*3.2. Turbulence Model Sensitivity*

In this subsection, the influence of the turbulence model on the shape of the sediment bed is investigated. The results using the *k* − *ε* and the *k* − *ω*2006 turbulence models are compared in Figures 4 and 5 in term of the time evolution of the maximum scour depth and the shape of the sediment bed.

**Figure 4.** Time evolution of the maximum scour depth from simulations with the *μ*(*I*) rheology using the *k* − *ε* (orange line) and *k* − *ω*2006 (red dotted line) turbulence models compared with the experimental data from Mao (1986) [2] (red dots).

**Figure 5.** Bed profiles at 25 s from simulations with the *μ*(*I*) rheology using the *k* − *ε* (orange line) and *k* − *ω*2006 (red dotted line) turbulence models compared with the experimental data from Mao (1986) [2] (red dots).

The time evolution and the equilibrium depth of the scour hole were significantly underestimated when using the *k* − *ω*2006 turbulence model. This turbulence model did not provide quantitative results in this configuration. However, the vortex shedding phenomenon was predicted, and the sediment bed downstream of the pipeline was eroded (see bed interface between x/D = 4 and x/D = 6).

The snapshot provided in Figure 6 confirms that the erosion was caused by the vortices in the wake of the cylinder. A strong sediment flux was associated with a vortex reaching the sand dune downstream of the pipeline. Therefore, the accretion of sediment visible using the *k* − *ε* model was no longer present when using the *k* − *ω*2006 turbulence model.

The *BSS* at 25 s using the *k* − *ω*2006 was 0.569, which was significantly lower than the one obtained with the *k* − *ε* model. The lee-wake erosion of the sand dune did not compensate the underestimation of the scour depth in the *BSS*.

**Figure 6.** Streamlines and sediment volumetric flux at 25 s for the simulation using the *k* − *ω*/2006 turbulence model.

A sensitivity analysis on the cross-diffusion term appearing in the dissipation equation through the coefficient *σd* was performed to identify the main differences between the two aforementioned models. The time evolution of the maximum scour depths from simulations using the *k* − *ω*2006, the modified *k* − *ε*, and the modified *k* − *ε* taking only the positive contribution of the cross-diffusion term is presented in Figure 7. It appears that removing the negative contribution of the cross-diffusion term in the modified *k* − *ε* turbulence model provided results closer to the ones obtained using the *k* − *ω*2006 turbulence model with an equilibrium erosion depth largely underestimated. The definition of the turbulent viscosity in the *k* − *ω*2006 turbulence model mainly affected the dilute regions, but the differences in terms of bed elevation visible in Figures 4 and 7 came from the cross-diffusion term.

**Figure 7.** Time evolution of the maximum scour depth from simulations with *μ*(*I*) rheology using the *k* − *ω*2006 (red dotted line) and modified *k* − turbulence model with (brown line) and without (brown dashed dotted line) the negative contribution of the cross-diffusion term compared with the experimental data from Mao (1986) [2] (red dots).

The cross-diffusion term was responsible for the behavior of *k* − *ε*. It became positive in free shear flows where the *k* − *ε* model is known to provide better predictions and became negative near solid boundaries where the classical *k* − *ω* model provides better predictions. In the *k* − *ω*2006 turbulence model, from Equation (30), only the positive contribution of the cross-diffusion term was incorporated

with a coefficient one order of magnitude smaller than in the *k* − *ε* model. The idea behind the *k* − *ω*2006 model is to incorporate the cross-diffusion term in free shear flows (positive contribution) to have a *k* − *ε* behavior and suppress it near solid boundaries (negative contribution) to have a classical *k* − *ω* behavior.
