Keywords: discontinuous, Taylor-Galerkin, Boltzmann-BGK.

In recent decades, the computational fluid dynamics community has largely focused on the application of the Euler and Navier-Stokes equation systems in finding solutions to fluid flow problems. This has been based on the continuum assumption that allows macroscopic variables such as pressure and temperature to be clearly defined. There are conditions, however, under which this continuum assumption breaks down. For example, nano-scale flows, rarefied gas dynamics and hypersonics are applications in which it is possible that insufficient molecular collisions are taking place to allow the gas to maintain equilibrium and for the continuum assumption to hold [1].

Continuum validity can be expressed in terms of the Knudsen number (K_n) which is the ratio of mean free molecular path to some typical length-scale of the flow and is a measure of the degree of rarefaction of the gas. Above K_n=0.1, we enter the transition regime in which the standard conservation equations no longer form a closed set due to our inability to express shear stresses and heat flux in terms of the lower order macroscopic variables. Once we reach the limit of validity of the continuum equations, we must take a step of abstraction back and solve the Boltzmann equation from which the Euler and Navier-Stokes systems are derived.

In this paper we present a discontinuous Taylor-Galerkin [2] two-step finite element procedure for solution of the Boltzmann-BGK equation [3]. This is a simplified version of the full Boltzmann equation. The term which poses the greatest mathematical difficulty in the full equation is the molecular collision term on the right-hand side and it is this term which is modified in the BGK version. This is done by making the assumption that the effect of molecular collisions is to force a non-equilibrium gas back to equilibrium over some timescale, t.

The Boltzmann-BGK solver allows us to analyse significantly non-equilibrium problems without contravening any of the assumptions that make our equation systems valid. In this paper we present some 2D gas flow examples including a shock tube problem, flow over a ridge and transonic flow over an aerofoil. Results are compared with analytical solutions where they exist and results from Navier-Stokes solvers and Direct Simulation Monte Carlo (DSMC) methods [1].

1

G.A. Bird, "Molecular Gas Dynamics and the Direct Simulation of Gas Flows", Clarendon Press, Oxford, 1994.

2

B. Cockburn, "Discontinuous Galerkin Methods", Journal of Applied Mathematics and Mechanics, 83, pp 731-754, 2002. doi:10.1002/zamm.200310088

3

P.L. Bhatnagar, E.P. Gross, M.Krook, "Model for collision processes in gases, I Small amplitude processes in charged and neutral one-component systems", Phys. Rev., 94, pp 511-524, 1954. doi:10.1103/PhysRev.94.511

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