Is the von Kármán constant affected by sediment suspension?

Authors


Abstract

[1] Is the von Kármán constant affected by sediment suspension? The presence of suspended sediment in channels and fluvial streams has been known for decades to affect turbulence transfer mechanism in sediment-laden flows, and, therefore, the transport and fate of sediments that determine the bathymetry of natural water courses. This study explores the density stratification effects on the turbulent velocity profile and its impact on the transport of sediment. There is as yet no consensus in the scientific community on the effect of sediment suspension on the von Kármán parameter,κ. Two different theories based on the empirical log-wake velocity profile are currently under debate: One supports a universal value ofκ = 0.41 and a strength of the wake, Π, that is affected by suspended sediment. The other suggests that both κ and Π could vary with suspended sediment. These different theories result in a conceptual problem regarding the effect of suspended sediment on κ, which has divided the research area. In this study, a new mixing length theory is proposed to describe theoretically the turbulent velocity profile. The analytical approach provides added insight defining κas a turbulent parameter which varies with the distance to the bed in sediment-laden flows. The theory is compared with previous experimental data and simulations using ak-εturbulence closure to the Reynolds averaged Navier Stokes equations model. The mixing length model indicates that the two contradictory theories incorporate the stratified flow effect into a different component of the log-wake law. The results of this work show that the log-wake fit with a reducedκ is the physically coherent approximation.

1. Introduction

[2] Is the von Kármán constant affected by sediment suspension? This is the major question under debate in this paper. Understanding the turbulent velocity profile in sediment-laden flows is of primary importance for understanding the mechanics of sediment transport in fluvial streams and channels. The sediment is kept in suspension due to the fluid turbulence. The transport and final fate of these particles depend on erosion and deposition processes, which finally constrain the bathymetry of the natural water ways. Suspended load is generally the largest fraction of sediment in transport [Hill et al., 1988]. Estimates of the turbulent velocity profile are required to quantify the amount of sediment transported in suspension [Garcia, 2008]. Therefore, accurate characterization of the turbulent velocity profile through a physically based theory for prediction of turbulent flow interaction with suspended sediment is relevant for understanding sediment transport [Smith and McLean, 1977; Gust and Southard, 1983; Villaret and Trowbridge, 1991; Yang, 2007; Garcia, 2008].

[3] Under clear-water turbulent flow conditions, the mean velocity profile is described in the vicinity of the wall by thelogarithmic law of the wall(henceforth designated log-law), i.e., flow velocity increases logarithmically with height above the channel bed. The shape of this function is characterized by the von Kármán parameterκ, which takes the universal value of 0.41 for clear water [Nezu and Rodi, 1986]. However, it has been extensively verified that the log-law does not hold at the outer-region of the boundary layer. For this region,Coles [1956] introduced the wake law, whose form depends on the strength of the wake parameter, Π. A complete approximation to the time-averaged velocity profile in turbulent flows can be obtained by coupling the log- and wake-laws [White, 1991; Guo and Julien, 2001]. The law of the wall also breaks down for shallow, steep flows with low particle submergence [Wiberg and Smith, 1991; Lamb et al., 2008]. Further, Gaudio et al. [2010] showed that the von Kármán constant is not universal in natural waterways. They found that κ is different from 0.41 in flows with either low submergence or with bed and suspended load transport.

[4] Sediment-laden turbulent velocity profiles have been examined by empirically fitting the log-law function to match experimental data [Vanoni, 1946; Einstein and Chien, 1955; Gust and Southard, 1983; Gust, 1984]. This approach indicates that the suspended sediment in the flow induces a damping effect on the turbulence momentum transfer, thereby reducing the von Kármán parameter κfrom its clear-water universal value [Vanoni, 1946; Einstein and Chien, 1955; Montes, 1973]. However, the log-law, theoretically, is only valid within the inner- or wall region, leaving without physical significance the previous data-matching in the whole stratified flow depth [Coleman, 1981, 1984, 1986]. Coleman [1981, 1984, 1986]conducted detailed experiments in a laboratory flume, concluding that velocity profiles in sediment-laden turbulent flow could be described by Coles' log-wake law. The von Kármán parameterκ and the strength of the wake Π must be estimated by profile matching. Coleman [1981, 1984, 1986]considered only near-wall data to determineκ. His fitted value of κ was very similar to the universal value of κ = 0.41 for clear-water flows, whereas fitted Π varied from 0.9, when the sediment concentration was high, to Π = 0.2 for clear-water flows [Nezu and Rodi, 1986]. The analysis by Coleman [1981, 1984], although physically reasonable, was severely criticized by Gust [1984], who proposed that the classical log-law is universal. One of the reasons for the critique was the lack of data in the near-wall region, typically four data points, which could lead to inaccurate fittings for the correspondingκ values. Subsequent work by Parker and Coleman [1986] corroborated the results presented by Coleman [1981, 1984, 1986] regarding the strength of the wake, but it did not provide enough additional information to support the value κ = 0.41. Itakura and Kishi [1980]applied the Monin-Obukhov theory for the length scale in sediment-laden flow and found a linear wake profile, which made their approach conceptually similar toColeman's [1981, 1984, 1986]. As an alternative to the log-wake law, a power law velocity profile for sediment suspension was considered byKarim and Kennedy [1987] and Woo et al. [1988]. Adopting the entropy concept Chiu et al. [2000] proposed yet another expression for the velocity profile.

[5] Lyn [1986, 1988] and Valiani [1988] further criticized the approach of Coleman [1981, 1984, 1986], arguing that more statistically rigorous log-wake fits suggest that bothκ and Π are affected by the suspended sediment. Valiani [1988]considered the log-wake model for sediment-laden flows and re-analyzedColeman's data [1986]. He proposed that, rather than determining κfrom the near-wall data, the entire velocity profile should be used, with values weighted inversely with respect to distance from the wall. Surprisingly, using this weighting,Valiani [1988] detected a strong effect of sediment suspension reducing the value of κ and a lesser effect on Π, which yielded smaller values from those obtained by Coleman [1986]. Guo and Julien [2001]proposed a novel wake function, which was coupled with the log-law, and they re-analyzedColeman's [1986] data. This wake function contained a wake parameter, Ω, that is different from Π. Their fitting analysis demonstrated that, in general, both κand Ω should be affected by sediment suspension. They found that for sediment-laden flowsκis greatly reduced below 0.41 and Ω is slightly larger than its clear-water value. Although the wake functions used byValiani [1988] and Guo and Julien [2001]are different, their comprehensive analyses are in general agreement, indicating that for sediment-laden flowsκ < 0.41 and the wake parameter is larger than its clear-water value, regardless of the specific wake function considered. However, bothValiani [1988] and Guo and Julien [2001]offered a characterization of the velocity profile using empirical fittings of log-wake laws to experiments. The log-wake law contains two free adjustable parameters (κand Π), which means that the log-wake results cannot add direct physical insight on the turbulent momentum transfer in sediment-laden flows. The results ofValiani [1988] and Guo and Julien [2001], therefore, relied on an empirical analysis rather than on a theory for the turbulent momentum transfer [Yang, 2007]. This unsolved issue is the basis of the present work.

