Toward twophase flow modeling of nondilute ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, F03015, doi:10.1029/2009JF001347, 2010

Toward two‐phase flow modeling of nondilute sediment transport in open channels Sanjeev K. Jha1,2 and Fabián A. Bombardelli1 Received 9 April 2009; revised 4 January 2010; accepted 23 March 2010; published 4 August 2010.

[1] In this paper, we generalize several models based upon the multiphase flow theory to address the nondilute transport of suspended sediment in open channels. These models naturally include the dilute condition as a special case. We assess the range of validity of models through simulations of the experimental tests by Vanoni (1946), Einstein and Chien (1955), Taggart et al. (1972), Coleman (1986), and Wang and Qian (1992). The K − " model is used to represent the turbulence in the carrier phase, and the kinetic theory of gases is employed for the closure of stresses due to inter‐particle collisions. We test various closures for the eddy viscosity of the disperse phase; we show the effects of increasing sediment concentration on the interaction forces between the two phases and on the stresses developed due to inter‐particle collisions. We determine the values of the Schmidt number required to approximate the experimental profiles of the volumetric concentration of sediment, suggesting that while the Schmidt number is smaller than one for dilute mixtures, it is larger than one for nondilute flows. Results indicate that nondilute models of increasing complexity are indeed necessary for flows having high solid volume fractions. Our results also suggest a more tested range of values for the threshold of sediment concentration of nondilute flows; this threshold can be located between 2 and 5%. It is observed that the virtual mass force becomes important in the simulation of nondilute flows. The eddy viscosity of the carrier phase is found to decrease with the increasing volumetric concentration in the flow. Citation: Jha, S. K., and F. A. Bombardelli (2010), Toward two‐phase flow modeling of nondilute sediment transport in open channels, J. Geophys. Res., 115, F03015, doi:10.1029/2009JF001347.

1. Introduction [2] A generalized model is required to capture the underlying physical mechanisms embedded in nondilute, sediment‐ laden flows. Such model is needed for the simulation of various industrial processes (e.g., pneumatic conveying, fluidized beds, slurry reactors, etc.) as well as natural phenomena (e.g., sediment transport in rivers and estuaries, volcanic eruptions, etc.). The specific problem considered in this work is the transport of suspended sediment of varying concentration in an open channel. [3] In recent years, various researchers have advocated the use of the two‐phase flow theory to simulate sediment‐laden, open‐channel flows [Drew, 1975; McTigue, 1981; Kobayashi and Seo, 1985; Cao et al., 1995; Villaret and Davies, 1995; Greimann et al., 1999; Greimann and Holly, 2001; Hsu et al., 2003a; Jiang et al., 2005; Hsu et al., 2004; Chauchat, 2007; Amoudry et al., 2008; Chauchat and Guillou, 2008; Toorman, 2008; Bombardelli and Jha, 2009; Jha and 1 Department of Civil and Environmental Engineering, University of California, Davis, California, USA. 2 California Department of Water Resources, Sacramento, California, USA.

Copyright 2010 by the American Geophysical Union. 0148‐0227/10/2009JF001347

Bombardelli, 2009]. Under an Eulerian‐Eulerian framework, the two‐phase flow theory considers that both phases (sediments and water in this case) are continua, and that they can be modeled separately. The interdependence of motions of both phases can be included in the model by considering the coupling among phases (one‐ to four‐way couplings) [see Loth, 2000]. Four‐way coupling refers to the case in which particles collide. Thus, the type of coupling in the flow plays a critical role in the definition of the concentration and the velocity of the sediments. It is worth mentioning here that in this paper we have interchangeably used the words “nondilute flow,” “dense flows,” and “flow carrying high concentration of sediments,” when we refer to flows in which four‐way coupling is important. We leave outside of the analysis, though, the cases of hyper‐concentrated flow in which rheological effects become important. In this regard, the minimum sediment concentration after which a yield stress is observed in the behavior of a mixture has been found to depend highly on the content of fines [Wan and Wang, 1994]; in this work we consider uni‐modal sediment in the sand range, for which such a limit should in principle exceed the values of maximum concentration analyzed herein (see below). [4] Current understanding of sediment‐laden, open‐channel flows mainly pertains to dilute cases. Comparatively, few

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JHA AND BOMBARDELLI: TWO‐FLUID MODELS FOR SEDIMENT TRANSPORT

studies have addressed open‐channel flows carrying relatively high amounts of sediment. Most previous studies which analyze the dilute or dense transport of suspended sediment can be classified into three approaches: (a) those which consider the flow as a mixture of sediment and water [Einstein and Chien, 1955; Vanoni, 1975; Coleman, 1986; Lyn, 1988, 1991]; (b) those which apply two‐phase flow theory [Cao et al., 1995; Greimann et al., 1999; Greimann and Holly, 2001; Hsu et al., 2003a; Jiang et al., 2005; Chauchat, 2007; Toorman, 2008]; and (c) those which apply the analogy between the sediment‐laden flow and the motion of gases (kinetic theory) [see Jenkins and Hanes, 1998; Ni et al., 2000]. From a mixture flow perspective, a brilliant yet simple derivation developed by Rouse [1937] is commonly used to predict the distribution of suspended sediment in cases of flow carrying low amounts of sediment. The Rousean formula is derived by balancing the gravitational settling of particles and the diffusive flux of the sediment due to turbulence, which maintains the particles in suspension: Dd

dC þ Ws C ¼ 0 dz

ð1Þ

In equation (1), Dd is the eddy diffusivity of the sediment; C is the volume concentration of the suspended sediment; Ws is the fall (settling) velocity; and z represents the vertical distance from the bed. In the Rousean formula, the sediment concentration is expressed in terms of the reference concentration at a certain distance from the bottom (usually taken at 5% of the water depth), and a power law exponent Z = Ws /( S U*), where is the von Kármán constant, U* denotes the shear (friction) velocity, and S refers to the “damping” or “attenuation” coefficient, which amounts to the inverse of the Schmidt number, Sc. The Sc is defined as the ratio between the eddy viscosity of the flow and the diffusivity of suspended sediment. Further, this equation is used in its so‐called standard form [García, 1999] (see also G. Parker, 1D sediment transport morphodynamics with application to rivers and turbidity currents, unpublished manuscript, available at http://vtchl.uiuc.edu/people/parkerg, 2004), assuming that the sediment has the same eddy diffusivity as the eddy viscosity of the flow, i.e., that S = 1. Many authors have remarked limitations of the standard Rousean expression even for dilute mixtures [Vanoni, 1975; Greimann et al., 1999; Cellino and Graf, 2002; Nikora and Goring, 2002; Nezu and Azuma, 2004; Muste et al., 2005; Lyn, 2008]. While using the semi‐logarithmic law of the wall approximates well the time‐averaged water velocity profile in an open channel, the distribution of sediment in suspension is predicted by the Rousean model with sizable differences with measured data (see for instance figures by Greimann et al. [1999]). Vanoni [1975] remarked in his classic manual that the Rousean distribution provides an adequate fit to the data, but that tuning is needed on the parameter Z through the damping parameter. The need for tuning S in Z in turn suggests that corrections are needed in the Rousean approach in order to capture certain aspects of the flow‐sediment interactions not included in Rouse’s balance. [5] Several studies have shown that when the flow carries relatively large amounts of sediment the following equation

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proposed by Hunt [1954] is more suitable than the Rousean balance (equation (1)): Dd

dC þ W s C ð1 C Þ ¼ 0 dz

ð2Þ

The factor (1 − C) represents the volume fraction of water, which in the case of dilute flows can be approximated as one. Based on Hunt’s diffusion equation, Woo et al. [1988], Tsai and Tsai [2000], and Mazumder and Ghoshal [2006] obtained the vertical distribution of (relatively large) concentrations of sediment in a steady and uniform two‐ dimensional turbulent flow. In their analysis, they took into account the effect of hindering (due to the high concentration of sediment) in the free fall of sediment. Tsai and Tsai [2000] and Mazumder and Ghoshal [2006] derived the gradient of the velocity of the water‐sediment mixture by expressing the turbulent shear stress in terms of Prandtl’s mixing length, which in turn they defined as a function of sediment concentration. Tsai and Tsai [2000] and Mazumder and Ghoshal [2006] then expressed Dd in terms of the mixing length and the velocity gradient of the water‐ sediment mixture. Thus, from equation (2) they obtained an expression for the distribution of sediment concentration. The predicted profiles of velocity and sediment concentration were compared with data of Einstein and Chien [1955], with good agreement. [6] Greimann and Holly [2001] derived a model to predict the distribution of suspended sediment in dense flows in open channels consisting of equations for the conservation of mass and the vertical momentum of the sediment phase, only. Greimann and Holly included an additional contribution to the shear stress in the momentum equation of the sediment phase, which results from inter‐particle collisions. The closure for this stress term was derived from the kinetic theory of gases (similar to the closure used by Balzer et al. [1996] and Enwald et al. [1996]; see Jenkins and Hanes [1998] for cases of gas‐solid flows). They considered drag and virtual mass as the dominant interaction forces between the two phases. The eddy viscosity and the turbulent intensity of the carrier phase were obtained from the empirical relationships of Nezu and Rodi [1986]. The relative velocity between both phases was defined in terms of the drift velocity, which was described as the correlation between the spatial distribution of particles and the fluctuating fluid velocity. Greimann and Holly [2001] found good agreement of model predictions for the concentration distribution with the experimental data of Einstein and Chien [1955] and Wang and Qian [1992]. Greimann and Holly acknowledged that their work lacks the modeling of the velocity distribution of the carrier phase, the velocity distribution of the disperse phase and modeling of the turbulence statistics. [7] Ni et al. [2000] combined the Boltzmann equation for gas molecules for the disperse phase with the continuum theory (for the carrier phase) to predict the vertical profiles of particle concentration in dilute and nondilute, solid‐liquid flows in pipes and open channels. In their test cases corresponding to open‐channel flows, Ni et al. compared model predictions with the experimental data of Wang and Qian [1992], with good agreement.

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[8] In related work, Hsu et al. [2004] simulated the slow flow of massive particles in the region of high concentration above the stationary bed. Starting from the two‐phase flow equations of Drew [1983] and based on a Favre average, they considered only the drag force as the interaction force between the two phases. Hsu et al. [2004] applied the eddy viscosity hypothesis for the closure of Reynolds stresses of both the carrier and sediment phases. For the fluid phase, they used a K − " model, where K and " denote the turbulent kinetic energy (TKE) and the dissipation rate of TKE (DTKE), respectively. The equation for the TKE was obtained from the model by Hsu et al. [2003a] and the equation for the DTKE was taken from the model of Elghobashi and Abou‐Arab [1983]. In these equations, the damping effects of sediment on the turbulence of the carrier fluid are incorporated. For the sediment phase, Hsu et al. [2004] defined an eddy viscosity in terms of the sediment mixing length and large‐scale sediment velocity fluctuations. The mixing length of sediment was considered to be equal to that of the carrier phase. A transport equation was defined to compute the large‐scale sediment velocity fluctuations. In order to model the inter‐particle stresses in the sediment phase, they employed the closure proposed by Jenkins and Hanes [1998] which uses the so‐called “granular temperature” as a measure of the strength of the particle velocity fluctuations. The granular temperature and the particle stresses were calculated based on the kinetic theory for dense inelastic spheres. Hsu et al. [2004] compared their results to the experimental data of Sumer et al. [1996] and Asano [1995] with good success. The two‐phase flow model proposed by Hsu et al. [2004] was developed for the transport of massive particles only. Later Amoudry et al. [2008] modified the model to study the nearshore sediment transport of both massive particles and small sand grains. Amoudry et al. [2008] argued that the fluid turbulence closure used by Hsu et al. [2004] was inadequate to represent the transport of smaller sediment particles, and that the coefficients used in the K − " model needed to be reformulated. Amoudry et al. [2008] proposed modified values of constants in the equation for the DTKE to properly account for the effect of particles on the flow turbulence. The proposed constants were based on previous studies of Squires and Eaton [1994] and Ahmed and Elghobashi [2000]. A good agreement was observed between the model prediction and the data for oscillatory flows and uniform steady flows. [9] Unlike nondilute flows, the literature on dilute mixtures is more abundant (discussed by Bombardelli and Jha [2009]). Villaret and Davies [1995] presented a review of several models, classifying them as “passive‐scalar‐ hypothesis” and two‐phase‐flow models. While in the first type of model a simple transport equation is solved for the sediment, in the second approach equations of mass and momentum for both phases are utilized. After simplified versions of models initially presented by Drew [1975] and McTigue [1981], the work by Kobayashi and Seo [1985] can perhaps be considered as the first two‐fluid model (TFM) for the sediment‐transport problem. Cao et al. [1995] used a two‐layered approach (suspension layer and bed layer) with a two‐phase model. Greimann et al. [1999], in turn, included in their analysis the drift velocity in addition to the relative velocity between phases to build a one‐dimensional

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model. Jiang et al. [2005] also combined the momentum equations for the carrier and the disperse phases and introduced simplifications to the resulting equation based on the assumption of dilute mixture and negligible particle inertia. Hsu et al. [2003a] used a semi‐analytical approach and a numerical solution of the TFM in combination with an extended K − " turbulence closure, addressing the fluctuations in the carrier fluid in dilute sediment‐transport studies. Chauchat and Guillou [2008] proposed a two‐dimensional (2D) vertical two‐phase model to represent sediment‐laden flows in an open channel, modeling fluid particle turbulent interactions. For the fluid phase, a K − " model was applied, while the solid phase was approached using two models: one based on the kinetic theory of granular flows (first‐order model) and the other assuming a homogeneous, isotropic, and steady fluid turbulence (algebraic model). In order to validate the numerical results the authors compared their outputs with experimental data. It was found that the model appropriately represented the main features of the sediment‐ laden flow, with no adjustment of empirical parameters, which the authors highlighted as an improvement with respect to classical sediment transport models. Finally, the algebraic model was found to be a good approach when compared to the first‐order model given the lack of complexity. Benavides and van Wachem [2008] employed a Favre average to develop a framework for turbulent gas‐ particle flows. The authors validated their model with data on gas‐solid flow in a vertical pipe. The model includes mass and momentum equations for the phases and convenient closures for the stresses and interaction terms; a K − " model was used to address the turbulence of the carrier phase and a transport equation for the granular temperature was employed to quantify the TKE of the disperse phase. Although Benavides and van Wachem [2008] focused on dilute flows for validation, some of the elements and closures of their models could be in principle applied to nondilute flows as well. [10] In the work of Bombardelli and Jha [2009] and Jha and Bombardelli [2009], we proposed a hierarchy of models to simulate the dilute transport of suspended sediment in open channels. In what follows we present a summary of those papers, since the models presented in this paper constitute a generalization of the models proposed and discussed therein. We developed three approaches composed by a standard sediment‐transport model (SSTM), a partial two‐fluid model (PTFM), and a complete two‐fluid model (CTFM). In the SSTM, it was assumed that the velocity of the sediment is the same as that of the water, and that there is no specific interaction between the sediment and the flow turbulence. In fact, it was also assumed that the sediment diffuses in the vertical direction as the fluid momentum does, following Rouse’s ideas. In the PTFM, the governing equations for the flow of a mixture are used as a surrogate for the motion of the carrier phase; the disperse phase is described separately. The CTFM is constituted by a set of equations representing the conservation of mass and momentum of each phase. The level of complexity increases from the SSTM to the CTFM. In the work of Bombardelli and Jha [2009] and Jha and Bombardelli [2009], we addressed several issues related to coupling between the two‐ phases, interaction forces, turbulence closures and turbulent diffusivities for dilute flows. We found that the PTFM and

