Abstract
In this paper, we consider the development of the two-dimensional discrete velocity Boltzmann model on a nine-velocity lattice. Compared to the conventional lattice Boltzmann approach for the present model, the collision rules for the interacting particles are formulated explicitly. The collisions are tailored in such a way that mass, momentum and energy are conserved and the H-theorem is fulfilled. By applying the Chapman–Enskog expansion, we show that the model recovers quasi-incompressible hydrodynamic equations for small Mach number limit and we derive the closed expression for the viscosity, depending on the collision cross-sections. In addition, the numerical implementation of the model with the on-lattice streaming and local collision step is proposed. As test problems, the shear wave decay and Taylor–Green vortex are considered, and a comparison of the numerical simulations with the analytical solutions is presented.
Highlights
In the kinetic theory, the distribution function of a rarefied gaseous system is governed by the Boltzmann equation or its models [1]
I = 1 . . . N, where f i (t, x) is the distribution function related to the particles with the velocity ci, eq i = 1 . . . N, τ is the relaxation time, f i is the local equilibrium, N is the number of the d discrete velocities, dt = ∂t + ci ∂r, r is the spatial variable
The collisions between the particles are described in a phenomenological way, i.e., it is postulated that, due to the collisions, the distribution function tends to the local equilibrium state at a eq rate proportional to f i − f i
Summary
The distribution function of a rarefied gaseous system is governed by the Boltzmann equation or its models [1]. Discrete velocities, dt = ∂t + ci ∂r , r is the spatial variable In this approach, the collisions between the particles are described in a phenomenological way, i.e., it is postulated that, due to the collisions, the distribution function tends to the local equilibrium state at a eq rate proportional to f i − f i. Broadwell equation in two dimensions has been investigated thoroughly [31,32,33,34,35,36], this model has correct collision invariants, but its discrete velocity set is too small and lacks isotropy [37]; the correct description of the hydrodynamics is impossible in the framework of this model. The numerical experiments show excellent agreement between the numerical simulation results and analytical solutions
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