[6] Sediment particles are expected to affect the turbulence structure near the wall [Smith and McLean, 1977], but there is no obvious reason to believe that they will greatly affect turbulence parameters in the outer region. The stratified flow depth by sediments may be ideally divided into two zones: near the bottom, where the concentration and its gradient are large and modify the flow characteristics to a large degree, and a main flow area with small sediment concentration where the sediment has little effect [Einstein and Chien, 1955; Montes, 1973]. Consequently, the debate on the effect of sediment suspension on the turbulent velocity profile remains open [Yang, 2007]. A comprehensive review of non-universality of von Kármánκ was given by Gaudio et al. [2010, 2011]. However, conflicting conclusions are found depending on the literature source, leading to uncertainty whether the von Kármán parameter is affected by the suspended sediment or not.

[7] This work proposes a physical interpretation of the turbulent momentum transfer in sediment-laden flows using a modified form of Prandtl's mixing length theory. This approach allows a more physically based characterization of the sediment-laden turbulent velocity profile. In this model, the von Kármán parameter is not a universal constant for a turbulent velocity profile in sediment-laden flows. It is a variable in terms of distance to the wall, which is determined by the damping of the mixing length in the turbulent momentum transfer, caused by the suspended sediment. This is the major difference between this proposal and either the log-wake model [Coleman, 1981; Valiani, 1988; Guo and Julien, 2001] or the log-law model [Vanoni, 1946; Montes, 1973; Gust, 1984; Wright and Parker, 2004a, 2004b], in which the von Kármán parameter is constant for a given velocity profile and determined by its adjustment by profile matching. The proposed theoretical model permits explanation of the turbulent momentum transfer mechanism in sediment-laden flows. When this theoretical profile is confronted to log-wake laws, it reveals the effects of the suspended sediment on the fitted parametersκand Π of the latter. The proposed theory has a major advantage over log-wake laws in that it does not require experimental data to adjust any empirical constant. It is a theoretical profile based on a physical interpretation of the turbulent momentum transfer. The present model was compared with RANS simulations and experimental observations, producing similar results. The proposed theory can therefore be used instead of the log-wake law to characterize the velocity profile with suspended sediment, thereby substituting fits to experimental data by a theoretical consideration on the turbulent momentum transfer.

2. Wall-Wake Velocity Profile for Clear-Water Flows

[8] The complete time-averaged velocity profile in wall-bounded turbulent boundary layers can be expressed in velocity defect form as [White 1991; Cebeci and Cousteix 2005]:

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where uis the time-averaged streamwise velocity component at elevationy above the channel bed, U is the free stream velocity, u* the shear velocity, u* = (τo/ρo)1/2, τo the boundary shear stress, ρothe mass density of clear-water,ω = ω(y/δ) the wake function, δ the boundary layer thickness, which, for fully developed free surface flows is equal to the water depth h, and Πothe wake parameter for clear-water flow.

[9] Equation (1) was analyzed by Nezu and Rodi [1986]for clear-water flows in open channels. They obtained mean values ofκ = 0.41 and Πo = 0.2. Several forms for the wake function ω have been proposed in the literature [Cebeci and Cousteix, 2005]. A widely used wake function is the law of Coles [1956]:

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This law is adopted in this work. The turbulent momentum transfer can be expressed in terms of Prandtl's mixing length lo [White, 1991] as

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where τis the Reynolds shear stress. For a steady-uniform two-dimensional flow, neglecting the viscous sublayer,τ is expressed as:

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where u′ and v′ are the fluctuations of the velocities from their time-averaged values in thex and y directions, respectively, and η = y/h is the normalized elevation. Solving equations (3) and (4) for τ, and computing du/dy using equation (1), results in a normalized mixing length as

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Equation (5) yields loin clear-water flows, once the values ofκ = 0.41 and Πo = 0.2 are assumed. If Πo = 0, equation (5)reduces to the mixing length distribution for the log-law, as indicated in the semi-empirical expression ofUmeyama and Gerritsen [1992]. Herein, equation (5)is used as a basis for a turbulent momentum transfer approach in sediment-laden flows.

3. Effect of Suspended Sediment on Turbulent Momentum Transfer

[10] In a steady two-dimensional turbulent flow, with a mean velocity profileu = u(y) (Figure 1a), the sediment suspension induces a mass density profile ρ = ρ(y) (Figure 1b). The momentum of the flow M = ρu at a certain elevation y above the channel bed can be expanded as a Taylor series, retaining just the first order term [Montes, 1973]:

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Similarly to Prandtl's theory, it is assumed that Δyin sediment-laden flows is the length scalel called the mixing lengthfor the sediment-laden flow. This mixing length is associated with the distance traveled by an elementary control volume in a direction normal to the wall, due to the fluctuationsv′, until its momentum is absorbed by the other layer. Thus, from equation (6), one can write:

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Further, ΔM can be evaluated from the velocity and mass density profiles to the first order as (see Figure 1b) [Montes, 1973]:

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where Δuis the velocity variation between sediment-laden layers separated by a distancel (Figure 1b), and the second order term ΔuΔρ has been neglected. The density gradient (Figure 1b) modifies the momentum transfer by its interaction with the velocity profile. Thus, from equations (7) and (8):

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It can be seen that equation (9) includes both the velocity and density gradient effects in the momentum transfer. For constant density, equation (9) simplifies to the classical theory proposed by Prandtl. The mass density variation could also be expanded in terms of l as:

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Substituting equation (10) into equation (9) produces:

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Equation (11), originally developed by Montes [1973], includes the effect of the suspension mass density on the momentum transfer. The turbulent momentum transfer is affected not only by the local values of density ρ and velocity u, but also by their gradients, indicating that in sediment-laden flows it is determined by a transport relationship. Therefore,equation (11) suggests that the effect of suspended particles must be relevant in regions with large gradients of mass and momentum transfer, such as near the bed, where both du/dy and /dy are large.

