**4. Discussion**

Close to the sediment bed, the cross-diffusion term became negative. Its contribution was incorporated in the *k* − *ε* model, but not in the *k* − *ω*2006 turbulence model. However, from Figure 7, the negative contribution of the cross-diffusion term played an important role in the time development of the scour hole depth.

The negative contribution of the cross-diffusion term seemed to be necessary to reproduce quantitatively the time development of the scour hole. Typical TKE (*k*) and the specific dissipation rate of the TKE (*ω*) profiles for free shear, boundary layer, and sediment transport flows are presented in Figure 8. For free shear flows, the peak value of *k* corresponds to the peak value of *ω*. The cross-diffusion term was always positive, and the *k* − *ω*2006 turbulence model behaved like a *k* − *ε*. For boundary layer flows, the peak value of *ω* was located at the wall, whereas the peak value of *k* was located further away. The gradient of *k* changed sign toward the boundary, as did the cross-diffusion term, which became negative. The negative cross-diffusion contribution was suppressed in the *k* − *ω*2006 model, having the effect of relaminarizing the flow close to the wall. When sediment transport was involved, the peak values of *k* and *ω* were offset, so that the cross-diffusion term became negative between the two peaks. Using the *k* − *ω*2006 model in this configuration suppressed the influence of the negative contribution of the cross-diffusion term, and the flow was relaminarized close to the sediment bed. This phenomenon was not physical and was responsible for the underestimation of the sediment erosion observed using the *k* − *ω*2006 turbulence model. Finally, our numerical results suggested that sediment transport shares more similarities with a free shear flow than with boundary layer flows. The negative contribution of the cross-diffusion term should therefore be incorporated to behave like a *k* − *ε* model near the sediment bed, while suppressed far from the bed to behave like the *k* − *ω*2006 model and allow vortex-shedding to develop.

**Figure 8.** Typical turbulent kinetic energy (*k*) and specific dissipation rate (*ω*) profiles for free shear flows, boundary flows, and sediment transport configurations.

Finally, even though the two-phase flow model relied on a more theoretical background than the classical single-phase flow models, empirical expressions are still needed, especially for the granular stress and turbulence models. However, these models are at a lower level of approximation in the sense that they have been developed and validated on other fluid and granular flow configurations. In this respect, they are more general and better describe the complex physics at work in sediment transport. From Section 3.1, the empiricism in the granular stress model did not seem to be a limitation since the dense granular flow rheology and the kinetic theory of granular flows provided accurate results. However, the available two-phase turbulence models did not fully take into account the complex interactions between the granular phase and the fluid turbulence. For this type of configuration, the coupling between the fluid turbulence and the sediment dynamics was crucial, and Reynolds averaged two-phase flow models showed their limitations.
