A Gradient Compression-Based Compact High-Order Gas-Kinetic Scheme on 3D Hybrid Unstructured Meshes

In this paper, a compact gas-kinetic scheme (CGKS) for compressible flow is constructed on hybrid unstructured mesh. Based on high-order gas evolution model at cell interfaces both cell-averaged flow variables and their gradients can be updated in CGKS and used in a compact third-order multi-resolution WENO reconstruction. In the precious CGKS, the flow variables at a cell interface are assumed to be continuous in space in the update of cell-averaged slopes. In order to improve the robustness of the scheme in discontinuous region in three-dimensional space, a compression factor for the cell-averaged gradients is proposed to take into account the possible discontinuity at cell interfaces. The accuracy of the scheme doesn't deteriorate with the implementation of the compression factor. Numerical tests from incompressible to hypersonic flow are presented to validate the high-order CGKS and demonstrate the effectiveness of the gradient compression factor for three-dimensional flow simulation on unstructured mesh.

[1]  R. Bouard,et al.  Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Part 1. Steady flow , 1977, Journal of Fluid Mechanics.

[2]  Sadatoshi Taneda,et al.  Experimental Investigation of the Wakes behind Cylinders and Plates at Low Reynolds Numbers , 1956 .

[3]  D. Tritton.  Experiments on the flow past a circular cylinder at low Reynolds numbers , 1959, Journal of Fluid Mechanics.

[4]  K. Xu,et al.  Gas-kinetic schemes for the compressible Euler equations: Positivity-preserving analysis , 1999 .

[5]  H. T. Huynh,et al.  A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods , 2007 .

[6]  M. Braza,et al.  Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder , 1986, Journal of Fluid Mechanics.

[7]  V. Schmitt,et al.  Pressure distributions on the ONERA M6 wing at transonic Mach numbers , 1979 .

[8]  Qian Wang,et al.  Compact high order finite volume method on unstructured grids III: Variational reconstruction , 2017, J. Comput. Phys..

[9]  Xiangxiong Zhang,et al.  On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations , 2017, J. Comput. Phys..

[10]  Chi-Wang Shu,et al.  High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments , 2016, J. Comput. Phys..

[11]  Chi-Wang Shu,et al.  Three-dimensional high-order least square-based finite difference-finite volume method on unstructured grids , 2020 .

[12]  A. Harten.  ENO schemes with subcell resolution , 1989 .

[13]  Compact High-Order Gas-Kinetic Scheme for Three-Dimensional Flow Simulations , 2021, AIAA Journal.

[14]  Fan Zhang,et al.  A direct discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids , 2015, J. Comput. Phys..

[15]  Michael Dumbser,et al.  Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations , 2010 .

[16]  Kun Xu,et al.  Direct modeling for computational fluid dynamics , 2015, Acta Mechanica Sinica.

[17]  Wei Shyy,et al.  Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations , 2019, Advances in Aerodynamics.

[18]  Vincent Mousseau,et al.  A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids , 2010, J. Comput. Phys..

[19]  C. Lévi-Strauss,et al.  Experimental investigation , 2013 .

[20]  Wei Shyy,et al.  A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh , 2018, J. Comput. Phys..

[21]  Kun Xu,et al.  A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method , 2001 .

[22]  Kun Xu,et al.  A Gas-Kinetic BGK Scheme for the Navier – Stokes Equations and Its Connection with Artificial Dissipation and Godunov Method 1 , 2001 .

[23]  Wei Shyy,et al.  A compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations , 2020, ArXiv.

[24]  R. P. Chhabra,et al.  Steady Flow of Power Law Fluids across a Circular Cylinder , 2008 .

[25]  Kun Xu,et al.  A third-order compact gas-kinetic scheme on unstructured meshes for compressible Navier-Stokes solutions , 2016, J. Comput. Phys..

[26]  Chi-Wang Shu,et al.  A new type of third-order finite volume multi-resolution WENO schemes on tetrahedral meshes , 2020, J. Comput. Phys..

[27]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[28]  A. S. Grove,et al.  An experimental investigation of the steady separated flow past a circular cylinder , 1964, Journal of Fluid Mechanics.

[29]  Wei Shyy,et al.  A compact fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations , 2017, J. Comput. Phys..

[30]  Wanai Li.  Efficient Implementation of High-Order Accurate Numerical Methods on Unstructured Grids , 2014 .

[31]  Jiequan Li,et al.  A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solvers II. High order numerical boundary conditions , 2018, J. Comput. Phys..

[32]  Kun Xu,et al.  High-order gas-kinetic scheme with three-dimensional WENO reconstruction for the Euler and Navier-Stokes solutions , 2019, Computers & Fluids.

[33]  Zhi J. Wang,et al.  Towards industrial large eddy simulation using the FR/CPR method , 2017 .

[34]  Zhi Jian Wang,et al.  On the accuracy and efficiency of discontinuous Galerkin, spectral difference and correction procedure via reconstruction methods , 2014, J. Comput. Phys..

[35]  Panagiotis Tsoutsanis,et al.  Assessment of high-order finite volume methods on unstructured meshes for RANS solutions of aeronautical configurations , 2017 .

[36]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[37]  Stéphane Clain,et al.  A high-order finite volume method for systems of conservation laws - Multi-dimensional Optimal Order Detection (MOOD) , 2011, J. Comput. Phys..

[38]  Yipei Chen,et al.  A three-dimensional unified gas-kinetic wave-particle solver for flow computation in all regimes , 2020, Physics of Fluids.

[39]  Jun Zhu,et al.  A new type of multi-resolution WENO schemes with increasingly higher order of accuracy , 2018, J. Comput. Phys..

[40]  Hong Luo,et al.  Robust Implicit Direct Discontinuous Galerkin Method for Simulating the Compressible Turbulent Flows , 2019 .

[41]  Andrew J. Christlieb,et al.  High-Order Multiderivative Time Integrators for Hyperbolic Conservation Laws , 2013, J. Sci. Comput..

[42]  Kun Xu,et al.  An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations , 2016, J. Comput. Phys..

[43]  Wei Shyy,et al.  Two-step multi-resolution reconstruction-based compact gas-kinetic scheme on tetrahedral mesh , 2021, ArXiv.

[44]  Jiequan Li,et al.  A Two-Stage Fourth Order Time-Accurate Discretization for Lax-Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws , 2015, SIAM J. Sci. Comput..

[45]  T. Nonomura,et al.  Investigation on subsonic to supersonic flow around a sphere at low Reynolds number of between 50 and 300 by direct numerical simulation , 2016 .

[46]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[47]  Hong Luo,et al.  A Parallel, High-Order Direct Discontinuous Galerkin Method for the Navier-Stokes Equations on 3D Hybrid Grids , 2017 .

[48]  J. Remacle,et al.  Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws , 2004 .