Master equation approach for modeling diatomic gas flows with a kinetic Fokker-Planck algorithm
Introduction
The design of spacecraft requires modeling of a wide range of rarefaction and non-equilibrium effects. The magnitude of non-equilibrium is in general characterized by the Knudsen number , where λ denotes the particle mean free path and l is a characteristic length scale of the regarded problem. When the local Knudsen number is small, many particle collisions occur, which leads to the assumption that the distribution of the thermal particle velocities is near to a local Maxwellian. In this case the gas flow can be modeled macroscopically by the well-known system of Navier-Stokes equations. On the other hand, a large local Knudsen number might lead to a non-equilibrium velocity distribution function. In such a case, the Navier-Stokes equations lose validity. Various applications require the modeling of internal energy modes of diatomic molecules. These can absorb a large amount of energy and thus have a major impact on the entire flow field. As an example, vibrational states of molecules are usually excited at temperatures of a few thousand Kelvin, which typically occur in gas flows around reentry vehicles. Because of the slow vibrational relaxation process, the vibrational energy modes are in general not in thermal equilibrium with the translational energy modes. Hence, it is necessary to simulate the exact evolution of the relaxation process. For the modeling of gases far away from translational and internal equilibrium, Chang and Uhlenbeck [1] proposed a generalization of the well-known Boltzmann-equation: where denotes the differential cross section for a collision between two particles with internal pre-collision states i and j and resulting post-collision states k and l, m means the particle mass, the relative particle velocity α and ψ describe the orientation of the post-collision relative particle velocity compared to the pre-collision relative particle velocity and refers to the particle distribution function for the discrete internal energy state i. The first integral sign in (1) relates to a three-dimensional integration over the entire velocity space. Note, that in this work three-dimensional integrations are indicated by a single integral, while missing limit values refer to an integration over the entire velocity space. Theoretically, Eqn. (1) can be used to accurately model gases with discrete internal energy structure at arbitrary Knudsen numbers, but due to the high dimensionality of and the complexity of the collision integral, the direct solution becomes a computational expensive task.
An alternative way to model non-equilibrium gas flows is the Direct Simulation Monte Carlo (DSMC) algorithm [2]. In the DSMC method the molecular particle motion is calculated directly by applying a stochastic simulation approach to simulate particle collisions. DSMC is proven to be consistent with the solution of the Boltzmann equation for the monatomic case [3] and has been validated for the diatomic case [4], [5]. In the last decades the algorithm has become a standard tool for modeling rarefied gas flows. An important requirement of each DSMC simulation is the proper resolution of molecular scales: The time step size shall be smaller than the local mean collision time and the grid cell size shall be smaller than the mean free path [6]. As a result, the computational effort for DSMC increases strongly as the Knudsen number decreases. This might become a challenge when modeling multiscale gas flows that feature a wide range of different local Knudsen numbers.
To solve this issue, it is common practice to couple DSMC with flow solvers which are less accurate in describing high Knudsen number flows, but more efficient in the low Knudsen number regime. One approach to such a hybridization is the coupling between the DSMC algorithm and Navier-Stokes solvers [7], [8]. However, combining DSMC and Navier-Stokes solvers becomes a challenging task because of the fluctuating boundary conditions for the Navier-Stokes solver, caused by the stochastic nature of the DSMC algorithm. Another approach is to couple DSMC with macroscopic particle methods [9], [10], [11], [12], which also calculate the motion of individual gas particles, but without modeling intermolecular collisions. A recent example for such a particle method is the kinetic Fokker-Planck model [9].
The fundamental approach of the kinetic Fokker-Planck model is to approximate the Boltzmann equation by a Fokker-Planck equation in velocity space. Instead of obtaining the particle distribution function by solving the Fokker-Planck equation directly, the motion of the underlying particles is modeled by an associated random process. This leads to a particle handling similar to the DSMC algorithm, which allows for a simple coupling of both methods. Jenny et al. [9] first introduced the linear model which leads to an incorrect Prandtl number for monatomic gases in the continuum limit. To fix this issue, various authors developed extensions to the linear model [13], [14], [15], [16]. A popular extension is the cubic model by Gorji et al. [13], which has been extended by methods to model gas mixtures [17], polyatomic species [18] and efficient integration algorithms [19]. Gorji and Jenny [20] also suggested a scheme to efficiently couple the cubic Fokker-Planck model with the DSMC algorithm.