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the CTFM provide better predictions of mean flow variables and sediment concentrations than the SSTM. We corroborated that in cases of dilute flows, the drag force is clearly the dominant interaction force between the two phases. We also observed that the eddy viscosity of the disperse phase can be neglected in the simulation of dilute flows. In our second paper, we applied several turbulence closures for the Reynolds stresses and fluxes in the carrier phase. We tested the performance of the K − " model, the K − w model, the algebraic stress model (ASM) and the Reynolds stress model (RSM). In both papers, we found that the sediment diffusivity had to be modified using the Schmidt number to obtain accurate predictions of the distribution of the sediment concentration. [11] Regarding the Schmidt number, there is no consensus in the literature on its value. Despite several studies using the Sc in the analysis of dilute flows, it is not clearly defined whether the Sc should be larger or smaller than one. For example, based on the regression presented by van Rijn [1984], it has been suggested that the value of the Schmidt number should not be larger than one. However, several studies have shown that, even in the case of dilute flows with different mass loadings, the values of the Sc can vary from one to two [Cellino and Graf, 2002; Abdel‐Fattah et al., 2004; Yoon and Kang, 2005]. Taggart et al. [1972] and Lees [1981] showed an increasing trend of the Sc with increasing suspended sediment concentration at the reference level above the bed. There have been some proposals to define the Sc in terms of sediment concentration [Amoudry et al., 2005] or turbulence statistics [Toorman, 2008]. In the work of Bombardelli and Jha [2009] and Jha and Bombardelli [2009] the values of the Sc were found to be less than one in all tests corresponding to dilute mixtures. [12] A detailed analysis of the papers presented above allows us to highlight the following issues: [13] 1) Previous studies did not agree on limits of the maximum value of the volumetric concentration of sediment (from the concentration profile in the wall‐normal direction, Cmax) to differentiate between dilute and nondilute flows. Cmax appears to be perhaps the most suitable variable to characterize the condition of the mixture. For example, most studies on gas‐particle flows considered that when Cmax in the flow is higher than 0.001, the inter‐particle interactions become important [Lumley, 1978; Nouri et al., 1987; Elghobashi, 1991, 1994]. Woo et al. [1988] argued that when Cmax > 0.04, the models developed for dilute flows do not work. Greimann and Holly [2001] and Mitter et al. [2004] adopted the value Cmax = 0.1 for the threshold beyond which the flow is nondilute. This raises the basic and very important question about the range of validity of the models developed for dilute and nondilute cases. [14] 2) In the application of the two‐phase flow theory to simulate the nondilute transport of suspended sediments in open channels, additional (unknown) correlations make the problem more difficult to solve. To the best of our knowledge, very few papers discuss how to model the diffusivity of sediments, stresses developed due to inter‐particle collisions, and the eddy viscosity of the disperse phase. [15] 3) The usefulness and accuracy of the kinetic theory of gases when modeling the inter‐particle stresses need to be verified through systematic comparisons of predicted mean

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values of the flow variables, turbulence statistics and the distribution of sediment concentration, simultaneously. Previous studies present comparison of either the mean velocity profiles or the distribution of sediment concentration, only. [16] 4) To the best of our knowledge, the values of the Sc are poorly known for nondilute flows. It is not clear whether the Sc varies with the concentration or whether a constant value of the Sc can be applied throughout the depth irrespective of local sediment concentration. [17] 5) The relative importance of the interaction forces such as lift, virtual mass and other forces with respect to drag are not clear for sediment‐laden, open‐channel flows. There is a lack of quantitative assessment in general, and especially for nondilute flows. [18] In this paper, we address most of the above issues by developing a theoretical/numerical framework for the nondilute transport of sediment in open channels, which has the dilute flows as a special case. To that end, we extend our framework and model equations presented in 2009 to simulate the nondilute transport in sections 2 and 3. Ranges of validity of the dilute and nondilute models in terms of the sediment concentration in the flow are defined. We investigate the performance of various closures for the eddy viscosity and diffusivity of the disperse phase. Estimations of the relative importance of interaction forces of drag, lift and virtual mass are also made. We assess the performances of the developed models by comparing simulation results with various test cases from the experimental data of Vanoni [1946], Einstein and Chien [1955], Coleman [1986], and Wang and Qian [1992] (sections 4 and 6). To the best of our knowledge, this is the first time that an overarching picture of the modeling of nondilute transport of suspended sediment in open channels using the two‐phase flow theory is presented.

2. Mathematical Models Based on the Two‐Fluid Model 2.1. Complete Two‐Fluid Model for Nondilute Flow [19] The general equations for two‐phase flows are based on the “two‐fluid” model (TFM). Drew and Passman [1999] presented a rigorous derivation of the balance equations of mass, momentum, and energy for two‐phase flows starting from the basic principles of continuum mechanics and applying an ensemble averaging to the local instantaneous equations of motion. The mass and momentum equations representing the most general equations to deal with a two‐ phase flow [Crowe et al., 1998; Drew and Passman, 1999; Prosperetti and Tryggvason, 2007] are as follows: Conservation of mass of the two‐fluid model (TFM) @ @ ~ j; p ¼ G p ~p U p ~p þ @t @xj

ð3Þ

Conservation of momentum of the two‐fluid model (TFM) ~ @ ~ i; p þ @ p ~p U ~ i; p U ~ j; p ¼ p @ P p ~p U @t @xj @xi i @ h ~ EA þ p T ij; p þ Tij; p þ p ~p gi ðFint Þi ð4Þ @xj

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where the subscript p refers to both phases (which could be c or d for the carrier and the disperse phases, respectively); ~ , and P ~ are ap denotes the volume fraction of phase p; ~, U the ensemble‐averaged density, velocity and pressure, respectively; T~ ij,p indicates the ensemble‐averaged deviatoric stresses of phase p; the stresses with the superscript EA constitute the remainder of the process of ensemble averaging; gi refers to the i‐th component of the acceleration due to gravity; (Fint)i involves all the interactive forces, typically, due to drag, lift, virtual mass, and turbulent dispersion; G is the mass transfer rate between phases; and x and t are the space and time coordinates, respectively. The indices i and j vary from 1 to 3 and the sum is implied in repeated indices (Einstein convention). These equations, which are also included in Appendix A as equations (A1) and (A2), are valid for both dilute or nondilute mixtures, provided that the adequate closures for the shear stresses, the inter‐phase forces and the stresses from inter‐particle collisions are given. [20] It is well known that length scales of turbulence range from the Kolmogorov length scale to the large length scales of the flow, determined by the flow depth or flow width [Gioia and Bombardelli, 2002]. Regarding the ensemble averaging, we believe that the length and time scales associated with it are smaller than the intermediate to large scales developed due to turbulence [Buscaglia et al., 2002; Bombardelli, 2004; Bombardelli et al., 2007; Bombardelli and Jha, 2009]. Therefore, we believe that the ensemble average encompasses scales of the order of the particle‐to‐ particle distance, only. This is a different approach to the interpretation of the ensemble average found in several works, which assumes that the ensemble average encompasses all scales [Drew and Passman, 1999]. As a consequence of this hypothesis, we consider that an additional turbulence (time) average of equations (3) and (4) is necessary, to account for the intermediate and large fluctuating scales (see also Hrenya and Sinclair [1997] and Lopez de Bertodano [1998], who have used a double average, and the discussion on chapter 8 of Prosperetti and Tryggvason [2007] on the nature of the averaging procedures). We have used this double averaging in our 2009 papers with success. The equations resulting from the double average are presented in equations (A6) and (A7). We observe that these equations involve several second‐ and third‐order correlations. From a theoretical point of view, different formulations can be provided for those correlations, as done by Elghobashi and Abou‐Arab [1983]. However, these theoretical correlations do not often have a solid experimental basis. In Appendix A, we make an attempt to provide analysis and scaling for some terms. We reduced the complete momentum equations obtained from time averaging (equation (A7)) into a simplified form as presented in equation (A8). Equation (A8) should be regarded as a model in itself. The model consisting of equations (A6) and (A8), and equations (5) and (6) below, is referred to as the complete two‐fluid model for nondilute flow (CTFMND): Conservation of mass of the CTFMND @ @ p p þ p p Uj; p þ p p0 uj;0 p ¼ 0 @t @xj

ð5Þ

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Conservation of momentum of the CTFMND @ p p Ui; p @ @P p p Ui; p Uj; p ¼ p þ @xj @xi @t i @ h þ p Tij; p þ &ij; p p; d þ Tij;Rep @xj þ p p gi ðFint Þi

ð6Þ

where overbars indicate time‐averaged variables and & ij,p models the stresses developed due to interparticle collisions (similar to Enwald et al. [1996]). This stress term becomes important in the momentum equations of the disperse phase while simulating the cases of nondilute flows. In equation (6), dp,d is 0 when p = c, and is 1 when p = d. [ 21 ] Similar approaches have been used by Dong and Zhang [1999], Hsu et al. [2003b], Mina and Sato [2004], Longo [2005], and Liu and Sato [2006], albeit with single averaging. It is worth pointing out that this model computes the vertical component of the velocity of the disperse phase, rather than imposing it (see below). 2.2. Partial Two‐Fluid Model for Nondilute Flow [22] In this paper, we extend to nondilute flows an approach we originally developed for bubble plumes [Buscaglia et al., 2002; Bombardelli et al., 2003; Bombardelli, 2004], and used afterwards in the dilute sediment transport problem. In this approach, mixture equations are derived by combining the mass and momentum equations of the two phases [see also Brennen, 2005]. The equations for the mixture are used as a surrogate for the carrier phase. We summarize the mathematical derivations of the mixture equations in Appendix B. Equations (B4) and (B5) of Appendix B are the mass and momentum conservation equations for the mixture, respectively. We can see that equations (B4) and (B5) neither involve the volume fraction, nor do they involve interaction forces between the phases. Therefore, this approach has an advantage in reducing the complexity involved in solving the nonlinear and inter‐dependent equations of the TFM. However, the last term of the (B5) contains the velocities of the carrier and the disperse phases. Thus, the two phases are connected through this term. [23] Following the hypothesis of addressing the intermediate and large scales of turbulence with another averaging operation, we also performed a time averaging of the governing equations of the mixture phase and the disperse phase. The steps involved in the time averaging of the ensembled averaged equations are presented in Appendix B. Finally, we consider equations (B7) and (B9) (equations (7) and (8) below) as the governing equations for the motion of the mixture. Conservation of mass for the mixture @m @ þ m Uj; m ¼ 0 @xj @t

ð7Þ

Conservation of momentum for the mixture

@P @ m Ui;m @ @ Re þ m gi m Ui;m Uj;m þ ¼ Tij;m þ Tij;m þ @t @xj @xi @xj

@ d d d d 1 Ui;m Ui;d Uj;m Uj;d m @xj ð8Þ

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Conservation of mass for the disperse phase @ ðd d Þ @ ðd d Wd Þ @ þ þ d d0 wd0 ¼ 0 @t @z @z

ð10Þ

Conservation of momentum for the carrier phase in the stream‐wise direction Figure 1. Schematic of sediment‐laden, open‐channel flow. The y axis points in the direction perpendicular to the page. where the subscript m indicates the mixture variables. To describe the motion of the disperse phase, we utilized equations (A6) and (A8) in their components for the disperse phase (p = d). We call this model partial two‐fluid model for nondilute flow (PTFMND) in its general version. [24] Instead of solving equation (8) in the wall‐normal direction, we decided to impose the vertical component of the velocity of the disperse phase via an algebraic model, keeping the computation of the stream‐wise component. This methodology, which as said follows previous usages in bubble plumes and sediment transport [Buscaglia et al., 2002; Bombardelli and Jha, 2009], produces a difference in the value of the vertical component of the velocity of the disperse phase. The algebraic model is expressed mathematically through equation (B10) in terms of a settling velocity, which can be obtained from empirical regressions such as that developed by Dietrich [1982]. Equation (B10) in Appendix B avoids the necessity to solve the full momentum equation of the disperse phase to obtain the vertical velocity of the disperse phase. We called this model the PTFMND in a restricted sense and this is the model we use in the computations in its 1D version.

ð11Þ

In equations (9) to (11), Uc denotes the mean velocity of the carrier in the stream‐wise direction; Wc and Wd indicate the components of the velocity of the carrier and disperse phases in the wall‐normal direction, respectively; Sb refers to the slope of the channel; and g is the acceleration of gravity. The full conservation equation of momentum for the carrier phase in the wall‐normal direction is given by: @ ðc c Wc Þ @ @Pc þ ðc c Wc Wc Þ ¼ c @t @z @z i @ h Re c Tzz; c þ Tzz; c c c g cos Fint; z ð12Þ þ @z

Solving (12) allows for the computation of Wc without any restriction. In the CTFM (dilute model) we adopted simply Wc = 0, a plausible representation of the physics of the problem. In this 1D case, instead of solving equation (12), we used an alternative approach to obtain Wc consisting in combining the mass conservation equations of both phases, as follows. The volume fractions of the carrier and disperse phases are related by: c þ d ¼ 1

ð13Þ

Assuming a quasi‐steady state condition and rc and rd as invariants, and approximating d0 wc0 ≈ d0 wd0 , we combined equations (9) and (10) to obtain (following Dong and Zhang [1999] and Mina and Sato [2004]):

3. Simulation of 1D Sediment‐Laden, Open‐Channel Flow [25] The flow scenario is presented in Figure 1. It is assumed that the slope of the channel (Sb) is small. Gradients of the variables in the stream‐wise and transverse @ @ 0; @y 0 in directions are considered negligible @x @ comparison to those in the wall‐normal direction @z . The sediment particles are assumed to have uniform size. In what follows, we pose the equations of the previous sections in one dimension in the wall‐normal direction. 3.1. 1D Complete Two‐Fluid Model for Nondilute Flows [26] In the case of the 1D complete two‐fluid model for nondilute flows (1D CTFMND), the mass conservation equations for the carrier and the disperse phases can be obtained by re‐writing equations (5) and (6) as follows, where we have eliminated overbars for the time‐averaged variables, except in the fluxes: Conservation of mass for the carrier phase @ ðc c Þ @ ðc c Wc Þ @ þ c d0 wc0 ¼ 0 @t @z @z

i @ ðc c Uc Þ @ @ h Re þ ðc c Uc Wc Þ ¼ c Txz; c þ Txz; c @t @z @z þ c c g Sb Fint;x

