Three-dimensional implementation of the Low Diffusion method for continuum flow simulations

Abstract

Concepts of a particle-based continuum method have existed for many years. The ultimate goal is to couple such a method with the Direct Simulation Monte Carlo (DSMC) in order to bridge the gap of numerical tools in the treatment of the transitional flow regime between near-equilibrium and rarefied gas flows. For this purpose, the Low Diffusion (LD) method, introduced first by Burt and Boyd, offers a promising solution. In this paper, the LD method is revisited and the implementation in a modern particle solver named PICLas is given. The modifications of the LD routines enable three-dimensional continuum flow simulations. The implementation is successfully verified through a series of test cases: simple stationary shock, oblique shock simulation and thermal Couette flow. Additionally, the capability of this method is demonstrated by the simulation of a hypersonic nitrogen flow around a 70°-blunted cone. Overall results are in very good agreement with experimental data. Finally, the scalability of PICLas using LD on a high performance cluster is presented.

Introduction

The simulation of gas flows is dominated by two approaches, classical Computational Fluid Dynamics (CFD) solving the Navier–Stokes equations and the particle-based Direct Simulation Monte Carlo (DSMC) method. CFD methods cover a wide range of gas flow regimes relevant for practical applications including especially near-equilibrium gases and flows near the thermal equilibrium. DSMC is applied for simulations of rarefied gas flows, which occur in space or near-vacuum technology and in micro-electro-mechanical systems. An advantage of the DSMC is that the method can treat non-equilibrium flows in a physically correct manner. Unfortunately, the DSMC method becomes computationally expensive for flows with small Knudsen numbers, thus making it practically unsuitable for the simulation of equilibrium flows. Consequently, there is a gap within numerical tools for applications in the transitional flow regime or in situations where continuum and rarefied flows appear simultaneously. The straight-forward solution, i.e. coupling CFD directly with DSMC, introduces challenges due to the very different underlying approaches of both methods. The deterministic CFD method is based on solving equation systems whereas the probabilistic DSMC method is particle-based and thus introduces statistical noise, which poses a great challenge for the information transfer to the deterministic CFD side.

Alternatives are equilibrium particle methods which are able to handle flows with small Knudsen numbers also in a computationally efficient manner, but can be coupled to the DSMC method in a more natural way. Therefore, different DSMC-based continuum methods have been developed in the last years, e.g. the Equilibrium Particle Simulation Method (EPSM) by Pullin [1] and the collision limiter scheme by Titov et al. [2].

The idea of the EPSM is to redistribute particle velocities by a direct resampling from a Maxwellian distribution at cell-averaged temperature and bulk velocity. This method is equivalent to the numerical solution of the Euler equations and is subject to large numerical diffusion errors and statistical noise [3]. The idea of the EPSM–DSMC-coupling has been modified and extended to a new method named Time Relaxed Monte Carlo (TRMC) by Pareschi and Russo [4]. This method promises to produce the same results as DSMC for flows with large Knudsen numbers while it will behave as a stochastic kinetic scheme for the underlying Euler equations in the limit of small Knudsen numbers [5]. However, Pareschi and Russo also emphasize that accuracy and robustness of the TRMC have still to be improved and that the method depends on coefficients which have to be chosen optimally and which is part of future research [4].

In the DSMC limiter scheme such as that of Titov et al., all particles within a cell may experience multiple collisions per time step instead of simulating binary collisions involving only a fraction of particles in each cell. It is also subject to large numerical diffusion errors and statistical noise [3]. Therefore, Burt and Josyula have improved the method, resulting in more accurate simulations by detecting and reducing the numerical diffusion [6]. Furthermore, the method has been extended recently in order to incorporate effects of physical diffusion for the simulation of viscous flows [7] and multiple species [8]. This new DSMC limiter scheme named Viscous Collision Limiter (VCL) is still subject of ongoing research and has yet to be extended to three-dimensional flows [8], where the efficiency in 3D simulations is still an open question.

Beside these two DSMC-based strategies (TRMC and VCL) for continuum flow simulations, another particle-based method is the Fokker–Planck solution algorithm. Here, the idea is to replace the velocity jump processes occurring during a particle collision by continuous stochastic differential equations of Fokker–Planck type. As a consequence, collision frequencies as well as mean free paths have not to be resolved to achieve physically accurate results [9]. However, the method was developed using Maxwell particles and becomes increasingly complex for more realistic particle interaction potentials (e.g. Hard Sphere) [10]. Furthermore, the complexity increases further for the simulation of gas mixtures instead of single species flows [11].

All particle-based continuum methods (TRMC, VCL and Fokker–Planck solution algorithm) are promising solutions for an “all-particle” hybrid algorithm but are prone to statistical noise due to their stochastic nature. The focus of this work is to revive the alternative idea of a deterministic particle-based continuum method. The Low Diffusion method (LD), firstly described by Burt and Boyd [3], represents such an alternative. The advantages of this method are low numerical diffusion behavior, significantly reduced statistical noise and less complexity. Here, the LD method is extended for highly parallelized full 3D flow simulations. In the following sections, the particle solver PICLas [12] as well as the implementation of LD in PICLas are described. This includes a description of the required modifications to LD for efficient 3D simulations. Finally, a series of test cases is presented to verify and validate the LD implementation and to demonstrate the overall accuracy of this new method.