (1).

Figure 1.

Comparison between different critical Shields functions derived for rough turbulent flows in flume experiments.

[5] Since these observations were mostly reported for low relative depth R/D corresponding to gravel initiation of motion on steep slopes, Bathurst et al. [1982] hypothesized that the traditional Shields approach (which assumes a constant value of approximately 0.04–0.06 at a high Reynolds number) could be based on the coincidence that most studies involved values of channel slope small enough that the real variation of θc with flow conditions has been too small to deserve comment. For instance, Shields himself considered the slope effects negligible for the low slopes (≤1%) examined in his analysis [Shields, 1936]. In addition, decreasing critical Shields stress could logically have been expected when the channel slope becomes very steep because of increased gravitational effects. The inverse was observed. This contradictory result was interpreted as the consequence of additional effects. For instance, Coleman and Nikora [2008] considered it the consequence of flow momentum exchanges through highly permeable gravel beds. It has also often been interpreted as the consequence of bed form drag [Buffington and Montgomery, 1997; Mueller et al., 2005]. This latter hypothesis also holds for flume results with near-uniform sediments, as most studies extrapolated to zero the bed load transport rate data even though associated flows are usually associated with the presence of small bed forms (undulating beds), whatever the initial planar bed surface considered. Recent work [Recking et al., 2008a, 2008b] investigated bed load and flow resistance interactions using a data set consisting of 1551 flume data. In these studies, the critical Shields value was defined by incipient deformation of an initially planar bed (recognized by changes in flow resistance with increasing discharge). The results [Recking, 2008] showed a strong correlation between θc and S, similar to those obtained in other studies (Figure 1), despite the exclusion of bed form drag in the approach used by Recking et al. [2008b]. Consequently, although bed forms can contribute to increasing Shields values in some circumstances, this does not explain the variations in θc observed with low relative depth on steep slopes. Another argument in favor of not considering the relative depth was proposed by Yalin [1977]. He argued that since the initiation of motion must depend only on the flow in the vicinity of the bed and since the flow in the vicinity of the bed is independent of the relative depth R/D, initiation of motion must be independent of R/D. This is true as long as the Nikuradse profile is valid with a well-developed logarithmic velocity profile close to the bed. However, for steep slopes and low relative depth flow conditions over gravel beds, several authors measured a deviation from the logarithmic profile both in flume experiments and in the field [O'Loughlin and Annambhotla, 1969; Christensen, 1971; Ashida and Bayazit, 1973; Mizuyama, 1977; Day, 1977; Nowell and Church, 1979; Marchand et al., 1984; Nakagawa et al., 1988; Bathurst, 1988; Jarrett, 1990; Aguirre-Pe and Fuentes, 1990; Robert, 1990; Wiberg and Smith, 1991; Tsujimoto, 1991; Pitlick, 1992; Ferro and Baiamonte, 1994; Byrd et al., 2000; Byrd and Furbish, 2000; Nikora et al., 2001; Katul et al., 2002; Franca, 2005]. Instead, a more complicated pattern was generally described, with a roughness layer close to the bed where the velocity profile was nearly constant, and a second zone (above the former), where the velocity profile was logarithmic. The roughness layer corresponds to zones of intense shear downstream of each roughness element (grains and particle clusters, producing Kelvin-Helmholtz instabilities) where the kinetic energy of the mean flow is transformed into turbulence energy. This turbulence thus produced intensifies the mixing or transfer of momentum, resulting in a continuous adjustment in the velocity profile close to the bed [O'Loughlin and Annambhotla, 1969]. Several studies showed that in these flows the mean flow velocity and turbulence intensity near the bed decreased with decreasing relative depth R/D [Bayazit, 1976; Tsujimoto, 1991; Wang et al., 1993; Dietrich and Koll, 1997; Carollo et al., 2005; Lamb et al., 2008] which contradicts Yalin [1977] argument.

[6] Grain movements are governed by the association of drag and lift forces, which are dependent, respectively, on the mean flow velocity and the velocity gradient in the vicinity of the grain. If the mean velocity and the velocity gradient are affected by low relative depths on steep slopes, as suggested above, this would have direct consequences on the force balance and θc. This led Tsujimoto [1991] to postulate that the effect of slope on θc is composed of two parts (equation (2)): Ψ1(S) as an effect of gravity itself and Ψ2(S) as an effect of the degeneration of velocity distribution owing to small relative depth.

