Implicit unified gas-kinetic scheme for steady state solution of hypersonic thermodynamic non-equilibrium flows

The unified gas-kinetic scheme (UGKS), designed to solve Boltzmann equation and its model equations, aims to accurately resolve multi-scale flow problems within a single computation framework. In this study, an implicit UGKS is proposed for the steady state solution of diatomic gas flows based on a multiple-temperature Boltzmann model equation with non-equilibrium among translational, rotational and vibrational modes (Zhang et al., 2023), utilizing a novel hybrid Cartesian-unstructured discrete velocity space (DVS) mesh approach. The multiple-temperature macroscopic equations corresponding to the Boltzmann model equation are simultaneously solved implicitly to address the stiffness and nonlinearity of the collision operator encountered when solving the Boltzmann model equation in isolation. Both macroscopic equations and Boltzmann model equation are solved by using the point relaxation symmetric Gauss–Seidel method to achieve a fast convergence rate. The efficiency of the implicit UGKS can be improved by about two orders of magnitude compared to the explicit one. Moreover, the hybrid Cartesian-unstructured DVS achieves a noteworthy reduction in both CPU-hours and memory consumption, requiring only 15%$\sim$23% of the advanced unstructured DVS. It effectively alleviates the dimensional crisis of UGKS in solving three-dimensional hypersonic flows, and can be extended to other DVS-based methods. As a result, UGKS extends its applicable regime to simulations of hypersonic thermodynamic non-equilibrium flows with Mach numbers up to 30, achieving three-dimensional flow simulations of complex geometrical configurations with relatively low computational resources. A series of test cases, including normal shock structures, two-dimensional hypersonic flows around cylinder, blunt wedge and flat plate, and three-dimensional hypersonic flows around a sphere and an X38-like configuration vehicle, are conducted to validate the implicit UGKS with hybrid Cartesian-unstructured DVS. Overall, the proposed method shows advantages of unified multi-scale methods in terms of computational efficiency and capturing multi-scale solution of hypersonic thermodynamic non-equilibrium flows.