ð9Þ

@ ½ð1 d ÞWc þ d Wd ¼ 0 @z

ð14Þ

This equation has also a physical interpretation and has been dubbed by Benavides and van Wachem (page 295) as the conservation of “volume.” On integrating equation (14), we considered that the mixture vertical velocity has a null value at the top boundary, and obtained the following equation: Equation for the carrier phase in the wall‐normal direction Wc ¼

d Wd ð1 d Þ

ð15Þ

In turn, the momentum equations for the disperse phase read: Conservation of momentum for the disperse phase in the stream‐wise direction i @ ðd d Ud Þ @ @ h Re þ ðd d Ud Wd Þ ¼ d Txz;d þ Txz;d þ &xz @t @z @z þ d d g Sb þ Fint;x ð16Þ

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Conservation of momentum for the disperse phase in the wall‐normal direction @ ðd d Wd Þ @ @Pc þ ðd d Wd Wd Þ ¼ d @t @z @z i @ h Re d Tzz; d þ Tzz; d þ &zz þ @z d d g cos þ Fint; z

Equations (16) and (17) follow from equation (A8). [27] The first term on the left‐hand side of equations (9) and (10) denotes the rate of change of mass in a point. The second and third terms on the left‐hand side of equations (9) and (10) represent the convective mass flux in the vertical direction and the transport by turbulent diffusion, respectively. The first and second terms on the left‐hand side of equations (11), (16) and (17) are the rate of change of specific momentum in a point, and the rate of transport of the specific momentum by convection in the vertical direction, respectively. The first term on the right‐hand side of equations (11) and (16), and the second term on the right‐ hand side of equation (17) are the deviatoric stress terms. The last term on the right‐hand side of equations (11), (16) and (17) refers to the interaction forces. The first term on the right‐hand side of equation (15) is the mean pressure gradient. In equations (16) and (17), & xz and & zz represent the stresses due to inter‐particle collisions [Enwald et al., 1996]. [28] The governing equations for the dilute version of this model developed by Bombardelli and Jha [2009] (the CTFM), can be obtained from the CTFMND model. For dilute conditions, the volume fraction of the sediment can be considered very small (ad 1) and therefore equation (15) can be approximated as Wc = 0, as before. Due to the dilute nature of the flow, the effect of inter‐particle collisions is very weak and, therefore, the resulting stresses (& xz and & zz) can be neglected in equations (16) and (17). From this analysis, it is possible to see that the closure associated with the stresses due to inter‐particle collisions, the use of a more sophisticated representation of the water velocity, and closures for those stresses are the differences between the CTFMND and the CTFM [Bombardelli and Jha, 2009; Jha and Bombardelli, 2009]. 3.2. 1D Partial Two‐Fluid Model for Nondilute Flows [29] The 1D partial two‐fluid model for nondilute flows (1D PTFMND) can be obtained from equations (A6), (A8), (B7), (B9), and (B10). The mass and momentum equations for the disperse phase are the same as those used in the CTFMND model. The following equations constitute the 1D PTFMND. The equation for the conservation of mass for the carrier phase is: @m @ ðm Wm Þ ¼0 þ @z @t

ð18Þ

which under steady state conditions leads to: Conservation of mass for the carrier phase @ ðm Wm Þ ¼0 @z

The conservation of mass for the disperse phase can be obtained from equation (A6) as: @ ðd d Þ @ ðd d Wd Þ @ þ þ d d0 wd0 ¼ 0 @t @z @z

ð17Þ

ð19Þ

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ð20Þ

Conservation of momentum for the carrier phase in the stream‐wise direction @ ðm Um Þ @ @ @ Re þ ðm Um Wm Þ ¼ Txz;m þ Txz;m þ m gSb @t @z @z @z

d d ½c Ud m Um þ d Ud ðd c Þ ðWd Þ d d 1 c ð1 d Þ m ð21Þ

which comes from (B9). Momentum equation for the carrier phase in the wall‐normal direction Wm ¼ 0

ð22Þ

Conservation of momentum for the disperse phase in the stream‐wise direction i @ ðd d Ud Þ @ @ h Re þ ðd d Ud Wd Þ ¼ d Txz; d þ Txz; d þ &xz @t @z @z ð23Þ þ d d g Sb þ Fint; x

Instead of solving the momentum equation for the disperse phase in the wall‐normal direction (which would look similar to equation (17)), we applied an algebraic model (equation (B10)) to obtain Wd = −Wnds, where Wnds is the fall (settling) velocity of multiple sediments. The subscript “nds” refers to the effect of the concentration of particles on the settling velocity of a group of particles [Lewis et al., 1949]. Thus, the final equation becomes: Equation for the disperse phase in the wall‐normal direction Wd ¼ Ws ð1 d Þn

ð24Þ

where Ws is the settling velocity of a single sediment particle; n is an exponent which depends upon the particle ffiReynolds qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi number (Rep is defined as Rep = ( g dp3 D=m )/n; dp is the diameter of the sediment particle; Dr = (rd − rm); rm is the density of the mixture and n refers to the kinematic viscosity of water) [Richardson and Zaki, 1954; Garside and Al‐Dibouni, 1977]. We obtained approximate values of n from a chart presented by Cheng [1997]. [30] It is worth noticing that the governing equations for the PTFM (which corresponds to dilute condition) can be obtained by applying the following approximations to the PTFMND: (a) for dilute conditions, the contribution of the last term in equation (21) can be neglected, since (ad rd /rm) 1; (b) the contribution of stresses due to inter‐particle collisions can be disregarded; i.e., in equation (23) & xz ≈ 0; and (c) (1 − ad) can be assumed to be identically equal to one in the equation (24). [31] It is worth mentioning that in both models the stream‐wise component of the velocity of the disperse phase is calculated through equations (16) (CTFMND) and (23) (PTFMND). This velocity is different from the stream‐ wise velocity of the water or carrier, as discussed by Muste

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et al. [2005, 2009], and we have modeled this difference successfully for dilute flows in our two previous papers. In this application, those velocities are also different, but we do not have data to compare our results to. 3.3. Closures for the Models [32] The governing equations require further closure. Closures are necessary for the terms like: −wd0 d0 , Txz,c , Txz,d, Re Re Re , Txz,d , Tzz,d , & xz,d, & zz,d, and Tm. We also need Tzz,d, Txz,c definitions for the interaction forces Fint. These closures are detailed in Appendix C. We present herein a summary of those closures: a) Solid‐fraction Reynolds fluxes are closed using the standard approach based on the gradient of time‐ averaged solid volume fractions; b) Reynolds stresses are approximated with the well‐known Boussinesq model, which is a linear (Newtonian in the general sense) model; c) the eddy viscosity of the carrier phase is obtained from the K − " model, including a buoyancy‐production term; d) the eddy viscosity of the disperse phase is modeled using three approaches, as follows: i) it is first assumed to be zero, ii) it is assumed to have a value equal to the eddy viscosity of the carrier in second place, and iii) it is finally modeled via the mixing‐length concept. When using the mixing‐length concept, the turbulence statistics of the disperse phase are computed in terms of the value of the mixing length. Inter‐ particle collisions are modeled with the approach proposed by Enwald et al. [1996] for fluidization, which was employed by Greimann and Holly [2001]. Although the models require a large number of closures, we believe to have selected adequate treatments for the variables needing closures. 3.4. Boundary Conditions [33] Similar to Bombardelli and Jha [2009], we evaluated the shear stress at the wall by computing the velocity in the first control volume with a semi‐logarithmic velocity law for a smooth or rough flow [Ferziger and Peric, 2002]. At the upper boundary, a zero Neumanm boundary condition was enforced for fluid and sediment horizontal velocities. [34] Regarding the boundary conditions for the sediment volume fraction at both ends of the computational domain (bed and free surface), we considered that our work focuses only on the sediment transport in suspension, so the boundary condition at the bottom needs to reflect the potential presence of bed load in some tests. In turn, the boundary condition at the free surface is complex in nature, considering the lack of adequate information therein. We used a Dirichlet boundary condition for the volume fraction of the disperse phase at the wall as: ad∣z=0 = ad,b. The value of ad,b was chosen from the experimental data. Most of the experimental data provide values of the sediment concentration at some distance from the bed. Therefore, we had to extrapolate the experimental data to obtain the value of the sediment fraction at the first node of our numerical model. At the upper boundary, we imposed a null total flux of sediment. To satisfy this boundary condition, we imposed ad∣z=h = 0 (to make zero the advective flux) and we enforced the diffusivity of the sediment to be zero at the top (Dd∣z=h = 0) to guarantee a zero diffusive flux. To obtain a null diffusivity at the upper boundary, a zero value of TKE is needed therein (see our 2009 papers). Jha and Bombardelli [2009] showed that using a zero Neumann condition for TKE (instead of a Diritchlet one) does not produce better agreement with data

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in the lower 75% of the water depth. For K and ", the standard wall functions (valid for single‐phase flows) were used at the bottom. At the top boundary, a null Neumann boundary condition was applied for ". [35] Summarizing, the boundary conditions applied in the present analysis were: Top boundary @Uc @Ud @Um @Wd @Wc ¼ ¼ ¼ ¼ ¼ Wm ¼ 0 @z @z @z @z @z Kjz¼h ¼

@"

¼0 @z z¼h

ð25Þ

ð26Þ

@d

C Ws þ Dd ¼0 @z z¼h

ð27Þ

d jz¼0 ¼ d;b

ð28Þ

U2 3=4 K 3=2 Kjz¼0 ¼ *1=2 ; "jz¼0 ¼ c d1 c

ð29Þ

Lower boundary

where d1 is the distance from the boundary to the center of the near‐boundary grid cell [Ferziger and Peric, 2002].

4. Experimental Data and Classification [36] There are abundant data available for cases of dilute sediment transport; however, except the data sets of Einstein and Chien [1955], Taggart et al. [1972], and Wang and Qian [1992] there are no data for the nondilute flow condition, to the best of our knowledge. These data sets only discussed the mean velocity of the mixture and the distribution of sediment across the flow depth. [37] We selected the following test cases: 18 and 19 from Vanoni [1946], S13, S14 and S16 from Einstein and Chien [1955]; 43 of Taggart et al. [1972]; 23 and 29 from Coleman [1986]; and SQ2 from Wang and Qian [1992]. The maximum concentration of the sediment in these tests ranges from 0.017% to 23%. The flow characteristics of the tests are presented in Table 1. [38] An important portion of our work consisted in assessing the validity of the developed models to explain the diverse data sets. To that end, we defined four categories based on a metric dependent upon the maximum volumetric concentration of sediment near the bed (ad,b): (I) very dilute flow: ad,b < 0.001; (II) dilute flow: 0.001 < ad,b < 0.01; (III) nondilute flow: 0.01 < ad,b < 0.1; and (IV) very nondilute flow: ad,b > 0.1 (please see Table 2). In Table 2, we also identify the category in which the selected experimental data lie. [39] It is worth pointing out here that values of the Stokes’ number (St) for the selected test cases are close to each other (see Table 1) and are significantly smaller than one. The Stokes’ number was calculated as the ratio between the adaptation time of the particle and the fluid‐time scale [see Greimann and Holly, 2001].

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Table 1. Summary of Flow Characteristics in the Selected Experimentsa

Reference

Test Case

Depth (cm)

Bed‐ Slope (× 10−3)

Particle Diameter (mm)

Settling Velocity (m/s)

Shear Velocity (m/s)

ad,bb

Repc

nd

Ste(× 10−3)

Coleman [1986] Coleman [1986] Taggart et al. [1972] Einstein and Chien [1955] Einstein and Chien [1955] Einstein and Chien [1955] Vanoni [1946] Vanoni [1946] Wang and Qian [1992]

23 29 43 S13 S14 S16 18 19 SQ2

17.0 16.8 7.4 13.41 12.36 11.9 14.08 7.19 8.0

2.0 2.0 8.0 18.6 25 25 1.25 1.25 10.0

0.210 0.210 0.15 0.274 0.274 0.274 0.134 0.1 0.137

0.023 0.023 0.019 0.038 0.038 0.038 0.0084 0.0082 0.019

0.041 0.041 0.069 0.105 0.121 0.125 0.041 0.041 0.074

0.00098 0.008 0.0413 0.1323 0.1456 0.2332 0.0012 0.00017 0.0795

12.24 12.24 7.39 18.25 18.25 18.25 6.24 4.02 6.43

NA NA 4.1 3.75 3.75 3.75 NA NA 4.3

0.91 0.91 2.70 3.83 4.67 4.16 0.4 0.78 2.49

a

The density of sediment was in all cases equal to 2650 kg/m3, except Wang and Qian’s [1992] test SQ2, in which rs = 2640 kg/m3. Here ad,b is the solid volume fraction at ffi the bottom. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c Rep is defined as Rep = ( g dp3 D=m )/n; dp is the diameter of the sediment particle; Dr = (rd − rm); rm is the density of the mixture and n refers to the kinematic viscosity of water). d Here n is the exponent in the equation for the velocity of the disperse phase, which depends on the particle Reynolds number Rep. e St is the Stokes number, which is the ratio of the particle adaptation time and the flow time scale [Crowe et al., 1998]. b

[40] Another important issue is given by the nature of the validation of our models. In spite of the fact that we validate our models with data on a flow which is essentially 1D, we believe that the general equations developed in this paper can be applied to flows with 3D gradients of sediment. The only possible limitation is with the turbulence model selected (the K − " model), which may not be suitable for flows with high anisotropy, zones with low turbulence intensity, and strong shear layers [Rodi, 1984]. More research is needed to fully validate these models in 3D, which is naturally out of the scope of this paper.

The final algebraic form of equation (30) is:

5. Numerical Technique

In the above definitions, i indicates the index of the control volume, which increases upwards; QU is the value of the dependent variable coming from the previous time step; and S and S′ are the components of the source term SQ. A standard tri‐diagonal Thomas algorithm is used to solve equation (32). The advection term (second term on the left hand side of equation (30)) was considered in the source term and was discretized using an upwinding scheme in an explicit manner as follows:

[41] For the numerical solution of the above models, the differential equations are expressed in the following general form [Svensson, 1998; Bombardelli and Jha, 2009]: @Q @ @ @Q þ ðW QÞ ¼ GQ þ SQ @t @z @z @z

ð30Þ

where Q is the dependent variable; GQ is the diffusion coefficient; and SQ is the source and sink term. Equation (30) was integrated over control volumes (horizontal area Ar, spatial step Dz, and time interval Dt) to obtain its finite difference form. By choosing uniform grid sizes, the diffusive term (first term on right‐hand side) was integrated [see Patankar, 1980; Svensson, 1998] in the following form:

1 1 1 G i þ ½ Q ð i þ 1 Þ Q ð i Þ G i ½ Q ð i Þ Q ð i 1 Þ Q Q Dz2 2 2

ð31Þ

DðiÞ QðiÞ ¼ AðiÞ Qði þ 1Þ þ BðiÞ Qði 1Þ þ C ðiÞ

ð32Þ

where A(i), B(i), C(i), and D(i) are matrix coefficients, defined as follows: AðiÞ ¼ GQ ði þ 1=2Þ=ArðiÞ; BðiÞ ¼ GQ ði 1=2Þ=ArðiÞ; C ðiÞ ¼ QU ðiÞ DzðiÞ=Dt þ DzðiÞS ðiÞ; and DðiÞ ¼ DzðiÞ=Dt þ ðGQ ði þ 1=2Þ þ GQ ði 1=2ÞÞ=ArðiÞ DzðiÞS 0 ðiÞ ð33Þ

t Wiþ1 Qtiþ1 Wit Qti @ ðW Q Þ ¼ Dz @z

ð34Þ

giving an overall first‐order accurate method. It is worth pointing out here that the terms corresponding to stresses due to inter‐particle collisions have also a similar form as the diffusion term in equation (30). Therefore, we have @ @ (ad & xz) and @z (ad & zz) in the discretized terms such as @z same way as the diffusive term discussed above (see equation (31)).

