Papers by François Renaud
Shock Waves, 2003
This paper is devoted to the modeling and numerical resolution of a non-dissipative compressible ... more This paper is devoted to the modeling and numerical resolution of a non-dissipative compressible turbulent plasma flow model involving three temperatures (turbulence, ions and electrons). The first step is to derive such a model. To do this, an analysis of the Reynolds averaged Euler equations (the k-model) is carried out. It is shown that thermodynamic requirements enable the derivation of an equation of state for turbulent variables. This equation of state is of the same type as those of an ideal gas. In this context, the various thermodynamic variables of turbulence can be obtained (energy, pressure, temperature etc.). This hyperbolic conservative model has exactly the same structure as the two temperatures plasma model of Zel'dovich. Thanks to the clear structure of these two models, the turbulent plasma model is derived and involves three temperatures. The second step is to derive an accurate numerical scheme for its solution. A linearized Riemann solver and a positive HLLC type solver are derived and embedded into a conventional Godunov scheme. It is shown that this method requires important corrections to preserve contact discontinuities and temperatures monotonicity. The corrections are based upon a non-conservative formulation of the turbulence and electrons energy equations, while total energy conservation is preserved. The modified method behaves correctly with contacts and shocks.
Journal of Fluid Mechanics, 2003
The aim of this paper is the derivation of a multiphase model of compressible fluids. Each fluid ... more The aim of this paper is the derivation of a multiphase model of compressible fluids. Each fluid has a different average translational velocity, density, pressure, internal energy as well as the energies related to rotation and vibration. The main difficulty is the description of these various translational, rotational and vibrational motions in the context of a one-dimensional model. The second difficulty is the determination of closure relations for such a system: the 'drag' force between inviscid fluids, pressure relaxation rate, vibration and rotation creation rates, etc. The rotation creation rate is particularly important for turbulent flows with shock waves. In order to derive the one-dimensional multiphase model, two different approaches are used. The first one is based on the Hamilton principle. This method gives thermodynamically consistent equations with a clear mathematical structure, coupling the various motions: translation, rotation and vibration. However the relaxation effects have to be added phenomenologically. In order to achieve the closure of the system and its numerical resolution we use the second approach, in which the pure fluid equations are discretized at the microscopic level and then averaged. In this context, the flow is considered to be the annular flow of two turbulent fluids. We also derive the continuous limit of this model which provides explicit formulae for the closure laws. The structure of this system of partial differential equations is the same as the one obtained by the Hamilton principle. The final issue is to determine the rate of energy (or entropy) rotation. We assume that all the entropy creation related to the various relaxation effects after the passage of the shock wave is converted to rotational motion. The one-dimensional model is validated by comparing its predictions with averaged two-dimensional direct numerical results. The problem on which this model is tested is the interaction of a shock wave propagating in a heavy gas with a light gas bubble. The results obtained by the one-dimensional multiphase model are in a very good agreement with the two-dimensional averaged results.
International Journal for Numerical Methods in Fluids, 2007
ABSTRACT This paper is devoted to the numerical approximation of a hyperbolic non-equilibrium mul... more ABSTRACT This paper is devoted to the numerical approximation of a hyperbolic non-equilibrium multiphase flow model with different velocities on moving meshes. Such goal poses several difficulties. The presence of different flow velocities in conjunction with cell velocities poses difficulties for upwinding fluxes. Another issue is related to the presence of non-conservative terms. To solve these difficulties, the discrete equations method (J. Comput. Phys. 2003; 186(2):361–396; J. Fluid. Mech. 2003; 495:283–321; J. Comput. Phys. 2004; 196:490–538; J. Comput. Phys. 2005; 205:567–610) is employed and generalized to the context of moving cells. The complementary conservation laws, available for the mixture, are used to determine the velocities of the cells boundaries. With these extensions, an accurate and robust multiphase flow method on moving meshes is obtained and validated over several test problems with exact or experimental solutions. Copyright © 2007 John Wiley & Sons, Ltd.
Shock Waves, 2003
This paper is devoted to the modeling and numerical resolution of a non-dissipative compressible ... more This paper is devoted to the modeling and numerical resolution of a non-dissipative compressible turbulent plasma flow model involving three temperatures (turbulence, ions and electrons). The first step is to derive such a model. To do this, an analysis of the Reynolds averaged Euler equations (the k-model) is carried out. It is shown that thermodynamic requirements enable the derivation of an equation of state for turbulent variables. This equation of state is of the same type as those of an ideal gas. In this context, the various thermodynamic variables of turbulence can be obtained (energy, pressure, temperature etc.). This hyperbolic conservative model has exactly the same structure as the two temperatures plasma model of Zel'dovich. Thanks to the clear structure of these two models, the turbulent plasma model is derived and involves three temperatures. The second step is to derive an accurate numerical scheme for its solution. A linearized Riemann solver and a positive HLLC type solver are derived and embedded into a conventional Godunov scheme. It is shown that this method requires important corrections to preserve contact discontinuities and temperatures monotonicity. The corrections are based upon a non-conservative formulation of the turbulence and electrons energy equations, while total energy conservation is preserved. The modified method behaves correctly with contacts and shocks.
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Papers by François Renaud