Figure 1.

(a) Definition sketch of sediment-laden free surface flow, whereu is the flow velocity, τ is the shear stress, C is the concentration and h is the flow depth. (b) Mass density ρ(y) and velocity u(y) profiles in a sediment-laden flow as components of the momentum,M = ρu, transferred between two layers separated by the mixing length l.

[11] The mixture mass density is given by:

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where C is the volumetric concentration of sediment, ρs is the density of solids and s = ρs/ρo. Differentiation of equation (12) with respect to y yields:

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The added-mass force or the virtual mass force is the inertia added to the particles because an accelerating or decelerating solid must displace some volume of the surrounding fluid as it moves through it. Therefore, the density of a solid particleρp is modified as [Montes, 1973; Liggett, 1994, section 3.11.2]

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with Kas the added-mass coefficient.Equation (12) can be modified using equation (14) as:

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where β = 1 − K. Sediment particles are maintained in suspension due to fluid turbulence. The vertical velocity has a random variation with time, provoking the particles to move upward and downward. This random unsteady particle movement associated with turbulent flow displaces the fluid surrounding the sediment. Therefore, a characteristic feature of the turbulent flow with suspended sediment is that the particles undergo a virtual mass force during their instantaneous unsteady movement. This effect is accounted for by the virtual mass coefficient β. It should be realized that the mixing length model developed here is a mixture flow model. Therefore, equation (15) is an improved definition accounting for the movement of the particles inside the fluid. Neglecting β is equivalent to assuming that the particles are static inside the mixture, which is by definition against the concept of turbulent flow.

[12] Differentiation with respect to y of equation (15) gives:

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Then, inserting equation (16) into equation (11) leads to:

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where the submerged specific gravity, R = s − 1, is introduced. On the other hand, the diffusion equation for the concentration of suspended sediments [Hunt, 1954] is

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where εs is the sediment diffusion coefficient and w the settling velocity. Differentiation of equation (18) with respect to y yields:

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Inserting equation (19) into equation (17) produces:

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The suspended sediment is stratified within the fluid according to the concentration gradient, dC/dy, which affects the density of the water–sediment mixture and the turbulent transfer of momentum. Coarser particles have larger values of w and, therefore, from equation (20), they will provide a greater damping on Δu. Equation (20) is a mixing length relationship which generalizes the classical theory of Prandtl for clear water. The density stratification effects are predominant in the region near the bed [Smith and McLean, 1977]. In order to simplify equation (20), εs may be approximated near the bed by [Montes, 1973]:

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Despite this near-bed simplification, it is reasonable to postulate its application to the whole water depth layer, because in the upper part of the water layer the transport of mass and momentum is less affected by suspended particles [Smith and McLean, 1977]. Although the sediment-laden mixing lengthldiffers from the clear-water valuelo, l = lo will be assumed. Whenever C→0, then dC/dy→0 as indicated by equation (18). Then, equation (20) tends to the classical Prandtl equation, Δu = lodu/dy, if llo. Therefore, lo will be considered as an estimator of l in equation (20). Using this approximation, as well as that given by equation (21), equation (20) is rewritten as:

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Equation (22) can be presented alternatively as:

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where ψ is a sediment damping factor defined by:

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In equation (23), ψ modifies the suspended mixing length lo and their product defines a modified mixing length for the sediment suspension, determined by lm = loψ. Inspection of equation (24) indicates that ψ is a function of the velocity, u(y), the concentration C(y), and their gradients.

[13] The present model for the turbulent momentum transfer, expressed by equations (23) and (24), implies a damping of the clear-water mixing length due to the suspended sediment concentration. This result agrees with previous studies in the literature, where the clear-water mixing length was modified by empirical damping functions [De Vantier and Narayanaswamy, 1989; Umeyama and Gerritsen, 1992; Kovacs, 1998; Mazumder and Ghoshal, 2006; Yang, 2007]. Umeyama and Gerritsen [1992] and Mazumder and Ghoshal [2006] used the empirical function

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where λ is a fitting parameter (positive) and Ca is a reference concentration. Equation (25) implies that ψ < 1, so that, indeed, ψ is a damping effect on the mixing length. It is an empirical relationship, whereas equation (24) was developed from physical principles. Kovacs [1998] proposed the empirical function:

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Yang [2007] presented another empirical relationship for ψ as a function of C.

[14] The present model is based on a theoretical approximation for density stratification effects expressed by equation (11). Previous work in the literature also considered the inclusion of density stratification effects on turbulent momentum transfer [Smith and McLean, 1977; Villaret and Trowbridge, 1991; Mazumder and Ghoshal, 2006; Wright and Parker, 2004a, 2004b]. In these studies, the momentum diffusivity εm was empirically modified to account for sediment damping as:

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and

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where α is a fitting parameter and Ri is the Richardson number. Smith and McLean [1977] indicated that εmo generally depends on Ri, but, however, it can be estimated by the clear-water eddy diffusivity [Smith and McLean, 1977; Villaret and Trowbridge, 1991; Mazumder and Ghoshal, 2006; Wright and Parker, 2004a, 2004b]. Equation (11) can be rewritten as

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or

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with the consequence that the damping in our model is also proportional to Ri. The mixing length lm and the momentum diffusivity εm are generally linked by εm = lm2du/dy [White, 1991; Cebeci and Cousteix, 2005]. Using the previous definition of lm = ψlo then:

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Thus, this mixing length model also implies that momentum diffusivity is damped by the suspended sediment. Using the second of equations (30), equation (31) can be rewritten as

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after using a first order Taylor series expansion. Equation (32) is the same as equation (27) by defining α = 1 − (2u/g)(du/dy)[2 − w(1 − 2C)/(2u*)]. The damping of the mixing length by suspended sediment is equivalent to a damping of the diffusivity, and, thus, our current approach is in agreement with arguments used in previous stratified models [Smith and McLean, 1977; Gelfenbaum and Smith, 1986; Villaret and Trowbridge, 1991; Mazumder and Ghoshal, 2006; Wright and Parker, 2004a, 2004b].

