Chengwen Zhong

The unified gas-kinetic scheme (UGKS) solves multi-scale flows within a single framework, but its explicit formulation is inefficient for unsteady simulations due to severe Courant–Friedrichs–Lewy (CFL) constraints arising from disparate mesh scales. We develop a time-accurate implicit UGKS with a simplified multi-scale flux for unsteady diatomic gas flows. The method employs a Crank–Nicolson scheme with a dual-time-stepping strategy, enabling CFL numbers hundreds of times larger than those of the explicit scheme and achieving an order-of-magnitude speedup. The simplified multi-scale flux recovers the Navier–Stokes limit in the continuum regime and accurately resolves non-equilibrium effects under rarefied conditions. Systematic calibration of four key numerical parameters (CFL number, weighting factor ε, point-relaxation symmetric Gauss–Seidel sweeps, and inner iterations) using Sod's shock tube establishes practical guidelines for balancing accuracy and efficiency. The method is validated on benchmark problems spanning from continuum to free-molecular regimes: laminar flow past circular and square cylinders, flow expansion between two connected cavities, and nozzle plume flow. Numerical results demonstrate that the method accurately captures the distinct vortex-shedding dynamics of square and circular cylinders, thermal non-equilibrium under rarefied conditions, and flow patterns of the nozzle plume under different degrees of ambient rarefaction.

An implicit simplified unified gas-kinetic scheme (ISUGKS) for diatomic gas flows is developed and implemented within the OpenFOAM framework. The solver integrates the Rykov kinetic model to characterize translational–rotational energy exchange and nonequilibrium effects, adopting an implicit macro–micro coupling strategy to eliminate the time-step limitation of explicit schemes. The macroscopic conservation equations are discretized using vector flux splitting, while the microscopic distribution function is solved with a first-order upwind scheme. Both systems are solved by a matrix-free point-relaxation symmetric Gauss–Seidel (PR-SGS) method for low cost and fast convergence. A dual-time-stepping approach is employed to preserve multiscale flow characteristics and accelerate steady-state convergence, and an error compensation technique is introduced to suppress conservation errors caused by velocity-space discretization. The interface flux is reconstructed via the right rectangular integration rule to simplify computation and improve efficiency. Validations are conducted through benchmark cases including a supersonic rarefied flow over a flat plate, a 3D supersonic/hypersonic flow over a sphere, an Apollo re-entry capsule, and a lifting-body vehicle, covering supersonic to hypersonic rarefied flow regimes. Numerical results show good agreement with experimental data, implicit UGKS, and DSMC solutions. Compared with the explicit counterpart, the proposed solver achieves more than one order of magnitude speedup under the same parallel configuration, demonstrating significantly improved efficiency for large-scale engineering simulations. The developed ISUGKS solver fills the gap of implicit multiscale kinetic solvers for diatomic gases in OpenFOAM and provides a reliable tool for hypersonic rarefied aerothermodynamic predictions.