^{1,a)}and Patrick Jenny

^{1}

### Abstract

A Fokker–Planck based kinetic model is presented here, which also accounts for internal energy modes characteristic for diatomic gas molecules. The model is based on a Fokker–Planck approximation of the Boltzmann equation for monatomic molecules, whereas phenomenological principles were employed for the derivation. It is shown that the model honors the equipartition theorem in equilibrium and fulfills the Landau–Teller relaxation equations for internal degrees of freedom. The objective behind this approximate kinetic model is accuracy at reasonably low computational cost. This can be achieved due to the fact that the resulting stochastic differential equations are continuous in time; therefore, no collisions between the simulated particles have to be calculated. Besides, because of the devised energy conserving time integration scheme, it is not required to resolve the collisional scales, i.e., the mean collision time and the mean free path of molecules. This, of course, gives rise to much more efficient simulations with respect to other particle methods, especially the conventional direct simulation Monte Carlo (DSMC), for small and moderate Knudsen numbers. To examine the new approach, first the computational cost of the model was compared with respect to DSMC, where significant speed up could be obtained for small Knudsen numbers. Second, the structure of a high Mach shock (in nitrogen) was studied, and the good performance of the model for such out of equilibrium conditions could be demonstrated. At last, a hypersonic flow of nitrogen over a wedge was studied, where good agreement with respect to DSMC (with level to level transition model) for vibrational and translational temperatures is shown.

The authors would like to acknowledge the constructive discussions with Professor M. Torrilhon and Professor S. V. Bogomolov. The authors would also like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

Funding for this research was provided by the Swiss National Science Foundation.

I. INTRODUCTION

II. KINETIC MODEL

A. Rotation

B. Vibration

C. Translation

1. Cubic model

2. Constitutive relations

III. ITŌ PROCESSES

IV. SOLUTION ALGORITHM

V. RESULTS

A. Heat-bath relaxation

B. Computational efficiency

C. Normal shock structure

1. Boundary conditions

2. Shock thickness of nitrogen

3. Rotational energy distribution

D. Hypersonic flow over a wedge

VI. CONCLUDING REMARKS

### Key Topics

- Boltzmann equations
- 67.0
- Diffusion
- 20.0
- Collision theories
- 13.0
- Numerical modeling
- 12.0
- Viscosity
- 10.0

## Figures

(a) Temporal evolution of normalized rotational energy in comparison with the LT solution (67) . For the FP simulation two different time steps of Δt = τ rot and Δt = τ rot /3 were used. (b) Distribution of the molecular rotational energy at equilibrium. The FP result is depicted by the solid line, where the Boltzmann distribution at temperature T is depicted by symbols.

(a) Temporal evolution of normalized rotational energy in comparison with the LT solution (67) . For the FP simulation two different time steps of Δt = τ rot and Δt = τ rot /3 were used. (b) Distribution of the molecular rotational energy at equilibrium. The FP result is depicted by the solid line, where the Boltzmann distribution at temperature T is depicted by symbols.

(a) Temporal evolution of normalized vibrational energy in comparison with the LT solution (67) . For the FP simulation two different time steps of Δt = τ vib and Δt = τ vib /3 were used. (b) Distribution of the molecular vibrational energy at equilibrium. The FP result is depicted by the solid line, where the Boltzmann distribution at temperature T is depicted by symbols.

(a) Temporal evolution of normalized vibrational energy in comparison with the LT solution (67) . For the FP simulation two different time steps of Δt = τ vib and Δt = τ vib /3 were used. (b) Distribution of the molecular vibrational energy at equilibrium. The FP result is depicted by the solid line, where the Boltzmann distribution at temperature T is depicted by symbols.

Planar Couette flow of nitrogen. The profile of the mean velocity U 2 of nitrogen is depicted along the x 1-coordinate (perpendicular to the flow direction). The solid lines represent FP results, circles DSMC results, and the dashed line the Navier–Stokes solution (i.e., the linear velocity profile).

Planar Couette flow of nitrogen. The profile of the mean velocity U 2 of nitrogen is depicted along the x 1-coordinate (perpendicular to the flow direction). The solid lines represent FP results, circles DSMC results, and the dashed line the Navier–Stokes solution (i.e., the linear velocity profile).

Computational cost comparison between FP simulations and DSMC. The ratio between computational time required for DSMC and the FP model, i.e., C ratio , is shown as a function of the Knudsen number.

Computational cost comparison between FP simulations and DSMC. The ratio between computational time required for DSMC and the FP model, i.e., C ratio , is shown as a function of the Knudsen number.