4. Proposed Mixing Length Model for Sediment-Laden Flows

[15] Based on the mixing length for clear-water flows, given byequation (5), consideration of the sediment-laden effect on the momentum transfer results in the normalized mixing length as

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where /dy is taken from equation (2). Therefore, a von Kármán parameter for sediment-laden flows can be defined by analogy with clear-water flows based onequation (33) as

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This equation indicates that the variable distribution κm = κm(y) depends on the function ψ = ψ(y). Therefore, ψmay be physically interpreted as a correction factor to the von Kármán constant for clear-water flows,κ = 0.41, introducing the concentration C = C(y) and velocity u = u(y) gradient effects in sediment-laden flows.

[16] For sediment-laden flow,equation (3) becomes [Umeyama and Gerritsen, 1992]:

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The Reynolds stress distribution, allowing for density stratification effects [Lyn, 1986; Mazumder and Ghoshal, 2006], is deduced from a vertical momentum balance as:

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Combining equations (35) and (36) produces a differential equation for the turbulent velocity profile:

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where u+ is the normalized velocity, u+ = u/u*.

[17] Many experimental studies have attempted to explain the effects of sediment suspension on the momentum transfer by a reduction of κ, that is, by the slope of the best fit line of the velocity profile over the entire water depth using semi-logarithmic scales. However, this technique is only a rough estimation of density stratification, which is not constant over the depth [Wright and Parker, 2004a, 2004b], as confirmed by ψ in equation (33). The velocity profile fitting technique with semilogarithmic scales, therefore, provides no insight into the mechanics of density stratification or any mechanistic interpretation of the reason underlying the decrease of fitted κ values. The present model introduces a local function given by equation (34), involving density stratification effects, in agreement with Wright and Parker [2004a, 2004b]. Therefore, this new formulation of the turbulent momentum transfer can physically explain the reduced fitted values of κ with semilogarithmic scaling. For illustrative purposes, consider flow near the bed, say η < 0.2 and assume Πo ≈ 0. Then, equation (33) yields the simplified form lm ≈ ψκy. If the term in brackets on the right hand side of equation (37) is neglected, its integration yields an estimate of the velocity profile in the wall region as:

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where subscript a refers to an arbitrary point in the wall region. Note that for ψ = 1 the clear-water log-law profile (withκ = 0.41) is regained from equation (38). However, in general ψ < 1, which means that if a log-law exists foru+, then the following relationship is satisfied:

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provided that the fitting parameter ϕ < 1. Note that given ηa, for different values of η, ϕis different as well, implying that a different log-law would strictly exist as an estimator for each pointu+ of the velocity profile near the bed. Equation (39)shows that a unique log-law for a velocity profile with suspended sediment can only exist in a mean sense, that is, a mean value ofϕ, inline image, can be fitted to estimate the velocity profile in a certain layer thickness. The discussion can be made more general by considering the complete equation (37), applicable to the whole water depth thickness h, rather than its simplified version of equation (38). However, the conclusions with regard to ϕand the existence of log-laws are identical. A mean inline image was proposed by Wright and Parker [2004a, 2004b]to fit log-laws to the whole water depth of sediment-laden flows.Cantero et al. [2009]used DNS (direct numerical simulation) to investigate density stratification effects in sediment-laden flows. They fitted log-laws in the wall layer to their DNS data and determined by best fit, an “apparent von Kármán constant” less than 0.41, inline image. On the basis of equation (39), the proposals of Wright and Parker [2004a, 2004b] and Cantero et al. [2009] agree with this model. The variation of density stratification with distance to the bed, ψ = f(y), can be measured in average with the damping parameter inline imagein a log-law that is fitted to modeled or measured velocity data. Nevertheless this approximation using inline image is only an empirical, indirect method to represent density stratification effects.

[18] From the works of Engelund [1970] and Montes [1973], pure suspension with w/u* much larger than 1 is not possible. Thus, for a sediment suspension with w/u* < 1 one can make the approximation w(1 − 2C)/(2u*) ≪ 2. Using this approximation, ψ of equation (24) is simplified to

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Current knowledge on sediment suspension and velocity profiles is mainly based on the hypothesis that sediment suspension affects the value of κ relative to that for the clear water [Gust, 1984]. However, the introduction of a modified von Kármán parameter can take account of variations in sediment concentration, C.This mathematical approach provides an explanation of the phenomena observed experimentally by other researchers, who fitted test data of velocity profiles to a log-law [Vanoni, 1946; Einstein and Chien, 1955; Montes, 1973; Gust and Southard, 1983; Gust, 1984; Wright and Parker, 2004a, 2004b]. They obtained diminished means of the von Kármán parameter as functions of any parameters indicating a bulk measure of density stratification, for example the depth-averaged concentration or a mean Richardson number. Furthermore, this model is based on a Taylor series expansion, and if needed, more terms of the series can be considered. A relationC = C(y) needs to be introduced into equation (40). Since the aim of this study is to describe the turbulent velocity profile, the precise definition of the concentration profile C(y) has not been considered. An exponential-type function corrected for the slope effects near the wall [Montes, 1973] is adopted here:

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where Cb is the maximum concentration, i.e., the concentration at the interface between the bed load and the suspended load, η = 0, and A and D are empirical coefficients. According to Montes [1973], D is approximately 30, while A can be estimated by fitting equation (41)to measured data of concentration profiles. The classical Ippen-Rouse equation [Montes, 1973] was first considered by the authors, following Umeyama and Gerritsen [1992]. However, this equation produced a poor approximation of the concentration gradients near the wall. It should be realized that the first factor of equation (41) forces it to get a null concentration gradient, dC/ = 0, at the wall, η = 0, which appears to be a reasonable boundary condition. The Ippen-Rouse equation induces the unreal wall boundary conditiondC/→∞. For a more accurate description of the concentration profile gradient near the wall, a finite-mixing length model is necessary [Nielsen and Teakle, 2004; Absi, 2010]. However, this considerably increases the complexity of the model, and, therefore, for the purpose of examining the turbulent velocity profile, the empirical characterization of the concentration profiles as given by equation (41) has been adopted in this work.

5. Evaluation of the Proposed Model Based on Experimental Data

5.1. Coleman [1981, 1984, 1986] Data Set

[19] Coleman [1981, 1984, 1986] conducted experiments in a laboratory flume 0.356 m wide with sand grains of diameter d = 0.105, 0.21 and 0.42 mm. His experimental results were re-analyzed using the proposed model. InColeman's [1986]experiments, the streamline of the maximum velocity lay at a certain distance below the free surface, due to so-called dip-phenomena, indicating the presence of a three-dimensional effect induced by the secondary current in the flow section. The proposed approach is restricted to two-dimensional uniform sediment-laden flows, so it cannot predict dip phenomena. No attempt has been made as yet to analyze this effect, thus a qualitative comparison was performed here, using a value of 0.2 for Πo based on data from Coleman's [1986]clear-water tests.