Table 2. List of Test Cases Selected to Validate Our Models and Classification of Tests Based on the Maximum Volumetric Concentration Near the Bed Category

Ranges

Selected Test Cases and Reference Data Source

Denominations

I II III IV

ad,b < 0.001 0.001 < ad,b < 0.01 0.01 < ad,b < 0.1 ad,b > 0.1

23 [Coleman, 1986], 19 [Vanoni, 1946] 29 [Coleman, 1986], 18 [Vanoni, 1946] 43 [Taggart et al., 1972], SQ2 [Wang and Qian, 1992] S13, S14, S16 [Einstein and Chien, 1955]

Very dilute Dilute Nondilute Very nondilute

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Table 3. Runs Performed Run

Model

Interaction Forcesa

Closure for n T,d

1

PTFM

Drag

n T, d = 0

2

CTFM

Drag

3

PTFMND

Drag + VM

n T, d = 0

2 dUd n T,d = lmd

Drag + VM Drag + VM Drag + VM Drag + Lift + VM

n T,d = n T,d = n T,c n T, d = 0 2 dUd n T,d = lmd dz

4 5 6 7

CTFMND CTFMND CTFMND CTFMND

Selected Test Cases for Validationb

dz

2 dUd lmd dz

Coleman 23, Coleman 29, Vanoni 18, Vanoni 19, Wang and Qian SQ2, and Taggart et al. 43 Coleman 23, Coleman 29, Vanoni 18, and Vanoni 19 Wang and Qian SQ2, Taggart et al. 43, Einstein and Chien S13, S14, and S16 Einstein and Chien S13, S14, and S16 Einstein and Chien S13, S14, and S16 Einstein and Chien S13, S14, and S16 Einstein and Chien S13, S14, and S16

a

VM represents the virtual mass force. See Table 1.

b

[42] In order to test the effect of using a first‐order scheme on the numerical results, we also discretized the convective term with a higher‐order upwinding scheme. We followed Fletcher [1997, p. 346] and implemented a third‐order QUICK scheme. The differences between results were insignificant, as expected.

6. Numerical Results and Discussion 6.1. Description of Runs Performed [43] We performed several runs, as summarized in Table 3. We also listed therein the selected test cases used for validation of models. Runs 1 and 2 use the models for dilute flows, the PTFM and the CTFM, respectively. Runs 1 to 4 were aimed to evaluate the capability of dilute and nondilute models to predict the mean velocity and distribution of sediment concentration for the test cases lying in the four categories defined in Table 2. Further, we performed Runs 4 to 6 to evaluate the effect of different formulations for n T,d. The relative importance of different interaction forces was addressed through Runs 4 to 7. It is worth pointing out here that while in Runs 1 and 2 we did not include the buoyancy terms in the K − " model (equations (C5) and (C6)), we did include them in Runs 3 to 7. [44] The steps in the procedure followed in each of the runs were: a) Data were gathered on the slope of the channel, and water depth (which sets the extension of the computational domain); b) the characteristics of the sediment in suspension were also identified (size, settling velocity and sediment volume fraction at the bottom, which was extrapolated to the bottom from the available data); c) the roughness height of the channel was either computed from fitting the semi‐ logarithmic profile to the data, or taken from the papers reporting the tests (if available); this serves to impose the boundary condition for the carrier phase at the bottom through the semi‐logarithmic velocity law; d) the turbulence closure was selected; e) the computed shear velocity was compared with the reported counterpart to check correctness of the prediction of the flow of the carrier phase; f) the value of the Schmidt number was tuned to match the data on sediment concentration. 6.2. Range of Validity of the Models [45] Figures 2–4 present the comparisons of predictions of the mean velocity of the carrier phase and the distribution of suspended sediment obtained from the simulations of cases

23 of Coleman [1986] and 19 of Vanoni [1946]. According to our classification, these tests fall under the category of very‐dilute flows (category I). It is clear that the predictions from the PTFM (Run 1) and the CTFM (Run 2) overlap. The results regarding the concentration of sediment correspond to particular, different values of the Sc, as detailed below. Bombardelli and Jha [2009] found similar results in simulating the test cases of Muste and Patel [1997] and Muste et al. [2005]. The values of ad,b for the test cases of Muste and Patel and Muste et al. are 6.53 × 10−5 and 1.62 × 10−3, which lie in the category of very dilute and dilute flows (categories I and II) respectively. In that paper, we were also able to validate our dilute models through comparisons of numerical results with data on turbulence statistics, which is not possible herein due to a lack of data of turbulence statistics in nondilute flows. [46] Next, the test cases pertaining to dilute flows (category II), i.e., cases 29 of Coleman [1986] and 18 of Vanoni [1946] were simulated. The simulation results are compared with the data in Figures 5–7. It can be observed from Figure 5 that when the ad,b is about 0.01 (as in test case 29 of Coleman [1986]), the velocity profiles obtained from Runs 1 and 2 showed differences in the upper portion of the water

Figure 2. Comparison of simulated values of the stream‐ wise component of the velocity obtained from the 1D PTFM and the 1D CTFM with data corresponding to the test cases 23 of Coleman [1986] and 19 of Vanoni [1946], which pertain to category I of Table 3. Note that the numerical results for the 1D PTFM and the 1D CTFM overlap.

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Figure 3. Comparison of simulated values of the concentration of sediment obtained from the 1D PTFM and the 1D CTFM with data corresponding to the test case 23 of Coleman [1986], which pertains to category I of Table 3. For comparison, the result given by the Rousean distribution (with a Schmidt number equal to 1) is shown. column. This difference in simulation results was of the order of 5%. However, the distribution of volumetric concentration obtained from the simulations are still overlapping. Thus, it can be argued that for dilute flows (categories I and II), there is no necessity of applying the CTFM as opposed to the more simple PTFM (this is congruent with the results of Bombardelli and Jha [2009]). [47] Further, we compared the performance of Runs 1 to 3 (PTFM, CTFM and PTFMND models) for the test cases of nondilute flows (category III). We used the experimental data 43 of Taggart et al. [1972] and SQ2 of Wang and Qian [1992] for this analysis. From Figure 8, it is clear that the velocity profiles from simulations are within 10% of each

other. However, the predictions of the profiles for volumetric concentration (Figures 9 and 10) show some differences in shape that cannot be eliminated by varying the Sc. By comparing the performance of Runs 1 and 2 (the PTFM and the CTFM), it seems that the latter provides better predictions. It is worth pointing out that Run 2 does not consider the effects of inter‐particle collisions. The PTFMND, in turn, seems to provide the best agreement with the data. Our computations show that modifying the PTFM by including an additional interaction term in the momentum equations; by adding the inter‐particle stresses; by using the K − " model with buoyancy terms; by including the virtual‐ mass force in addition to the drag force; and by employing

Figure 4. Comparison of simulated values of the concentration of sediment obtained from the 1D PTFM and the 1D CTFM with data corresponding to the test case 19 of Vanoni [1946], which pertains to category I of Table 3. 11 of 27

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Figure 5. Comparison of simulated values of the stream‐wise component of the velocity obtained from the 1D PTFM and the 1D CTFM with data corresponding to the test cases 29 of Coleman [1986] and 18 of Vanoni [1946], which pertain to category II of Table 3. the mixing length to determine n T,d, a better match of simulation results with the experimental data is obtained. It seems also from Figure 9 that for any value of the Sc, the simulation results obtained from Run 1 (PTFM) do not approximate closely the experimental data. Therefore, these results suggest that for maximum concentration values of, say, between 2 and 5%, models developed for dilute flows are not adequate. [48] Next, we developed Runs 3 and 4 for the test cases of very nondilute flows (category IV). The comparisons between simulation results and experimental data are presented in Figures 11–14. It is possible to see in Figure 11 that the results obtained from Run 3 (PTFMND) show under‐prediction of the mean velocity of the carrier phase in

the upper half depth of the channel. For smaller concentrations (e.g., for test case S13), velocity profiles obtained from both runs are almost overlapping, but for the test cases with relatively higher concentration in the flow, the differences in the predictions from Runs 3 (PTFMND) and 4 (CTFMND) are pronounced. It is of note that Run 3 provides the mean velocity of the mixture whereas Run 4 provides the mean velocity of the carrier phase. [49] Figures 12–14 show comparisons between simulation results (Runs 3 and 4), experimental data, and results obtained with Hunt’s model for the concentration of sediment. The coefficients Bs and ks in Hunt’s model (see Appendix C) were chosen in such a way that the simulation results approximated the experimental data. We used

Figure 6. Comparison of simulated values of the concentration of sediment obtained from the 1D PTFM and the 1D CTFM with data corresponding to the test case 29 of Coleman [1986], which pertains to category II of Table 3. 12 of 27

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Figure 7. Comparison of simulated values of the concentration of sediment obtained from the 1D PTFM and the 1D CTFM with data corresponding to the test case 18 of Vanoni [1946], which pertains to category II of Table 3. values of Bs equal to 0.996, 1.0 and 0.995 for test cases S13, S14 and S16 of Einstein and Chien, respectively. The value of ks for all test cases was assumed to be 0.2. From Figures 12–14, it is apparent that Hunt’s model shows accurate predictions of the volumetric concentration only in the region close to the bed, and that it shows over‐prediction in the upper region of the water column. On the other hand, we observed that the simulation results from Runs 3 and 4 are close to the experimental data throughout the distance for which data is available. It is possible to argue, though, that the CTFMND provides a prediction which seems to be closer to the experimental points by a small amount. Root mean square error (RMSE) values were computed, providing support for this effect in some occasions (see captions of

Figures 12–14). However, given the fact that there is only data for the lower 40% of the water depth, these RMSE values should be taken with caution. This seemingly better prediction capability of the CTFMND adds to the better prediction of velocities of the carrier phase. Based on these considerations, the performances of both the CTFMND and PTFMND suggest that the former would be a better model than the latter in the simulation of tests pertaining to the category of very nondilute flows. Naturally, this conclusion should be revisited in the future with more experimental evidence, including turbulence statistics. [50] We calculated U* using different methods. In Table 4, we compare the range of values of U* obtained using diverse theoretical expressions with those obtained from simula-

Figure 8. Comparison of simulated values of the stream‐wise component of the velocity obtained from the 1D PTFM, the 1D CTFM, and the 1D PTFMND with data corresponding to the test cases SQ2 of Wang and Qian [1992] and 43 of Taggart et al. [1972], which pertain to category III of Table 3. 13 of 27

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Figure 9. Comparison of simulated values of the concentration of sediment obtained from the 1D PTFM, the 1D CTFM, and the 1D PTFMND with data corresponding to the test case SQ2 of Wang and Qian [1992], which pertains to category III of Table 3. Notice that the PTFMND better approximates the shape of the data points. tions. We observed that all values of U* obtained from different runs lie within those ranges. 6.3. Variation of the Sc [51] We observed in this work that for the test cases corresponding to dilute mixtures (categories I and II), the obtained values of the Sc were always less than one (see also van Rijn [1984] and our 2009 papers). We also found that the values of the Sc were lower when using the CTFM than those obtained with the PTFM in agreement with Jha and Bombardelli [2009]. The final values of the Sc obtained from various dilute and nondilute test cases are listed in Table 4. The differences shown in Table 4 in the Sc can be attributed to different settling velocities and shear velocities used or obtained in the different runs. The values of Ws computed using the CTFM were larger than those obtained from the experiments and used in the PTFM (which in turn are not “measured,” but rather calculated as well, introducing uncertainty), while the values of shear velocity were quite similar, giving a larger Ws /U* ratio. As a consequence of this fact, the values of the Sc are lower, following the same physical behavior suggested by van Rijn more than twenty years ago (1984). [52] Next, we assessed the feasibility of formulations such as equation (C16) to provide the Schmidt number in nondilute mixtures (categories III and IV). Through fitting the data we found a value of s1 equal to 1 for the tests of nondilute flows (category III), whereas the value of s1 was found to be 2 for cases of very nondilute flows (category IV). This indicates that with increasing sediment concentrations in the flow, the Sc presents higher values. In other words, the sediment diffusivity is smaller. The sediment diffusivity can be expected to be lower in nondilute flows as opposed to the case of dilute flows, because of inter‐particle collisions. This result suggests an interesting analogy between the diffusivity in sediment‐laden flows and the diffusivity in

gases; this diffusivity is reduced when the mass density of the gas increases. This analogy is also in agreement with the success of the use of the kinetic theory of gases in the definition of the shear stresses due to inter‐particle collisions as shown in Figures 8–14 (see also discussion below). [53] In Figure 15, we plotted the values of Sc found in our models for cases of nondilute and very nondilute flows (Runs 3 and 4), with open circles (unfilled symbols). Figure 15 also contains our previous results presented by Jha and Bombardelli [2009] showing values of the Sc corresponding to cases of dilute flows. We notice in Figure 15 that there is a large difference in the values of Sc obtained numerically for dilute and nondilute cases. We observe that

Figure 10. Comparison of simulated values of the concentration of sediment obtained from the 1D PTFM, the 1D CTFM, and the 1D PTFMND with data corresponding to the test case 43 of Taggart et al. [1972], which pertains to category III of Table 3. Note that the PTFMND better approximates the shape of the data points.

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Figure 11. Comparison of simulated values of the stream‐wise component of the velocity obtained from the 1D PTFMND and the 1D CTFMND with data corresponding to the test cases S13, S14, and S16 of Einstein and Chien [1955], which pertain to category IV of Table 3. The profiles have been shifted to fit into one plot. even in the case of dilute flows, the maximum values of Sc coming from experiments lie close to 1.5 (Cellino [1997] and field data of Coleman [1970]). The numerical values of Sc for nondilute cases are always higher than those for dilute cases, which is in agreement with the findings of Hsu et al. [2003b]. [54] It is also possible to see that the computed values of Sc for nondilute cases are in the range of 1.5 to 2.12. This finding is in line with observations of Umeyama [1992], who used values of the Sc in the range of 1.25 to 2.5 to obtain good agreement between theoretical predictions and

experimental data of Einstein and Chien [1955]. We also observed that the values of Ws /U* obtained from Runs 3 and 4 were smaller than 0.3, while it varied in the range of 0.2 to 0.7 for Runs 1 and 2. This variation is the result of the behavior of U* and Ws, as explained above in this section.

