



Twophase modeling of turbulence in dilute sedimentladen, openchannel
flows. In a previous paper in this special issue, we addressed several aspects of the modeling of sediment transport, focusing on the phase interactions at the waterdepth scale for dilute flows. We proposed, for the first time to the best of our knowledge, a framework constituted by three modeling approaches involving different complexities: (1) A standard sedimenttransport model (SSTM), (2) a partial twofluid model (PTFM), and (3) a complete twofluid model (CTFM). While the first approach represents a relatively simple attempt to model the motion of the two phases, the last one is described by a set of equations representing the conservation of mass and momentum of each phase within a fixed elemental volume of the mixture. The main assumption used in deriving the equations for the CTFM is that both phases behavemacroscopically as fluid continua [21–23]. This hypothesis is customary in particle laden flows in open channels (see for instance [11–13,20,31,30,32,34,37,49,67,70]). By posing the SSTM, the PTFM and the CTFM in one dimension (1D) in the wallnormal direction, we were able to test in the companion paper the accuracy of the models against data by Lyn [45], Muste and Patel [50], and Muste et al. [51], in terms of the prediction of the turbulenceaveraged water velocity, the averaged velocity of the disperse phase and flow turbulence statistics. We tested the effect of neglecting the eddy viscosity of the disperse phase (the sediment); we considered diverse models for the Schmidt number; and we reported those values of the Schmidt number that best approximated the data. We additionally tested the importance of certain terms added by some authors to the K–ε model (the model used to treat the turbulence of the carrier phase, where K and ε denote the turbulent kinetic energy (TKE) and the dissipation rate of turbulent kinetic energy (DTKE), respectively), concluding that those terms do not produce major changes in the predictions in any of the framework approaches described above. These framework models were tested with just one possible turbulence closure (the K–ε model); there is naturally a need to rigorously test them with other turbulence closures. Turbulence in openchannel flows is nonhomogeneous and nonisotropic [5,27–29,52,57], which puts forward an important question on the choice of the most appropriate turbulence closure for modeling when the ReynoldsAveraged NavierStokes (RANS) equations are used. This issue is still relevant in spite of the wide use of the K–ε model. In addition, the use of the Reynolds stress model (RSM) in twophase flows has been scarce, and reasonable questions arise as to which values of the vertical component of the velocity of the disperse phase and the Schmidt number such closure would yield when paired with the CTFM. Finally, Choi and Kang [15] found that the RSM improved the description of most flow variables in open channels with vegetation (in 1D), as opposed to the K–ε model. Thus, a fair issue surfaces as to whether the application of the RSM to sedimentladen, openchannel flows can yield improvement in the model predictions. We keep our focus on dilute flows. In the case of dilute flows, the volume fraction of sediment is small (αd ≈ 10−3 to 10−2 as a rough estimate [22]). The dilute flow is defined as the flow in which particle–particle interactions can be neglected [32,44]. Based on the above concepts, this paper evaluates the effect of various turbulence closures in the simulation of openchannel, sediment transport, under different approaches of modeling discussed in [6]. To the best of our knowledge, this is the first time that the focus on modeling in sediment transport engineering is shifted completely to address the performance of different turbulence closures against common datasets (cf. [46]). The paper is organized as follows. In Sect. 2, we describe very recent contributions on the turbulence modeling in single and twophase flows using the RSM, the algebraic stress model (ASM) and the K–ε and K–ω models (where ω denotes the specific dissipation rate of TKE per unity TKE; see [72]). In Sect. 3, we briefly review the developed framework of models for dilute sedimentladen, openchannel flows presented in [6], and discuss singlephase turbulence models. In Sect. 4, we introduce the extensions of the RSM and the K–ω model to twophase flows. We present the group of numerical tests that we performed in Sect. 5, and we discuss the comparisons between those results and data in Sect. 6, for both meanflow variables, flow turbulence statistics, and for sediment distributions. 
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& Environmental Engineering  University of California, Davis 2001 Ghausi Hall, One Shields Ave., Davis, CA 95616 