Multi-level lattice Boltzmann model on square lattice for compressible flows

Abstract A compressible lattice Boltzmann model is established on a square lattice. The model allows large variations in the mean velocity by introducing a large particle-velocity set. To maintain tractability, the support set of the equilibrium distribution is chosen to include only four directions and three particle-velocity levels in which the third level is introduced to improve the stability of the model. This simple structure of the equilibrium distribution makes the model efficient for the simulation of flows over a wide range of Mach numbers and gives it the capability of capturing shock jumps. Unlike the standard lattice Boltzmann model, the formulation eliminated the fourth-order velocity tensors, which were the source of concerns over the homogeneity of square lattices. A modified collision invariant eliminates the second-order discretization error of the fluctuation velocity in the macroscopic conservation equation from which the Navier–Stokes equation and energy equation are recovered. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Two-dimensional shock-wave propagations and boundary layer flows were successfully simulated. The model can be easily extended to three-dimensional cubic lattices.

[1]  Connection Between the Lattice Boltzmann Equation and the Beam Scheme , 1998 .

[2]  Chenghai Sun,et al.  Three-dimensional lattice Boltzmann model for compressible flows. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  L. Munn,et al.  Red blood cells initiate leukocyte rolling in postcapillary expansions: a lattice Boltzmann analysis. , 2003, Biophysical journal.

[4]  P. Sweby.  High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws , 1984 .

[5]  Chenghai Sun,et al.  Simulations of Compressible Flows with Strong Shocks by an Adaptive Lattice Boltzmann Model , 2000 .

[6]  W. Shyy,et al.  Viscous flow computations with the method of lattice Boltzmann equation , 2003 .

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

[8]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[9]  Bruce J. Palmer,et al.  Lattice Boltzmann Algorithm for Simulating Thermal Flow in Compressible Fluids , 2000 .

[10]  Kun Xu,et al.  A new class of gas-kinetic relaxation schemes for the compressible Euler equations , 1995 .

[11]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[12]  Y. Qian.  Fractional Propagation and the Elimination of Staggered Invariants in Lattice-BGK Models , 1997 .

[13]  Antony Jameson,et al.  Gas-kinetic finite volume methods, flux-vector splitting, and artificial diffusion , 1995 .

[14]  Thermal Lattice Boltzmann Model for Compressible Fluid , 2000 .

[15]  Matthaeus,et al.  Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[16]  Balasubramanya T. Nadiga,et al.  A Method for Near-Equilibrium Discrete-Velocity Gas Flows , 1994, comp-gas/9501007.

[17]  Chenghai Sun,et al.  Lattice-Boltzmann models for high speed flows , 1998 .

[18]  Su Cheng-hai.  Lattice Boltzmann Model for Compressible Fluid on a Square Lattice , 2000 .

[19]  S. Osher,et al.  Uniformly High-Order Accurate Nonoscillatory Schemes. I , 1987 .

[20]  Chenghai Sun,et al.  Adaptive lattice Boltzmann model for compressible flows: Viscous and conductive properties , 2000 .

[21]  Zhao,et al.  Lattice boltzmann method for compressible flows with high mach numbers , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  B. Nadiga.  An Euler solver based on locally adaptive discrete velocities , 1995, comp-gas/9501010.

[23]  Lattice-Boltzmann model for compressible perfect gases , 2000 .

[24]  Feng Xu,et al.  A Thermal LBGK Model for Large Density and Temperature Differences , 1997 .

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

[26]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .

[27]  Rakesh K Jain,et al.  Red blood cells augment leukocyte rolling in a virtual blood vessel. , 2002, Biophysical journal.

[28]  Chen,et al.  Lattice Boltzmann model for compressible fluids , 1992 .

[29]  A. Ecer,et al.  Parallel Computational Fluid Dynamics, '91 , 1992 .

[30]  Parallel Computing of Transonic Cascade Flows Using the Lattice-Boltzmann Method , 2002 .

[31]  Theo G. Theofanous,et al.  The lattice Boltzmann equation method: theoretical interpretation, numerics and implications , 2003 .

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

[33]  R. H. Sanders,et al.  The possible relation of the 3-kiloparsec arm to explosions in the galactic nucleus , 1974 .

[34]  P. Colella.  Multidimensional upwind methods for hyperbolic conservation laws , 1990 .