A compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations

Abstract

Following the development of a third-order compact gas-kinetic scheme (GKS) for the Euler and Navier-Stokes equations (Journal of Computational Physics 410 (2020) 109367), in this paper an even higher-order compact GKS up to sixth order of accuracy will be constructed for the shock and acoustic wave computation on unstructured mesh. The compactness is defined by the physical domain of dependence for an unstructured triangular cell, which may involve the closest neighbors of neighboring cells. The compactness and high-order accuracy of the scheme are coming from the consistency between the high-order initial reconstruction and the high-order gas evolution model under GKS framework. The high-order evolution solution at a cell interface provides not only a time-accurate flux function, but also the time-evolving flow variables. Therefore, the cell-averaged flow variables and their gradients can be explicitly updated at the next time level from the moments of the same time-dependent gas distribution function. Based on the cell averages and cellaveraged derivatives, both linear and nonlinear high-order reconstruction can be obtained for macroscopic flow variables in the evaluation of local equilibrium and non-equilibrium states. The current nonlinear reconstruction is a combination of WENO and ENO methodology, which is specifically suitable for compact GKS on unstructured mesh with a high-order (≥ 4) accuracy. The initial piecewise discontinuous reconstruction is used for the determination non-equilibrium state and an evolved smooth reconstruction for the equilibrium state. The evolution model in gas-kinetic scheme is based on a relaxation process from non-equilibrium to equilibrium state. The time-accurate gas distribution function in GKS provides the Navier-Stokes flux function directly without separating the inviscid and viscous terms, which simplifies the numerical method on unstructured mesh. Based on the timeaccurate flux solver, the two-stage fourth-order time discretisation can be applied to get a fourth-order time-accurate solution with only two stages, which reduces two reconstructions in comparison with the same time-accurate method with Runge-Kutta time stepping. The current high-order GKS can uniformly capture acoustic and shock waves without identifying trouble cells and implementing additional limiting procedure. In addition, the fourth-up to sixth-order compact GKS can use almost the same time step as a second-order shock capturing scheme. The fourth-order GKS on unstructured mesh will be used in the computations 1 from low speed incompressible viscous flow to the high Mach number shock interaction. The accuracy, efficiency, and robustness of the scheme have been validated. The main conclusion of the paper is that beyond the first-order Riemann solver, the use of high-order gas evolution model seems necessary in the development of high-order schemes.

A p-multigrid compact gas-kinetic scheme for steady-state acceleration

“…Due to the physically reliable evolution process and efficient time discretization, the CGKS achieves remarkable success for unsteady compressible flow simulation, such as in computational aeroacoustics [41] and implicit large eddy simulation [14]. The CGKS has been constructed up to the eighth-order of accuracy in space on structured mesh [40], and fourth-order accuracy on triangular mesh with the possible use of a large CFL number, such as CF L ≈ 0.8 [42,43]. The computation of hypersonic flow around a space vehicle on 3-D hybrid mesh demonstrates its robustness in flow simulation with complex geometry [13].…”

Two-stage Fourth-order Gas Kinetic Solver-based Compact Subcell Finite Volume Method for Compressible Flows over Triangular Meshes

“…The reconstruction is implemented for each component of conserved variables Q. To achieve fourth-order space accuracy, Q over each main cell Ω i is approximated by a solution In comparison, Figure 2 also shows the stencil used in a GKS-based FV method and the classical k-exact FV 36 . Note that for the GKS-based FV, cell-averaged slopes are updated and participate in the reconstruction, which helps to avoid the large stencil for the k-exact FV.…”

“…It is difficult to achieve the high resolution for shock waves as in the SCFV method. In addition, for comparison with traditional FV-GKS, a recently proposed fourth-order FV-GKS based on WENO reconstruction is considered 36 . As shown in Figure 11, with the same mesh size h = 1/120, the result obtained by the current scheme is much more accurate than that obtained by FV-GKS owing to the subcell resolution of the current scheme.…”

A gradient-compression-based compact high-order gas-kinetic scheme on three-dimensional hybrid unstructured mesh

“…The two-stage fourth-order (S2O4) temporal discretization is adopted here as that in the previous CGKS [43,16]. Following the definition of Eq.…”

“…( 3). The details can be found in [43]. A fourth-order temporal accuracy for the Euler equations can be achieved for the conservative flow variables on arbitrary mesh by Eq.…”