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The former would be a decreasing function of S while the latter would be an increasing function of S. Increasing θc with increasing slope implies that Ψ2 grows faster than Ψ1 when the slope is increased. Considering equation (1), this means that the changing flow hydraulics (mean velocity and turbulence profiles) with increasing slopes results in critical relative depths (i.e., R/D measured at incipient motion) declining more slowly than the rate of slope increase. This could be demonstrated using an analytical model on the basis of a force balance, but the problem is estimating the appropriate near-bed flow velocity and its variations with relative depth on steep slopes. Tsujimoto [1991] first successfully reproduced these opposite influences of decreasing relative depth and simultaneously increasing slope using a near bed flow velocity function fitted on his own data. Despite large uncertainties associated with calibrating the model, it produced increasing critical Shields stress values with increasing slope values. Armanini and Gregoretti [2005] successfully reproduced incipient motion experiments [Gregoretti, 2000] on steep slopes (21% < S < 36%) by replacing the logarithmic profile with a linear profile [Nikora et al., 2001]. Their objective was to study effects of grain protrusion and the linear profile used in the model was fitted from the experimental data set considered. By correcting the logarithmic Nikuradse function with a coefficient (a function of the ratio between flow depth and roughness layer thickness) deduced from a turbulence model [Bezzola, 2002] for the mean flow velocity and adding a turbulence-induced pressure fluctuation, Vollmer and Kleinhans [2007] also reproduced opposite influences of decreasing relative depth and simultaneously increasing slope but finally concluded that both effects were partly compensated as soon as the relative depth was higher than 2 (slopes less than 6%), producing a constant Shields value. Lamb et al. [2008] developed a near-bed mean flow velocity profile by matching Nikuradse [1933] logarithmic profile with a quadratic profile obtained by integrating a constant mixing length within the roughness layer (which they assumed proportional to the bed roughness ks). They also considered a fluctuating turbulent component from data available in the literature. Thus, they obtained a variation of θc with increasing slope much similar to the equations presented in Figure 1.

[7] Whatever the approach used, one major difficulty consists in evaluating the model's ability to reproduce the roughness layer velocities and its variations with relative depth because available roughness layer velocity measurements are scarce. On the other hand, a large number of flume and field measurements are available for depth average velocities, covering a wide range of relative depth and slope. These data were analyzed in the work of Recking et al. [2008b]. In this paper, they are used to fit a velocity profile incorporating a roughness layer [Aguirre-Pe and Fuentes, 1990]. This reproduced a confident variation of the near-bed flow velocities with varying relative depth R/D. This flow velocity function was incorporated into a critical Shields stress model on the basis of a force balance. Finally, the results were compared with flume and field critical Shields stress data.

2. Theoretical Analysis

[8] The objective of this section is not to develop an exact analytical model for gravel incipient motion but rather to investigate whether it is possible to reproduce, at least qualitatively, the observed variation of θc considering available flow resistance data. The forces acting on a cohesionless particle are lift (FL), drag (FD), buoyancy and gravity (W). At the threshold of motion, forces acting tangential (Ft) and normal (Fn) to a particle must satisfy the relation:

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where ϕ is the intergranular friction angle. Considering the angle a of the channel bed slope Sb (Sb = tana), when rearranged with forces this yields:

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The forces can be expressed as:

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where CD and CL are drag and lift coefficients, D is the grain diameter, Ax is the cross-sectional area of the particle, which is perpendicular to and exposed to the flow, Vp is the volume of the particle (used to calculate the gravity force ρsgVp), Vps is the submerged volume of the particle (used to calculate the buoyancy force ρgVps) and uD is an integration of the velocity profile over the exposed height of the grain. We will approximate uD by the fluid velocity calculated at the center height of the particle y = D/2 [Ikeda, 1982]. Very sophisticated models have been proposed in the past [Wiberg and Smith, 1987; Bridge and Bennett, 1992] by integrating the grain position and its protrusion into the bed. As the purpose of this study was to test the adequacy between measured flow resistance and critical Shields variations, a simpler model was considered here. For simplicity reasons, particles are assumed to be spherical and fully exposed (Figure 2), with the location of the virtual origin of the velocity profile located in the vicinity of the bottom of the grain particles considered (see discussion by Ikeda [1982] for an explanation of this choice). It is necessary to consider a decrease in Ax and Vps when the flow relative depth R/D is less than 1 on very steep slopes [Lamb et al., 2008]. This was done by writing Ax = ξAA and Vps = ξVVp (Figure 2), where A = π (D/2)2 is the total grain surface, Vp = 4/3π (D/2)3 is the total grain volume, and ξA and ξV are geometrical coefficients, which are unity when R/D > 1 and a function of R/D when R/D < 1. Combining equations (4)(10) yields the following expression for the critical Shields number:

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with ξA and ξV defined by (see appendix for further detail):

