An efficient direct simulation Monte Carlo method for low Mach number noncontinuum gas flows based on the Bhatnagar–Gross–Krook model

Abstract

The direct simulation Monte Carlo ͑DSMC͒ method is the preferred approach for simulating rarefied gas flows in complex geometries. However, the standard DSMC method becomes inefficient in the limit M = Ū / ͗c͘ → 0 when the thermal velocity fluctuations which scale with the speed of sound ͗c͘ are much larger than characteristic ensemble-averaged flow speed Ū . In this paper, we propose a modified DSMC algorithm which simulates the linearized Bhatnagar-Gross-Krook ͑BGK͒ approximation to the Boltzmann equation. The particles in this method are uniformly distributed in space and have a rest-state, equilibrium Maxwellian velocity distribution. The deviations from the equilibrium state are captured by weightings, indicating that each particle represents a noninteger expectation for finding gas molecules with the simulation particle's velocity in the spatial cell where it is found. The BGK approximation makes the implementation of such a method simple and efficient because the BGK collision rule can be implemented by adjusting particle weightings without the need to create new particles during the simulation. The method can be incorporated into existing DSMC simulation programs with minimal changes. We apply the new algorithm to two test problems-pressure-driven flow through a nanochannel and flow around an isolated sphere. Results obtained with the new method are in excellent agreement with previous theories and simulations.

Noncontinuum drag force on a nanowire vibrating normal to a wall: Simulations and theory

“…As discussed in Ref. 23, the BGK relaxation process does not conserve momentum and energy exactly at every time step but does so only in an average sense. To improve statistical accuracy, we strictly enforce conservation of mass, momentum, and energy at every time step by adjusting the postrelaxation weightings of all particles in the cell according to the procedure described in Ref.…”

“…Although the simulation method is very general and can be used to study low speed flow in any geometry, below, we present some of the details of the simulation method in the context of the nanowire problem. It should be emphasized that a thorough description of the method is beyond the scope of this article and for more details we refer the reader to our earlier publication 23 where we have also documented the success of this method in studying test problems for which analytical results are available.…”

“…As explained in Ref. 23, such a boundary condition offers improved convergence when compared to a boundary condition in which the simulation domain is assumed to be in equilibrium with a stationary gas reservoir. The JeffreyOnishi boundary condition is implemented by ensuring that any time a particle strikes the outer boundary, it is specularly reflected back into the simulation domain with a postreflection weighting given by 23…”

“…We therefore turn to our newly developed BGK-LM-DSMC simulation method, 23 an efficient modified DSMC algorithm that simulates the linearized BGK approximation to the Boltzmann equation. Here, each simulation particle is assumed to represent N m ͑1+ MW i ͒ molecules where M, as before, represents the Mach number and W i represents a variable noninteger particle weighting.…”

High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation

“…The insight afforded by this model equation is sufficient to describe the qualitative properties of the (true) flow in many cases of practical interest (Vincenti & Kruger 1965;Cercignani 2000;Sone 2007). Furthermore, the DSMC and LVDSMC methods discussed for the Boltzmann equation have been applied to study of the Boltzmann-BGK equation (Ramanathan & Koch 2009;Hadjiconstantinou, Radtke & Baker 2010). Other numerical schemes, such as finite differencing techniques and the lattice Boltzmann (LB) method, have been used to investigate a number of canonical gas flows (Loyalka, Petrellis & Storvick 1979; High frequency asymptotic analysis of the Boltzmann-BGK equation 3 Chen et al 2003;Yu, Girimaji & Luo 2005;Loyalka & Tompson 2009;Shi & Sader 2010;Yap & Sader 2012).…”