*2.1. Governing Equations*

The two-phase flow model for sediment transport, SedFoam, developed by Cheng et al. (2017) [16] and Chauchat et al. (2017) [17], was used as a starting point. SedFoam is an open-source and multi-version solver implemented in the open-source CFD toolbox OpenFOAM and proposes several granular stress and turbulence models (https://github.com/SedFoam/sedfoam). SedFoam is designed to study three-dimensional sediment transport.

In the two-phase flow formulation, both solid and fluid phases are described by Eulerian equations. The fluid and solid mass conservation equations are written as:

$$\frac{\partial \phi}{\partial t} + \frac{\partial \phi u\_i^s}{\partial x\_i} = 0 \tag{1}$$

$$\frac{\partial(1-\phi)}{\partial t} + \frac{\partial(1-\phi)u\_i^f}{\partial x\_i} = 0 \tag{2}$$

where *φ* is the sediment phase concentration, *usi* and *ufi* are the sediment and fluid phase velocities, respectively, and *i* = 1, 2 are the stream-wise and vertical components.

The momentum equations for the solid and fluid phases are given by:

$$\begin{split} \frac{\partial \rho^s \phi \mathbf{u}\_i^s}{\partial t} + \frac{\partial \rho^s \phi \mathbf{u}\_i^s \mathbf{u}\_j^s}{\partial \mathbf{x}\_j} = -\phi \frac{\partial p}{\partial \mathbf{x}\_i} - \frac{\partial p^s}{\partial \mathbf{x}\_i} + \frac{\partial \tau\_{ij}^s}{\partial \mathbf{x}\_j} + \phi \rho^s g\_i + \phi (1 - \phi) \mathbf{K} (\mathbf{u}\_i^f - \mathbf{u}\_i^s) \\ -(1 - \phi) \frac{1}{\sigma\_c} \mathbf{K} \mathbf{v}\_t^f \frac{\partial \phi}{\partial \mathbf{x}\_i} \end{split} \tag{3}$$

$$\begin{split} \frac{\partial \rho^{f} (1 - \phi) u^{f}\_{i}}{\partial t} + \frac{\partial \rho^{f} (1 - \phi) u^{f}\_{i} u^{f}\_{j}}{\partial x\_{j}} &= -(1 - \phi) \frac{\partial p}{\partial x\_{i}} + \frac{\partial \pi^{f}\_{ij}}{\partial x\_{j}} + (1 - \phi) \rho^{f} g\_{i} \\ &- \phi (1 - \phi) \mathcal{K} (u^{f}\_{i} - u^{s}\_{i}) + (1 - \phi) \frac{1}{\mathcal{C}\_{\mathbb{C}}} \mathcal{K} \nu^{f}\_{i} \frac{\partial \phi}{\partial x\_{i}} \end{split} (4)$$

with *ρs* and *ρf* the solid and fluid density, *gi* the acceleration of gravity, *p* the fluid pressure, *ps* the solid phase normal stress, and *τsij* and *τ fij* the solid and fluid phase shear stresses. The solid phase shear stress closure model is detailed in Section 2.2, and the fluid phase shear stress is expressed as:

$$
\pi\_{ij}^f = \rho^f (1 - \phi) \left[ 2\nu\_{Eff} S\_{ij}^f - \frac{2}{3} k \delta\_{ij} \right]. \tag{5}
$$

*Sfij* = 1/2 *<sup>∂</sup>ufi* /*∂xj* + *∂ufj* /*∂xi* − 1/3 *<sup>∂</sup>ufk*/*∂ufk* is the deviatoric part of the fluid strain rate tensor; *k* is the turbulent kinetic energy (TKE); and *<sup>ν</sup>Eff* is the effective velocity defined by *<sup>ν</sup>Eff* = *νft* + *νmix* with *νft* the eddy viscosity calculated by a turbulence closure model (see Section 2.3) and *νmix* the mixture viscosity following the model proposed by Boyer et al. (2011) [18]:

$$\frac{\nu\_{\rm mix}}{\nu^f} = 1 + 2.5 \phi \left( 1 - \frac{\phi}{\phi\_{\rm max}} \right)^{-1},\tag{6}$$

where *φmax* = 0.635 is the maximum value for the solid phase concentration and *νf* is the fluid kinematic viscosity.

The last two terms of the right-hand side of both momentum equations represent the drag force coupling the two phases. *σc* is the Schmidt number, and *K* is the drag parameter modeled according to Richardson and Zaki (1954) [19]:

$$K = 0.75 \mathbb{C}\_d \frac{\rho^s}{d} \parallel \boldsymbol{\mu}^f - \boldsymbol{\mu}^s \parallel (1 - \phi)^{-h\_{\text{Exp}}}.\tag{7}$$

*d* is the particles' diameter; *hExp* is the hindrance exponent controlling the drag increase with increasing solid concentration; and *Cd* is the drag coefficient calculated by the empirical formula given by Schiller and Naumann (1933) [20]:

$$\mathcal{C}\_d = \begin{cases} \begin{array}{c} \frac{24}{Re\_p}(1 + 0.15 Re\_p^{0.687}), & Re\_p \le 1000 \\ 0.44, & Re\_p > 1000 \end{array} \end{cases} \tag{8}$$

where *Rep* is the particulate Reynolds number defined by: *Rep* = (1 − *φ*) - *uf* − *u<sup>s</sup>* - *d*/*νf* .