Figure 12. Comparison of simulated values of the concentration of sediment obtained from the 1D PTFMND and the 1D CTFMND with data corresponding to the test case S13 of Einstein and Chien [1955], which pertains to category IV of Table 3. Results obtained with Hunt’s model are also included. The RMSE values for the runs with the PTFMND and the CTFMND were 0.0141 and 0.006, respectively (to be taken with caution).

Figure 13. Comparison of simulated values of the concentration of sediment obtained from the 1D PTFMND and the 1D CTFMND with data corresponding to the test case S14 of Einstein and Chien [1955], which pertains to category IV of Table 3. Results obtained with Hunt’s model are also included. The RMSE values for the runs with the PTFMND and the CTFMND were 0.01628 and 0.01145, respectively (to be taken with caution).

6.4. Effect of Various Models for the n T,d [55] In Figure 16, we present simulation results of Runs 4 to 6 and compare them with the data set S‐16 of Einstein and Chien for the same Schmidt number and the CTFMND. It is clear from Figure 16 that the application of n T,d = n T,c

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1997]. It can be concluded that the mixing length theory can be used to study the distribution of volumetric concentration for nondilute, water‐solid flows as well. It is worth pointing out here that for the tests S13, S14 and S16, the values of the constant b used in the calculation of lmd (please see equation (C7)) are the same than those employed by Mazumder and Ghoshal [2006] (Table 4). In their analysis, Mazumder and Ghoshal found that the estimated value of b lie in the range of 1–18 for fine grained sand. For the test cases of SQ2 of Wang and Qian and 43 of Taggart et al., we assumed a constant value of b equal to 18. The results presented are virtually independent from the values selected of b within the range b = 4–18 (not shown).

Figure 14. Comparison of simulated values of the concentration of sediment obtained from the 1D PTFMND and the 1D CTFMND with data corresponding to the test case S16 of Einstein and Chien [1955], which pertains to category IV of Table 3. Results obtained with Hunt’s model are also included. The RMSE values for the runs with the PTFMND and the CTFMND were 0.044 and 0.021, respectively (to be taken with caution). leads to over‐prediction of the experimental data. When n T,d was considered negligible as it is commonly assumed in dilute flows [see Hsu et al., 2003a; Bombardelli and Jha, 2009; Jha and Bombardelli, 2009], we observed that the simulation results are closer to the experimental data but that they do not fit the data points. For the same value of the Sc in all three approaches, the best match between the simulation results and experimental data was when we

observed

d , originally conapplied the third approach n T,d = l2md dU dz ceived for gas‐solid, nondilute flows [Hrenya and Sinclair,

6.5. Relative Importance of Interaction Forces [56] Figure 17 examines the importance of the lift and virtual mass forces in comparison to the drag force in the simulation of nondilute flows. The simulation results are obtained from Run 4, i.e., with the CTFMND. The magnitude of the forces gradually decreases in the direction away from the bed. From Figure 17, it is clear that the lift force is of the order of 3 to 4% of the drag force close to the wall, while the virtual mass is of the order of 25% of the drag force at that location. We also observed that the relative magnitude of the virtual mass is much larger than the relative magnitude of the lift force at any depth in the flow. Therefore, the lift force can be disregarded in the analysis. This conclusion is in agreement with the assumptions of Greimann et al. [1999] and Greimann and Holly [2001]. We noticed in addition that these computed relative weights of the forces do not show much variation in terms of the sediment concentration. This could be attributed to the fact that the interaction forces of lift, drag and virtual mass are increasing simultaneously; therefore, the relative magnitude of forces does not vary much.

Table 4. Values of Variables Obtained From the Simulationsa

Run

Test Case

U* Obtained From Various Runs (m/s)

Range of U* Obtained From Using Diverse Expressions (m/s)

Sc

s 1b

bc

0.041–0.058 0.60 ‐ ‐ 0.041–0.057 0.70 ‐ ‐ 0.036–0.042 0.88 ‐ ‐ 0.027–0.029 0.85 ‐ ‐ 0.072–0.089 0.98 ‐ ‐ 0.066–0.076 0.95 ‐ ‐ CTFM 0.041–0.058 0.53 ‐ ‐ 0.041–0.057 0.65 ‐ ‐ 0.036–0.042 0.55 ‐ ‐ 0.027–0.029 0.80 ‐ ‐ 0.072–0.089 1.20 ‐ ‐ 0.066–0.076 1.15 ‐ ‐ PTFMND 0.072–0.089 ‐ 1.0 18 0.066–0.076 ‐ 1.0 18 0.094–0.156 ‐ 2.0 18 0.108–0.174 ‐ 2.0 18 0.110–0.171 ‐ 2.0 4 CTFMND 0.094–0.156 ‐ 2.0 18 0.108–0.174 ‐ 2.0 18 0.110–0.171 ‐ 2.0 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi a Values of the shear velocity were obtained by using: g Rb Sb , g Rb Se , g h Sb , g h Se , where Rb: Hydraulic radius; Sb: slope of the channel bed; Se: slope of the energy line. In some test cases, due to lack of information, we considered the bed slope and the slope of the energy line to be equal. b Here s1 indicates the factor in equation (C16). c Here b is a factor in the exponent of the expression for the mixing length. PTFM

23 29 18 19 SQ2 43 23 29 18 19 SQ2 43 SQ2 43 S13 S14 S16 S13 S14 S16

0.058 0.057 0.042 0.029 0.089 0.076 0.056 0.044 0.040 0.029 0.080 0.072 0.083 0.075 0.137 0.156 0.159 0.128 0.131 0.123

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Figure 15. Comparison of values of the Schmidt number (Sc) obtained from our runs with the K − " model (for the dilute and nondilute models) and those obtained from observations (adapted from Lyn [2008] and Jha and Bombardelli [2009]). The van Rijn formula and the Einstein and Chien model are also included. Open symbols indicate numerical results while solid symbols refer to laboratory or field data. The values of the Sc pertaining to nondilute conditions were found to be larger than one and are located in the upper left corner, indicated with unfilled circles. In the tests with a variable Sc in the depth in the wall‐normal direction, we included in the graph only the maximum value observed. 6.6. Effect of Concentration on Stresses Due to Inter‐particle Collisions and on Turbulence Statistics [57] Figure 18 illustrates the effect of the sediment concentration on the magnitude of the stresses due to inter‐particle

collisions (obtained from Run 4, i.e., with the CTFMND). It can be observed that in the region close to the bed, the contribution of stresses due to inter‐particle collisions is more important than the contribution away from the bed, as

Figure 16. Comparison of simulated values of the concentration of sediment obtained from the 1D CTFMND for various formulations of the eddy viscosity of the disperse phase with data corresponding to the test case S16 of Einstein and Chien [1955]. 17 of 27

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Figure 17. Comparison of simulated values of the relative weight of the lift and virtual mass forces obtained with the 1D CTFMND for increasing sediment concentration in the flow. Test cases correspond to S13, S14, and S16 of Einstein and Chien [1955], of maximum volumetric concentrations equal to 0.133, 0.146, and 0.227, respectively. expected. Beyond 40% of the depth, the effect of the inter‐ particle stresses is insignificant. We also observed that as sediment concentration increased from S13 to S16, the magnitude of the stresses due to inter‐particle collisions increases (as expected). This indicates that the closure based on the kinetic theory of gases possesses the expected behavior in capturing the influence of the inter‐particle collisions. It can be also seen that for very nondilute flows the former stresses amount to 85 to 95% of the “total” stresses close to the bed. [58] Regarding the turbulence statistics, we observed from results of Run 4 (CTFMND) that both the TKE and DTKE increased with an increase in concentration in the flow (not

shown herein). These results are in line with the findings of Mueller [1973], but they disagree with results obtained by Wang and Qian [1992]. More research is needed to address this issue further. However, it is important to remark that our model is able to capture the variation of n T,c, as explained below. Instead of plotting the results of TKE and DTKE, we decided to plot the variation of n T,c which is taken directly from the numerical result without any postprocessing. Figure 19 shows predictions obtained with the CTFMND from Run 4 for the nondimensional n T,c; Figure 19 allows for the comparison of results corresponding to cases of the very nondilute category, S13, S14, and S16. The nondimensional n T,c decreases with increasing sediment con-

Figure 18. Assessment of simulated values of the stresses due to inter‐particle collisions obtained from the 1D CTFMND for increasing sediment concentration in the flow. Test cases correspond to S13, S14, and S16 of Einstein and Chien [1955], of maximum volumetric concentrations equal to 0.133, 0.146, and 0.227, respectively. 18 of 27

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Figure 19. Comparison of simulated values of the eddy viscosity of the carrier phase as a function of increasing sediment concentration in the flow. Test cases correspond to S13, S14, and S16 of Einstein and Chien [1955], of maximum volumetric concentrations equal to 0.133, 0.146, and 0.227, respectively. The word “modified” refers to the adjustment of a constant in the Smith and McLean equation to approximate the numerical results (filled symbols), as explained in text (section 6.6).

centrations in the flow, as expected. We also calculated the values of n T,c using the formula proposed by Smith and McLean [1977] (equation (C15)). It is important to emphasize herein that we did not include this regression in the model; we developed the comparisons after running the models. Interestingly, the values of n T,c given the Smith and McLean [1977] regression were found to be smaller than those obtained from Run 4. The maximum differences of n T,c were of the order of 20%, corresponding to the test case S16. We tried to minimize this difference by tuning the multiplying factor (4.7) which appears in the formula of Smith and McLean. Smith and McLean adopted this value directly from another paper by Businger et al. [1971] whose work was related to stratification in atmospheric surface layers. We observed that by reducing this value from 4.7 to 2.0, the nondimensional n T,c obtained from the Smith and McLean’s formula are now close to our simulation results (see open symbols in Figure 19; the word “modified” indicates that the original factor was reduced from 4.7 to 2). This result suggests that: a) the buoyancy terms included in the K and " equations of our models are able to capture the effect of density stratification in the flow, and b) that formulations coming from atmospheric problems can be adapted/modified to water‐solid flows. 6.7. Effect of Concentration on the Velocity of Sediment in the Wall‐Normal Direction [59] Profiles for the nondimensional settling velocity (Wd /U*) of the sediment particles obtained from Run 4 (CTMFND) are shown in Figure 20. It can be seen that the simulated Wd /U* increases with height in the lower 15% of the water depth and is almost constant in the remaining 85%. We also observed that the value of the settling velocity decreases with increasing concentrations of sediment in the flow, as expected. Due to the higher concentration of sus-

pended sediment, the motion of particles is affected by the motion of neighboring particles (three‐way coupling).

7. Conclusions and Final Remarks [60] We believe to have presented for the first time, to the best of our knowledge, an integrated approach to examine the range of validity of dilute and nondilute models based upon the maximum amount of sediment concentration in the flow. Simulation results were compared with the published experimental data for the velocity distribution of the carrier as well as the profiles of concentration of sediment in the flow for various sediment loads. From this analysis, we can draw the following conclusions:

Figure 20. Assessment of simulated values of the settling velocity obtained from the 1D CTFMND for increasing sediment concentration in the flow. Conditions correspond to tests S13, S14, and S16 of Einstein and Chien [1955].

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[61] 1) In the simulation of cases of very dilute and dilute flows (ad,b < 0.01), the predictions from the PTFM and CTFM for velocities and the distribution of volumetric concentration overlap. Thus, instead of applying the more complex model (the CTFM), the PTFM provides accurate results for such flows. When the magnitude of ad,b in the flow exceeds the value of the order of 0.01, nondilute models are necessary. In such flows, the predictions obtained from the PTFMND are more accurate than those obtained with the PTFM or the CTFM. The CTFMND seems to be required to simulate tests in which the flow is of very nondilute nature (ad,b > 0.1), where the PTFMND does not seem to provide accurate predictions of the velocity of the carrier phase. These findings provide recommendations for the modeler to select the most accurate approach with the adequate level of complexity in each opportunity. Compared to models discussed by Villaret and Davies [1995], our framework proposes an alternative view with the PTFMND, which has an interesting range of validity as shown in this paper. [62] 2) In nondilute flows, the relative magnitude of the lift and virtual mass forces with respect to the drag force are smaller than 5% and 25%, respectively, for the test cases studied in this work. While in the simulation of dilute flows only the drag force accounts for most of the interaction forces, this work shows that for nondilute flows the virtual mass force also plays an important role. The influence of the lift force was observed to be relatively small in comparison to the drag force, and therefore lift can be neglected in the simulation of nondilute flows. [63] 3) For the prediction of the distribution of volumetric concentration of sediment, a good agreement between simulated results and experimental data can be obtained by choosing an appropriate value of the Sc for a wide set of models. For tests corresponding to dilute flows, the value of the Sc was found to be always less than one. In the case of nondilute flows, a formulation in terms of the local sediment concentration provided good results, showing an increase of the Schmidt number (smaller eddy diffusivity) for nondilute flows. The maximum value of the Sc found in this work is 2.12 for nondilute cases, which is closed to empirical values reported elsewhere [Cellino and Graf, 2002], and values reported by Umeyama and Gerritsen [1992]. The values of the Sc for nondilute cases were found to be always larger than one, implying that the diffusivity of the sediment is reduced in a sizable way with respect to the flow eddy viscosity. [64] 4) Our results suggest that the use of an expression for the n T, d based on the mixing length concept provides more accurate results of sediment concentration than considering n T, d negligible or assuming n T, d = n T, c, for a range of values of b ranging from 5 to 18. [65] 5) For concentrations ad, b > 0.01, the inter‐particle collisions become important. Our findings suggest that the threshold of ad, b values to classify a flow as nondilute varies between 2 and 5%, which is close to the value of 4% proposed by Woo et al. [1988]. Our study indicates that the classification “map” of Elghobashi [1991, 1994], which refers to “dense suspension” when the volumetric concentration exceeds 0.001 would need to be revisited, at least for water‐solid flows. [66] It is interesting to note that our model produces values of the eddy viscosity of the carrier phase (n T, c) which

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decrease with increasing sediment concentration in the flow. This is a very important result, which is in agreement with the methodology used by many researchers who modify n T, c using the Richardson number to include the effect of stratification in the flow. In our runs, we have not used any of those regressions, and our findings demonstrate that the buoyancy terms added to the standard K − " model used are able to capture the effects of stratification in the flow successfully. [67] Finally, our work clearly underscores the need for more experimental data regarding turbulence in nondilute mixtures to validate further the models presented herein, and generalize its conclusions.