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For a given bed slope (a constant) and material (characterized by D and ϕ), θc will depend on CD, CL and the velocity profile. Usually this equation is solved by ignoring the lift force (CL = 0), but without any proper justification [Vanoni et al., 1966] because both analytical and experimental studies have confirmed its presence, with a slight effect on incipient motion [Einstein and El-Samni, 1949; Inokuchi and Takayama, 1973]. The slope is also often ignored (a = 0). The term (u/u*)D is usually calculated from the Nikuradse logarithmic velocity profile (equation 11):

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where Br = 8.5 for rough flows (Re* > 70) and ks is the equivalent Nikuradse bed roughness. In its simplest form, Sb = 0, ϕ = 52° [Buffington et al., 1992], CD = 0.45, CL = 0, ks = D and (u/u*) calculated at y = D/2, the model produces a constant critical Shields value θc ≈ 0.06.

Figure 2.

Exposed grain surface reduction when the grain is partially submerged.

[9] However, as discussed in the introduction, when flows occur at low relative depths, the velocity profile was observed to progressively deviate from the classical logarithmic profile. Instead, a more complicated profile was described (Figure 3a), made up of a first zone close to the bed (called the roughness layer), where the mean velocity is almost uniform, and a second zone located above it, where the logarithmic velocity function (equation (11)) is valid. More precisely, two sublayers were considered [Nowell and Church, 1979; Nikora et al., 2001], with a first zone corresponding to the flow below the top of roughness elements and essentially controlled by the roughness density and form drag induced by each element, and a second zone, above it, produced by the wakes shed from roughness elements (also called the wake zone). The roughness layer thickness is the grain diameter in order of magnitude [Nowell and Church, 1979; Tsujimoto, 1991; Carollo et al., 2005; Manes et al., 2007], but it is likely to vary with slope and relative depth, especially for the wake zone because the wake's frequency and size are related to the mean flow velocity and sediment size (through the Strouhal number). The roughness layer also varies with the concentration of protruding sediments (the ratio between the number of grains and the maximum number of grains that can be arranged in the reference area). An optimum concentration was observed over which the roughness layer effects stabilized [Carollo et al., 2005] or even decreased with bed smoothing [Nowell and Church, 1979]. However, the measurements available indicate that the roughness layer still exists with a maximum concentration, i.e., with a gravel bed of near-uniform sediment distribution [Aguirre-Pe and Fuentes, 1990; Tsujimoto, 1991; Wang et al., 1993; Manes et al., 2007]. This velocity profile can be described by the following equations [Aguirre-Pe and Fuentes, 1990]:

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where β is the ratio between the roughness layer thickness and the grain diameter, α is a grain diameter factor in the Nikuradse equivalent roughness used for the logarithmic part of the profile. Figure 3b shows the Nikuradse profile and its modification after addition of a roughness layer using equations (12) and (13) (in this example α was fitted for the case β = 1 and R/D = 3).

Figure 3.

(a) Velocity profile measured over gravel bed (from Marchand et al. [1984], Lake Creek) and (b) the standard logarithmic profile with its approximation with a roughness layer for a given mean velocity U.

[10] A function α(R/D) must be determined to calculate (u/u*) in the roughness layer. This was done using flow resistance measurements. Aguirre-Pe and Fuentes [1990] integrated equations (12) and (13) to obtain the following flow resistance equation:

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This flow resistance equation can be compared to measured flow resistance values (8/f)1/2 to determine α. Recking et al. [2008b, Figure 4] presents a data set comprising 612 flume and field flow resistance values obtained without bed load (this restriction made it possible to exclude bed form drag and the other additional effects described in the work of Recking et al. [2008b]. Keulegan's law [Keulegan, 1938], which is a flow depth integration of the Nikuradse profile (equation (11)) with ks = D, is also plotted for comparison. The deviation from Keulegan's law for R/D approximately lower than 25 was interpreted in the work of Recking et al. [2008b] as a progressive departure from the Nikuradse profile (equation (12)), with an increased influence of the roughness layer (equation (13)) as R/D decreased. The flume data obtained with near-uniform sediments of diameter D merge with field data obtained over poorly sorted sediments when the measured field flow depths were scaled with D84 (the size such that 84% of the sediment is finer). This occurs because in the presence of a nonuniform roughness height, the development of the roughness layer was found to be controlled essentially by larger elements protruding from the bed [White, 1940; Nowell and Church, 1979; Wiberg and Smith, 1991]; consequently, large particle diameters would be the best to scale the hydraulic radius in such turbulent flows. When R/D84 < 1, the flow could no longer interact with the top of larger grains and the wake effect was damped, producing a change in the flow resistance behavior. Consequently, we distinguished three flow types, as was observed by Bathurst [1985]: large-scale roughness (R/D84 < 1.4), intermediate-scale roughness (1.4 < R/D84 < 25), and small-scale roughness (25 < R/D84). The Figure 4 data were best fitted by the following flow resistance equation [Recking et al., 2008b]:

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where the subscript RL designates the roughness layer. Setting equations (14) and (15) equal to one another allows us to deduce the coefficient α:

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Used in equation (13), (u/u*)D in the roughness layer can be calculated. Since we have no idea of the appropriate value for β, two possible situations were considered:

Figure 4.

