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Improvement of the flow model
Some works aim at obtaining an expression for the shifted shear stress without « manually » imposing the shift length. They do so by using an improved description of the water flow. An inherent limitation of the Shallow Water equations is that they poorly account for what happens in the water near the surface of the sediment layer. As the Shallow Water equations are depth-integrated equations, they do not make any distinction between the boundary layer, where the water velocity changes quickly, and the rest of the fluid. Yet interaction between the fluid and the sediment layer happens at the surface of the sediment layer. Refining the model in the water layer to better describe the zone near the sediment surface is then an interesting option. The Triple Deck model [102, 105, 132] is one of the models aiming at providing such a better description. As the name indicates, the water layer is divided in three « decks ». In the « lower deck », a viscous problem is solved. The perturbations in the « lower deck » act on the « main deck » as a perturbation of the stream lines. The deflection of the stream lines is transmitted to the « upper deck », which is an ideal fluid layer. In the « upper deck », a pressure disturbance is created, and this pressure disturbance is transmitted back to the lower deck, which promotes the velocity perturbations. The Triple Deck theory is a more refined model than the Shallow Water equations, but it avoids resolving the full Navier-Stokes equations.
In [87], the water flow above an erodible bump is resolved using the Triple Deck theory. First, a linearized steady triple deck problem is solved. Then, a mass transport equation giving the solid flux is solved, and finally, the evolution of the topography is computed. The stability of the erodible bed is investigated both numerically and analytically; the results are in good agreement. Dune growth is obtained from initially small perturbations of the surface of the erodible material. A phase lag between the top of the sediment bump and the maximum shear stress is observed. Though the modeling simplifications are many, the importance of a good description of the boundary layer is clearly shown.
Another flow description was recently proposed in [81]. The water model consists in a layer of ideal fluid on top of a layer of viscous fluid. The two layers interact strongly with one another. Again, a phase lag between the shear stress and the topography is achieved. The authors of [81] compare the numerical results obtained with their model to numerical results obtained with the multilayer Saint-Venant model [22] and they report that the phase lag can also be achieved with the multilayer model. In [61], a modified water flow description is used as well, and as before, the phase shift is obtained, not imposed. A non-local solid flux with an integral term is obtained. Linearizing the equations even gives an explicit expression for the shear stress, which features a non-local (integral) term = fsv2 1 b + Z +1 0 1=3@xb(x ; t)d .
Non-local models for other applications
We present here a few non-local models, used for other applications, that present some similarities with the non-local model (1). The behavior of the solution of (1) is probably very different from the solution of the non-local models below. Our point here is that the research articles on these non-local models highlight the difficulty of analyzing them and of proposing robust numerical schemes to solve them. There exist many non-local models. They are developed for a large variety of applications: chemotaxis, traffic flow, plasma physics…
In plasma physics, a non-local electron conduction model is used for the simulation of laser-driven Inertial Confinement Fusion experiments. While a local theory, the Spiter- Härm theory, was formerly used, some experiments have evidenced that the electron heat flow is non-local. This flux depends not only on the local conditions, but also on the portion of the temperature profile enclosed in a few hundreds of mean free paths. The heat flux is therefore expressed as Q(t; x) = Z R3 W(x; x0)QSH(t; x0)dx0.
Numerical resolution of the Shallow Water-Exner system
In the present thesis, a numerical method for the resolution of the system (1) coupled to the Shallow Water equations is proposed. A finite-volume discretization is used. We propose here a short overview of the existing finite-volume schemes for the resolution of the Shallow Water-Exner system. The numerical treatment of the Shallow Water-Exner system is discussed only for classical, local solid flux formulae. Two main approaches exist for solving the system. The first approach is said to be « uncoupled ». The equations for the water layer and those for the sediment layer are solved separately. This approach is justified by the fact that in many cases, the water layer and the sediment bed evolve at very different time scales. The bed evolution is typically much slower, meaning that for instance, one could take a larger time step in the sediment layer than in the water layer. The hydrodynamic and morphodynamic unknowns are exchanged at some specific time instants only. This approach is adopted in the industrial codes [121, 123, 127]. However, when the characteristic time scales for the two layers are closed, the uncoupled approach is not suitable and stability issues appear. More specifically, the uncoupled approach cannot deal with supercritical flows [45].
In the « coupled » approach, the complete Shallow Water-Exner system is solved. A possible strategy is to approximate the eigenvalues of the Jacobian matrix of the full system. The computation of the eigenvalues is relatively easy when the solid flux is given by the law (1.7), but this is not the general case. This is the technique used in [94]. In [78], the authors use a flux-limited version of Roe’s scheme to solve several formulations of the Shallow Water-Exner system. An approach based on the Roe scheme is adopted in [28] to propose a 1D and 2D scheme for unstructured grids. In [25], a relaxation approach is proposed. The water pressure and the sediment flux are relaxed; the computation of the eigenvalues is easy.
Finally, let us mention an intermediate approach introduced in [24]. A three-wave approximate Riemann solver for the Shallow Water-Exner system is proposed. The hydraulic and morphodynamic intermediate states are computed in a decoupled way, but the wave velocities of the full system are evaluated – they are approximate values of the eigenvalues of the Jacobian matrix of the Shallow Water-Exner system.
