ED Mathématiques et Informatique
Design and development of innovative numerical methods to solve the 3D Navier-Stokes equations to model hypersonic flows on hybrid meshes
by Vincent DELMAS (IMB - Institut de Mathématiques de Bordeaux)
The defense will take place at 14h00 - Salle de conférence IMB, Bâtiment A33, Université de Bordeaux, 351 cours de la libération, 33405 Talence, France
in front of the jury composed of
- Raphaël LOUBERE - Directeur de recherche - Université de Bordeaux - Directeur de these
- Panagiotis TSOUTSANIS - Professeur des universités - Cranfield University - Rapporteur
- Pierre-Henri MAIRE - Directeur de recherche - CEA DAM - CoDirecteur de these
- Rémi ABGRALL - Professeur des universités - Université de Zürich - Examinateur
- Heloïse BEAUGENDRE - Professeure des universités - Bordeaux INP - Examinateur
- Paola CINNELLA - Professeure des universités - Université de Paris Sorbonne - Examinateur
- Michael DUMBSER - Professeur des universités - Université de Trente - Examinateur
- Phlippe VILLEDIEU - Directeur de recherche - ONERA - Rapporteur
The objective of this PhD thesis is the development of robust and accurate numerical methods for the simulation of three-dimensional hypersonic flows around vehicles with complex geometries. The flow regimes considered correspond to flight altitudes for which the continuum hypothesis is valid, and the fluid behavior is described by the compressible Navier–Stokes equations, expressing the conservation of mass, momentum, and energy. Hypersonic flows, characterized by velocities far exceeding the speed of sound, involve extreme physical phenomena. In particular, strong detached shock waves produce abrupt discontinuities in the flow variables, leading to an intense conversion of kinetic energy into internal energy and to very high temperatures. In addition, a very thin boundary layer develops along the vehicle surface, where strong velocity and temperature gradients normal to the wall generate significant momentum and energy transfers. These effects result in severe aerodynamic and aerothermal loads, whose accurate numerical prediction—especially of wall heat fluxes—remains a major challenge in the design of thermal protection systems for hypersonic and space vehicles. This thesis presents the development of a robust and accurate subface-based, cell-centered finite volume method for solving the three-dimensional compressible Navier–Stokes equations in hypersonic conditions on hybrid unstructured meshes. While unstructured grids are essential for handling complex three-dimensional geometries, they impose stringent requirements on numerical robustness, stability, and accuracy. A major difficulty lies in the simultaneous resolution of strong bow shocks ahead of the vehicle and thin boundary layers along its surface. Sufficient numerical dissipation is required to stabilize shock waves, whereas excessive dissipation degrades the prediction of viscous stresses and heat transfer. One of the central objectives of this work is to balance these conflicting requirements. The first part of the manuscript is devoted to the discretization of the hyperbolic, inviscid part of the governing equations, namely the Euler equations. A novel positivity-preserving, cell-centered finite volume discretization of the multidimensional Euler system is proposed. The method relies on the partitioning of each cell face into subfaces associated with mesh nodes. Subface fluxes are computed using an approximate Riemann solver that depends on the mean states of adjacent cells and on a nodal velocity projected onto the subface normal. Global conservation is enforced by requiring that the sum of subface fluxes around each node vanishes, which uniquely determines the nodal velocities and eliminates classical numerical instabilities such as odd–even decoupling and carbuncle phenomena. The second part of the thesis addresses the discretization of viscous and heat-conducting terms. A multipoint approximation of the viscous stress tensor and heat flux is developed within the same framework. Auxiliary nodal variables are locally eliminated by enforcing flux continuity at mesh vertices, leading to a symmetric positive definite diffusion operator and a natural treatment of boundary conditions. Finally, a hybrid meshing strategy is adopted, combining structured-like elements near the wall for accurate boundary-layer resolution with unstructured elements in the outer flow. The robustness and accuracy of the proposed numerical methods are demonstrated on several three-dimensional hypersonic test cases.