Data set of 612 flow resistance values obtained over flat beds without bed load transport (348 were obtained with nonuniform sediments, 260 of which were from field experiments). From Recking et al. [2008b].

[11] 1. No logarithmic part (R/D < β): u(y)/u* is constant and equal to its integration over the entire flow depth U/u* (equation (15)).

[12] 2. A logarithmic part is present (R/D > β): equation (13) is used with α defined by equation (17).

[13] This produces a two-part solution (with 1 < αRL < 3.5):

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Velocity ratios (u/u*)RL calculated with equation (18) for several values of β are plotted in Figure 5 as a function of the relative flow depth R/D and are compared to results obtained with the Nikuradse logarithmic profile (equation (11) with y = D/2 and ks = D). Figure 5 indicates a decrease in (u/u*)RL with decreasing R/D, which is in accordance with experimental observations [Tsujimoto, 1991]. This variation is also obtained with β = 0 (logarithmic profile with y = D/2 and ks = αRL), which is not surprising because most information concerning the roughness layer are supposed to be contained in flow resistance data through αRL. Including a roughness layer (β > 0) only allows one to modify the flow velocity distribution within the profile.

Figure 5.

Near-bed velocity ratio (u/u*) deduced from velocity profiles with and without a roughness layer of thickness βD.

[14] An expression for the critical Shields parameter that accounts for the roughness layer can be obtained from equation (8) with (u/u*)D defined from equation (18) and with R/D in equation (18) replaced by θc(s − 1)/S determined by rearranging equation (1):

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This two-part expression for θc results from the two-part solution of equation (18) where the respective R/D ranges were replaced by corresponding θc ranges. θc can be solved iteratively for different slopes. Note that both the channel slope Sb = tana and energy slope S are concerned in the calculation. This is not a problem for uniform flow conditions (which is the case for most flume experiments) for which S = Sb, but in the field the energy slope can differ significantly from the geometric bed slope during flooding [Meirovich et al., 1998; Smart, 1999]. The model is compared with field and flume data in the following section.

3. Comparison of the Model With Experimental Results

3.1. Comparison With Near-Uniform Sediment Data

[15] A data set of 270 values (Table 1) was constructed using almost all the available data verifying the following conditions: uniform or near-uniform natural sediments, flume studies (closed-conduit experiments were not considered), rough turbulent flows, i.e., verifying Re* > 70 [Julien, 1995; Graf and Altinakar, 2000], and slope available (if R/D was available instead of S, the energy slope was estimated using equation (1)). Six values from Aguirre-Pe and Fuentes obtained with a sediment diameter of 17.5 mm were considered as outliers and were not used (they were associated with very high critical Shields values in the 0.09–0.11 range for slopes between 0.02 and 0.04). Günter [1971] data were as reported in the work of Abdel-Aal [1975]. Data from Ashida and Bayazit [1973] were the same as those used later by Mizuyama [1977]. Data from Suszka [1991] includes data from Bathurst et al. [1987]. Data from Everts [1973], Pang-Yung [1939], Neil [1967], Gilbert [1914], Mavis et al. [1937], Meyer-Peter and Muller [1948] and Wolman and Brush [1961] were as reported in the work of Buffington and Montgomery [1997]. When the authors proposed θc values on the basis of a slope correction [Ashida and Bayazit, 1973; Bathurst et al., 1987], the values were recalculated with equation (1). The final data set (270 values) was plotted in Figure 6. Most data correspond to fully submerged grains (H > D). The data are considerably scattered when plotted on the Shields diagram (Figure 6a). The scatter is greatly reduced when they are plotted as a function of the slope (Figure 6b). The best fit (r2 = 0.85) was obtained with a linear function (equation (20)):