[20] Figures 24 compare results obtained from the proposed model with data from three experiments with reference concentrations Ca = 9·10−3, 9.5·10−3 and 2.6·10−3 m3m−3, where Ca is the concentration at a given reference elevation above the channel bed in the experiments. The computed velocity profile u+ = u+(η) from equation (37) is plotted in Figures 2a, 3a, and 4a for runs 9, 29 and 40 of Coleman [1981, 1986]. Figures 2b, 3b and 4b contain the fitted equation (41) for runs 9, 29 and 40. Equation (37)is solved in velocity-defect form,dZ/ = −du+/, where Z = (U − u)/u*, using equation (33)for the mixing length distribution in sediment-laden flows andequation (40) for ψ. C(y) and dC/dy are estimated from equation (41), once A is determined by fitting to concentration test data. The term /dy in equation (33) is evaluated using equation (2). The approximate value β = 0.5 is taken from Montes [1973]. Equation (37)is solved numerically using a standard 4th-order Runge–Kutta method, with the free surfaceZ(η = 1) = 0 as boundary condition. To avoid a singularity, η ≈ 0.999 is used as the initial value for the computation. Equation (37) is implicit in du+/dη. The du+/ value in a given computational step for the evaluation of ψ is used from the previous step. The computational step is successively reduced until no significant variations are detected between the assumed du+/ in ψ and the value computed from equation (37). To transform the defect profiles into absolute velocities, the relation u+ = (U/u*) − Z is used, where U/u* is obtained from the corresponding experiment. Equation (41) was found to provide a good fit to the test data for η < 0.2, that is, within the wall region. The proposed model is not able to recognize the dip-phenomena, as expected. However, in general, there is good agreement between predictions and observations. To further assess the robustness of the proposed theory, recent computational fluid dynamics (CFD) results have been considered, which were obtained byYoon and Kang [2005]solving the two-dimensional Reynolds averaged Navier–Stokes (henceforth RANS) equations with ak-εturbulence closure for sediment-laden flows. Corresponding results for the same test cases are included inFigures 24. It is apparent that the results of the proposed model are also close to those of the RANS equations. Thus, the proposed model is a good estimator of turbulent velocity profiles with suspended sediments given its general qualitative agreement with experimental evidence and CFD results.

Figure 2.

(a) Comparison of results obtained from equation (37) for the velocity profile u+ = u+(η) with test data by Coleman [1981], RUN 9, and with CFD results by Yoon and Kang [2005]. (b) Comparison of fitted equation (41) for the concentration profile C = C(η) with test data by Coleman [1981].

Figure 3.

(a) Comparison of results obtained from the equation (37) for the velocity profile u+ = u+(η) with test data by Coleman [1981], RUN 29, and with CFD results by Yoon and Kang [2005]. (b) Comparison of fitted equation (41) for the concentration profile C = C(η) with test data by Coleman [1981].

Figure 4.

(a) Comparison of results obtained from equation (37) for the velocity profile u+ = u+(η) with test data by Coleman [1981], RUN 40, and with CFD results by Yoon and Kang [2005]. (b) Comparison of fitted equation (41) for the concentration profile C = C(η) with test data by Coleman [1981].

[21] To explore the influence of sediment suspension on the velocity profile, the functions κm = κm(η) and lm = lm(η) for these runs are plotted in Figures 5a and 5b. The first relevant result is the sediment damping effect on κm. The parameter κm was far from constant, less than 0.41, and its distribution greatly affected by the suspended sediment. The κmfunction has a semi-normal distribution that declines toward a minimum value as one moves away from the bed, followed by a recovery of the near-bed valueκm ≈ 0.41 as one approaches η = 1. This specific distribution is fixed by u+(η), C(η), du+/(η) and dC/(η) in equation (40). Of particular importance in the evaluation of ψ is the function dC/(η). Note from equation (41) that dC/→0 for η→0, and, thus, κm→0.41. For η→1, equation (41) indicates that dC/ is proportional to C. In the usual case C(η→1) ≈ 0 and, therefore, ψ→1 and κm→0.41. The Ca value is inversely related to the minimum of κm, designated κmM (see depth distributions κm = κm(η) in Figure 5a). For run 29, in which Ca = 9.5 × 10−3 m3m−3, κmM ≈ 0.34. The fact that κm tends toward 0.41 near the bed (Figure 5a) may initially appear to be consistent with Coleman's [1981, 1986] promotion of the universal use of κ = 0.41 in log-wake models. However, there is a large variation ofκmin the wall-region (η < 0.2) (Figure 5a). The reduced κ values found by Valiani [1988]in sediment-laden flows are an indirect bulk estimation of the flow density stratification in the wall zone, dictated by the curveκm = κm(η). Any fitted κin a log-wake model will be influenced by the density stratification effect given byκm = κm(η). Density stratification effects may increase away from the wall-layer as in RUN 9 (Ca = 9.5 × 10−3 m3m−3) of Figure 5a. Two observations are of interest: (i) If the fitting technique in a log-wake law results inκ ≈ 0.41 the log-law component of the log-wake law catches only the flow conditions very near the wall. This clear-water log-term leaves the main density stratification effects given byκm = κm(η) for the wake component; (ii) If the fitting technique results in κ < 0.41, the density stratification effect indicated by κm = κm(η) is partially incorporated into the log-law component, whereas the remainder needs to be incorporated, of necessity, into the wake component. The reduction of the mixing length caused by suspended sediment (Figure 5b) agrees with results obtained by Umeyama and Gerritsen [1992], Kovacs [1998] and Yang [2007]. Measurements in both clear water and sediment-laden flows [Nikora and Goering, 2000] indicate that the Reynolds stress matches the linear shear stress distribution only for η greater than about 0.1. Using the linear Reynolds stress distribution (1 − η) in equation (37) therefore introduces some error close to the wall in the variation predicted for κm = κm(η).

Figure 5.

Depth-distribution of: (a) von Kármán parameterκm = κm(η) and (b) mixing length lm/h = lm/h(η), for RUNS 9, 29 and 40 of Coleman [1981], based on the numerical solution of equation (37).

