Didier VIGNERON

Numerical simulation of weakly compressible flows in deforming domains

Abstract

This thesis is dedicated to the design and the analysis of numerical schemes for the simulation of three-dimensional weakly compressible fluid flows with moving bound- aries. For treating this kind of problems in the framework of a cell-centered finite volume method, three main themes are addressed: the discretization schemes on moving meshes, the preconditioning techniques for weakly compressible flows and a multi-block mesh strategy for adapting the mesh to the domain deformation. For the discretization of the Navier-Stokes equations, the primitive variables and their gradients must be reconstructed to the cell faces to perform the flux integration. For the advective fluxes, the variables are commonly reconstructed from the cell centroids with a second-, or even, a third-order Taylor expansion. For the viscous fluxes, the gradients are often reconstructed to first-order only, which yields a inconsistent discretization on irregular meshes. We propose a second-order gradient reconstruction by means of a least- square technique applied at the cell faces. We then extend this face-based strategy to the variables so as to perform both the variable and gradient reconstructions together in a single procedure. On moving meshes, the Arbitrary Lagrangian-Eulerian term is treated in a way that is consistent with the Discrete Geometric Conservation Law. A preconditioning technique that modifies the eigenvalues and the eigenvector struc- ture of the system is applied to ensure the convergence of the parallel pseudo-transient Newton-GMRES iterations at low Mach numbers. Nevertheless, we show that it is not always possible to optimize both the eigenvalues and the eigenvectors. Contrary to some existing methods, we suggest to rather optimize the eigenvectors, which yields a better convergence rate and permits to avoid the use of user-defined parameters. The analy- sis of the preconditioning techniques is extended to low cell Reynolds numbers and to high cell Strouhal numbers. The scaling of the pressure and velocity dissipation terms of the AUSM + -up scheme is modified to preserve the accuracy in the incompressible limit. We also show that these two terms should be scaled independently for high cell Strouhal numbers. Some 3D compressible manufactured solutions are designed to measure the accuracy. Finally, preconditioned non-reflecting boundary conditions are proposed for weakly compressible flows. For moving boundary flows, the mesh deformation is computed through a Laplacian equation with a sliding condition for some of the boundary vertices. With the aim to avoid any re-meshing step, we demonstrate the feasibility of a multi-block strategy that consists in partitioning the mesh into blocks that can be deformed independently. The treatment of the interface between two 3D unstructured mesh blocks sliding on each other is done by means of an algorithm that defines a new common surface mesh for the possibly non- planar interface. This algorithm is also able to automatically identify boundary walls. The different methods designed in this thesis are based on rigorous analytic develop- ments and their efficiency is demonstrated in terms of accuracy and robustness on several targeted test-cases.

Jury members

• B. DEWALS, University of Liège, President 
• V. TERRAPON, University of Liège, Advisor 
• G. DIMITRIADIS, University of Liège 
• K. HILLEWAERT, University of Liège 
• Mme I. LEPOT, CENAERO
• R. PECNIK, TU Delft (The Netherlands)

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