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## Erosion of granular soils and cohesive soils

The numerical modelling of erosion has been the subject of many studies over the past twenty years. Two erosion modelling methods have been validated for laminar flows on granular beds. The first empirical formula was proposed by [Vardoulakis et al. 1996]. It introduced a third fluidised solid phase between the fluid and solid phases. This phase is resolved by Darcy-Brinkman equations. The erosion of the solid phase is described by a source term. This term induces exchanges of mass between the soil and fluid phases in the mass conservation equations. The second empirical formula was proposed by [Ouriemi et al. 2009]. The solid and fluid phases interact through quantity of movement exchanges. The equations of biphasic media developed by [Jackson 2000] form the basis of the transport equations used in this model. These two methods and their adaptability to our configurations are discussed in paragraph 1.2.2.

In the framework of modelling the erosion of a fine cohesive soil by a turbulent flow, the solid/fluid interface can be considered singular, and not as a third fluidised solid phase. Each phase is biphasic: a compact assembly of particles containing water and water containing particles in dispersed phase. The difficulty therefore lies in representing the mobile interface and precisely calculating the mechanical quantities on the interface. Two approaches can be distinguished in the framework of the numerical simulation of flows in the presence of interfaces: capturing and monitoring the interface. The first, called the Eulerian approach, consists in defining the media (water-soil) in a given domain (fixed mesh) and determining its evolution. The second, a mixed Eulerian-Lagrangian approach, consists in displacing the frontier through time and adapting the mesh.

**Different approaches to model interfaces**

Modelling the interface separating a fluid and a solid is highly complex, since each medium is usually described through very different approaches. Fluid is more naturally described using Eulerian models and the soil by Lagrangian models. The mixed Eulerian-Lagrangian approaches introduced by [Donea et al. 1982] achieve a good compromise between the two descriptions. The two media are defined within a mobile mesh while the Eulerian and Lagrangian resolutions of the solid and fluid behaviours, respectively, are independent. The Eulerian equations of the fluid are first calculated independently of the Lagrangian model of the solid. The equations governing the behaviour of the solid are resolved on the basis of the results found for the fluid. The meshing of the fluid zone is deformed in correspondence with the results obtained for the solid. The equations of the fluid are calculated within the updated mesh, and so forth. The mixed Eulerian-Lagrangian methods ensure great simplicity in the formulation of the equations governing the behaviours of the fluid and the solid and they lead to considerable gains in calculation time. Furthermore, the advantage of the method is that it provides a fine description of the flow variables at the interface. Its disadvantage is that it is limited by the distortion of the mesh, making it necessary to consider remeshing methods. Remeshing can introduce major problems, especially in 3D geometry, and often leads to a huge increase in calculation time.

Completely Eulerian methods with immersed boundaries have also been developed [Angot 2005, Peskin 1977]. The equations of the solid are written so that they resemble the fluid equations as much as possible. The main characteristic of these methods is that there are no fixed boundaries between domains, which often introduces complex meshing. The flow presenting an immersed obstacle can, for example, be calculated with a fixed Cartesian mesh independent of changes in the shape of the object. A VOF (Volume of Fluid) [DeBar 1974] or Level-Set [Osher and Sethian 1981] method can be used to localise the different domains and describe their evolution. The advantage of this method compared to mixed Euler-Lagrange models is that it requires considerably less remeshing. Although the meshing of the zones close to the interface must be refined, the extent and frequency of the remeshing required for this method are not so great. The major disadvantage of this method, however, is that it is difficult to obtain a fine resolution of the flow variables at the interface. Interpolations are required to ensure the representativeness of the position of the portion of the interface within the mesh. [Lachouette et al. 2008] developed a numerical erosion model based on this method. It permits resolving a viscous incompressible laminar flow with erosion in 2 and 3-D geometry. The interface is represented by the fictitious domains method and its evolution is described by the Level-Set method within a fixed Cartesian grid (cf. paragraph 1.2.3). The method is validated in the case of piping erosion in laminar regime. The results of numerical modelling of the HET for a laminar flow agree well with those obtained with the analytical model of [Bonelli et al. 2006].

**Single phase modelling and slow erosion kinetics**

This work follows on from the research works of [Lachouette et al. 2008]. The originality of the modelling method developed by [Lachouette et al. 2008] is that only two domains, fluid and solid, are considered. They are separated by a singular interface and not by a 3rd fluidised solid domain. Each phase is biphasic. The soil is a compact assembly of grains containing water. The flow contains grains in dispersed phase. The concentrations in minority phases are inversely proportional to the distance to the interface. The hypothesis of diluted flow permits neglecting from the model the presence of particles in the flow.

