Relaxation time simulation method with internal energy exchange for perfect gas flow at near-continuum conditions

TL;DR: In this paper, an internal energy exchange scheme for the relaxation time simulation method (RTSM) which solves the BGK equation for the perfect gas flow at near-continuum region discrete rotational energies are introduced to model the relaxation of internal energy modes.

About: This article is published in Communications in Nonlinear Science and Numerical Simulation.The article was published on 2007-10-01 and is currently open access. It has received 6 citations till now. The article focuses on the topics: Perfect gas & Internal energy.

Summary

1. Introduction

• Rarefied gas flow is an important problem in aeronautics and astronautics.
• In resent years, the rapid development of MEMS technique has brought us an exigent requirement for the investigation of gas flow in micro systems [1, 2] .
• The DSMC method is suitable for gas flows with a high Knudsen number.
• In micro flows the gas can be dense despite a high Knudsen number due to the small characteristic length [4] .
• No collisions between particles are calculated and the effect of collisions is simulated by redistributing the total momentum and energy of all the particles in each cell at each time step amongst all the particles in the cell.

2.1 Relaxation Time simulation method

• The BGK approximation simplifies the collision term on the right hand side of the Boltzmann equation by using a relaxation time that poses the greatest mathematical difficulties and needs modify.
• The best known model equation is called the BGK equation after Bhatnager, Gross and Krook [10, 11] .
• That is to say, the distribution function relaxes towards equilibrium with a time interval of τ for all velocities.
• The relaxation time τ is a little different from the collision time.

2.3 Redistribution procedures

• After the local viscosity µ and the rotational relaxation number R Z , which are function of local temperature in cells, are calculated, the probabilities for translational relaxation and rotational relaxation can be obtained from Eqs. (5-8).
• Both the translational energy and rotational energy, which need redistribution, form a thermal energy pool with a total energy tot E .
• Thus the characteristic temperature of the re-distributed energy can be calculated EQUATION.
• All the rescaling procedures for either the translational redistribution or the rotational redistribution will be based on this characteristic temperature.
• It is clear that this temperature equals to the sampled temperature in one cell for Pullin's EDSM [5] .

3. Results and Discussion

• The present algorithm was performed in FORTRAN based on the standard DSMC code [3] in UNIX system.
• The INDEX technique of DSMC is introduced to avoid tracking each particle so that little additional memory is required.
• To verify the new models, a 2D gas flow in a micro channel is simulated and the results are compared with the standard DSMC method.

Fig. 1 Channel flow with freesteam incoming gas

• The channel walls are isothermal and the temperature is 300 K.
• Full diffuse model is used to calculate the collisions between the molecules and the walls.
• When the gas density is larger than a certain value, the RTSM method will be more efficient than the DSMC method.

4. Conclusions

• The Relaxation Time simulation method (RTSM) was modified and improved by introducing the internal energy exchange scheme.
• The Larsen-Borgnakke model with discrete rotational energies is introduced to model the energy exchange between the translational and internal modes.
• The developed RTSM agrees better with the standard DSMC with little additional computational cost.
• The present results show a possibility of a hybrid RTSM/DSMC code for the continuum/rarefied gas flow.

Figures

Fig. 1 Channel flow with freesteam incoming gas

Fig. 3 (b) Temperature contours. RTSM, Scheme A

Fig. 3 (b) Temperature contours. RTSM, Scheme A

Fig. 2 Velocity and pressure along the midline of the channel

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Relaxation time simulation method with internal energy exchange for perfect gas flow at near-continuum region" ?

This paper presents an internal energy exchange scheme for the Relaxation Time simulation method ( RTSM ) which solves the BGK equation for the perfect gas flow at near-continuum region. The LarsenBorgnakke model with discrete rotational energies is introduced to model the energy exchange between the translational and internal modes. 

Q2. What future works have the authors mentioned in the paper "Relaxation time simulation method with internal energy exchange for perfect gas flow at near-continuum region" ?

Although the Prandtl number in the RTSM is still overestimated, the present results show a possibility of a hybrid RTSM/DSMC code for the continuum/rarefied gas flow. 

Q3. What is the standard computational method for the highKnudsen-number flows?

Bird’s directsimulation Monte Carlo (DSMC) method[3] is the standard computational method for the highKnudsen-number flows, where the governing equation is the Boltzmann equation. 

Q4. What is the particle simulation method for DSMC?

Pullin[5] proposed a particle simulation method called the Equilibrium Particle Simulation Method (EPSM) as the infinite collision rate of DSMC for a given cell network and number of simulator particles. 

Q5. What is the effect of collisions between particles in DSMC?

no collisions between particles are calculated and the effect of collisions is simulated by redistributing the total momentum and energy of all the particles in each cell at each time step amongst all the particles in the cell. 

Q6. What is the exact solution of Eq. 2?

(2)The exact solution of Eq. (2) is0 0( ) ( (0) )exp( / )f t f f t fτ= − − − , (3)where (0) ( 0)f f t= = is the particle velocity distribution established by the convection phase of thesimulation before the effect of collision is simulated. 

Q7. What is the particle number for velocity distribution determined from the local relaxation time?

The particle number for velocity distribution determined from the local relaxation time can be derived from the cell density and temperature and any desired viscosity law. 

Q8. What is the BGK approximation for the collision term?

Based on this relaxation time BGK approximation the collision term can be approximated as0( ) coll nf n f f t τ ∂⎡ ⎤ = −⎢ ⎥∂⎣ ⎦ . 

Q9. What is the key part of the RTSM method?

Therefore when the Larsen-Borgnakke model is introduced into the RTSM method, the crucial part is the determination of the probability of the inelastic parts. 

Q10. What is the common method of RTSM?

The results show that the rotational relaxation in scheme A improves the RTSM results with better agreement with the standard DSMC results, when comparing with the scheme B and non-rotational relaxation RTSM. 

Q11. How can the re-distributed energy be calculated?

Thus the characteristic temperature of the re-distributed energy can be calculated as2 / 3 2 tot ct rE kT N N = +(9)All the rescaling procedures for either the translational redistribution or the rotationalredistribution will be based on this characteristic temperature. 

Q12. What is the RTSM method for near continuum flow?

These comparisonsshow that for the near continuum flow the RTSM method is more efficient than the DSMCmethod and could replace DSMC in that region.