## 1. Introduction

The scour phenomenon is a major cause of the rupture or self-burying of submarine pipelines. This phenomenon has to be accurately predicted during the design process of these submarine structures. An important amount of experimental and numerical studies has been published in the literature to characterize both the shape and the time development of the scour hole. The erosion under a submarine pipeline can be decomposed into three steps: (1) the onset; when the current around the cylinder is strong enough, it generates a pressure drop, which liquefies the sediments underneath the cylinder; (2) the tunneling stage; when a breach is formed between the cylinder and the sediment bed, it expands due to the strong current in the breach; (3) the lee-wake erosion stage; when the gap is large enough, vortices are shed in the wake of the cylinder, leading to erosion downstream of the scour hole [

1]. Mao (1986) [

2] performed extensive experiments for which the shape of the sediment bed around the cylinder was recorded for various flow conditions. The data collected by Mao (1986) [

2] have been widely used as a benchmark for sediment transport models applied to the scour phenomenon.

The numerical simulation of scour under a pipeline has been extensively investigated during the last few decades. The first numerical simulations were based on single-phase models and differed mainly by the way turbulence was modeled. Models based on potential flow theory like Chiew (1991) [

3] or Li and Cheng (1999) [

4] tend to predict correctly the final shape of the upstream part of the scour hole and its maximum depth. However, they fail to reproduce the downstream part because, according to Sumer (2002) [

1], the erosion in this region is dominated by the wake of the cylinder and vortex shedding in the sediment bed. Other single-phase flow models using

$k-\epsilon $ turbulence models like Leeuwenstein and Wind (1984) [

5] were used to determine the shape of the scour hole, but also failed to reproduce the equilibrium stage. Indeed, in early numerical models, the role played by the suspended sediment was underestimated, and only the bed load transport component of sediment transport was taken into account. Clear water cases were better predicted than live-bed cases because suspension plays a more important role in live-bed conditions. Liang et al. (2004) [

6] presented a numerical model able to simulate the time development of the scour hole in live-bed and clear-water conditions. The authors’ model took into account bed load and suspended load transport, and a

$k\u2013\epsilon $ turbulence model was used. Furthermore, the influence of the local bed slope on the critical shear stress was incorporated. This means that the threshold Shields parameter

${\theta}_{c}$ was adjusted to a higher value for sediments moving up slope and to lower value for sediments moving down slope. The mathematical model presented by Liang et al. (2004) [

6] showed fairly good agreement with the experimental data from Mao (1986) [

2], but the accumulation of sediment behind the pipeline was over-predicted for both live-bed and clear-water scour cases. Liang et al. (2004) [

6] argued that this over-prediction was caused by the choice of the

$k\u2013\epsilon $ turbulence model. Indeed, the

$k\u2013\epsilon $ model tends to smooth out the fluctuations in the wake of the pipeline, and therefore, the interaction with the sediment bed can be altered.

The mutual interactions between the fluid and the sediments are more complex than a simple local shear stress relation, as is classically assumed in single-phase flow models. During the past two decades, a new modeling approach has emerged, two-phase flow models [

7,

8,

9]. One of the main advantages over classical models is that the two-phase flow approach does not require the use of the empirical sediment transport rate and erosion-deposition formulas. The physical grounds on which this new generation of sediment transport models is based should allow improving scour modeling. The two-phase flow approach has already been applied to the scour below a submarine pipeline configuration. Zhao and Fernando (2007) [

10] used a two-phase model implemented in the CFD software FLUENT with a

$k\u2013\epsilon $ turbulence model. The temporal evolution of the maximum scour hole was captured in clear-water conditions, but they found that sediments were still in motion in the “fixed” sediment bed layer. It was the first time that a two-phase flow model was applied to the case of scour under a pipeline. At the time, their results were encouraging given the complexity of the phenomenon.

Bakhtiary et al. (2011) [

11] also simulated scour under a pipeline in clear-water conditions with a

$k\u2013\epsilon $ turbulence model. The tunneling stage of scour was correctly reproduced, and the shape of the sediment bed corresponded to the experimental data from Mao (1986) [

2]. Nevertheless, the upstream part of the scour hole seemed to be better predicted than the downstream part.

More recently, Lee et al. (2016) [

12] used a two-phase flow model with a

$k\u2013\epsilon $ turbulence model and the

$\mu \left(I\right)$ rheology for the granular phase [

13]. The authors performed simulations of scour under a pipeline in live-bed conditions. The undisturbed Shields number was higher than in the previous simulations, and no comparison can be made between this model and the other models cited previously. Nevertheless, the time evolution of the scour hole corresponded to the experimental data. They found a strong influence of the turbulence model parameters on the final morphology. Two-phase flow models reached the level of performance of the classical models. Lee et al. (2016) [

12] also pointed out the limitation of the

$k\u2013\epsilon $ model and the necessity to use a

$k\u2013\omega $-type turbulence model for prediction of the final shape of the scour hole downstream of the pipeline.

The revisited

$k\u2013\omega $ turbulence model from Wilcox (2006) [

14] adapted for two-phase flow by Nagel (2019) [

15] (referred to as

$k\u2013\omega $2006 in this paper) can reproduce vortex-shedding phenomenon (see

Appendix A). It should therefore be able to simulate the lee-wake erosion stage.

In

Section 2, the two-phase flow model is presented. The numerical setup is detailed in

Section 2.4, and the results are presented and discussed in

Section 3.

## 4. Discussion

Close to the sediment bed, the cross-diffusion term became negative. Its contribution was incorporated in the

$k\u2013\epsilon $ model, but not in the

$k\u2013\omega $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 (

$\omega $) 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

$\omega $. The cross-diffusion term was always positive, and the

$k\u2013\omega $2006 turbulence model behaved like a

$k\u2013\epsilon $. For boundary layer flows, the peak value of

$\omega $ 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\u2013\omega $2006 model, having the effect of relaminarizing the flow close to the wall. When sediment transport was involved, the peak values of

k and

$\omega $ were offset, so that the cross-diffusion term became negative between the two peaks. Using the

$k\u2013\omega 2006$ 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\u2013\omega $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\u2013\epsilon $ model near the sediment bed, while suppressed far from the bed to behave like the

$k\u2013\omega $2006 model and allow vortex-shedding to develop.

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.

## 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 $\mu \left(I\right)$ 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\u2013\epsilon $ turbulence model.

The turbulence model however had a significant influence on the bed morphology. On the one hand, the $k\u2013\epsilon $ 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\u2013\epsilon $ 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\u2013\omega $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\u2013\epsilon $ and the $k\u2013\omega $2006 revealed the importance of the cross-diffusion term responsible for the behavior of $k\u2013\epsilon $. The negative and positive contribution of the cross-diffusion term were incorporated in the $k\u2013\epsilon $ model, whereas only the positive contribution was incorporated in $k\u2013\omega $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\u2013\omega $2006 behavior in the outer regions and a $k\u2013\epsilon $ 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.