Appendix A: Derivation of General Equations for the CTFMND [68] The basic governing equations for two‐phase flows, i.e., the two‐fluid model (TFM) can be written as follows [Drew and Passman, 1999]: Conservation of mass @ @ ~ j;p ¼ G p ~p þ p ~p U @t @xj

ðA1Þ

Conservation of momentum ~ @ ~ i; p þ @ p ~p U ~ i; p U ~ j; p ¼ p @ P p ~p U @t @xj @xi i @ h ~ þ p T ij; p þ Tij;EAp þ p ~p gi ðFint Þi @xj ðA2Þ

In the above equations, the subscript p stands for both phases (which could be c or d for the carrier and the disperse phases, respectively); ap represents the volume fraction of ~ , and P ~ represent the ensemble‐averaged denphase p; ~, U sity, velocity and pressure, respectively; T~ ij, p refers to the ensemble‐averaged deviatoric stresses of phase p; the stresses with the superscript EA are the remainder of the process of ensemble averaging; gi is the i‐th component of the acceleration due to gravity; (Fint)i includes all the interactive forces, typically, due to drag, lift, virtual mass, and turbulent dispersion; G is the mass transfer rate between phases; and x and t are the space and time coordinates, respectively. The indices i and j vary from 1 to 3 and the sum is implied in repeated indices (Einstein convention). [69] We performed time averaging of equations (A1) and (A2); this involves the introduction of the Reynolds decomposition to each instantaneous variable, followed by time averaging. ~ i; p ¼ Ui; p þ ui;0 p ; P ~ ¼ P þ p0 ; p ¼ p þ p0 ; U 0 0 Þi T~ ij; p ¼ Tij; p þ tij;0 p ; Tij;EAp ¼ Tij;EAp þ tij;EAp ; ðFint Þi ¼ Fint i þðFint ðA3Þ

where the upper‐case letters with overbar represent the mean components of the variable, and the lower case letters with the superscript ’ indicate the fluctuations of the variable. By

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inserting equations (A3) into equations (A1) and (A2), we obtain: Conservation of mass i i @h @ h ~p p þ p0 þ ~p p þ p0 Uj; p þ uj;0 p ¼ 0 @t @xj

ðA4Þ

Conservation of momentum i @h ~p p þ p0 Ui; p þ ui;0 p @t i @ h ~p p þ p0 Ui; p þ ui;0 p U j; p þ uj;0 p þ @xj @ ¼ p þ p0 P þ p0 @xi i @ h EA0 p þ p0 Tij; p þ tij;0 p þ Tij;EAp þ tij; þ p @xj 0 þ ~p p þ p0 gi Fint i Fint i

ðA5Þ

By performing averaging over equation (A4), the following equation results (we have removed the symbol of ensemble average from the equation): i @ @ h p p þ p p Uj; p þ p p0 uj;0 p ¼ 0 @t @xj

ðA6Þ

Similarly, by performing averaging over equation (A5), we obtained: i @h 0 p p Ui;p þ p p0 ui;p @t 2 3 0 0 p p Ui;p Uj;p þ p p ui;p uj;p 7 @ 6 0 7 6 þ U 0 u 0 þ p Uj;p p0 ui;p þ 5 @xj 4 p i;p p j;p 0 0 þp p0 ui;p uj;p " # EA 0 @P @ p Tij;p þ p Tij;p 0 @P p þ ¼ p 0 þ 0 t EA0 @xi @xi @xj þp0 tij;p p ij;p þ p p gi Fint i

ðA7Þ

order correlations…” The implication of this statement is that the neglected terms are small, but no scaling or physical explanation was provided back then. Elghobashi and Abou‐ Arab [1983] also approximated the transient terms (partial derivatives with respect to time) collecting them in a single counterpart. In our case, a) We collapsed the two time derivatives of (A7) into one term following the procedure of Elghobashi and Abou‐Arab [1983]; 0 0 uj;p as b) We neglected the third‐order correlations p0 ui;p opposed to the second‐order correlations, also following Elghobashi and Abou‐Arab [1983]; 0 0 uj;p + rp c) We considered that the terms rp p ui;p 0 0 0 0 Ui;p p uj;p + rp Uj;p p ui;p indicated the Reynolds stresses and Reynolds solid fluxes; therefore, they would naturally Re , since this is the intrinsic physical constitute the term Tij,p meaning of Reynolds stresses on one hand, and we represented the solid fluxes through a Schmidt number (associated with the eddy viscosity of the carrier) on the other hand; d) We assumed that the term including the pressure fluctuation was negligible, but we could not find information to undertake a thorough scaling. EA + 0 t 0 + 0 t EA0 [70] We assumed that the terms p Tij;p p ij;p p ij;p would be included in the Reynolds stresses as well, because these terms involve time averages of variables which are related to the particle‐to‐particle distance (from the notion of ensemble average discussed before). [71] It is worth pointing out here that in equation (A8), we assume that the components Tij and TijRe contain contributions of the viscous deviatoric stresses, the ensemble‐averaged stresses as well as the contribution from the turbulent fluctuations. In the above equations, & ij,p has been included to represent the stresses developed due to interparticle collisions (similar to Enwald et al. [1996]). This stress term becomes important in the momentum equations of the disperse phase while simulating the cases of nondilute flows. In equation (A8), d p,d is 0 when p = c, and is 1 when p = d.

Appendix B: Derivation of General Equations for the PTFMND [72] The mixture variables used in the PTFMND are defined below:

Equation (A7) can be approximated as: @ p p Ui;p @ @P p p Ui;p Uj;p ¼ p þ @xj @xi @t i @ h Re þ p Tij;p þ &ij;p p;d þ Tij;p @xj þ p p gi ðFint Þi

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ðA8Þ

We obtained (A8) following the work of Elghobashi and Abou‐Arab with the equations of the TKE and DTKE. Elghobashi and Abou‐Arab [1983] started with theoretically exact equations for K and " composed by 38 and 67 terms, respectively, to conclude with equations containing about 10 terms each. Unfortunately, in spite of the advance in this regard from 1983 to this day, still we are unable to give accurate models for the correlations of fourth order, for instance. Elghobashi and Abou‐Arab stated on page 934 of their paper “We decide at the outset on neglecting all fourth‐

~ c þ d ~d U ~d m Um ¼ c ~c U

ðB1Þ

m ¼ c ~c þ d ~d ;

ðB2Þ

Tm ¼ c T~ c þ d T~ d ;

ðB3Þ

where the subscript m stands for mixture and ∼ denotes ensemble averaging. The governing equations for the PTFMND are obtained from equations (A1) and (A2). We use the variables defined in equations (B1) to (B3) to obtain the following equations for mixture (for more details, please see Buscaglia et al. [2002] and especially Bombardelli [2004]): Conservation of mass for the mixture

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@m @ þ m Uj;m ¼ 0 @xj @t

ðB4Þ

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Conservation of momentum for the mixture @P @Tij;m @ m Ui;m @ þ m Ui;m Uj;m þ ¼ þ m gi @xj @xi @t @xj

@ d ~d ~ ~ j;c U ~ i;d U ~ j;d U i;c U d ~d 1 @xj m ðB5Þ

The last term of equation (B5) arises as a consequence of the nonlinearity of the convective term in the equations for the two phases; it was disregarded by Buscaglia et al. [2002], Bombardelli [2003], Bombardelli et al. [2003], and Bombardelli [2004] on grounds of the mixture being dilute. It was also disregarded in our previous papers on sediment transport. As discussed in the Appendix A, we performed time averaging of equations (B4) and (B5) by decomposing the variables in their mean and fluctuation components, as follows: 0 p ¼ p þ p0 ; Ui;m ¼ Ui;m þ ui;m ; Tm ¼ Tm þ tm0

ðB6Þ

where the upper‐case letters with overbar represent the mean component of the variable and the lower case letters with the superscript ’ indicate fluctuating components of the variable. The resulting equations are (we have removed the symbol of ensemble average from the equations): Conservation of mass for the mixture @m @ m Uj;m ¼ 0 þ @xj @t

ðB7Þ

them as opposed to second‐order correlations, the number of terms reduces to 32, still a very large number of terms. Addressing the importance of each term would amount to a paper on its own. The equation was then drastically reduced to the 8 current terms, eliminating the fluctuating components. Therefore, we propose to model the momentum equation in the following simplified form:

@P @ m Ui;m @ @ Re þ m Ui;m Uj;m þ ¼ Tij;m þ Tij;m þ m gi @t @xj @xi @xj

@ d d d d 1 Ui;m Ui;d Uj;m Uj;d @xj m ðB9Þ

Further simplification to the PTFMND model is introduced by employing the following: Wd ¼ Wm Ws

ðB10Þ

where Ws is the fall velocity of the disperse phase. This equation replaces the momentum equation of the disperse phase (A8). Therefore, equations (A6), (A8), (B7), (B9), and (B10) are the equations of the PTFMND.

Appendix C C1.

Closures for the Models

C1.1. Closure for the Solid‐Fraction Reynolds Fluxes [73] Following our previous papers on this topic [Bombardelli and Jha, 2009; Jha and Bombardelli, 2009], we used the following definition for the Reynolds fluxes, which is based on the gradient of the time‐averaged solid volume fractions:

Conservation of momentum for the mixture

d0 wc0 ¼ d0 wd0 ¼ Dd

@d @z

@P @T @ m Ui;m @ 0 0 ij;m þ m Ui;m Uj;m þ m ui;m uj;m þ ¼ þ m gi @xj @t @xj @xi ( ) 0 i d d þ d h @ 0 0 0 0 0 d d þ d 1 Ui;c þ ui;c Ui;d ui;d Uj;c þ uj;c Uj;d uj;d @xj m

Equation (B8) is really involved and contains terms which require further closure and add uncertainty and complexity to the models. Expanding the last term of (B8), it is possible to obtain: 8 9 2 2 2d 0 > 0 2 d > 0 2 d < = d d d d þ d þ 2 d d þ d @ m m m @xj > > : ; 0 0 0 0 Ui;c þ ui;c Ui;d ui;d Uj;c þ uj;c Uj;d uj;d

ðB8Þ

T ;c where Dd = Sc , n T,c is the eddy viscosity of the carrier phase, and Sc is the Schmidt number.

C1.2. Closures for the Reynolds Stresses [74] We employed the Boussinesq model to define the Reynolds stresses of the carrier and disperse phases, as follows: TijRe

which gives 80 terms. A simple scaling indicates that the last two terms of the first factor (in terms of the solid volume fraction fluctuations and density of the disperse phase) can be disregarded as opposed to the first three ones. In turn, the second term leads to third‐order correlations. Neglecting

ðC1Þ

@Ui @Uj 2 ij K ¼ c T þ @xj @xi 3

ðC2Þ

where K is the turbulent kinetic energy (TKE). This amounts to assuming that the nondilute condition of the flow does not compromise the “Newtonian‐type” behavior of the phases. In addition, we assumed that the turbulence of the disperse

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phase is in equilibrium with that of the carrier phase. We employed components of equation (C2) to define closures for the stresses appeared in equations (11), (12), (16), (17), (21), and (23) in the following way: Re Txz;c þ Txz;c c T ;c Re ¼ d T ;d Tzz;d

@Wd : @z

@Uc @Ud Re ; Txz;d þ Txz;d ; d T;d @z @z ðC3Þ

where the eddy viscosities of the carrier and the disperse phases (n T,c and n T,d respectively) require further closure. We utilized the K − " model to define the eddy viscosity of the carrier phase as: T ;c ¼ C

K2 "

ðC4Þ

where Cm is a constant with a value equal to 0.09; and " is the dissipation rate of TKE (DTKE) of the carrier phase. The 1D transport equations for the TKE and DTKE are given below. We added a buoyancy production term along with the shear production term in the equations for TKE and DTKE, in order to include the stratification effects driven by the nondilute nature of the mixture (see below). @Uc 2 2 T;c @K @ C K @K @z ¼ þ @t @z k " @z shear production d c T ;c @d " g c Sc @z þ buoyancy production @Uc 2 " @" @ C K @" @z K ¼ þ @t @z " " @z shear production d c T ;c 2 C"1 g c Sc @d " C"2 " þ K buoyancy production @z K

2

ðC5Þ

[see, e.g., Kobayashi and Seo, 1985; Hsu et al., 2003a, 2004; Amoudry et al., 2008], ranging from expression of the eddy viscosity (Chen and Wood 1986), the adoption of a null value for the stresses [Hsu et al., 2003a], or the use of the eddy viscosity of the disperse phase equal to the eddy viscosity of the carrier [Kobayashi and Seo, 1985]. Since there is no guideline available in the literature on how to model the n T,d in the case of nondilute flows, we employed three possible approaches: (a) n T,d = n T,c [Kobayashi and Seo, 1985; Hsu et al., 2003a]; (b) n T,d = 0 [Hsu et al., 2003a; Bombardelli and Jha, Jha and Bombardelli,

2009;

d 2009]; and (c) n T,d = l2md dU dz , where lmd is the mixing length. Approach (c) was used by Hrenya and Sinclair [1997] to simulate the vertical transport of nondilute gas‐ solid suspensions in a pipe. Following Umeyama and Gerritsen [1992] and Mazumder and Ghoshal [2006], lmd can be defined as: z 0:5ð1þ d =d;b Þ lmd ¼ z 1 h

Kd ¼

T ;d C lmd

2 ;

ðC8Þ

3=2

ðC6Þ

C1.3. Closures for the Eddy Viscosity of the Disperse Phase (n T, d) [76] Previous researchers have employed diverse models to address the stresses of the disperse phase in dilute flows

ðC7Þ

where b is a constant that needs to be determined from experimental data, and ad,b is the volumetric concentration of the disperse phase at the reference height. Mazumder and Ghoshal adjusted the value of b to fit the velocity profile obtained from their simulation with the experimental data of Einstein and Chien [1955]. [77] In approaches (a) and (b), the TKE and DTKE of the disperse phase are considered to be in equilibrium with the TKE and DTKE of the carrier phase as discussed above. In approach (c), the TKE and DTKE of the disperse phase are obtained from the following relationships:

C"1 T;c

The values of the constants in equations (C5) and (C6) are: sk = 1.4, s" = 1.3, C"1 = 1.44, C"2 = 1.92. It is worth pointing out here that we have used the same values of the model constants which were originally developed for single‐ phase flows. Ideally, these constants should be different and their values should be obtained from experimental or theoretical investigations. However, due to the lack of detailed experimental data sets on this topic or appropriate Direct Numerical Simulation (DNS) results, researchers have used the unmodified values of the constants in the turbulence model [see, e.g., Hsu et al., 2003a; Longo, 2005; Amoudry et al., 2008]. In equations (C5) and (C6), the buoyancy terms are negative due to the derivative of the solid fraction with respect to z and, therefore, they act as a sink term for TKE and DTKE (cf. G. Parker, unpublished manuscript, 2004). [75] Similarly, the boundary conditions at the wall are adopted as in a single‐phase flow.

