High order spectral difference lattice Boltzmann method for incompressible hydrodynamics

Abstract This work presents a lattice Boltzmann equation (LBE) based high order spectral difference method for incompressible flows. In the present method, the spectral difference (SD) method is adopted to discretize the convection and collision term of the LBE to obtain high order (≥3) accuracy. Because the SD scheme represents the solution as cell local polynomials and the solution polynomials have good tensor-product property, the present spectral difference lattice Boltzmann method (SD-LBM) can be implemented on arbitrary unstructured quadrilateral meshes for effective and efficient treatment of complex geometries. Thanks to only first oder PDEs involved in the LBE, no special techniques, such as hybridizable discontinuous Galerkin method (HDG), local discontinuous Galerkin method (LDG) and so on, are needed to discrete diffusion term, and thus, it simplifies the algorithm and implementation of the high order spectral difference method for simulating viscous flows. The proposed SD-LBM is validated with four incompressible flow benchmarks in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the lid-driven cavity flow without singularity at the two top corners–Burggraf flow; and (c) the unsteady Taylor–Green vortex flow; (d) the Blasius boundary-layer flow past a flat plate. Computational results are compared with analytical solutions of these cases and convergence studies of these cases are also given. The designed accuracy of the proposed SD-LBM is clearly verified.

[1]  Zhaoli Guo,et al.  Explicit finite-difference lattice Boltzmann method for curvilinear coordinates. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  L. Luo,et al.  Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation , 1997 .

[3]  Dhiraj V. Patil,et al.  Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh , 2009, J. Comput. Phys..

[4]  Jianzhong Lin,et al.  Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element , 2003 .

[5]  Ernst Rank,et al.  High‐order finite elements applied to the discrete Boltzmann equation , 2006 .

[6]  Abed Zadehgol,et al.  A nodal discontinuous Galerkin lattice Boltzmann method for fluid flow problems , 2014 .

[7]  Bernardo Cockburn,et al.  The hybridizable discontinuous Galerkin methods , 2011 .

[8]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[9]  A. Chorin.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[10]  John H. Kolias,et al.  A CONSERVATIVE STAGGERED-GRID CHEBYSHEV MULTIDOMAIN METHOD FOR COMPRESSIBLE FLOWS , 1995 .

[11]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[12]  C. Shu,et al.  Lattice Boltzmann Method and Its Applications in Engineering , 2013 .

[13]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[14]  David A. Kopriva,et al.  A Staggered-Grid Multidomain Spectral Method for the Compressible Navier-Stokes Equations , 1998 .

[15]  Zhaoli Guo,et al.  Discrete unified gas kinetic scheme for all Knudsen number flows: low-speed isothermal case. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  K. Hejranfar,et al.  Chebyshev collocation spectral lattice Boltzmann method for simulation of low-speed flows. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Li-Shi Luo,et al.  An implicit block LU-SGS finite-volume lattice-Boltzmann scheme for steady flows on arbitrary unstructured meshes , 2016, J. Comput. Phys..

[18]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[19]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases : an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases , 1954 .

[20]  Peng Wang,et al.  Performance evaluation of the general characteristics based off-lattice Boltzmann scheme and DUGKS for low speed continuum flows , 2017, J. Comput. Phys..

[21]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[22]  Chunlei Liang,et al.  A comparison of computational efficiencies of spectral difference method and correction procedure via reconstruction , 2013, J. Comput. Phys..

[23]  Kazem Hejranfar,et al.  Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates , 2014, J. Comput. Phys..

[24]  Taehun Lee,et al.  A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows , 2011, J. Comput. Phys..

[25]  Rémi Abgrall,et al.  High‐order CFD methods: current status and perspective , 2013 .

[26]  Eleuterio F. Toro,et al.  ADER: Arbitrary High Order Godunov Approach , 2002, J. Sci. Comput..

[27]  Li-Shi Luo,et al.  Finite Volume Lattice Boltzmann Method for Nearly Incompressible Flows on Arbitrary Unstructured Meshes , 2016 .

[28]  O. Burggraf.  Analytical and numerical studies of the structure of steady separated flows , 1966, Journal of Fluid Mechanics.

[29]  H. Schlichting.  Boundary Layer Theory , 1955 .

[30]  Bernardo Cockburn,et al.  An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations , 2011, J. Comput. Phys..

[31]  Timothy J. Barth,et al.  Recent developments in high order K-exact reconstruction on unstructured meshes , 1993 .

[32]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[33]  Bernardo Cockburn,et al.  A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations , 2010 .

[34]  Zhaoli Guo,et al.  Discrete unified gas kinetic scheme for all Knudsen number flows. II. Thermal compressible case. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  G. Taylor,et al.  Mechanism of the production of small eddies from large ones , 1937 .

[36]  S. Succi,et al.  Lattice Boltzmann method on unstructured grids: further developments. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Taehun Lee,et al.  A characteristic Galerkin method for discrete Boltzmann equation , 2001 .

[38]  Wanai Li,et al.  High‐order k‐exact WENO finite volume schemes for solving gas dynamic Euler equations on unstructured grids , 2012 .

[39]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

[40]  James D. Sterling,et al.  Accuracy of Discrete-Velocity BGK Models for the Simulation of the Incompressible Navier-Stokes Equations , 1993, comp-gas/9307003.

[41]  Bernardo Cockburn.  Discontinuous Galerkin methods , 2003 .

[42]  Taehun Lee,et al.  An Eulerian description of the streaming process in the lattice Boltzmann equation , 2003 .

[43]  D. Kopriva.  A Conservative Staggered-Grid Chebyshev Multidomain Method for Compressible Flows. II. A Semi-Structured Method , 1996 .

[44]  L. Luo,et al.  Lattice Boltzmann Model for the Incompressible Navier–Stokes Equation , 1997 .

[45]  E. LeBoeuf,et al.  Least-squares finite-element scheme for the lattice Boltzmann method on an unstructured mesh. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  Shi Jin,et al.  Physical symmetry and lattice symmetry in the lattice Boltzmann method , 1997 .

[47]  Guido Kanschat,et al.  Local Discontinuous Galerkin Methods for the Stokes System , 2002, SIAM J. Numer. Anal..

[48]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[49]  Raytcho D. Lazarov,et al.  Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems , 2009, SIAM J. Numer. Anal..

[50]  Antony Jameson,et al.  A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy , 2010, J. Sci. Comput..

[51]  L. Luo,et al.  A priori derivation of the lattice Boltzmann equation , 1997 .

[52]  Manfred Krafczyk,et al.  An upwind discretization scheme for the finite volume lattice Boltzmann method , 2006 .