The configuration of the erosion tests (JET and HET) are such that the hypothesis of a diluted flow can be applied. This implies that the mass of eroded particles in the flow must be less than 1% (up to 10% in the literature). In all the cases that we have treated experimentally with JETs and HETs, the concentrations measured were close to one per thousand. The hypothesis of a soil with very low permeability permits omitting the influence of water particles in the solid phase. The solid phase will be modelled with characteristic parameters of the displacement at the interface governed by the erosion law used. The hypothesis of slow erosion kinetics as regards flow velocity permits considering a stationary flow. The equations governing the fluid and the interface are uncoupled: the fluid is stationary with regard to the equations of the interface and the interface is immobile for the solution of the flow.

**Resolution by DNS or LES**

Direct numerical simulations (DNS) or large scale simulations (LES) can lead to better knowledge of the interaction mechanisms between the fluid and the soil, and thus of the process involved in erosion. However, using such solution methods cannot be considered for our application for two reasons. The first concerns the Reynolds numbers of the flows and the dimensions of the JET and the HET. High Reynolds numbers and a large calculation domain would lead, at each time step, to very long calculation times. Therefore an entire erosion study cannot be performed with such a model. The second concerns the modelling of the particle transport. With a model as fine as that provided by a DNS or LES, it is necessary to model the entire process of erosion and grain transport. Besides the fact that the modes of erosion of a cohesive soil remain an open question, DNS and LES use Stokes time steps to model particle transport. This means that the diameter and density of the particles cannot be dissociated.

**Choice of RANS models**

It is necessary to choose a simplified flow solution model whose pertinence and calculation time are pertinent and proven. The RANS (Reynolds Average Navier Stokes) method has all the qualities to permit the modelling of a cohesive soil by a turbulent flow. In conformity with the RANS method, the non stationary flow is converted into a stationary one with the main fluctuations averaged statistically. Thus the Reynolds stresses introduce fluctuations of velocity that cannot be obtained by the RANS method. This introduces the closing problem of RANS equations. This closure problem caused by averaging must be resolved by a turbulence model. A direct solution can be considered by transport equations or eddy viscosity models which introduce a turbulent viscosity. Direct solution by transport equations can be done with an RSM (Reynolds Stress Model) turbulence model. The introduction of a turbulent viscosity concerns in particular two equation models, such as the k − ε and k − ω turbulence models.

**Geometric singularity induced by the erosion law**

According to the standard erosion law, Eq. (1.1), the displacement of a point of the interface depends only on the shear stress exerted by the fluid on the soil at this point. In the case of a normal flow at the soil surface, the mean shear stress is null at the stagnation point. It then increases up to its maximum and decreases again by moving further from the stagnation region, as shown in In the case of erosion by a turbulent jet flow, this singularity is not observed experimentally. On the contrary, a symmetrical scour with a maximum scour depth at the jet centreline is observed, as can be seen in Figure 2.2.

#### Smoothing of the non eroded soil peak

Three hypotheses can be considered to explain the smoothing of this theoretical peak of non eroded soil.

The first is related to the non-homogeneities of a real soil. The occurrence of a soil peak like that shown in Figure 2.1 is only possible for very cohesive and very fine soils. Otherwise, the instability of such a singularity and the presence of coarser particles would cause collapse and very rapid smoothing of the singularity.

The second hypothesis is related to the governing variables of erosion. In law (1.1), only the influence of the shear stress is considered. It is nonetheless possible that erosion is governed by other variables of the flow (cf. paragraph 5.2). If these variables are non null at the stagnation point, the singularity caused by the law (1.1) will no longer appear.

The third hypothesis is to take into account that the shear stress considered in the erosion law is averaged. The fluctuations of instantaneous values due to the turbulence at the jet stagnation region, and a weak pulse of the jet linked to large scale turbulent structures in 3-D geometry may explain the smoothing of the non eroded soil peak.

The literature contains different elements that can be used to take this path of investigation further. The study performed by [Geers et al. 2006] on jets impinging a flat plate show that in the case of laminar flows, the stagnation point of the jet is fixed. It is also fixed for turbulent flows, in the case where the distance separating the nozzle from the plane is shorter than the potential core. Otherwise, fluctuations of the position of the stagnation point are observed. [Hadziabdic and Hanjalic 2008] performed LES numerical simulations to validate the experimental observations of [Geers et al. 2006], cf. Figure 2.3 and Figure 2.4. The oscillation of the stagnation point is attributed to the pulse of the jet or to a precession movement of the jet. These instabilities are due to the large scale of the turbulence.

