An implicit gas-kinetic scheme for internal and external flows

The gas-kinetic scheme(GKS) is a promising computational fluid dynamics (CFD) method for solving the Navier-Stokes equations. It is based on the analytical solution of the BGK equation, which enables accurate and robust simulations. While GKS has demonstrated excellent properties (e.g., unified treatment of inviscid and viscous fluxes, inherent adaptive dissipation control), its application to classical engineering problems, such as aerodynamic flows and fluid machinery, remains underdeveloped compared to conventional CFD methods. This study bridges this gap by advancing GKS capabilities for real-world engineering challenges. First, the GKS is extended to a rotating coordinate frame, enabling efficient simulations of internal flows in turbomachinery. Second, the computational inefficiency of explicit GKS is addressed through an implicit time discretization using the generalized minimal residual method. The Jacobian matrices for inviscid/viscous fluxes are approximated using the first-order kinetic flux vector splitting scheme and the thin shear layer approximation to enhance robustness and computational efficiency further. Third, the shear-stress transport turbulence model is coupled to expand GKS's applicability to industrial turbulent flows. Numerical tests, including internal compressor rotor flow and external flow over a 3-D wingbody, validate the proposed method's accuracy and efficiency. Via our implicit scheme, the force coefficients of the 3-D wing-body flow with about five million mesh elements can converge after 500 steps. This work represents a practical advancement of GKS, demonstrating its potential to compete with established CFD solvers in high-Reynolds-number external and internal turbulent flows.