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"d ¼

C3 Kd : lmd

ðC9Þ

which are used in the closure of the stresses due to inter‐ particle collisions (see next section). C1.4. Closures for Stresses Due to Inter‐particle Collisions [78] There is no universal model to represent the inter‐ particle collisions. The original model proposed by Enwald et al. [1996] and used by Greimann and Holly [2001] is used in this work. Such model is as follows: 4 2 d go ð1 þ eÞ 5 d rffiffiffi 4 Mij ij þ 2d d dp go ð1 þ eÞ 5

@ ðUd Þm 2ðSd Þij þ ij @xm

d ðd Þij ¼ 2 2d d go ð1 þ eÞ ij

ðC10Þ

where Mij is the second order moment of the particle phase and (Sd)ij is the strain rate tensor. Mij is defined as: Mij = @ ðU Þ t dij − 2 n T,d((Sd)ij − 13( @xdm m )dij) and (Sd)ij is defined as:

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h i @ ðUd Þ @ ðU Þ (Sd)ij = 12 @xdj i þ @xi j . Specific components of the tensor given by equation (C10) read: d &xz ¼

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4 2 @Ud d go ð1 þ eÞT ;d @z 5 d rffiffiffi 4 2 @Ud þ d d dp go ð1 þ eÞ 5

@z

d &zz ¼ 2d d go ð1 þ eÞ rffiffiffi 16 @Wd 12 @Wd T ;d dp þ 2 þ 15 5

@z @z

CD ¼ ðC11Þ

24 1 þ 0:15 Re0:687 r Rer

ðC14Þ

jU U jd

where Rer = c c d p is the explicit particle Reynolds number (G. Parker, unpublished manuscript, 2004), and m is the dynamic viscosity. ðC12Þ

where go = (1 − ad /ad, max)−2.5ad, max; t is the granular temperature, which can be defined as t = 2 Kd /3; Kd is the TKE of the disperse phase, as discussed above; ad,max is the maximum packing concentration which has the value of 0.53 as an upper limit; e is the coefficient of restitution of particle collision. The value of e depends upon the interacting particles and on the flow conditions. Niño and García [1998] showed that for natural sediment saltating close to the bed in a open‐ channel flow, the value of e varies between 0.5 and 0.2 depending upon flow conditions. Schmeeckle et al. [2001] obtained a value of 0.65 from experiments in a turbulent channel with natural sediments. Tsuji et al. [1987] considered a value of 0.8 for polystyrene particles moving in a gas flow. Shen et al. [1989] employed a value of 0.95 for nonspherical particles. Greimann and Holly [2001] assumed a value of 0.9. A detailed analysis of the effects of these coefficients in the saltating motion of particles close to the bed of a channel can be found in the work of González [2008]. Considering the wide range of values for the e, we have used e = 0.9 in this study. C1.5. Closures for the Interaction Forces [ 79] For nondilute flows, Greimann and Holly [2001] considered the drag and virtual mass forces, while Villaret and Davies [1995] considered drag and lift as the most important interaction forces among the phases. In this study, we considered all the above forces and performed an assessment of their individual importance in the predictions. The expressions for the forces of drag, lift, and virtual mass utilized are as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ! FD ¼ d m CD ðUm Ud Þ2 þðWm Wd Þ2 4dp n o k ðUm Ud Þ~i þ ðWm Wd Þ~

@Um ~ @Um ~ ! k FL ¼ CL d m Wd i þ ðUd Um Þ @z @z

@ ðUm Ud Þ @Ud ~ ! d m i FVM ¼ Wd 2 @z @t @Wd @Wd ~ þ Wd k @t @z

while it is replaced by the velocity of the carrier in the CTFMND. In the runs, the drag coefficient was computed as follows [Crowe et al., 1998]:

C2.

C2.1. Effects of Stratification [80] There are mainly two approaches used by researchers to include the effect of stratification in the simulation of nondilute flows. One approach is to simulate the stratification effect through the buoyancy terms in equations (C5) and (C6) [see, e.g., Hsu et al., 2003b]. We adopted this approach herein, which means that we did not use any regression, allowing the model to compute the resulting eddy viscosity of the flow. These terms in the turbulence model consider the damping of turbulence due to the vertical gradient of suspended sediment concentration that exists in a turbulent open‐channel flow, as explained above. A different approach is to apply a correction factor to n T,c [Smith and McLean, 1977; Gelfenbaum and Smith, 1986; Villaret and Trowbridge, 1991; Ghoshal and Mazumder, 2005; G. Parker, unpublished manuscript, 2004]. In this work, we compared predictions using the formula by Smith and McLean with predictions with our model (equations (C5) and (C6)). The formula of Smith and McLean reads: 2

T ;c

3 dC 7 Rg z 6 6 dz 7 ¼ U* z 1 61 þ 4:7 2 7 h 4 dUc 5

ðC15Þ

dz

where R = dc − 1; Uc = Uc(z); C = C(z); and is von‐ Kármán constant. C2.2. Alternative Formulation for the Sc [81] We determined the value of the Sc by comparing the predicted distributions of sediment concentration with data. In addition to testing a unique value of Sc in the entire water depth, we also used the following approach to obtain the Schmidt number as a function of the distance from the bottom (similar to Amoudry et al. [2005]): d d 2 þ Sc ¼ 1 1 d;b d;b

ðC13Þ

where CD indicates the drag coefficient; CL is the lift coefficient; and ~i and ~ k denote the unit vectors in the stream‐wise and wall‐normal directions, respectively. In the PTFMND, the subscript m refers to the mixture velocity,

Extended Analyses

ðC16Þ

where s1 and s2 are empirical constants. We assumed a value of 0.5 for s2 following Amoudry et al. [2005], and estimated the values of s1 by tuning it to achieve the closest possible predictions of the distribution of suspended sediment from simulations against experimental data.

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S damping parameter; it is the inverse of the Schmidt number. Dz computational grid in space (in the vertical direction). Dt computational time step. Dr difference in density of both phases. & stress due to inter‐particle collisions. dij Kronecker delta. t granular temperature. angle between the bed of the channel and a horizontal line. m dynamic viscosity. n T eddy viscosity. Subscripts c relative to the carrier phase. d relative to the disperse phase. m relative to the mixture. p phase. Superscripts ` fluctuation. Re explicit particle Reynolds number. n exponent used in the definition of settling velocity.

C2.3. Hunt’s Model [82] Hunt’s [1954] model is expressed by: 80 1 0 19q z 1=2 a1=2 > > < = 1 B 1 C 1 Ca C B B s hC h ¼ @ A @ A a z 1=2 > > 1C Ca : 1 ; Bs 1 h h

ðC17Þ Ws where q = ; a represents a distance of reference ks Bs ðg h Sb Þ1=2 from the bed; Ca is the concentration of sediment at the reference height; and Bs and ks are model parameters. Hunt compared his simulation results with the experimental data of Vanoni [1946]. The values of Bs and ks were chosen to fit the experimental data obtaining 0.991–1.005 and 0.296– 0.444, respectively. More details about Hunt’s model can be found in the work of Jha [2009].

Notation Ar C Dd d1 dp e Fint g h K lmd P Rb S Sb Se St T U u′ U* W w′ WS Wnds

area of the control volume. concentration of suspended sediment. vertical diffusivity for sediment. height of first grid cell. diameter of particles. coefficient of restitution. interaction forces. acceleration due to gravity. depth of the channel. turbulent kinetic energy. mixing length. pressure. hydraulic radius of the channel. source term. slope of the channel bed. energy slope. stokes number. deviatoric stresses. velocity. velocity fluctuations in the stream‐wise direction. wall‐friction (shear) velocity. mean velocity in the wall‐normal direction. velocity fluctuations in the wall‐normal direction. settling velocity. fall velocity of the sediment in nondilute flow. stream‐wise coordinate. transverse coordinate. wall‐normal coordinate. time coordinate.

x y z t Greek symbols Sc Schmidt number. von‐Kárman constant. Q general variable used in the numerical scheme of the code. " dissipation rate of turbulent kinetic energy. a volume fraction. r density. G mass sources (or sinks) due to the convective or molecular fluxes.

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[83] Acknowledgments. The authors are grateful for fruitful discussions regarding several aspects of the sediment‐transport problem with David Schoellhamer from the Civil and Environmental Engineering Department and with W. Kollmann from the Mechanical Engineering Department at the University of California, Davis. This research has been completed thanks to the support of the Cooperative Institute for Coastal and Estuarine Environmental Technology (CICEET), awarded to Stefan Wuertz and Fabián Bombardelli and thanks to an award from the California Department of Water Resources to Fabián Bombardelli.

References Abdel‐Fattah, S., A. Amin, and L. van Rijn (2004), Sand transport in Nile River, Egypt, J. Hydraul. Eng., 130(6), 488–500, doi:10.1061/(ASCE) 0733-9429(2004)130:6(488). Ahmed, A., and S. Elghobashi (2000), On the mechanisms of modifying the structure of turbulent homogeneous shear flows by disperse particles, Phys. Fluids, 12(11), 2906–2930, doi:10.1063/1.1308509. Amoudry, L., T. J. Hsu, and P. L. Liu (2005), Schmidt number and near‐ bed boundary condition effect on a two‐phase dilute sediment transport, J. Geophys. Res., 110, C09003, doi:10.1029/2004JC002798. Amoudry, L., T. J. Hsu, and L. F. Liu (2008), Two‐phase model for sand transport in sheet flow regime, J. Geophys. Res., 113, C03011, doi:10.1029/2007JC004179. Asano, T. (1995), Sediment transport under sheet‐flow conditions, J. Waterw. Port Coastal Ocean Eng., 121(5), 239–246, doi:10.1061/ (ASCE)0733-950X(1995)121:5(239). Balzer, G., A. Boelle, and O. Simonin (1996), Eulerian gas–solid flow modelling of dense fluidized bed, in Fluidization VIII: Proceedings of the Eighth Engineering Foundation Conference on Fluidization, May 14–19, 1995, Tours, France, pp. 1125–1134, Eng. Found., New York. Barton, J. R., and P. N. Lin (1955), A study of the sediment transport in alluvial streams, Rep. 55JRB2, Civ. Eng. Dep., Colo. State Univ., Fort Collins. Benavides, A., and B. van Wachem (2008), Numerical simulation and validation of dilute turbulent gas‐particle flow with inelastic collisions and turbulence modulation, Powder Technol., 182(2), 294–306. Bombardelli, F. A. (2003), Characterization of coherent structures from parallel, LES computations of wandering effects in bubble plumes, paper presented at World Water and Environmental Resources Congress, Am. Soc. Civ. Eng., Philadelphia, Pa. Bombardelli, F. A. (2004), Turbulence in multiphase models for aeration bubble plumes, Ph.D. thesis, Dep. of Civ. and Environ. Eng., Univ. of Ill. at Urbana‐Champaign, Urbana. Bombardelli, F. A., and S. K. Jha (2009), Hierarchical modeling of the dilute transport of suspended sediment in open channels, Environ. Fluid Mech., 9(2), 207–235, doi:10.1007/s10652-008-9091-6. Bombardelli, F. A., G. C. Buscaglia, and M. H. García (2003), Parallel computations of the dynamic behavior of bubble plumes, in ASME

25 of 27

F03015


2003 Pressure Vessels and Piping Conference, pp. 185–202, Am. Soc. of Mech. Eng., Cleveland, Ohio. Bombardelli, F. A., G. C. Buscaglia, C. R. Rehmann, L. E. Rincón, and M. H. García (2007), Modeling and scaling of aeration bubble plumes: A two‐phase flow analysis, J. Hydraul. Res., 45(5), 617–630. Brennen, C. E. (2005), Fundamentals of Multiphase Flow, 368 pp., Cambridge Univ. Press, Cambridge, U. K. Buscaglia, G. C., F. A. Bombardelli, and M. H. García (2002), Numerical modeling of large scale bubble plumes accounting for mass transfer effects, Int. J. Multiphase Flow, 28, 1763–1785, doi:10.1016/S03019322(02)00075-7. Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley (1971), Flux profile relationships in the atmospheric surface layer, J. Atmos. Sci., 28(2), 181–189, doi:10.1175/1520-0469(1971)0282.0.CO;2. Cao, Z., L. Wei, and J. Xie (1995), Sediment‐laden flow in open channels from two‐phase flow viewpoint, J. Hydraul. Eng., 121(10), 725–735, doi:10.1061/(ASCE)0733-9429(1995)121:10(725). Cellino, M. (1997), Experimental study of suspension flow in open channels, Ph.D thesis, Civ. Eng. Dep., Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. Cellino, M., and W. H. Graf (2002), Suspension flow in open channels: Experimental study, J. Hydraul. Res., 15, 435–447. Chauchat, J. (2007), Contribution to two‐phase flow modeling for sediment transport in estuarine and coastal zones (in French), Ph.D. thesis, Univ. of Caen, Caen, France. Chauchat, J., and S. Guillou (2008), On turbulence closure for two‐phase sediment‐laden flow models, J. Geophys. Res., 113, C11017, doi:10.1029/2007JC004708. Chen, C. P., and P. E. Wood (1986), Turbulence closure modeling of the dilute gas‐particle axisymmetric jet, AIChE J., 32(1), 163–166, doi:10.1002/aic.690320121. Cheng, N. S. (1997), Effect of concentration on settling velocity of sediment particles, J. Hydraul. Eng., 123(8), 728–731, doi:10.1061/ (ASCE)0733-9429(1997)123:8(728). Coleman, N. L. (1970), Flume studies of the sediment transfer coefficient, Water Resour. Res., 6, 801–809, doi:10.1029/WR006i003p00801. Coleman, N. L. (1986), Effects of suspended sediment on the open‐channel distribution, Water Resour. Res., 22, 1377–1384, doi:10.1029/ WR022i010p01377. Crowe, C. T., M. Sommerfeld, and Y. Tsuji (1998), Multiphase Flows With Droplets and Particles, 496 pp., CRC Press, Boca Raton, Fla. Dietrich, W. E. (1982), Settling velocity of natural particles, Water Resour. Res., 18, 1615–1626, doi:10.1029/WR018i006p01615. Dong, P., and K. Zhang (1999), Two‐phase flow modeling of sediment motions in oscillatory sheet flow, Coastal Eng., 36, 87–109, doi:10.1016/S0378-3839(98)00052-0. Drew, D. A. (1975), Turbulent sediment transport over a flat bottom using momentum balance, J. Appl. Mech., 42, 38–44. Drew, D. A. (1983), Mathematical modeling of two‐phase flow, Annu. Rev. Fluid Mech., 15, 261–291, doi:10.1146/annurev.fl.15.010183.001401. Drew, D., and S. Passman (1999), Theory of Multicomponent Fluids, Appl. Math. Sci., vol. 135, 308 pp., Springer, New York. Einstein, H. A., and N. Chien (1955), Effects of heavy sediment concentration near the bed on velocity and sediment distribution, MRD Sed. Ser. Rep. 8, Univ. of Calif., Berkeley. Elghobashi, S. (1991), Particle‐laden turbulent flows: Direct numerical simulation and closure models, Appl. Sci. Res., 48, 301–314, doi:10.1007/BF02008202. Elghobashi, S. (1994), On predicting particle‐laden turbulent flows, Appl. Sci. Res., 52, 309–329, doi:10.1007/BF00936835. Elghobashi, S., and T. W. Abou‐Arab (1983), A two‐equation turbulence model for two‐phase flows, Phys. Fluids, 26, 931–938, doi:10.1063/ 1.864243. Enwald, H., E. Peirano, and A. E. Almsted (1996), Eulerian two‐phase flow theory applied to fluidization, Int. J. Multiphase Flow, 22, 21–66, doi:10.1016/S0301-9322(96)90004-X. Fletcher, C. A. J. (1997), Computational Techniques for Fluid Dynamics, 256 pp., Springer, New York. Ferziger, J. H., and M. Peric (2002), Computational Methods for Fluid Dynamics, 423 pp., Springer, New York. García, M. H. (1999), Sedimentation and erosion hydraulics, in Hydraulic Design Handbook, edited by L. W. Mays, chap. 6, pp. 6.1–6113, McGraw‐Hill, New York. Garside, J., and M. R. Al‐Dibouni (1977), Velocity‐voidage relationships for fluidization and sedimentation in solid‐liquid systems, Ind. Eng. Chem. Res. Process Des. Dev., 16(2), 206–214, doi:10.1021/ i260062a008. Gelfenbaum, G., and J. D. Smith (1986), Experimental evaluation of a generalized suspended‐sediment transport theory, in Shelf and Sandstones,