**Table of contents :**

**Chapter 1. State of the art **

1.1 Erosion in hydraulic structures

1.1.1 Context

1.1.1.1 Erosion at the scale of the structure

1.1.1.2 Estimation of soil erodibility

1.1.1.3 Erosion parameters

1.1.2 HET and JET erosion tests

1.1.2.1 The Hole Erosion Test

1.1.2.2 The Jet Erosion Test

1.1.3 Erosion laws

1.1.3.1 Rate of soil removal

1.1.3.2 Determination of critical shear stress

1.1.3.3 Correlation of erosion coefficient and critical shear stress

1.2 Numerical modelling of erosion

1.2.1 Context

1.2.1.1 Erosion of granular soils and cohesive soils

1.2.1.2 Different approaches to model interfaces

1.2.2 Biphasic and triphasic models

1.2.2.1 The approach of Papamichos and Vardoulakis (2005)

1.2.2.2 The approach of Ouriemi et al. 2009

1.2.3 Singular interface

1.3 Conclusions on the state of the art

**Chapter 2. Modelling method **

2.1 Hypotheses

2.1.1 Single phase modelling and slow erosion kinetics

2.1.2 Analysis of orders of magnitude

2.2 Flow modelling

2.2.1 RANS modelling and the closure problem

2.2.1.1 Navier-Stokes Equations

2.2.1.2 Resolution by DNS or LES

2.2.1.3 Choice of RANS models

2.2.2 Turbulence models

2.2.2.1 Eddy viscosity models

2.2.2.2 Reynolds Stress Model

2.3 Erosion modelling

2.3.1 Classical erosion law

2.3.1.1 Definition of the eroded mass flux

2.3.1.2 Shear stress

2.3.2 The standard erosion law adapted to impinging jets

2.3.2.1 Geometric singularity induced by the erosion law

2.3.2.2 Smoothing of the non eroded soil peak

2.3.2.3 Adaptation of the erosion law

2.4 Numerical model

2.4.1 Global numerical method

2.4.2 Flow discretization

2.4.2.1 Solution of Navier-Stokes equations

2.4.2.2 Wall laws

2.4.2.3 Taking roughness into account

2.4.3 Updating the position of the interface

2.4.3.1 Interface displacement code

2.4.3.2 Remeshing

2.5 Conclusions on the modelling method

**Chapter 3. Results obtained on impinging flows **

3.1 Independency of results regarding mesh density and turbulence models

3.1.1 Independency of results in relation to the meshing

3.1.2 Influence of the turbulence model

3.2 Erosion modelling

3.2.1 Comparison of results of the semi-empirical model

3.2.2 Study of the sensitivity of the model to erosion parameters

3.2.3 Discussion

3.3 Validation of the JET interpretation model

3.3.1 Characterization of the soils tested

3.3.2 JET modelling results

3.3.3 Discussion

3.4 Conclusions on the application to jet flows

**Chapter 4. Results obtained on tangential flows **

4.1 Validation of the numerical model in a 2D Poiseuille flow configuration

4.1.1 Theoretical solution

4.1.2 Numerical results

4.2 Concentrated leak erosion in turbulent flow

4.2.1 Independence from meshing and turbulence models

4.2.1.1 Independence of results from mesh density

4.2.1.2 Influence of the turbulence model

4.2.2 Results with erosion

4.2.3 Study of the model’s sensitivity to erosion parameters

4.2.4 Discussion

4.3 Modelling the HETs

4.3.1 Characterisation of the soils modelled

4.3.2 HET modelling results

4.3.3 Discussion

4.4 Conclusions of the application to piping flows

**Chapter 5. Study of the erosion law **

5.1 Differences between JET and HET for erosion parameters

5.1.1 Experimental and literature data

5.1.2 Dispersion of results

5.1.3 Influence of flow parameters

5.2 Variables susceptible to influence erosion

5.2.1 Possible explanations for JET and HET differences?

5.2.2 Flow signature

5.2.2.1 Stresses and forces exerted by the flow on the plane

5.2.2.2 Pressure gradient

5.2.2.3 Turbulence variables

5.2.3 Flow variables susceptible to influence erosion

5.3 Paths for developing the erosion law

5.3.1 Flow variables of the JET and HETs

5.3.2 Taking fluctuations into account in the stagnation region

5.3.3 Taking into account the pressure gradient in the erosion law

5.4 Conclusions on the study of the erosion law

Conclusion

Outlook for further research

**References**