Gas-Kinetic Harmonic Balance Method for Unsteady Supersonic/Hypersonic Flows with Oscillating Motions

In this paper, a novel gas-kinetic harmonic balance method (HBM) is developed for unsteady flows with oscillating motions. Particularly, the emphasis here is on supersonic and hypersonic flows. Because of the presence of an additional source term in the harmonic balance equations, usually very small time steps are needed for explicit algorithms, and some factorization-based implicit algorithms may fail to converge in the case of higher frequencies. To achieve better convergence performance and robustness, a Jacobian-free Newton-Krylov method is adopted. By constructing a simplified preconditioning matrix, the lower-upper symmetric Gauss-Seidel scheme is employed as a preconditioner, so that the computational cost and memory requirement can be reduced. Considering that conventional spatial discretization schemes may encounter numerical difficulties and low robustness in high-Mach-number flows, a more general gas-kinetic scheme (GKS) is extended for the harmonic balance equations. As the HBM solves a mathematically steady problem, the present GKS focuses on evaluation of the time-independent fluxes at each cell interface and each subtime level. The gas-kinetic HBM is applied to a supersonic oscillating flat plate cascade and an oscillating panel in hypersonic flow. By comparison with other methods, the advantages of the developed method are demonstrated.