Hervé Guillard

On the behaviour of upwind schemes in the low Mach number limit

This paper presents an asymptotic analysis in power of the Mach number of the flux difference splitting approximation of the compressible Euler equations in the low Mach number limit. We prove that the solutions of the discrete system contain pressure fluctuations of order of the Mach number while the continuous pressure scales with the square of the Mach number. This explains in a rigorous manner why this approximation of the compressible equations fails to compute very subsonic flow. We then show that a preconditioning of the numerical dissipation tensor allows to recover a correct scaling of the pressure. These theoretical results are totally confirmed by numerical experiments.

On the significance of the geometric conservation law for flow computations on moving meshes

The objective of this paper is to establish a firm theoretical basis for the enforcement of discrete geometric conservation laws (D-GCLs) while solving flow problems with moving meshes. The GCL condition governs the geometric parameters of a given numerical solution method, and requires that these be computed so that the numerical procedure reproduces exactly a constant solution. In this paper, we show that this requirement corresponds to a time-accuracy condition. More specifically, we prove that satisfying an appropriate D-GCL is a sufficient condition for a numerical scheme to be at least first-order time-accurate on moving meshes.

Godunov type method on non-structured meshes for three-dimensional moving boundary problems

Abstract This paper presents a numerical method for the computation of compressible flows in domains whose boundaries move in a well defined predictable manner. The method uses the space-time formulation by Godunov while the discretization is conducted on non-structured tetrahedral meshes, using Roes approximate Riemann solver, an implicit time stepping and a MUSCL-type interpolation. The computation of the geometrical parameters required to take into account the movement of the boundaries is described. Examples including the calculation of the flow in the cylinder of an internal combustion engine illustrates the possibilities of the method.

A second order defect correction scheme for unsteady problems

Using the defect correction method (DeC), we propose an implicit scheme that is second order accurate both in time and space, but that uses only first order jacobian. We start a theoretical analysis of the truncation error of the scheme and then perform a linear stability analysis of it. We present some numerical experiments over simple test-cases. Finally, the capability and accuracy of this new scheme is outlined by the analysis of a complex unsteady flow in a 2-D model of a piston engine.

Numerical Investigations of the Tulip Flame Instability—Comparisons with Experimental Results

Abstract A two-dimensional adaptive finite-element code is used to numerically investigate the propagation of a laminar premixed flame in a closed rectangular chamber giving rise to the so-called tulip instability. The physical model includes a single one-step chemical reaction where the physical parameters involved in the model are chosen in order to adequately represent a stoichiometric methane-air flame. Attention is focused on the shape of the flame and the flowfield generated by the combustion process. A detailed comparison between the numerical results and available experimental data shows a very good agreement, for various sizes of the combustion chamber.

Adaptive spectral methods with application to mixing layer computations

This paper reports some experiments on the use of adaptive Chebyshev pseudospectral methods for compressible mixing layer computations. Different functionals measuring the optimality of the polynomial approximation are discussed and compared. In particular, we address the problem of the practical computation of the various functionals. The utility of the self-adaptive method is then demonstrated by some examples from compressible mixing layer calculations.

A Darcy law for the drift velocity in a two-phase flow model

This work deals with the design and numerical approximation of an Eulerian mixture model for the simulation of two-phase dispersed flows. In contrast to the more classical two-fluid or Drift-flux models, the influence of the velocity disequilibrium is taken into account through dissipative second-order terms characterized by a Darcy law for the relative velocity. As a result, the convective part of the model is always unconditionally hyperbolic. We show that this model corresponds to the first-order equilibrium approximation of classical two-fluid models. A finite volume approximation of this system taking advantage of the hyperbolic nature of the convective part of the model and of the particular structural form of the dissipative part is proposed. Numerical applications are presented to assess the capabilities of the model.

Iterative methods with spectral preconditioning for elliptic equations

Abstract We derive two preconditioners for the iterative solution of the linear system arising from Chebyshev approximation of a generalized Helmholtz problem. These preconditioners are constructed as full spectral approximations of a differential problem close in some sense to the original one. The analysis and numerical experiments show the efficiency of these iterative schemes and indicate that they appear as valuable alternative to the usual finite difference or finite element preconditionings.

Some Finite-Element Investigations of Stiff Combustion Problems: Mesh Adaption and Implicit Time-Stepping

The purpose of this paper is to present some preliminary numerical experiments illustrating a global approach to the solution of multidimensional compressible reactive flows. The emphasis is put on solutions to a class of stiff combustion problems, namely thin flame propagation phenomena in medium or low Mach number flows. In combustion problems, stiffness is due both to the presence of disparate characteristic length scales and disparate characteristic time scales.

Multigrid Strategies for CFD Problems on Non-Structured Meshes

For solving a problem on a fine mesh, the multigrid technology requires the definition of coarse levels, coarse grid operators and inter-grid transfer operators. For non-structured 3–D meshes in CFD, two major MG techniques have emerged in the last years. The first one relies on the use of non-nested triangulations while the second technique is associated to finite volume discretization and agglomeration/aggregation techniques. In this paper, we first present some automatic ways to coarsen 3-D meshes and show that these geometrical methods result in efficient multigrid algorithms. Then, we briefly describe the volume agglomeration method and shows an example of its application in a 3-D industrial CFD code.