Other numerical schemes
In this section, we present and investigate the behavior of other numerical schemes for the sediment layer. The rationale for the design of different numerical schemes is the following:
Even if the main objective of this work is to illustrate the influence of the viscosity operator on the behavior of the sediment layer, in practice, the flow in the nonlocal model should not be very different from the flow in the local model. The local model can be reformulated as a nonlinear convection-diffusion equation. It is then a natural idea to use this structure to design a scheme for the local model, and then try to extend this scheme for the non-local model. The scheme presented in §2.2.2 does not rely on the features of the local model.
In the scheme (2.17), (2.18), the link between the continuity equation and the equation on the velocity is poorly exploited. Indeed, the continuity equation is discretized regardless of the nature of the velocity – the fact that the velocity v is itself a function of the sediment depth b is not used.
Therefore, we present here numerical schemes relying on the properties of the local model. For reasons that will be explained below, these numerical schemes are not satisfactory. We merely report the attempts that we have made. A scheme (2.50) for the local model is first presented. It is then immediately extended into a scheme (2.53) for the non-local model. A dissipative entropy balance is proved for the scheme for the non-local model. The same proof can be used to show that the scheme for the local model satisfies a dissipative entropy balance.
The layer-averaged Euler system with variable density
In order to describe and simulate complex flows where the velocity field cannot be approximated by its vertical mean, multilayer models have been developed [14, 19, 20, 34, 40, 41]. Unfortunately these models are physically relevant for non miscible fluids. In [21, 22, 57, 116], some authors have proposed a simpler and more general formulation for multilayer model with mass exchanges between the layers. The obtained model has the form of a conservation law with source terms. The layer-averaged approximation of the 3d Navier-Stokes system with constant density is studied in [6]. Compared to the constant density case, when considering the density variations, additional source terms appear, see remark 12. Notice that in [21] the hydrostatic Navier-Stokes equations with variable density is tackled but only in the 2d context.
With respect to commonly used Euler or Navier-Stokes approximations, the appealing features of the proposed multilayer approach are the easy handling of the free surface, which does not require moving meshes (e.g. [47]), and the possibility to take advantage of robust and accurate numerical techniques developed in extensive amount for classical one-layer Saint-Venant equations.
We consider a discretization of the fluid domain by layers (see Fig. 3.2) where the layer contains the points of coordinates (x; y; z) with z 2 L(t; x; y) = (z1=2; z+1=2) and fz+1=2g=1;:::;N is defined by z+1=2(t; x; y) = zb(x; y) + P j=1 hj(t; x; y); 2 [0; : : : ;N].
Table of contents :
Abstract
Contents
List of Figures
Introduction
Context
Sediment transport
Variable density flows
Contributions
Modeling and simulation of sediment transport
Modeling and simulation of variable density flows
Outline of the conclusion and perspectives
Modeling and simulation of sediment transport
Modeling and simulation of variable density flows
Common perspectives
I Sediment transport
1 Existing models based on the Shallow Water equations
1.1 The Shallow Water equations
1.2 The Exner equation
1.2.1 The transport threshold
1.2.2 Some bed load transport formulae
1.2.3 Critics made to the bed load transport formulae
1.3 Possible improvements to the Exner model
1.3.1 Necessity of a phase shift
1.3.2 Improvement of the flow model
1.3.3 Improved description of the sediment layer
1.3.4 Non-local models for other applications
1.4 Numerical resolution of the Shallow Water-Exner system
2 A non-local sediment transport model
2.1 Overview of the water-sediment system
2.1.1 Bilayer Navier-Stokes equations
2.1.2 Introduction of a threshold for the onset of motion
2.2 The sediment layer integrated model
2.2.1 Vertically averaged models
2.2.2 Numerical scheme
2.2.3 Numerical validation
2.3 Coupled water and sediment system
2.3.1 Modeling of the coupled system
2.3.2 Numerical strategy for the coupled system
2.3.3 Numerical results for the coupled system
2.4 Other numerical schemes
2.4.1 A scheme for the local model
2.4.2 Extension for the non-local model
2.5 Conclusions and perspectives
List of main symbols used in Chapter 2
II The Navier-Stokes system with temperature and salinity for free-surface flows
3 Low-Mach approximation & layer-averaged formulation
3.1 Introduction
3.2 The 3d Navier-Stokes-Fourier system
3.2.1 The compressible Navier-Stokes-Fourier system
3.2.2 Boundary conditions
3.2.3 The incompressible limit
3.2.4 The Navier-Stokes-Fourier system with salinity
3.2.5 The Euler-Fourier system
3.2.6 The hydrostatic assumption
3.2.7 The Boussinesq assumption
3.3 The layer-averaged models
3.3.1 The layer-averaged Euler system with variable density
3.3.2 The layer-averaged Navier-Stokes-Fourier system
3.4 Conclusion
Acknowledgments
4 Numerical scheme and validation
4.1 Introduction
4.2 The layer-averaged models
4.2.1 The multilayer Navier-Stokes-Fourier model
4.2.2 The layer-averaged Euler-Fourier system
4.3 Numerical scheme for the layer-averaged Euler-Fourier system
4.3.1 Strategy for the time discretization
4.3.2 Semi-discrete (in time) scheme
4.3.3 Finite volume formalism for the Euler part
4.3.4 Kinetic fluxes
4.3.5 Discrete entropy inequality
4.4 Numerical scheme for the layer-averaged Navier-Stokes-Fourier system .
4.4.1 Semi-discrete (in time) scheme
4.4.2 Spatial discretization of the diffusion terms
4.5 Numerical validation
4.5.1 Analytic solution
4.5.2 Lock exchange
4.5.3 Diffusion
4.6 Conclusion
Acknowledgments
List of main symbols in Chapters 3 and 4
Bibliography
