DSMC Collision Frequency Traditional & Sophisticated Alejandro L. Garcia Dept. Physics, San Jose State Univ. Center for Comp. Sci. & Eng., LBNL

Lucky Number 7 Graeme’s notes on sophisticated DSMC say that accuracy of collision rate depends on number of particles per cell. Lucky 7 No such dependence in traditional DSMC. Why?

Collision Frequency From basic kinetic theory, collision frequency (number of collisions per particle per unit time in a volume V) is So the total number of collisions in a time step is

DSMC Collisions DSMC uses this result to determine the number of attempted collisions in a cell as Attempted collisions are accepted with probability,

Traditional DSMC Collisions In traditional DSMC, the average number of collisions is This gives the correct result since for Poisson,

Alternative Formulation In Graeme’s 1994 book he uses This also gives the correct result since, As mentioned in his notes for this meeting, the approach is now obsolete.

Nearest Particle Selection In traditional DSMC, collisions partners are drawn at random in a cell. In sophisticated DSMC, the nearest particle in the cell is used as the collision partner (unless those two particles recently collided). Does this introduce a bias in average relative velocity if number of particles in a cell is small? Preliminary 1D runs indicated that it does not bias the acceptance rate or collision frequency.

Sophisticated DSMC In sophisticated DSMC the time step and cell size vary dynamically so now Dt and V are also random variables.

Sophisticated DSMC Collisions In sophisticated DSMC, the average number of collisions is If N, V, and Dt are correlated then equality does not hold.

Simple Example Suppose we dynamically make the cell sizes such that the number of particles in a cell is exactly N0 This simple example is not sophisticated DSMC yet it illustrates the effect of a dynamically variable cell volume.

Collisions in Simple Example Since the number of particles in a cell is exactly N0 the average number of collisions is Two problems:

Results Simple Example Quick calculation estimates that number of collisions will be lower by a factor of <N> Prediction Simulation 32 1.00 1.00 16 1.00 1.00 8 0.98 0.99 0.94 0.95 0.75 0.77

Quick “Fix” in Simple Example Since the number of particles in a cell is exactly N0 we might think that instead we should compute the number of attempted collisions as so that

Results for Quick “Fix” Quick calculation estimates that number of collisions will be higher by a factor of <N> Prediction Simulation 32 1.03 1.03 16 1.06 1.06 8 1.12 1.13 1.25 1.27 1.55 1.57

Conclusion Sophisticated DSMC is a powerful and useful extension to traditional DSMC. For many reasons we SHOULD NOT be thinking of returning to traditional DSMC. The development of traditional DSMC benefitted from theoretical analysis. Sophisticated DSMC is more complex so this analysis will be more difficult, but still needed.