# A collective collision operator for DSMC

Abstract: A new scheme to simulate elastic collisions in particle simulation codes is presented. The new scheme aims at simulating the collisions in the highly collisional regime, in which particle simulation techniques typically become computationally expensive. The new scheme is based on the concept of a grid-based collision field. According to this scheme, the particles perform a single collision with the background grid during a time step. The properties of the background field are calculated from the moments of the distribution function accumulated on the grid. The collision operator is based on the Langevin equation. Based on comparisons with other methods, it is found that the Langevin method overestimates the collision frequency for dilute gases.

## Summary (2 min read)

### INTRODUCTION

- Particle simulation methods and in particular the Direct Simulation Monte Carlo method have proved very successful in simulating complicated physical phenomena in a raretied environment.
- Many problems of practical interest involve dense flows that make DSMC simulation computationally very: demanding, if not impossible.
- Even in the most favorable problems, particle simulation methods have the disadvantage of being computationally expensive.
- These collisions keep reproducing the equilibrium distributions without adding any new information to the flow.
- In a previous paper by Gallis and Torczynski (1), the BGK equation ( 2) and Cercignani's extension (3) w'ere examined as possible collision operators.

### DISCLAIMER

- Portions of this document may be illegible in electronic image products.
- Images are produced from the best available original document.
- For the highly collisional regime, and when used with the Chapman-Enskog expansion, it leads to the Navier-Stokes equations.
- The term rrv~represents the collisions replenis.tig the equilibrium distribution f., and the term mfrepresents the collisions depleting the distributionT he physical interpretation of the assumption that the replenishing number of collisions is given by nvfo is that post-collision molecules are selected from a -elli~distribution at the local average velocity and temperature.
- Several modifications to the BGK equation have been proposed at the expense of the simplicity of the model.

### I LANGEVIN'S EQUATION

- Langevin proposed a method for Brownian motion to model collisions between particles (4).
- It can be shown that the Langevin equation forms the basis for deriving all the physically significant relations concerning the motion of Brownian particles.
- In the DSMC method a fraction of the particles modi@ their properties signtilcantly in every time step while the rest-remain unchanged, but in the Langevin model every particle in a cell receives a small modification of its velocity in every time step.
- The modification constitutes a small change in their properties.
- In cases where the temperature of I'(1 -e ) collision frequency is very low, the probabilistic contribution O adds a vev small vector to the particle velocity.

### APPLICATION TEST CASE

- The flow travels from Iefl to right and the sides of the wedge are assumed to be specularly reflecting.
- Figure 2(a-c ) in the same fashion presents the corresponding translational temperature contour lines.
- At .x = Y = O,the location of the leading edge of the wedge, the shock layer forms, A quantitative comparison indicates that all three simulations are in agreement about the downstream conditions and the shock angle.
- Comparing the figures in more detail, it is seen that the Langevin method predicts a very thin shock compared to the DSMC and BGKC solutions, which are seen to be in remarkable agreement throughout the domain.
- An attempt was made to circumvent this problem by artificially decreasing the collision frequency.

### CONCLUSIONS

- From this comparison, it is clear that the Langevin approach overestimates the coilisiomlity of the flow with respect to the DSMC and BGKC methods.
- As mentioned earlier, the paradigm of the Langevin approach is quite different from that of the Boltzmann equation.
- The Langevin approach maybe suitable for very high densities and charged flows, in which the collisions are so frequent that they can be replaced by a Brownian force acting on the particles.
- Its application to dilute gases does not appear to be appropriate.

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##### Citations

6 citations

### Cites background from "A collective collision operator for..."

...Recently, a few new models were reported to modified the BGK equation by changing the equilibrium distribution [15,16], which were expected to give a realistic Prandtl number and improve the heat transfer modeling....

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##### References

3,171 citations

2,797 citations

### "A collective collision operator for..." refers background or methods in this paper

...Ay =[(2RT/Pr)dij-2(l-Pr)Pij/(rPr)]-(1) (3)...

[...]

...In a previous paper by Gallis and Torczynski (1), the BGK equation (2) and Cercignani's extension (3) were examined as possible collision operators....

[...]

...Cercignani (3) proposed a modification to the BGK equation that would allow it to reproduce the same viscosity and thermal conductivity as the full Boltzmann equation, here referred to as the BGKC method....

[...]

2,088 citations

### "A collective collision operator for..." refers background or methods in this paper

...The generalization of the BGK equation is obtained by the replacement of the Maxwellian distribution with a local anisotropic three-dimensional Gaussian, referred to as the ellipsoidal statistical (ES) model: 3 I (2) /0=p- (3)/(2)(detA)exp...

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...In a previous paper by Gallis and Torczynski (1), the BGK equation (2) and Cercignani's extension (3) were examined as possible collision operators....

[...]

126 citations

44 citations

### "A collective collision operator for..." refers background or methods in this paper

...= n n ( f 0 - f ) (1) J collision where f0 is the Maxwellian distribution corresponding to the local temperature and average velocity, / is the initial distribution of the particles, n the number density, and v is the collision frequency....

[...]

...In a previous paper by Gallis and Torczynski (1), the BGK equation (2) and Cercignani's extension (3) were examined as possible collision operators....

[...]

...Recently this method was introduced into particle simulations with reasonable success (1)....

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