Implicit Gas-Kinetic Scheme for Internal and External Flows

While the gas-kinetic scheme (GKS) has demonstrated excellent properties, e.g., unified treatment of inviscid and viscous fluxes, its application has been largely confined to idealized external flows, leaving its potential for complex engineering applications significantly underdeveloped. This study extends the GKS to engineering flow problems. First, this work presents the formulation and implementation of GKS for internal rotating flows, enabling efficient simulations of highly complex industrial configurations like turbomachinery rotors. Second, the severe time-step constraints typical of explicit kinetic schemes are overcome by constructing a robust, implicit defect-correction framework with two linear-solver options: a symmetric Gauss–Seidel (SGS) iteration and a restarted generalized minimal residual method (GMRES) method equipped with a one-sweep SGS-type preconditioner. The Jacobian matrices are approximated using the first-order kinetic flux vector splitting scheme and the thin shear layer approximation to enhance robustness. Third, the shear-stress transport turbulence model is adopted in the current GKS. Numerical tests, including an internal compressor rotor and external flow over a 3-D wing–body, on highly complex engineering geometries thoroughly validate the scheme. Via our implicit scheme, the force coefficients of the 3-D wing–body flow converge after only 500 steps. This work advances the GKS into industrial-level engineering practices to match and compete with established conventional Navier–Stokes solvers.