[22] The numerical predictions of equation (37) for u+are now used to fit a log-wake law following the method ofColeman [1981, 1986]. The asymptotic value of equation (1)in the wall-region is:

display math

where Πs is the strength of the wake for a flow with sediment suspension, U+ is U/u* and κfitthe von Kármán coefficient fitted in a log-wake model.κfit is a constant whereas κm is a complex turbulent variable. Equation (42) was fitted with data computed with equation (37) near the bed, 0.001 < η < 0.01. Plotting U+ − u+ against lnη the value of κfit is fixed by the slope of the regression line using equation (42), and Πs by its intersection with the (U+ − u+) axis. This computation should not be considered as a rigorous attempt to investigate conditions near the wall, since the proposed theoretical approach considers neither the viscous nor the buffer layer, usually found when η < 0.015 [Coleman, 1992; Nezu and Azuma, 2004]. Instead, it is intended to add insight to, and extend, Coleman's [1986] findings.

[23] Figures 6a and 6b show κfit and κmM values obtained for d = 0.105 mm, together with κfit computed by Coleman [1981, 1984, 1986], Valiani [1988] and Guo and Julien [2001], respectively, against the mean Richardson number. The mean Richardson number inline image is a bulk measure of density stratification effects and is given by [Coleman, 1981, 1984, 1986; Valiani, 1988]:

display math

where inline image is the mean density in the flow depth h. Inspecting Figure 6a the values of κfit are close to 0.41, as observed by Coleman [1981, 1984, 1986]. Further, in Figure 6b, κmM follows essentially the same trend as the κfit values obtained by Valiani [1988]. Valiani [1988]included the complete velocity profile to fit log-wake models, but used weighting factors that are inversely proportional to the distance from the wall. He did not consider solely the influence of data restricted to the wall-region (asColeman [1986] did), thereby obtaining a value of κfit smaller than 0.41 due to the incorporation of density stratification effects by fitting κ to the entire flow depth. The fact that κfit varies with inline image (Figure 6b) further emphasizes that Valiani's [1988]approach for fitting the log-wake law incorporates density stratification effects in the wall-zone; that is, the damping of the mixing length as indicated by the distribution ofκm(η) (Figure 5a). Moreover, these results suggest that κvalues in log-wake laws should be determined from detailed statistical analysis across the complete water depth, instead of using only near-bed data. Otherwise, there is no guarantee that theκ value will correctly reflect the density stratification within the fluid. Because κfit and κmM are both influenced by the degree of density stratification within the flow, a correlation between the two is expected, as shown in Figure 6b. The results for sand with grain sizes of d = 0.21 and 0.42 mm, are presented in Figures 6c–6e, respectively. The trends are essentially as previously discussed on the basis of data shown in Figures 6a and 6b, supporting the present deliberation. The values of κfit obtained by Guo and Julien [2001] are included in Figure 6. They found κfit < 0.41 in agreement with Valiani [1988], although the drop below 0.41 in their log-wake model was larger. This may be attributed to the different fitting technique and wake function. However, bothValiani [1988] and Guo and Julien [2001] are in agreement that κfit < 0.41, which, in turn, arises from density stratification effects, as displayed from our model results of κmM.

Figure 6.

(a, c, e) Comparisons of κfit obtained by fitting equation (42) to the numerical results of equation (37) with the κfit obtained by Coleman [1981] (the results for the 40 runs in Coleman's paper are displayed) for d = 0.105 mm (Figure 6a), 0.211 mm (Figure 6c), and 0.42 mm (Figure 6e). (b, d, f) Comparisons of κmM obtained from the numerical solution of equation (37) with the κfit obtained by Valiani [1988] and by Guo and Julien [2001] for d 0.105 mm (Figure 6b), 0.211 mm (Figure 6d), and 0.42 mm (Figure 6f). The independent variable is the mean Richardson number, inline image, which is a measure of the density gradient effect in a given run (the results for the 40 runs in Coleman's paper are displayed).

[24] In addition, we used the near-bed data to analyze the hypothesis of a constant value ofκ = 0.41, independent of density stratification, in the wall region [Coleman, 1981, 1984, 1986]. Because if κfit ≈ 0.41, any actual variation κm = κm(η) in the wall region will be incorporated into the parameter Πs when equation (42)is applied. However, this effect does not necessarily imply that the suspended sediment affects the outer-layer turbulence structure of the boundary layer. It is an effect of the fitting technique. Thus, this critical perspective calls for caution when concluding that the sediment suspension results in an alteration of the turbulence structure of the outer-layer (η > 0.2), typically associated with Πs. It can be noted that on the basis of our mixing length model, Πs is determined by application of equation (42), and its numerical value is influenced by the effect of suspended sediment on the wall region, but not necessarily by its effects on the outer-layer.

[25] The proposed approach results in values of κmM which are similar in order of magnitude to κfit (Figure 6). The κmM values are obtained from a theoretical velocity profile based on an improved definition of the mixing length. By contrast, κfit is the result of a fit to experimental data, without any physical support. Therefore, the comparison of these results (Figure 6) has shed light on the empirical approach of other researchers. Turbulent momentum transfer reduces the mixing length, which introduces the new theoretical parameter κm = κm(η) investigated here.

[26] Modeled values of Πs for d = 0.105, 0.21 and 0.42 mm are plotted in Figure 7 against the values obtained by Coleman [1986] applying equation (42) to his experimental velocity profiles. The Πs values of Coleman [1986] are higher than the values obtained here, although the trends observed in Figure 7 for d = 0.105 mm are very similar. The discrepancies may be attributed to dip-phenomena, becauseColeman [1986]considered the elevation of the maximum velocity streamline as the boundary layer thickness to calibrate his wall-wake model. However, the differences, although significant for Πs, appear to be irrelevant for practical considerations, given the reasonable agreement between the predictions and the observations of velocity profiles (see Figures 3 and 4 for runs 29 and 40 and sand diameters 0.21 and 0.42 mm, respectively). The results of the proposed model are not far from Coleman's [1986] data, despite the evident deviation of Πs predictions in Figure 7. In conclusion, Coleman's [1986] Πsvalues include the effect of the damping of turbulence near the wall, because the log-law is assumed to be that of a clear-water flow, whereas the analyses ofValiani [1988] and Guo and Julien [2001] incorporate this effect mainly in κfit. Both parameters reflect the same effect, and the differences are based on a predominant effect of applying the fitting method to a different term of the log-wake model. In addition to its physical significance, an additional advantage of the proposed velocity profile is that it does not require any experimental measurements to adjust the value of empirical constants. Further, the computation based on the ODEequation (37) is simpler than RANS or DNS solutions.

