A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion

Abstract

In this paper, a stochastic model is presented to simulate the flow of gases, which are not in thermodynamic equilibrium, like in rarefied or micro situations. For the interaction of a particle with others, statistical moments of the local ensemble have to be evaluated, but unlike in molecular dynamics simulations or DSMC, no collisions between computational particles are considered. In addition, a novel integration technique allows for time steps independent of the stochastic time scale.

The stochastic model represents a Fokker–Planck equation in the kinetic description, which can be viewed as an approximation to the Boltzmann equation. This allows for a rigorous investigation of the relation between the new model and classical fluid and kinetic equations. The fluid dynamic equations of Navier–Stokes and Fourier are fully recovered for small relaxation times, while for larger values the new model extents into the kinetic regime.

Numerical studies demonstrate that the stochastic model is consistent with Navier–Stokes in that limit, but also that the results become significantly different, if the conditions for equilibrium are invalid. The application to the Knudsen paradox demonstrates the correctness and relevance of this development, and comparisons with existing kinetic equations and standard solution algorithms reveal its advantages. Moreover, results of a test case with geometrically complex boundaries are presented.

Introduction

In this paper, a new stochastic modeling approach for monatomic gas flow is presented. The motivation is to improve our understanding regarding the applicability of the Navier–Stokes equation in situations with extreme gradients, e.g. in the presence of very strong shocks or micro-scale geometries. In these situations, lack of sufficient collisions between particles leads to strong non-equilibrium and the assumptions of the Navier–Stokes model are violated.

One way to address the validity of the Navier–Stokes equations is to use the concept of irreversible thermodynamics, where transport equations for the molecular stress tensor and heat flux are considered. This approach yields extended fluid dynamic models, which are successful, but still limited to moderate non-equilibrium, see e.g. [28], [30] and references therein. Another approach is to use the equations of kinetic gas theory from which the equations for fluid- and thermodynamics can be derived. In general, the basis is provided by the Boltzmann equation, which is derived for rarefied gases and describes the evolution of the distribution function for particle velocities. The Boltzmann equation can be used in numerical computations by direct discretization. However, due to the high dimensionality of the distribution function and the complexity of the collision operator, the enormous computational requirements are a limiting factor.

Computational costs can be reduced by considering simplified models that replace Boltzmann’s equation like the BGK model, see [20]. Still, in the BGK model an equation for the distribution function needs to be discretized. However, the collision operator is largely simplified. Another pragmatic procedure to solve the Boltzmann equation is given by the direct simulation Monte-Carlo (DSMC) method. In DSMC, particle paths and velocities are computed and local pair-wise collisions of the particles are introduced in a stochastic way. This method is successful, but only applicable for steady high speed flow problems; see [5].

In this paper, the approximation of the Boltzmann collision operator by a Fokker–Planck model, which was already published by Kirkwood [14], [15] for liquids and by Heinz [9], [10] for one-atomic gas, is analyzed and compared with other approximations. This Fokker–Planck model is proposed as an approach for monatomic gas flow, which may be viewed at the same time as a stochastic model for molecular motion. Moreover, a consistent and efficient numerical solution algorithm is devised. The mass density function (MDF) of molecules in the joint physical–velocity–space is consistently approximated by a cloud of stochastic particles, where the particle number density represents the mass density. Since the evolution of the particle positions and velocities by stochastic differential equations (SDEs) is relatively cheap, a very efficient and flexible numerical method can be derived. An important component is a novel time stepping scheme to integrate the SDEs for particle positions and velocities independent of the relaxation time. Note that this is crucial to also deal efficiently with very low Knudsen number flows, where the relaxation time (which scales linearly with the mean free path length) is typically much smaller than the time step size allowed by a simple CFL criterion. The method also features exact preservation of fluctuation energies. Since this particle algorithm is a consistent solution method for the Fokker–Planck equation, it can be employed to rigorously investigate the Fokker–Planck model in the context of kinetic gas theory. We would like to clarify that the goal of our method is to provide an approximation to the Boltzmann equation in a similar way as DSMC does.

It is easy to recognize the analogy of the stochastic model with probability density function (PDF) modeling of turbulent flows. The advantage of PDF methods is that both, convection and turbulence-reaction interaction are represented exactly without modeling assumptions [24]. They have successfully been applied in modeling several chemically inert [2], [7], [21] and reactive [22], [18], [26] flows. In terms of solution algorithms, particle methods are usually preferred, which is due to the high dimensionality of the PDF transport equation [13], [12], [25]. Although conceptually similar, here the algorithmic task is different. On one hand, there is no mean pressure coupling and on the other hand kinetic energy is conserved. In this paper we present a highly accurate, energy conserving particle tracking scheme, which is novel. Moreover, a consistent formulation of isothermal wall-boundary conditions is devised.

The strategy of this paper is the following: In Section 2, we formulate fundamental conditions for approximative models of Boltzmann’s collision operator and derive the Fokker–Planck operator as such a model. Then, the accuracy of this model is investigated by studying the evolution equations of the fluid quantities induced by the model and by comparison to Boltzmann. After establishing the model, we transform the Fokker–Planck equation into stochastic differential equations for particle paths. Section 3 presents the discussion of the relation to other stochastic particle models and the comparison to DSMC. Section 4 derives the numerical method by exact integration of the stochastic equations with frozen coefficients for a full time step. Algorithmic details of how to extract and interpolate averaged quantities are given. In Section 5, we present numerical experiments that verify the preservation properties of the new method. Realistic cases like channel flow and two-dimensional external flow are investigated to demonstrate the capabilities of the new model.