F03015

Can. Soc. Pet. Geol. Mem., vol. 11, edited by R. J. Knight and J. R. McLean, pp. 133–144, Calgary. Ghoshal, K., and B. S. Mazumder (2005), Sediment‐induced stratification in turbulent open‐channel flow, Environmetrics, 16, 673–686, doi:10.1002/env.729. Gioia, G., and F. A. Bombardelli (2002), Scaling and similarity in rough channel flows, Phys. Rev. Lett., 88(1), 014501, doi:10.1103/PhysRevLett. 88.014501. González, A. E. (2008), Numerical modeling of sediment transport near the bed using a two‐phase flow approach, Ph.D. thesis, Dep. of Civ. and Environ. Eng., Univ. of Calif., Davis. Greimann, B. P., and F. M. Holly (2001), Two‐phase flow analysis of concentration profiles, J. Hydraul. Eng., 127(9), 753–762, doi:10.1061/ (ASCE)0733-9429(2001)127:9(753). Greimann, B. P., M. Muste, and F. M. Holly (1999), Two‐phase formulation of suspended sediment transport, J. Hydraul. Res., 37, 479–500. Hrenya, C. M., and J. L. Sinclair (1997), Effects of particle‐phase turbulence in gas‐solid flows, AIChE J., 43(4), 853–869, doi:10.1002/ aic.690430402. Hsu, T., J. T. Jenkins, and P. L.‐F. Liu (2003a), On two‐phase sediment transport: Dilute flow, J. Geophys. Res., 108(C3), 3057, doi:10.1029/ 2001JC001276. Hsu, T., H. Chang, and C. Hsieh (2003b), A two‐phase flow model of wave‐induced sheet flow, J. Hydraul. Res., 41(3), 299–310. Hsu, T. J., J. T. Jenkins, and P. L.‐F. Liu (2004), On two‐phase sediment transport: Sheet flow of massive particles, Proc. R. Soc. A, 460(2048), 2223–2250, doi:10.1098/rspa.2003.1273. Hunt, J. N. (1954), The turbulent transport of suspended sediment in open channels, Proc. R. Soc. A, London, 224(1158), 322–335, doi:10.1098/ rspa.1954.0161. Jenkins, J. T., and D. M. Hanes (1998), Collisional sheet flows of sediment driven by a turbulent fluid, J. Fluid Mech., 370, 29–52, doi:10.1017/ S0022112098001840. Jha, S. K. (2009). Theoretical and numerical modeling of suspended sediment transport in open channels using an Eulerian‐Eulerian two‐phase flow approach, Ph.D. thesis, Dep. of Civ. and Environ. Eng., Univ. of Calif., Davis. Jha, S. K., and F. A. Bombardelli (2009), Two‐phase modeling of turbulence in dilute sediment‐laden, open‐channel flows, Environ. Fluid Mech., 9(2), 237–266, doi:10.1007/s10652-008-9118-z. Jiang, J., A. W. Law, and N. S. Cheng (2005), Two‐phase analysis of vertical sediment laden jets, J. Eng. Mech., 131(3), 308–318, doi:10.1061/ (ASCE)0733-9399(2005)131:3(308). Kobayashi, N., and S. N. Seo (1985), Fluid and sediment interaction over a plane bed, J. Hydraul. Eng., 111(6), 903–919, doi:10.1061/(ASCE)07339429(1985)111:6(903). Lees, B. J. (1981), Relationship between eddy viscosity of seawater and eddy diffusivity of suspended particles, Geo Mar. Lett., 1, 249–254, doi:10.1007/BF02462442. Lewis, W. K., E. R. Gilliland, and W. C. Bauer (1949), Characteristic of fluidized particles, Ind. Eng. Chem., 41, 1104–1117, doi:10.1021/ ie50474a004. Liu, H., and S. Sato (2006), A two‐phase flow model for asymmetric sheetflow conditions, Coastal Eng., 53, 825–843, doi:10.1016/j.coastaleng. 2006.04.002. Longo, S. (2005), Two‐phase flow modeling of sediment motion in sheet‐ flows above plane beds, J. Hydraul. Eng., 131(5), 366–379, doi:10.1061/ (ASCE)0733-9429(2005)131:5(366). Lopez de Bertodano, M. A. (1998), Two fluid model for two‐phase turbulent jets, Nucl. Eng. Des., 179, 65–74, doi:10.1016/S0029-5493(97) 00244-6. Loth, E. (2000), Numerical approaches to dilute two‐phase flow, Prog. Energy Combust. Sci., 26(3), 161–223, doi:10.1016/S0360-1285(99) 00013-1. Lumley, J. L. (1978), Two‐phase and non‐Newtonian flows, in Turbulence, edited by P. Bradshaw, pp. 289–324, Springer, New York. Lyn, D. A. (1988), A similarity approach to turbulent sediment‐laden flows in open channels, J. Fluid Mech., 193, 1–26, doi:10.1017/S0022112088002034. Lyn, D. A. (1991), Resistance in flat‐bed sediment‐laden flows, J. Hydraul. Eng., 117(1), 94‐114. Lyn, D. A. (2008), Sedimentation Engineering: Theories, Measurements, Modeling and Practice, ASCE Manuals Rep. Eng. Pract., vol. 110, edited by M. H. García, Am. Soc. of Civ. Eng., Reston, Va. Mazumder, B. S., and K. Ghoshal (2006), Velocity and concentration profiles in uniform sediment‐laden flow, Appl. Math. Model., 30, 164–176, doi:10.1016/j.apm.2005.03.015. McTigue, D. F. (1981), Mixture theory for suspended sediment transport, J. Hydraul. Eng., 107(HY6), 659–673.

26 of 27

F03015


Mina, K. M., and S. Sato (2004), A transport model for sheet flow based on two‐phase flow, Coastal Eng. J., 46(3), 329–367, doi:10.1142/ S0578563404001099. Mitter, A., J. P. Malhotra, and H. T. Jadeja (2004), The two‐fluid modeling of gas‐particle transport phenomenon in confined systems considering inter particle collision effects, Int. J. Numer. Methods Heat Fluid Flow, 14(5), 579–605, doi:10.1108/09615530410539937. Mueller, A. (1973), Turbulence measurement over a movable bed with sediment transport by laser anemometry, paper presented at 15th Congress, Int. Assoc. Hydraul. Res., Istanbul, Turkey. Muste, M., and V. C. Patel (1997), Velocity profiles for particles and liquid in open‐channel flow with suspended sediment, J. Hydraul. Eng., 123(9), 742–751, doi:10.1061/(ASCE)0733-9429(1997)123:9(742). Muste, M., K. Fujita, I. Yu, and R. Ettema (2005), Two‐phase versus mixed‐flow perspective on suspended sediment transport in turbulent channel flows, Water Resour. Res., 41, W10402, doi:10.1029/ 2004WR003595. Muste, M., I. Yu, K. Fujita, and R. Ettema (2009), Two‐phase flow insights into open‐channel flows with suspended particles of different densities, Environ. Fluid Mech., 9(2), 161–186, doi:10.1007/s10652-008-9102-7. Nezu, I., and R. Azuma (2004), Turbulence characteristics and interaction between particles and fluid in particle‐laden open channel flows, J. Hydraul. Eng., 130(10), 988–1001, doi:10.1061/(ASCE)0733-9429(2004) 130:10(988). Nezu, I., and W. Rodi (1986), Open‐channel flow measurements with a laser doppler anemometer, J. Hydraul. Eng., 112(5), 335–355, doi:10.1061/(ASCE)0733-9429(1986)112:5(335). Ni, J. R., G. Q. Wang, and A. G. L. Borthwick (2000), Kinetic theory for particles in dilute and dense solid‐liquid flows, J. Hydraul. Eng., 126(12), 893–903, doi:10.1061/(ASCE)0733-9429(2000)126:12(893). Nikora, V., and D. G. Goring (2002), Fluctuations of suspended‐sediment concentration and turbulent sediment fluxes in an open‐channel flow, J. Hydraul. Eng., 128(2), 214–224, doi:10.1061/(ASCE)0733-9429 (2002)128:2(214). Niño, Y., and M. H. García (1998), Experiments on saltation of fine sand in water, J. Hydraul. Eng., 124(10), 1014–1025, doi:10.1061/(ASCE)07339429(1998)124:10(1014). Nouri, J. M., J. H. Whitelaw, and M. Yianneskis (1987), Particle motion and turbulence in dense two‐phase flows, Int. J. Multiphase Flow, 13(6), 729–739, doi:10.1016/0301-9322(87)90062-0. Patankar, S. V. (1980), Numerical Heat Transfer and Fluid Flow, 197 pp., Taylor and Francis, Hemisphere, New York. Prosperetti, A., and G. Tryggvason (2007), Computational Methods for Multiphase Flow, Cambridge Univ. Press, Cambridge, U. K., doi:10.1017/CBO9780511607486. Richardson, J. F., and W. N. Zaki (1954), Sedimentation and fluidization: Part 1, Trans. Inst. Chem. Eng., 32, 35–53. Rodi, W. (1984), Turbulence Models and Their Application in Hydraulics. A State‐of‐the‐Art Review, IAHR, Delft, Netherlands. Rouse, H. (1937), Modern conception of the mechanics of turbulence, Trans. ASCE, 102, 463–543. Schmeeckle, M., J. Nelson, J. Pittlick, and J. Bennett (2001), Interparticle collisions of natural sediment grains in water, Water Resour. Res., 37, 2377–2391, doi:10.1029/2001WR000531. Shen, N., T. Tsuji, and Y. Morikawa (1989), Numerical simulation of gas‐ solid two‐phase flow in horizontal pipe, Jpn. Soc. Mech. Eng., 55, 2294– 2301.

F03015

Smith, J. D., and S. R. McLean (1977), Spatially averaged flow over a wavy surface, J. Geophys. Res., 82(12), 1735–1746, doi:10.1029/ JC082i012p01735. Squires, K. D., and J. K. Eaton (1994), Effect of selective modification of turbulence on two‐equation models for particle‐laden turbulent flows, J. Fluids Eng., 116, 778–784. Sumer, B. M., A. Kozakiewicz, J. Fredsoe, and R. Deigaard (1996), Velocity and concentration profiles in sheet‐flow layer of movable bed, J. Hydraul. Eng., 122, 549–558, doi:10.1061/(ASCE)0733-9429(1996)122:10(549). Svensson, U. (1998), Program for boundary layers in the environment‐ system description and manual, SMHI Rep. Oceanogr. 24, Norrköping, Sweden. Taggart, W. C., C. A. Yermoli, S. Montes, and A. T. Ippen (1972), Effects of sediment size and gradation on concentration profiles for turbulent flow, Rep. 152, Mass. Inst. of Technol., Cambridge. Toorman, E. A. (2008), Vertical mixing in the fully developed turbulent layer of sediment‐laden open‐channel flow, J. Hydraul. Eng., 134(9), 1225–1235, doi:10.1061/(ASCE)0733-9429(2008)134:9(1225). Tsai, C. H., and C. T. Tsai (2000), Velocity and concentration distributions of sediment‐laden open channel flow, J. Am. Water Resour. Assoc., 36(5), 1075–1086, doi:10.1111/j.1752-1688.2000.tb05711.x. Tsuji, Y., T. Morikawa, T. Tanaka, N. Nakatsukasa, and M. Nakatani (1987), Numerical simulation of gas‐solid two‐phase flow in a two‐ dimensional horizontal channel, Int. J. Multiphase Flow, 13(5), 671– 684, doi:10.1016/0301-9322(87)90044-9. Umeyama, M. (1992), Vertical distribution of suspended sediment in uniform open‐channel flow, J. Hydraul. Eng., 118(6), 936–941, doi:10.1061/(ASCE)0733-9429(1992)118:6(936). Umeyama, M., and F. Gerritsen (1992), Velocity distribution in uniform sediment‐laden flow, J. Hydraul. Eng., 118(2), 229–245, doi:10.1061/ (ASCE)0733-9429(1992)118:2(229). Vanoni, V. A. (1946), Transportation of suspended sediment by water, Trans. ASCE, 111, 67–133. Vanoni, V. A. (1975), Suspension of sediment, in Sedimentation Engineering, edited by V. A. Vanoni, pp. 39–53, Am. Soc. Civ. Eng., New York. van Rijn, L. C. (1984), Sediment transport. Part II: Suspended load transport, J. Hydraul. Eng., 110(11), 1613–1641, doi:10.1061/(ASCE)07339429(1984)110:11(1613). Villaret, C., and A. G. Davies (1995), Modeling sediment‐turbulent flow interactions, Appl. Mech. Rev., 48(9), 601–609, doi:10.1115/1.3023148. Villaret, C., and J. H. Trowbridge (1991), Effects of stratification by suspended sediments on turbulent shear flows, J. Geophys. Res., 96(C6), 10,659–10,680, doi:10.1029/91JC01025. Wan, Z., and Z. Wang (1994), Hyperconcentrated Flow, IAHR Monogr. Ser., 230 pp., Balkema, Rotterdam, Netherlands. Wang, X., and N. Qian (1992), Velocity profiles of sediment laden flow, Int. J. Sediment Res., 7(1), 27–58. Woo, H. S., P. Y. Julien, and E. V. Richardson (1988), Suspension of large concentration of sands, J. Hydraul. Eng., 114(8), 888–898, doi:10.1061/ (ASCE)0733-9429(1988)114:8(888). Yoon, J. Y., and S. K. Kang (2005), A numerical model of sediment‐laden turbulent flow in an open channel, Can. J. Civ. Eng., 32, 233–240, doi:10.1139/l04-089. F. A. Bombardelli and S. K. Jha, Department of Civil and Environmental Engineering, University of California, One Shields Ave., 2001 EU III, Davis, CA 95616, USA. ([email protected]; [email protected])

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