Two-dimensional, two-phase granular sediment transport model with applications to scouring downstream of an apron

Abstract

We present a two-dimensional, two-phase model for non-cohesive sediment transport. This model solves concentration-weighted averaged equations of motion for both fluid and sediment phases. The model accounts for the interphase momentum transfer by considering drag forces. A collisional theory is used to compute the sediment stresses, while a two-equation (k–ε) fluid turbulence closure is implemented. A benchmark sediment transport problem concerning the scouring downstream of an apron is carried out as an example and numerical results agree with existing experimental data.

Introduction

Sediment transport is a complex, dynamic, multidimensional and multi-scale process. Models used to describe this process can generally be divided into single phase models and two-phase models. The former assume that the presence of sediment has no direct influence on the hydrodynamics (fluid velocity and pressure), while the latter consider equations of motion for fluid and sediment phases with consideration of inter-phase interactions (Drew, 1983). In single phase models, the sediment concentration is commonly obtained by solving an advection and diffusion equation. The interaction between sediment particles and fluid turbulence can be neglected (Savioli and Justesen, 1996, Wu et al., 2000) or can be accounted for, to some extent, through a “buoyant energy production” term that is similar to that of density stratified flows (Hagatun and Eidsvik, 1986, Gessler et al., 1999).

For one-dimensional flows such as sediment transport under sheet flow conditions, the two-phase flow approach has become increasingly popular (Kobayashi and Seo, 1985, Asano, 1990, Dong and Zhang, 1999, Hsu et al., 2004, Longo, 2005, Amoudry et al., 2008). As mentioned previously, this approach solves equations of motion (conservation of mass and momentum for both phases). Momentum transfer between the two phases is taken into account and is usually specified by considering the forces of the fluid on the sediment particles (Drew et al., 1979, Drew and Lahey, 1979). The turbulent fluid stress can be obtained by using a variety of fluid turbulence closures, most of them based on the turbulent viscosity hypothesis. The effect of sediment particles on the carrier fluid turbulence also has to be considered and has been studied in chemical and mechanical engineering (e.g. Gore and Crowe, 1989, Squires and Eaton, 1994, Crowe et al., 1996). These studies show that the presence of particles does impact the carrier fluid turbulence and can increase or decrease turbulent intensity. Finally, sediment phase stresses need to be appropriately specified, which is usually done either by using empirical formulae (e.g. Bagnold, 1954, Savage and McKeown, 1983) or by using theoretical expressions (e.g., Jenkins, 1998 for collisional regime, Carpen and Brady, 2002 when the fluid viscosity dominates).

So far, two-phase models have focused mainly on one-dimensional situations (i.e., sheet flow), and only very recently has the two-phase approach been used in numerical modeling of scour (e.g. Zhao and Fernando, 2007). Most multi-dimensional models are still using the single phase approach for which sediment is considered as a tracer (that may influence the fluid density) and for which the sediment concentration is determined by solving an advection–diffusion equation (e.g., Zhang et al., 1999, Harris and Wiberg, 2001 for two-dimensional continental shelf models, Brørs, 1999 for a two-dimensional scour model, Gessler et al., 1999, Lesser et al., 2004 for fully three-dimensional sediment transport models). The bathymetric changes in such models are usually calculated by solving the Exner equation, which is the conservation of sediment mass between the water column and the sediment bed (e.g. Brørs, 1999, Zhang et al., 1999, Harris and Wiberg, 2001). However, near bed quantities, such as the bed load transport and the erosion flux/pickup rate, need to be explicitly prescribed instead of being part of the model's solutions and diverse empirical relationships are usually employed to that end.

This results in a modeling deficiency in that few multi-dimensional models include complete and detailed descriptions of the near-bed physical processes. We seek to address this issue by developing a model based on a two-phase approach, for which concentration-weighted averaged equations of motion are solved for a sediment and a fluid phase. The two phases are assumed to only interact through drag forces. The correlations between fluctuating quantities are modeled using the turbulent viscosity and the gradient diffusion hypotheses. The fluid turbulent stresses are calculated using a modified k–ε model that accounts for the two-way particle-turbulence interaction, and the sediment stresses are calculated using a collisional granular flow theory. We also perform a two-dimensional test case: scouring downstream of an apron. Scour is a natural phenomenon that is the result of the interaction of structures (usually man made) with fluid flows and with erodible beds. It is of particular importance because of the possible damage incurring to the structure. Several experiments on scouring downstream of structures were undertaken in the 60s (e.g. Breusers, 1965, Breusers, 1967, Dietz, 1969) and were summarized in Breusers and Raudkivi (1991). More recent studies on the subject include the work by Buchko et al. (1987), Hoffmans and Booij (1993) and Hoffmans and Pilarczyk (1995). Scour downstream of an apron is chosen here for its simplicity as a two-dimensional sediment transport problem. It is also similar to the turbulent flow over a backward facing step, which is commonly used as a benchmark test for turbulence models.

In this paper, we will first introduce the two-dimensional, two-phase model's formulation, discuss its validity and present some of its numerical characteristics. We then present an example of the application of this model by studying scour downstream of an apron and we focus on the downstream boundary condition and initial condition used. The numerical results obtained for this example are also compared favorably with experimental data.