Only a few papers address the modeling of polyatomic gas flows within the Fokker-Planck ansatz. In analogy to the modeling of the translational modes, Gorji and Jenny [18] assumed the internal molecular energy structure to be a continuous scalar and modeled the relaxation process by a Fokker-Planck equation with associated random process. A similar approach is used by Mathiaud and Mieussens [21] to extend the ellipsoidal Fokker-Planck model [15] to the calculation of polyatomic species. Pfeiffer et al. [12] adapted an approach which was originally invented for the kinetic BGK model [22] to the ellipsoidal Fokker-Planck model to describe rotational relaxation within an nozzle expansion flow. Mathiaud and Mieussens [23] derived an “reduced” Fokker-Planck model to handle diatomic gas species with discrete vibrational energy levels. The H-theorem as well as the correct continuum limit is proofed for their model, but no application is shown in the paper. In summary, to authors knowledge, the modeling of discrete internal energy structures of molecules within Fokker-Planck is only discussed theoretically or not taken into account. Also only little attention is paid so far to the relaxation of the internal energy distribution function.
In this work we present a scheme for extending arbitrary Fokker-Planck models to describe gases with excited continuous and discrete internal energies. We use Gorjis cubic model [13], [18] to implement the proposed scheme and to perform calculations demonstrating the accuracy of the model. Particular attention is paid to the relaxation of the internal energy distribution function.
The structure of this paper is as follows: Section 2 gives a short review of the cubic Fokker-Planck model and the DSMC algorithm, which is used to perform reference calculation for test cases considered in this paper. Section 3 proposes a collision operator that combines the Fokker-Planck and Master equation in order to model internal energy relaxation. Section 4 introduces reasonable rate coefficients for the Master-equation, while the performance of the proposed models is discussed in Sec. 5. Section 6 derives stochastic processes in order to solve the Master equation on a kinetic level. In Sec. 7, different test cases are investigated. Conclusions are given in Sec. 8. In Appendix A and Appendix B the conservation equations and the H-Theorem for the model are proven.
Section snippets
DSMC
In the DSMC algorithm [2] the distribution function is approximated by a set of computational particles, typically representing a large number of real atoms or molecules. Alternately the particles are moved through the domain and the particle velocities are updated by modeling molecular collisions. The domain is divided into grid cells and collisions are only performed between particles which are in the same cell. Due to the splitting between particle motion and collisions, the cell
Modeling internal degrees of freedom
In this section a Master equation ansatz to model internal energy relaxation is described. All calculations presented here are performed assuming only one discrete internal energy mode, which is modeled by the distribution function . In principle, all derivations could also be performed by assuming a continuous internal energy mode. In this case, the discrete distribution function would be replaced by a continuous function and the sum would be replaced by an
Models for rate coefficients
In this section we suggest three different models for the rate coefficients of the Master equation (39).
Discussion of the model
In Sec. 3 the kinetic model (27) was introduced in order to describe gases featuring internal energies. This section is dedicated to evaluating the performance of this model based on theoretical considerations. The section is structured into two parts. Section 5.1 discusses physical limitations of our model that are induced by the splitting of the distribution function as described by Eq. (36). Section 5.2 tries to eventuate the performance of our model by investigating moment equations, which
Solving stochastic equations of motions
This section discusses the simulation of the stochastic jump process described by the Master equation (39). Depending on the expressions for the rate coefficients, three different algorithms are developed to calculate the temporal evolution of the internal energy state of a particle over a Fokker-Planck time step Δt. The algorithms are derived from Gillespie's “direct simulation method” [41], which is briefly recalled here.
The “direct simulation method” provides an algorithm to simulate the
Test cases
In this section the suggested models are applied to various test cases. All simulations are performed for a molecular nitrogen model with variable hard sphere (VHS)-collision parameters , and and a characteristic vibrational temperature of . To calculate the viscosity, which is needed to evaluate the Fokker-Planck relaxation time τ, the VHS-power law is used: where denotes the reference viscosity [2] consistent with
Conclusion
A scheme to model polyatomic gases within the kinetic Fokker-Planck approach is presented. The scheme can be easily adapted to different Fokker-Planck models and is implemented for example in the cubic Fokker-Planck model. The H-Theorem is proven for the scheme on the premise that the underlying Fokker-Planck model satisfies the H-Theorem. Three different models are presented to describe the internal particle energies as a continuous scalar or as a set of discrete levels. All models are
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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