Figure 7.

Predicted strength of the wake (Пs,th) based on equation (42) fitted to the numerical results for u+(η) obtained using equation (37), and fitted values obtained by Coleman [1981] in his experiments (Пs,exp) using equation (42).

5.2. Montes [1973] and Lyn [1986] Data Sets

[27] To further check the proposed model, experiments from two additional and independent data sets available in the literature [Montes, 1973; Lyn, 1986] are considered. Runs 20 and 23 of Montes [1973] are included in Figure 8. Montes's experiments were conducted in a flume of 0.487 m width. In the experiments described by Montes [1973]there were no secondary currents, thus the flow could be considered two-dimensional, and in his clear-water control tests the approximate values ofκ and Πo were = 0.41 and 0, respectively. Run 20 of Montes [1973] is plotted in Figures 8a and 8b. For this experiment the interface values were Ca = 0.0246 and ηa = 0.0308. The fitted concentration profile C(η) using equation (41) is plotted in Figure 8b and the prediction of u+(η) using equation (37) is plotted in Figure 8a. The modeled velocity profile approaches closely to experimental data. The function κm(η) obtained from equation (40) is represented in Figure 8c. As previously mentioned, the variation of κm(η) is influenced by some of the simplifying assumptions of the theory. Therefore, an approximate experimental estimate of κm(η) was sought from equation (38) as

display math

Using the measured velocity profiles, equation (44) was evaluated and the corresponding experimental variation of κm(η) is depicted in Figure 8c. Despite data scatter, the reduction of κm toward the wall can be appreciated. The theoretical prediction using equation (40)is in fair agreement with the experimental estimates. The present model provides, therefore, an advanced mechanistic interpretation of stratified velocity profiles. To further prove the improvement of this theory over previous statistical analysis, a log-lawu+ = U++(1/κfit)lnη deduced from equation (42) was fitted to experimental data. The statistical value obtained for κfit is 0.349, which is marked in Figure 8c, and the fitted log-law is plotted inFigure 8aas a dotted line. The fitted log-law is close to the theoretical prediction usingequation (37). κfit = 0.349 is only an empirical, at best, average and indirect indication of the density stratification effects.

Figure 8.

(a) Comparison of results obtained from the equation (37) for the velocity profile u+ = u+(η) with test data by Montes [1973], RUN 20. (b) Comparison of fitted equation (41) for the concentration profile C = C(η) with test data by Montes [1973]. (c) Comparison of κm = κm(η) from equation (40) with test data by Montes [1973]; (d, e, f) Analogous to Figures 8a–8c but for RUN 23.

[28] In addition, a depth averaged value of κm was obtained from equation (37) as

display math

For run 20, the variation of κs is significant up to ηmax = 0.3 (Figure 8c). Taking this limit for depth-averaging, inline image was found, a value very close to κfit = 0.349. This computation further reveals that the fitted κfit incorporates empirically a mean density stratification effect. Run 23 of Montes [1973] is plotted in Figures 8d–8f. For this experiment, Ca = 0.05 and ηa = 0.0296. Taking ηmax = 0.4 (Figure 8f) a value inline image results comparable to κfit = 0.325 (Figure 8f). The results are analogous to run 20 and therefore confirm the interpretation offered here.

[29] The experiment ST2A from Lyn [1986] is plotted in Figures 9a and 9b. For this experiment, Ca = 0.0032 and ηa = 0.071, and the value Πo = 0.2 was used. The concentration profile is plotted in Figure 9b, whereas Figure 9a compares predicted and measured velocity profiles. The predicted velocity profile is very close to the observations, supporting the predictions of the proposed model.

Figure 9.

(a) Comparison of results obtained from the equation (37) for the velocity profile u+ = u+(η) with test data by Lyn [1986], ST2A. (b) Comparison of fitted equation (41) for the concentration profile C = C(η) with test data by Lyn [1986].

6. Effect of Velocity Profile on Suspended Load

[30] The turbulent velocity profile in flows with sediment suspension has been a subject of major theoretical and practical interest in the past 50 years. One of the major reasons is that the velocity and concentration profiles allow the prediction of the sediment discharge, qs, [McLean, 1992; Garcia, 2008]

display math

Since the velocity u affects the computation of qs, equation (46), a precise definition of the turbulent velocity profile of sediment-laden flows is relevant for the description of earth surface processes. For example, river bed structures like ripples, dunes or antidunes are influenced by the action of the turbulent stream. Therefore, an improved description ofu not only provides insights into the mechanics of sediment suspension but also permits a more accurate and rational computation of qs. The normalized suspended load qN is

display math

The elevations are η1 and η2, and Δqs is the suspended sediment load between them. To investigate the effect of u+ on qN, representative runs from the data sets of Coleman [1986], Montes [1973] and Lyn [1986] were selected (Table 1, runs 29, ST2A and 23, respectively). For each experiment, the layer η1 was selected as the first elevation at which experimental measurements were taken, ηa. The upper limit of integration η2 was selected in order to have simultaneous measurements of velocity and concentration profiles. Equation (47) was evaluated numerically using the measured u+ and C/Co for each run. The computation are presented in Table 1 and labeled as qNexp. Computations using velocity profiles u+ predicted by equation (37) are also shown in Table 1 as qNth. The clear-water log-law velocity profile

display math

is frequently used for computation of qs [Guo and Wood, 1995]. Equation (48) was used into equation (47) for the same 3 experiments and the results are presented in Table 1. Results presented in Table 1confirm that the proposed velocity profile model produces suspended load estimations reasonably close to the experimental values, with a maximum deviation of −4.11%. In contrast, the log-law model induces a maximum deviation of −26.66%.