(a) Shock thickness comparison between FP results and experiments by Alsmeyer. 33 The FP results are shown by symbols and experiments are shown by solid line. At the reciprocal shock thickness for cases with T up = 30 K and 3000 K are shown by squares. (b) Normalized density profiles of Ma up = 4 in nitrogen at different upstream temperatures using FP.

(a) Shock thickness comparison between FP results and experiments by Alsmeyer. 33 The FP results are shown by symbols and experiments are shown by solid line. At the reciprocal shock thickness for cases with T up = 30 K and 3000 K are shown by squares. (b) Normalized density profiles of Ma up = 4 in nitrogen at different upstream temperatures using FP.

(a) Shock profile for in nitrogen. FP results are shown by lines and experiments of Robben and Talbot 23 by symbols. (b) PDF of rotational energy for a shock in nitrogen. FP results are shown by symbols and experiments of Robben and Talbot 23 by dashed lines. The solid lines show the equilibrium distributions at upstream and downstream conditions. In between, two different locations were considered, i.e., x/L* ∈ {0, 4.1}.

(a) Shock profile for in nitrogen. FP results are shown by lines and experiments of Robben and Talbot 23 by symbols. (b) PDF of rotational energy for a shock in nitrogen. FP results are shown by symbols and experiments of Robben and Talbot 23 by dashed lines. The solid lines show the equilibrium distributions at upstream and downstream conditions. In between, two different locations were considered, i.e., x/L* ∈ {0, 4.1}.

(a) Shock profile for in nitrogen. FP results are shown by lines and experiments of Robben and Talbot 23 by symbols. (b) PDF of rotational energy for a shock in nitrogen. FP results are shown by symbols and experiments of Robben and Talbot 23 by dashed lines. The solid lines show the equilibrium distributions at upstream and downstream conditions. In between, four different locations were considered, i.e., x/L* ∈ {−12.9, −3.5, 0, 2.3}.

(a) Shock profile for in nitrogen. FP results are shown by lines and experiments of Robben and Talbot 23 by symbols. (b) PDF of rotational energy for a shock in nitrogen. FP results are shown by symbols and experiments of Robben and Talbot 23 by dashed lines. The solid lines show the equilibrium distributions at upstream and downstream conditions. In between, four different locations were considered, i.e., x/L* ∈ {−12.9, −3.5, 0, 2.3}.

(a) Shock profile for in nitrogen. FP results are shown by lines and experiments of Robben and Talbot 23 by symbols. (b) PDF of rotational energy for shock in nitrogen. FP results are shown by symbols and experiments of Robben and Talbot 23 by dashed lines. The solid lines show the equilibrium distributions at upstream and downstream conditions. In between, four different locations were considered, i.e., x/L* ∈ { − 6.3, −4.23, −1.06, 3.17}.

(a) Shock profile for in nitrogen. FP results are shown by lines and experiments of Robben and Talbot 23 by symbols. (b) PDF of rotational energy for shock in nitrogen. FP results are shown by symbols and experiments of Robben and Talbot 23 by dashed lines. The solid lines show the equilibrium distributions at upstream and downstream conditions. In between, four different locations were considered, i.e., x/L* ∈ { − 6.3, −4.23, −1.06, 3.17}.

Hypersonic nitrogen flow over a wedge using the FP model. Translational temperature at top, rotational temperature at middle, and vibrational temperature at bottom.

Hypersonic nitrogen flow over a wedge using the FP model. Translational temperature at top, rotational temperature at middle, and vibrational temperature at bottom.

Temperature contours for hypersonic nitrogen flow over a wedge using different approaches. (a) Vibrational temperature contours from the FP model. (b) Translational temperature contours from the FP model. (c) Vibrational temperature contours from DSMC using level to level transition model. (d) Translational temperature contours from DSMC using level to level transition model. Figs. 10(c) and 10(d) are reprinted with permission from I. D. Boyd, Phys. Fluids 3, 1785–1791 (1991). Copyright 2008, American Institute of Physics.

Temperature contours for hypersonic nitrogen flow over a wedge using different approaches. (a) Vibrational temperature contours from the FP model. (b) Translational temperature contours from the FP model. (c) Vibrational temperature contours from DSMC using level to level transition model. (d) Translational temperature contours from DSMC using level to level transition model. Figs. 10(c) and 10(d) are reprinted with permission from I. D. Boyd, Phys. Fluids 3, 1785–1791 (1991). Copyright 2008, American Institute of Physics.

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