A grid-independent particle pairing strategy for DSMC

doi:10.1016/j.jcp.2013.03.018

Journal of Computational Physics

Volume 246, 1 August 2013, Pages 28-36

https://doi.org/10.1016/j.jcp.2013.03.018 Get rights and content

Abstract

A new particle pairing strategy is presented for the Direct Simulation Monte Carlo (DSMC) method. By combining Nearest Neighbour (NN) and Quad Tree Sort (QTS) algorithms, this modified DSMC method becomes largely independent of grid cell size. Therefore, the method possesses considerable advantages in terms of coupling with other grid-based methods in comparison to classical DSMC methods, which have to resolve the mean free path in each cell. The Green–Kubo theory is used to estimate the effects of cell size on transport coefficients. First, the Green–Kubo theory is reviewed for the calculation of the viscosity coefficient as a function of cell size in the classical DSMC method. Next, this theory is extended to the case of using the Nearest Neighbour pairing strategy. Plane Couette flow simulations are used to demonstrate the success of the new particle pairing strategy in becoming independent of the cell size.

Introduction

The coupling of a Direct Simulation Monte Carlo (DSMC) particle code with other simulation methods (e.g. Particle in Cell (PIC), Low Diffusion) often leads to conflicting requirements for the simulation grid. For example, in a coupled PIC-DSMC code for the simulation of rarefied and partially charged gases, the individual requirements for the grid lead to a cell size in the range of the Debye length for the PIC method [1], whereas the DSMC method needs cells of the size of the mean free path $λ$ [2]. These two values can differ by several orders of magnitude depending on the gas conditions. Hence, the optimal grid for the PIC method can vary significantly from the optimal grid for the DSMC method. The use of different grids for each method, assisted by a subsequent coupling of these grids is one possible solution. However, the additional effort of applying two, sometimes adaptive, grids and the additional computational costs are problematic. This strategy also leads to higher complexity, which is unfavourable for an efficient parallelization. Finally, a grid for every coupled method is unfeasible, if more than two methods are coupled.

A strategy is suggested here as a step towards a grid independent DSMC method. A Quad Tree Sort method [3] in combination with a Nearest Neighbour algorithm [4] is discussed in section two. This combination is used in order to achieve a DSMC method largely independent of the grid. In section three, the Green–Kubo theory [5], [6], [7] is reviewed and extended to the NN algorithm. Finally, the grid independence and the Green–Kubo theory are verified by a Couette flow simulation [8] in section four.

Section snippets

Nearest Neighbour (NN)

The basic idea of the DSMC is to simulate gas flows as a particle stream. Simulating each real particle is impossible due to the vast particle number $N_{real}$ . Thus, a weighting factor W $W = \frac{N_{real}}{N_{sim}}$ is used leading to a number of simulated particles $N_{sim}$ representing a number of real particles $N_{real}$ . These particles move without intermolecular interactions through the computational domain for a certain time $Δ t$ and perform collisions with each other after $Δ t$ , for which collision pairs have to be

Cell size dependence of transport coefficients

Specific constraints of the DSMC method are the required time and space discretization. It is assumed that the time step $Δ t \to 0$ , so the error in time discretization can be neglected. For the spatial discretization, the Green–Kubo theory may be used to estimate the cell size effects on the transport coefficients. In the following, the calculation of the viscosity coefficient as a function of the cell size of the regular DSMC pairing method is reviewed and afterwards extended to the case of the

Results

Before discussing the cell size effect, the NN and the QTS algorithms are verified. Additionally, the benefit of the QTS is demonstrated. For the discussion of the cell size effects on viscosity, the simulation of a plane Couette flow is chosen for both the classical and the new DSMC method. Here, a stationary gas between two plane surfaces moving in opposite x directions with the constant velocity $v_{wall} = 100 \frac{m}{s}$ is simulated. Both walls are defined as diffuse reflecting surfaces at a temperature

Conclusion and outlook

This article describes a method that allows physically correct pairing of particles for collisions in DSMC simulations, independent of the simulation grid. For this, a combination of methods is used, comprising a Nearest Neighbour (NN) search and a Quad Tree Sort (QTS) algorithm. The NN algorithm leads to results almost independent of the grid, while the QTS is used to reduce the computational costs to a more practicable level. The NN algorithm and its consequences for the calculation of the

Acknowledgments

The authors gratefully thank the Deutsche Forschungsgemeinschaft (DFG) for funding this research within the project “Kinetic Algorithms for the Maxwell–Boltzmann System and the Simulation of Magnetospheric Propulsion Systems”.

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A grid-independent particle pairing strategy for DSMC

Abstract

Introduction

Section snippets

Nearest Neighbour (NN)

Cell size dependence of transport coefficients

Results

Conclusion and outlook

Acknowledgments

Journal of Computational Physics

Plasma Physics via Computer Simulation

Molecular Gas Dynamics and the Direct Simulation of Gas Flows

Calculation of hard-sphere viscosity by means of correlation functions

Journal of Chemical Physics

Cell size dependence of transport coefficients in stochastic particle algorithms

Physics of Fluids