Table 1. Effect of Velocity Profile u+ in Suspended Load Computations
RunqNexpqNthError (%)
equation (37)equation (48)equation (37)equation (48)
29 [Coleman, 1986]2.882.853.151.04−9.47
ST2A [Lyn, 1986]1.291.341.65−4.11−26.66
23 [Montes, 1973]0.550.530.642.23−16.64

7. Discussion of Results in the Context of Velocity Profile Modeling

[31] Early studies [Vanoni, 1946; Einstein and Chien, 1955] proposed that suspended sediment induces a damping of turbulence based on indirect evidence. An indirect bulk measure of the turbulence damping was given by the von Kármán parameter κ, understanding it as the inverse of the slope of the best fit line to experimental data in the plane u+-lnη. However, Coleman [1981, 1986], based on his experiments, concluded that κ = 0.41 and that sediments affect the outer layer turbulence structure by an increased wake function. Currently, the research community is divided on this point. A more rational framework to investigate the turbulent velocity profile relies on understanding the turbulent momentum transfer in sediment-laden flow [Yang, 2007]. Consideration of the density gradient effects on the turbulence momentum transfer by a generalized mixing length theory enables the formulation of a more general turbulent velocity profile for sediment-laden flows without any pre-assumed hypothesis aboutκ. The mixing length formulation proposed in this work allows the definition of a function which is variable with its distance to the wall and dependent on density stratification. This κ function can be defined as a modified von Kármán parameter, κm. This does not mean that a priori it is assumed that κvaries with distance to the wall, and it is a theoretical result different from previous log-wake studies. Note that the log-law [Montes, 1973] or the log-wake law [Guo and Julien, 2001] are velocity profile functions that are empirically fitted to experimental data. In contrast, the proposed equation (37)is a mechanistic model based on the turbulent momentum transfer with suspended sediment. It therefore introduces rationality in the computations based on physical arguments. Further, the use of the log-law [Montes, 1973] or the log-wake law [Guo and Julien, 2001] requires experimental measurements. In contrast, the modeled velocity profile for sediment-laden flows uses the concentration profile as an input but itsequation (37) is free from other fitting parameters. The model provides a theoretical computation of the velocity defect profiles Z with no need for velocity data. An estimation of U+ from flow resistance equations permits the determination of the absolute velocity u+ = U+ − Z without resorting to experimental measurements. Thereafter, the proposed model permits the theoretical simulation of the velocity profile, thereby simplifying practical computations. In addition, previous stratified flow models [Smith and McLean, 1977; Villaret and Trowbridge, 1991; Mazumder and Ghoshal, 2006; Wright and Parker, 2004a, 2004b] obtained the parameter αfor the damping of eddy-viscosity inequation (27) by fitting it to experimental data. In contrast, our model provides an analytical predictor for it (see equation (32)).

8. Conclusions

[32] In this study, the effects of sediment suspension on turbulent momentum transfer have been investigated using a new mixing length model. A new local damping factor ψarose from the modified mixing length model for sediment-laden flows. This damping factor is given by a function of the distance to the wall that depends on the velocity and concentration profiles and their gradients. The damping factor allows one to define a von Kármán parameter as a local variable function of the distance to the wall. The velocity profile obtained is based on theoretical considerations, and it is the major difference as compared to the log-wake model [Coleman, 1981; Valiani, 1988] or the log-law model [Montes, 1973; Gust, 1984], where the name “von Kármán constant” is given to a free adjustable parameter determined by fitting to observations.

[33] The prediction of velocity from the proposed model agrees well with experimental observations. Thus, the defined von Kármán parameter as a function of the distance to the wall describes the turbulent momentum transfer with suspended sediment. The proposed model is further compared with a more general model using CFD simulation of turbulent sediment-laden flows, again resulting in good agreement. The model proposed is a good estimator of the velocity profile, as confirmed by experimental data. This requires integration of a first order differential equation, which can be achieved by using standard and well-known techniques. The model application is more complex than the use of a simple log-wake model, but, in contrast, the latter requires the calibration of theκfitparameter whereas our model simulates density stratification by theoretical means, which is an advantage. The numerical solution of our first order differential equation is simpler than programming the solution of the two-dimensional RANS equations using thek-ε closure, making the proposed model suitable for a broader audience. The improved velocity profile proposed can also be used to obtain more accurate estimations of the suspended load.

[34] The theoretical results obtained with the proposed model have been applied to analyze log-wake data fitting methods. It was found that if near-wall data are used, the conclusions ofColeman [1981, 1984] are generally supported, resulting in κ ≈ 0.41 and a wake strength Πs dependent on inline image. However, it is shown that this result does not imply that suspended sediment affects the turbulent structure of the outer layer. Further, it has been observed that when the whole wall-layer is considered, the fitted von Kármán parameter in a log-wake model abruptly drops below 0.41. This is a measure of density stratification effects, also reflected by the minimum values of the depth-distribution curve ofκm = κm(η) of the numerical model.

[35] The new density stratification theoretical results presented back up the experimental findings of Valiani [1988] and Guo and Julien [2001] that κfit < 0.41. Their results are an average measure in statistical terms of the density stratification of the flow, whereas this model provides a physical interpretation of their results. The density stratification effect causes a damping of the mixing length which mainly affects the momentum transfer in the wall-layer. This effect is theoretically approximated by the new von Kármán parameter distribution across the flow depth. The empirically fitted reducedκfit found by other researchers can be mechanistically interpreted in the light of our new von Kármán parameter: κfitis below 0.41 when the log-wake law is fitted to experimental data because of the damping of turbulence near the wall, that is,ψ < 1 or κs = (0.41ψ) < 0.41. This indicates that the log-wake model might be considered reliable for sediment-laden flows if it is carefully calibrated by using test data in the whole stratified water depth.

[36] The present results overcome the conceptual problem regarding the von Kármán parameter and its relation with suspended sediment: Does the suspended sediment affect κ, or not? From our model results we highlight that a von Kármán parameter can be defined as a turbulence variable in a mixing length differential model of the velocity profile. The effect of κm = κm(η) on log-wake law matching to test data is indirectly accounted for by settingκfit = 0.41 and Πs( inline image), following Coleman [1986], or κfit( inline image) and Πs( inline image), according to Valiani [1988] and Guo and Julien [2001]. The Coleman [1986]approximation is not a coherent solution for the velocity profile, and it is, therefore, not recommended for practical applications on sediment transport. Our model results show that a physically coherent log-wake fit should haveκfit < 0.41, and, in short, the suspended sediment affects κ.

Acknowledgments

[37] The good improvements and suggestions offered by the Chief Editor A. Densmore and the Associate Editor J.M. Buffington are gratefully acknowledged. Four anonymous reviewers and Junke Guo provided useful critiques and comments to improve the manuscript. The critical review and kind support received from Neville Robinson, Flinders University, Australia, is greatly appreciated. This research was partially funded by the project P08-AGR-03925 (Junta de Andalucía).