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The data from Table 1 were used to test the predicted values of θc from equation (19). Uniform flow conditions were assumed (S is replaced by sin(atan(Sb)) in equation (19)) with κ = 0.4. Measurements with natural sediments [Buffington et al., 1992; Gregoretti, 2000] indicated that in the field a mean value of 52° would be appropriate for the intergranular friction angle ϕ (whose value is much higher than the mass angle of repose value 32° usually considered for an en masse failure of a bed volume). For flows over natural sediments CD = 0.45 is usually considered. However, Coleman [1967] measured a decrease in both drag and lift coefficients with an increasing particle Reynolds number Re* [Vollmer and Kleinhans, 2007] and proposed an asymptotic value of CD = 0.25 for Re* higher than 5.104, whereas other measurements indicated that CD could also increase up to 0.9 for low relative depth (these aspects are discussed in the work of Armanini and Gregoretti [2005] and Lamb et al. [2008]). The same uncertainties exist for lift force. It is usually considered through a ratio CL/CD = 0.85 [Chepil, 1958; Wiberg and Smith, 1987; Seminara et al., 2002; Armanini and Gregoretti, 2005; Lamb et al., 2008]. Lift force corresponds to a pressure gradient generated by the mean flow velocity gradient in the y-direction close to the bed and could be reduced when the flow velocity profile becomes uniform close to the bed. Patnaik et al. [1994] measured a decreasing lift coefficient CL with decreasing relative submergence δ/D (where δ is the boundary layer thickness) in wind tunnel experiments. However, to the best of my knowledge there was no flume investigation of the lift coefficient at low relative depths and the relation CL/CD = 0.85 was maintained.

Figure 6.

Data set obtained in flume for rough turbulent flows with near-uniform sediment mixtures (no slope correction) (a) presentation on the Shields diagram and (b) comparison with slope.

Table 1. Range of Available Laboratory Data for Incipient Motion in Rough Turbulent Flows With Near-Uniform Sediments
SourceMethodaValuesSD (mm)Re*
  • a

    Abbreviations are as follows: E, extrapolation to zero; R, reference to a nonzero transport rate; V, visual; NS, not specified.

  • b

    As reported in the work of Buffington and Montgomery [1997].

Ashida and Bayazit [1973]E150.01–0.26.4–22.5500–3500
Saad [1989]V140.001–0.00161.2–9.585–703
Suszka [1991]R270.0017–0.093.3–44.5200–9000
Aguirre-Pe et al. [2003]NS150.03–0.0638.6–416500–8500
Gregoretti [2000]V270.21–0.3623–344700–8500
Tabata and Ichinose [1971] 230.0045–0.2216.5–108926–40,592
Everts [1973]bV30.002–0.0093.57123–146
Ho [1939], Pang-Yung [1939]bV20.003–0.0053.26–5.7110–275
Günter [1971] 90.001–0.0053.8–4.3150–215
Neil [1967]bV260.0016–0.265–20221–2100
Dey and Raju [2002]V330.01–0.0174.9–9.520–5800
Gilbert [1914]bV140.005–0.0251.7–770–587
Mavis et al. [1937]bV160.001–0.011.4–4.375–235
Meyer-Peter and Muller [1948]bV210.0005–0.012–8.581–686
Wolman and Brush [1961]bV60.002–0.01271–94
Yang and Wang [2006]R190.005–0.0251.5–40126–6000

[16] Figure 7a presents the model's results with β = 1, CD = 0.45, CL/CD = 0.85, and different values for the intergranular friction angle ϕ. Several observations can be made. First, the proposed velocity profile function allows us to fit the data fairly well with a force balance model. Second, the model prediction tends to zero, as expected, when the bed slope angle equals the intergranular friction angle. Third, the measured mean value ϕ = 52° is appropriate. The roughness layer thickness was varied through β in Figure 7b to improve this result. No roughness layer was simulated by calculating (u/u*) at y = D/2 using a logarithmic profile with ks = αRL (which must be distinguished from the Nikuradse profile consisting in using the logarithmic profile with ks = D). Changing β did not improve the results on very steep slopes, but Figure 7b suggests that on gentle slopes the roughness layer thickness is on the order of magnitude of the grain diameter (β=1), which is in good accordance with measurements. A higher value could be appropriate for steep slopes. Figure 7c presents the model simulations with ϕ = 52°, β = 1 and CD = 0.45 and 0.9. The model fit the data well with CD = 0.45 on gentle slopes and CD = 0.9 on steep slopes, which argues for an increased drag coefficient with decreasing R/D. To finish, the same force balance was used with Nikuradse's law (equation (11)) and the results are plotted on Figure 7d for comparison. This illustrates that the relationship observed between θc and S could not be obtained with the Nikuradse profile, even when considering a reduced drag force resulting from reduced grain submergence (as was concluded by Lamb et al. [2008]). Figure 7d also indicates that the grain surface and volume correction for low submergence did not influence the result for comparison with available data.

