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Lattice Boltzmann method and gas-kinetic BGK scheme in the low-Mach number viscous flow simulations. (English) Zbl 1236.76052
Summary: Both lattice Boltzmann method (LBM) and the gas-kinetic BGK scheme are based on the numerical discretization of the Boltzmann equation with collisional models, such as, the Bhatnagar–Gross–Krook (BGK) model. LBM tracks limited number of particles and the viscous flow behavior emerges automatically from the intrinsic particle stream and collisions process. On the other hand, the gas-kinetic BGK scheme is a finite volume scheme, where the time-dependent gas distribution function with continuous particle velocity space is constructed and used in the evaluation of the numerical fluxes across cell interfaces. Currently, LBM is mainly used for low Mach number, nearly incompressible flow simulation. For the gas-kinetic scheme, the application is focusing on the high speed compressible flows. In this paper, we are going to compare both schemes in the isothermal low-Mach number flow simulations. The methodology for developing both schemes will be clarified through the introduction of operator splitting Boltzmann model and operator averaging Boltzmann model. From the operator splitting Boltzmann model, the error rooted in many kinetic schemes, which are based on the decoupling of particle transport and collision, can be easily understood. As to the test case, we choose to use the 2D cavity flow since it is one of the most extensively studied cases. Detailed simulation results with different Reynolds numbers, as well as the benchmark solutions, are presented.

MSC:
76M28 Particle methods and lattice-gas methods
76D05 Navier-Stokes equations for incompressible viscous fluids
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
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