**5. Conclusions**

This paper presented a numerical investigation of the scour phenomenon below a submarine pipeline. SedFoam, a two-phase flow model for sediment transport applications, was used to study the sensitivity of the scour hole formation and of the bed morphology to the granular stress and the turbulence closure. The quality of the different simulations was measured using the Brier Skill Score. The granular stress model was not sensitive, and similar results were obtained between simulations using *μ*(*I*) rheology and the kinetic theory for granular flows. Both models provided a quantitative time evolution of the erosion depth and of the bed morphology when coupled with the *k* − *ε* turbulence model.

The turbulence model however had a significant influence on the bed morphology. On the one hand, the *k* − *ε* model provided the right equilibrium maximum erosion depth, but overestimated the bed elevation downstream of the pipeline. This accretion phenomenon was explained by the incapacity of the *k* − *ε* model to reproduce the vortex-shedding phenomenon and the lee-wake erosion stage of scour. Therefore, a turbulence model able to reproduce vortex shedding should be used. The *k* − *ω*2006 model, which can reproduce the vortex-shedding, strongly underestimated the erosion depth, but allowed qualitatively reproducing the lee-wake stage of scour.

An in-depth analysis of the *k* − *ε* and the *k* − *ω*2006 revealed the importance of the cross-diffusion term responsible for the behavior of *k* − *ε*. The negative and positive contribution of the cross-diffusion term were incorporated in the *k* − *ε* model, whereas only the positive contribution was incorporated in *k* − *ω*2006. The numerical results showed that the negative contribution of the cross-diffusion term was required near the sediment bed to reproduce quantitatively the time development of the scour hole.

An improved URANS two-phase flow turbulence model should have a *k* − *ω*2006 behavior in the outer regions and a *k* − *ε* behavior near the sediment bed. Such a turbulence model would allow providing accurate results in conditions where the interactions between the fluid vortices and the sediment bed are important.

The coupling between the sediments dynamics and the turbulence is a very complex phenomenon, and it should be investigated in detail using large eddy simulations. It would allow better understanding the interactions between the turbulent wake of the cylinder and the sediment bed downstream of the pipeline. This is beyond the scope of the present paper, and it is left for future research.

**Author Contributions:** Funding acquisition, writing—review, project administration and supervision, J.C.; methodology, software, validation, A.M., T.N., C.B. and J.C.; writing—original draft preparation A.M. and J.C.

**Funding:** The work presented in this manuscript was financially supported by the French national research agency ANR projects SegSed (ANR-16-CE01-0005-03) and SheetFlow (ANR-18-CE01-0003) and the European Community's Horizon 2020 Program through the Integrated Infrastructure Initiative HYDRALAB + FREEDATA (654110).

**Acknowledgments:** We would like to thank Cheng-Hsien Lee, Tian-Jian Hsu, and Zhen Cheng for fruitful discussions around multiphase flow modeling of scour. Most of the computations presented in this paper were performed using the GENCI infrastructure under Allocation A0060107567 and the GRICAD infrastructure. We are also grateful to the developers involved in OpenFOAM.

**Conflicts of Interest:** The authors declare no conflict of interest.