Figure 7.

Comparison between the theoretical model and the near-uniform sediment data set (a) β = 1, CD = 0.45, CL/CD = 0.85 and varying ϕ; (b) ϕ = 52°, varying β; (c) ϕ = 52°, varying CD; and (d) comparison with a force balance model including the Nikuradse profile (dashed lines correspond to no grain surface and volume correction with ξA = ξV = 1).

[17] Finally, the model fit the data fairly well whatever values were used for coefficients. Most particularly, very satisfactory results were obtained with coefficients which are in good agreement with available measurements: ϕ = 52°, CD = 0.45 (with a progressive increase to 0.9 with decreasing relative depth), CL/CD = 0.85 and β = 1.

3.2. Comparison With Nonuniform (Flume and Field) Sediment Data

[18] Most available incipient motion data were obtained in flume experiments with near-uniform materials. Gravel bed river sediments are not uniform, so what is the relevance of this research to field problems? Defining incipient motion conditions for poorly sorted sediments can be difficult because all individual size fractions in a mixture may not behave identically for a given bed shear stress τ. More particularly, two effects, one absolute and one relative [Wilcock and Southard, 1988], complicate the phenomenon. The absolute size effect (ratio of driving to resisting forces) makes the smaller grains easier to move, whereas the relative size effect (hiding fine sediments and overexposure of coarser sediments) produces the opposite effect. On the basis of flume and field measurements or theoretical analysis, some researchers [Parker and Klingeman, 1982; Andrews, 1983; Wiberg and Smith, 1987; Kuhnle, 1992; Wilcock, 1993] considered that both effects nearly compensate each other (relative protrusion of bed particles into the flow compensates for the differences in particle weight) and that consequently particles of various sizes have equal mobility, i.e., are entrained at about the same mean bed shear stress (or the same flow discharge). Others observed a selective size entrainment with increasing shear stress [Komar, 1987; Ashworth and Ferguson, 1989; Lisle, 1995; Lanzoni, 2000].

[19] All these studies proposed a hiding function (equation (21)) giving the critical Shields number θci associated with diameter Di from the critical Shield stress θc50 associated with median diameter D50:

equation image

where b is a coefficient whose value is 0 if the critical shear stress is simply proportional to individual particle size (constant critical Shields value) or −1 in case of complete equal mobility (values were generally proposed between −0.6 and −1). Subscript ‘c’ is commonly replaced by ‘r’ when authors referred to a nonzero reference bed load transport [Parker and Klingeman, 1982] instead of a critical value (zero transport).

[20] Figure 8 presents 47 reference Shields values θr50 deduced from bed load fractionwise measurements by different authors (D50 corresponds to the bed surface material or to the laboratory mixture). When used with data from Mueller et al. [2005] (45 values), they allowed a comparison with the theoretical model and with near-uniform sediments data (Figure 9). All data confirm a variation with slope. Two observations can be deduced from Figure 9.

Figure 8.

Reference Shields values for median diameter D50 obtained from bed load fractionwise calculation. Abbreviations are as follows: L, laboratory; F, field.

Figure 8.


Figure 9.

Comparison between field values (data are from Figure 8 and from Mueller et al. [2005]) and near-uniform sediment data (Table 1).

[21] 1. Mueller et al.'s [2005] data are lower than data obtained from fractionwise calculation (Figure 8) on slopes less than 1%.

[22] 2. Field data plot above the near-uniform sediment data.

[23] The differences with Mueller et al.'s [2005] data can be explained because these authors used the median diameter D50 to scale the flow depth responsible for a low transport rate value (as defined by Parker et al. [1982]) whatever the actual diameter of the transported sediment. If the transported mixture was finer than D50 (which was the case in several runs) the corresponding flow depths used for calculations may have been lower than what would have been expected for transportation of D50, producing low critical Shields values when scaled with D50.

[24] Differences between field and flume values can be explained from the flow properties. Incipient motion of a grain depends on the resistance this grain exerts to the flow. One important feature that characterizes flows over near-uniform sediments is that the grain diameter D used to scale the critical shear stress in the Shields number is the one responsible for the total flow resistance (assuming the idealized case where no additional flow resistance is to be considered, which is the case in flume experiments). With poorly sorted sediments, only protruding elements are responsible for the roughness layer development and the flow resistance equation was observed to behave like near-uniform sediments when the flow depth was scaled with D84 (Figure 4). Thus, for comparison with near-uniform sediments, D84 should also logically be used to scale the shear stress in the Shields number. This hypothesis is supported by comparing θr84 values (calculated with equation (21) and the values presented in Figure 8) and the near-uniform sediment data set in Figure 10.

