Fabián A. Bombardelli, Ph.D.
Associate Professor

Hierarchical modeling of the dilute transport of suspended sediment in open channels.
Fabián A. Bombardelli and Sanjeev K. Jha

Traditional approaches to quantify the dilute transport of sediments in open channels have consisted in representing the solid phase as a scalar field with concentration C [54,71]. In other words, the carrier fluid (water) has been usually considered as a quasi single-phase flow in those approaches, and the transport of sediments has been characterized via an advectiondiffusion equation for C [37]. Further, the sediment phase has been customarily assumed to travel at the same stream-wise velocity of the mixture fluid [17,24,43,44,71] and its diffusivity in the wall-normal direction has been specified in terms of a balance between turbulent diffusion and the effect of gravity. These hypotheses, which lead to the well-known Rousean distribution of sediments under equilibrium conditions [58] do not work well in all cases. In fact, it has been reported repeatedly by various researchers that the profile of water velocity in open-channel flows deviates only slightly from the standard semi-logarithmic law [14,30,49,50; 51, p. 996] but, more importantly, that the distribution of the concentration of sediments in the wall-normal direction differs appreciably from that obtained through the standard Rousean equation [14, 28, p. 488; 50, p. 18, 51, p. 994].

Several attempts have been made to correct the above “standard” sediment-transport model. For instance, some authors have proposed to alter the semi-logarithmic law by modifying the value of the von-Kármán’s constant [3,16,30,44,50; 51, p. 996]. In addition, some authors have suggested improving the advection-diffusion equation through the inclusion of a mechanism called “drift” [19], which would result from additional particle velocity fluctuations due to the uneven distribution of solid concentration in the flow. Another modification proposed by diverse researchers has consisted in simply adjusting the values of the Schmidt number (the ratio between the eddy viscosity of the flow and the diffusivity of suspended sediment) in the advection-diffusion equation, or in the standard Rousean equation to improve the agreement with data [14, 71, p. 81; 72].

From amulti-component flowperspective, Drew[22] employed the two-phase, turbulenceaveraged mass and momentum equations for water and sediments to analyze sediment transport in open channels. Buoyancy and drag were considered to be the only interaction forces. Drew applied the mixing length theory to represent the Reynolds stresses. By using order-ofmagnitude estimates, assuming a dilute mixture, and neglecting the relative velocity between the phases, Drew derived differential equations for the distributions of stream-wise velocity and suspended-sediment concentration in the wall-normal direction. Through numerical solutions of the equations, he comparedmodel predictions with experimental data, and found that although the overall prediction of the model was good in terms of flow velocities, it could not accurately predict the near-bottom sediment concentrations. McTigue [47], in turn, combined the turbulence-averaged mass and momentum equations for solids and water in the stream-wise and wall-normal directions. Using scaling of the terms in the resulting momentum equation in the wall-normal direction, he recovered the classical balance between the downward flux of settling particles due to gravity and the diffusive flux due to turbulence. He applied three models for the diffusivity of the disperse phase, and compared the analytical predictions of the distribution of sediment concentration with experimental data. McTigue analyzed the behavior of the solution for the concentration in two layers characterized by different flow length scales. He concluded that different models show satisfactory predictions in both layers. Kobayashi and Seo [35] were perhaps the first in working with the complete twofluid model (CTFM; see below) using six equations: two for the mass balance of each phase and four for the momentum balance of each phase in the wall-normal as well as the streamwise directions. The interaction between fluid and sediment was represented through the forces of buoyancy and drag. Kobayashi and Seo investigated the regions of suspended-load 123 Environ Fluid Mech (2009) 9:207–235 209 and bed-load separately. In the bed-load region, they considered the interaction of the solids and the bed. Kobayashi and Seo derived equations for the profiles of sediment concentration and for the relative velocity between water and sediment (which was ignored in previous contributions). The models were solved numerically and analytically, and predictions were compared with experimental data with partial success. Following Kobayashi and Seo [35], Cao et al. [11] also used a two-layered approach (suspension layer and bed layer). They employed mixture equations for the carrier phase and mass and momentum equations for the disperse phase. In their analysis, they emphasized that the selection of closure models for the turbulent diffusivity and eddy viscosity plays a critical role in the final result for the velocity and concentration profiles. They compared the analytically-predicted mean velocities and distribution of sediment concentration in the wall-normal direction with experimental data, obtaining satisfactory agreement. Villaret and Davies [73] presented a review of sediment transport models ranging from simplified approaches based on the passive scalar hypothesis to complex two-phase flow models. Drag and lift were considered as interaction forces in the two-fluid model. Villaret and Davies compared the performance of those models in predicting the mean velocity of the water and the distribution of the concentration of sediment. Based on their observations, they advocated the use of two-phase flow models to reduce the empiricism involved in the classical models.

