Pierre-Henri Maire

The numerical simulation of hypersonic flows remains a critical area of research, not only for the design of hypersonic flight vehicles but also for the accurate prediction of their aerodynamic and aerothermal loads. The hypersonic regime is characterized by intense shock and rarefaction waves, along with other complex phenomena, making numerical robustness and accuracy particularly challenging.

This work presents the development of a robust and accurate subface-based, cell-centered finite volume (FV) scheme for solving the three-dimensional compressible Navier–Stokes (NS) equations, specifically designed for simulating hypersonic flows around complex 3D vehicles. The use of unstructured grids is essential for such applications, as they simplify the meshing of intricate geometries. However, they also impose stringent requirements on the robustness and accuracy of the numerical methods. Despite decades of advances in numerical algorithms, accurately simulating hypersonic flows remains highly demanding. A key challenge lies in capturing the strong, curved bow shock forming ahead of a vehicle nose: stabilizing this shock requires sufficient numerical dissipation in the inviscid fluxes, while resolving the sharp gradients of temperature and velocity within the boundary layer (BL) demands both refined near-wall grid spacing and consistent discretization of the viscous and conductive fluxes. Moreover, the dissipation introduced to stabilize the shock must be carefully controlled to avoid degrading the prediction of heat transfer within the BL.

We first focus on the hyperbolic part of the NS equations — the Euler equations — and present a novel, positivity-preserving, cell-centered FV discretization of the multidimensional Euler system based on partitioning each cell face into subfaces that meet at the grid nodes [1]. The associated subface flux approximation is derived from an approximate Riemann solver that accounts for both the mean values of the adjacent cells and the velocity of the node from which the subface originates. The projection of the nodal velocity onto the unit normal of the subface serves as a parameter in this solver. Because of this, the computed subface flux is not unique, leading to a non-classical, non-face-based conservation. Conservation is recovered by enforcing that the subface fluxes around each node sum to zero, which uniquely determines the nodal velocities. This multipoint flux approximation eliminates common numerical pathologies of classical face-based FV methods, such as odd–even decoupling and the carbuncle instability.

We then briefly describe the multipoint stress/flux approximation for the viscous and heat-conducting terms of the NS equations, following [2], to complete the FV discretization framework. To meet the computational challenges of hypersonic flows, we employ a hybrid meshing strategy: a structured-like grid (prisms and hexahedra) near the vehicle surface to accurately resolve the BL, combined with an unstructured grid (tetrahedra and pyramids) in the outer domain. The robustness and accuracy of the resulting NS solver are demonstrated on a range of representative test cases, as discussed in [3].

References :
[1] V. Delmas, R. Loubère and P.-H. Maire, A node-conservative cell-centered Finite Volume method for solving multidimensional Euler equations over general unstructured grids, J. Comput Phys., 539, 2025.

[2] P. Jacq, Finite Volume methods on unstructured gris for solving anisotropic heat transfer and compressible Navier-Stokes equations. Ph.D. thesis Bordeaux university 2014.

[3] J. Annaloro et al., Space debris atmospheric entry prediction with spacecraft-oriented tools. 7th European Conference on Space Debris, 7, ESA Space Debris Office, 2017.