Figure 10.

Comparison between θr50, θr84 and near-uniform sediment data.

[25] Figure 11 plots θr16, θr50 and θr84 from the data presented in Figure 8. All fractions can be approximated by a linear function of the slope:

equation image
equation image
equation image

There is more scatter for θr16 and θr84 because corresponding grain diameters are generally not defined or poorly defined (moreover, D90 is often used instead of D84). The best fit was obtained for θr50 and a general function for θri can be proposed when equation (21) is used with equation (23):

equation image

The value for b is more uncertain. The values reported in Figure 8 vary between −0.5 and −1, and are a good illustration of how difficult it is to conclude whether or not there is equal mobility. When equation (25) is used with a mean value b = −0.93, all θri values are reproduced to within ±50% (Figure 12).

Figure 11.

The θri values versus the slope. All fraction sizes can be approximated by a linear function θri = aS + b.

Figure 12.

Comparison between measured and calculated θri values with b = −0.93 (dashed lines correspond to the envelope ± 50%).

[26] Improving this result would require a better understanding of the physical processes controlling b. The slope effect described for θri may contribute to the scatter observed in b. This coefficient does not necessarily vary with the slope (R2 = 0.1). However, the origin of the scatter could be explained, because to cover a large range of shear stress (especially for motion of larger diameters), the data sets used to fit equation (21) were obtained by varying the slope in many experiments [Wilcock and Southard, 1988; Kuhnle, 1993; Petit, 1994]. This means that the slope effect described in this paper may have affected the value of b in the fitting process. For instance, considering D84/D50 = 2, if a value of b = −0.9 is obtained when all measurements are taken on a constant slope, the same experiment measuring θr50 on slope 0.001 and θr84 on slope 0.003 would have led to b = −0.8 when considering equation (23). Inversely, measuring θr50 on a 0.003 slope and θr84 on a 0.001 slope (i.e., eroding the bed) would have given b = −1. This problem also concerns field measurements since only a single average geometric bed slope is generally considered for all measurements, whereas the following points are true.

[27] 1. The energy slope can significantly differ from the geometric slope during flooding [Meirovich et al., 1998; Smart, 1999], i.e., for incipient motion of the larger elements.

[28] 2. The Shields stress can vary widely with the slope locally: for the same river reach, Sear [1996] measured θr50 values in riffles (steep slope) that were twice as high as values measured in pools (gentle slope). A similar result was obtained by Church and Hassan [2002].

[29] However, in addition to slope effects, many questions remain unresolved on the incipient motion mechanisms in these fluvial environments, especially in relation to the role of nonuniform sediments, defining the critical Shields value, and most notably the real influence of river bed sediment structure at the microscale and mesoscale on the exponent value b of equation (25). For instance, bed material entrainment in gravel bed rivers is often controlled by patches of fine sediments. Several phases have been described in the literature as occurring in this type of sedimentary microfeature [Garcia et al., 2007], and recent technical developments [Gibbins et al., 2007; Vericat et al., 2007, 2008] have aided in examining the interaction between hydraulics and bed load under such marginal transport conditions (i.e., very low bed load transport rates). These rates are also very sensitive to changing hydraulic conditions [Church and Hassan, 2005]. Under such circumstances bed material moves only partially and thus entrainment is size-selective, i.e., it departs from the exponent −1 [e.g., Wilcock and McArdell, 1993], still far from equal-mobility conditions that may occur when the armor layer progressively becomes unstable and all the local bed material participates in transport (i.e., the fully mobile phase).

4. Discussion

4.1. Comparison With Contradictory Experimental Results

[30] Several experimental results, all obtained in closed conduit experiments [Luque and Van Beek, 1976; Chiew and Parker, 1994; Dey, 2003], confirmed the decreasing theoretical curve in Figure 7d (obtained with the Nikuradse velocity profile and with no grain surface and volume correction). Two explanations can be given for these differences.

[31] The first explanation concerns the bed shear stress calculation. For all these experiments, grains were always fully submerged and the bed shear stress was calculated from the measured mean flow velocity U and the Darcy-Weisbach equation:

equation image

with the friction coefficient f computed with a standard equation such as the Colebrook-White equation after sidewall correction. These flow resistance equations use a constant roughness for all rough flow conditions and may not be valid when flow differs widely from the conditions used to established the equation, especially when the roughness layer is non negligible or in presence of bed load transport [Recking et al., 2008b]. For instance, if the data from Tabata and Ichinose [1971] were reconsidered using this method, by replacing the measured values by τ calculated with equation (26), with f computed from Keulegan's equation (ks = D), it would produce almost constant critical Shields values and the slope effect measured by the authors (Figure 6) would not have been observed.

