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Simulations of hypersonic reentry flows remain a challenging task when it comes to developing robust yet accurate numerical methods. The presence of shock waves and the stiffness of the governing equations regarding thermochemistry are very demanding numerically. Furthermore, the different length and time scales of associated physical phenomena bring the need for efficient methods to make computational costs affordable. In particular, Finite Volume methods have benefited from years of development and form the basis of most recent numerical codes when it comes to predicting flows in the continuum regime. These methods rely on a spatial discretization that influences the accuracy of the numerical solution. Recent advances in mesh adaptation for aeronautical flows are now being adopted in the hypersonic flow community.
The following work proposes to extend mesh adaptation to steady laminar hypersonic reactive flows. The use of unstructured meshes allows us to consider complex geometries while capturing anisotropic flow features such as shock waves. The anisotropic mesh adaptation follows an automatic procedure that intends to reach mesh convergence based on user-input tolerances.
The starting point of this work is the MUSCL-V4 scheme, a robust spatial discretization scheme built on top of the dual of the mesh. The gradient reconstruction makes use of the finite element shape functions that allow handling highly anisotropic meshes while conserving second-order accuracy in space. Time integration is done using a backward Euler method followed by a linearization of the implicit terms. The evaluation of the jacobians matrix stemming from the linearization is a crucial step in the elaboration of robust methods for steady-state applications. The present work investigates an exact linearization of the convective terms as well as the chemical source term. Then, the resulting linear system is solved using a Symmetric Gauss-Seidel solver.
The mesh adaptation is performed following the Feature-Based framework. Hence, the hessian matrix of a given sensor field serves as input to define the error estimates, which provides a metric field to the remesher in order to adapt the mesh. Within an inner loop, several mesh adaptations are carried out for a given number of nodes, with the aim of converging the mesh-solution couple. The outer loop of the mesh adaptation procedure increases the number of nodes in the domain at each iteration to automatically achieve mesh convergence.
First results are obtained on two-dimensional simple configurations to assess the behavior of the mesh adaptation procedure in the context of hypersonic reactive flows. It is observed that solving the system of equations in a coupled way is mandatory to ensure the robustness of the approach on highly anisotropic meshes. As compared to the calorically perfect gas model, more iterations are needed, but the numerical solving benefits from an already converged solution obtained on the previously adapted meshes.
This work is part of an effort to extend the approach to Goal-Oriented mesh adaptation that accounts for the sensitivity of an output functional (wall heat flux). Thus, diffusive effects will be considered in further developments.