Greimann et al. [28] included in their analysis a “drift velocity” in addition to the relative velocity between phases. They developed an analytical expression for the sediment concentration profile and the relative velocity in dilute flows from the complete two-fluid equations. They considered the following interaction forces between water and sediment: drag, added mass and lift. For turbulence closure, Greimann et al. employed the expressions of turbulence intensities developed by Nezu and Rodi [53] for clear water flows. Greimann et al. compared their analytical model with experimental datasets, obtaining satisfactory results. Greimann and Holly [29] extended the above model to non-dilute conditions by addressing the importance of particle–particle interactions, and the effect of particle inertia. Jiang et al. [33] also developed an analytical model for the distributions of sediment concentration and relative velocity. They 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. Jiang et al. [33] applied the expressions of Nezu and Nakagawa [52] to represent the turbulence intensities and obtained a final equation for the distribution of sediment concentration similar to that of Greimann et al. [28]. Comparisons of results with experimental data showed more accurate predictions for the distribution of sediment concentration than those coming from the Rousean model and from the models of Greimann et al. [28] and Griemann andHolly [29]. Hsu et al. [31] obtained, in turn, the profile of sediment concentration in the wall-normal direction by using a semi-analytical approach and a numerical solution of the CTFM in combination with an extended K–ε turbulence closure. These authors were, to the best of our knowledge, the first to employ a turbulence treatment based on a two-equation model to address the fluctuations in the carrier fluid in dilute sediment-transport studies (see below the enumeration of contributions about sheet flows). In their comparison with experimental data, Hsu et al. [31] focused on the flow near the bed only, and found the predictions to be closer to the experimental data than the Rousean formula. They explained that the better prediction was due to the use of additional terms in the K–ε turbulence model; however, no comparison of turbulence statistics with data was presented.

Recent studies concerning sediment motion in very dense sheet flows have also used the CTFM equations (see for example [21,32,39,40]). Most of those authors have used K–ε type turbulence closures, but no comparison of turbulence statistics with data has been presented. In addition to solid-water flows, there is a wealth of models for bubble-water flows, which use different levels of the TFM (see, for instance, [5] and [61]). Table 1 summarizes recent models for solid-water flows. A detailed analysis of the above contributions reveals the following issues:

1. Different formulations have been adopted to address the dilute sediment-transport problem; however, no general guideline exists on the theory to use in any given case. For example, it is not clear whether the CTFM is needed for an accurate simulation of sediment transport in open channels, or whether the mixture equations are enough. While some authors have not considered other models than the CTFM and/or multi-group models in bubbly flows [67,75], other authors have shown that good predictions can be obtained with relatively simple models [5,60]. This discussion has not been pursued in sediment-transport problems.

2. The relative importance of the diverse forces in the TFM has not been determined for sediment-laden, open-channel flows yet. Related to this issue is the so-called “turbulent dispersion force” [23], which may be included in the momentum equation for both phases instead of the diffusive term in the mass balance equation (see [5]).

3. In the case of solid-water flows, different authors have compared model predictions with data in terms of mean-flow carrier velocities and the distribution of suspended-sediment concentration, but except in studies on oscillatory sheet flow [39] the velocity profile of the disperse phase has not been presented in papers on dilute sediment transport.

4. To the best of our knowledge no study has compared simulated turbulence statistics with data, considering the sediment transport as a two-phase flow [46]. This is relatively easy to understand in terms of the difficulties in measuring turbulence statistics in two-phase flows, some of which have been overcome only recently. As said, only Hsu et al. [31,32] have presented predicted turbulence statistics in dilute sediment-transport studies, but no comparisons with measurements have been reported. This is a crucial issue that provides strong motivation for the research presented herein.

5. Several terms have been added in some models to the equations of turbulent kinetic energy (K) and dissipation rate of turbulent kinetic energy (ε) in order to account for the role exerted by the disperse phase on the flow turbulence. However, these results need to be verified through additional systematic comparisons with recent datasets.

The main objective of this paper is then to introduce, discuss and validate a theoretical and numerical framework constituted by a set of models for the prediction of flow variables in dilute sediment-laden, open-channel flows. We address all points mentioned above and validate the framework with data in the range of fine sands. In Sects. 2 and 3, we present and analyze the theoretical models for general two-phase flows. The framework includes the 1D standard sediment transport model (1D SSTM) in addition to the 1D CTFM and the 1D PTFM. In Sect. 4, we detail the numerical treatment of the equations, and in Sect. 5 we describe the datasets selected for testing the models. Finally, in Sect. 6 we present the numerical results and the comparisons of model predictions with the datasets.

Department of Civil & Environmental Engineering - University of California, Davis
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