[32] The second reason could be physical, since most of these data were obtained with low flow velocities (lower than 0.5 m/s) over fine sediments (grain diameters were smaller than 2–3 mm). Since the wake's frequency and size are related to the size of sediments and the mean flow velocity, the roughness layer may not develop for these flow conditions. For instance, Nowell and Church [1979] considered it would not develop for sand grain roughness smaller than approximately 1 mm.

4.2. Critical Shields Variation With Other Physical Parameters

[33] This paper only investigated the critical Shields dependence on slope and associated relative depth. Dependence on the particle Reynolds number Re* (Figure 6a) and the grain diameter were also hypothesized in the literature [Shvidchenko et al., 2001]. By analyzing their own data set, Lamb et al. [2008] concluded that the trend of increasing θc with Re* stemmed from the dependence of Re* on slope S and should not be considered important. Similar conclusions could be drawn for the dependence on grain diameter since a correlation exists between grain diameter and slope for both flume and field data (Figure 13). However, additional research is necessary to definitively conclude on these questions.

Figure 13.

Grain size variation with slope (D50 was considered for field data).

5. Conclusion

[34] During the last few decades, both field and flume measurements have shown that the critical Shields number θc was increased for low relative depth R/D associated with steep slopes S. Recent analytical models confirmed this trend considering experimental observations that the Nikuradse profile was not valid over gravel beds on steep slopes. However, despite these results, the critical Shields number for rough turbulent flows usually continues to be considered constant, or even to be decreased because of increased gravitational effects on steep slopes. This may have very important consequences on bed load prediction.

[35] This paper proposed a new theoretical model that adequately reproduced the critical Shields stress increase with increasing slope. In particular, it was demonstrated that the measured critical shear stress variations could not be obtained with the classical Nikuradse velocity profile. Instead, when available flow resistance data obtained without bed load transport over flat beds (for a wide range of slopes and relative depths) are used to fit a velocity profile, it necessarily produces an increased critical Shields number with increasing slope when used in a force balance model. It was explained by the existence of a roughness layer, which corresponds to a low and uniform velocity profile close to the bed that may affect both the lift and the drag forces. However, additional investigations are still needed to fully describe this layer. For instance, Figure 7b suggests that the appropriate roughness layer thickness to be considered at the incipient motion flow condition could increase with increasing slope (which could be explained because the wake frequency and size depend on the grain diameter and mean flow velocity). The roughness layer's turbulence and mean flow velocity distribution also deserve additional research. Near-bed peak turbulent intensity was measured for varying flow conditions by many authors, which allowed Lamb et al. [2008] to propose a function relating these flow properties to relative depth. A similar data set relating the near bed flow velocity to the flow relative depth would be needed to validate the velocity function proposed in this paper (equation (18)).

[36] Field data were used to discuss the relevance of this research to field problems. Because the roughness layer is produced by larger elements protruding from the bed, diameter such as D84 were found to be most representative for scaling both flow depth and bed shear stress for comparison with near-uniform sediment data. A hiding function was proposed for field application taking into account the slope effect. Uncertainties remain, however, on the exponent value b associated with a wide scatter in the available data set. This scatter could partly be explained by the slope effect. Spatially averaged equations [Coleman and Nikora, 2008] could also help better understand the complexity of microscale and macroscale features controlling sediment entrainment in the natural environment.

[37] At this time, several aspects need to be investigated by additional research: (1) the roughness layer properties, (2) the drag and lift coefficient variation with relative depth and slope and (3) the effects of slope on the hiding function for field applications with nonuniform sediments.

Appendix A

[38] This appendix presents the coefficients used to introduce partial submergence in the model.

A1. Coefficient ξA

[39] The objective is to express the grain surface below the free surface in the form: Ax = ξAA, where A is the total grain surface (Figure A1).

equation image

This equation also holds for R < D/2.

Figure A1.

Grain surface partition.

A2. Coefficient ξV

[40] The objective is to express the grain volume below the free surface in the form Vps = ξVVp, where Vp is the total grain volume (Figure A2).

equation image

This equation also holds for R < D/2.

Figure A2.

Grain volume partition.


[41] This study was supported by the Cemagref and the ECCO-PNRH program from ANR/INSU ANR-05-ECCO-015. The author would like to thank J. M. Buffington, R. J. Batalla, and a third anonymous reviewer, who greatly contributed to this work by providing helpful reviews of an earlier version of this manuscript. Thanks are also extended to Editor A. Porporato and Associate Editor G. Grant.