Two-stage fourth-order subcell finite volume method on hexahedral meshes for compressible flows

Abstract

As an extension of the two-stage fourth-order subcell finite volume (SCFV) method that we developed for two-dimensional compressible flows [C. Zhang et al., “Two-stage fourth-order gas kinetic solver based compact subcell finite volume method for compressible flows on triangular meshes,” Phys. Fluids 33, 126108 (2021)], this study continues our efforts toward three-dimensional (3D) simulations on hexahedral meshes. The two components of subcell divisions and two-stage fourth-order time stepping are utilized to improve efficiency and enhance compactness, which are crucial for 3D simulations. In particular, the current method subdivides each cell into a set of subcells or control volumes (CVs) to increase the degrees of freedom for high-order reconstruction, which involves only face-neighboring cells. For traditional finite volume (FV) methods, high-order reconstruction is performed on each CV individually. In contrast, the reconstruction of SCFV is shared by a set of CVs belonging to the same cell, which can be much more efficient and compact. Moreover, the SCFV framework is combined with the high-order flux evolution by adopting a robust and time-dependent gas-kinetic flux solver and an efficient two-stage fourth-order temporal discretization. The multi-stage Runge–Kutta (RK) method is thus avoided. The coupling of inviscid and viscous terms in the gas-kinetic flux enables us to directly simulate viscous flows. To capture shocks, a limiting procedure by hierarchical reconstruction is developed for effectively preserving the accuracy in smooth flow regions and suppressing numerical oscillations near flow discontinuities. Several benchmark cases are tested. The high-order accuracy and efficiency of this scheme are validated and compared to the k-exact FV method and the traditional Riemann solver combined with a multi-stage RK method. In particular, the simulation of the supersonic Taylor–Green vortex problem demonstrates the good performance of this scheme in compressible turbulence with the presence of shock waves.