DSMC Collision FrequencyTraditional & Sophisticated

Alejandro L. GarciaDept. Physics, San Jose State Univ.Center for Comp. Sci. & Eng., LBNL

Lucky Number 7Graeme’s notes on sophisticated DSMC say that accuracy of collision rate depends on number of particles per cell.

No such dependence in traditional DSMC. Why?

Lucky 7

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 isV

vNf r

V

tvNtfNM r

2

2

21

DSMC Collisions

DSMC uses this result to determine the number of attempted collisions in a cell as

Attempted collisions are accepted with probability,

V

tvNNM MAXr

TRY 2

)1(

MAXr

rACCEPT v

vP

Traditional DSMC Collisions

In traditional DSMC, the average number of collisions is

This gives the correct result since for Poisson,

MAXr

rMAXr

ACCEPTTRY

v

v

V

tvNN

PMM

2

)1(

22)1( NNNNN

Alternative FormulationIn 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.

2NNNNN

ee

V

tvNNM MAXre

TRY 2

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 DSMCIn sophisticated DSMC the time step and cell size vary dynamically so now t and V are also random variables.

Sophisticated DSMC CollisionsIn sophisticated DSMC, the average number of

collisions is

V

tvNN

V

tvNN

v

v

V

tvNN

PMM

rr

MAXr

rMAXr

ACCEPTTRY

2

)1(

2

)1(

2

)1(

?

If N, V, and t are correlated then equality does not hold.

Simple ExampleSuppose 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 ExampleSince the number of particles in a cell is exactly N0

the average number of collisions is

Two problems:

V

tvNNM r 1

2

)1( 00

VV

NNNN

11

)1(22

000

Results Simple ExampleQuick calculation estimates that number of

collisions will be lower by a factor of

<N> Prediction Simulation32 1.00 1.0016 1.00 1.008 0.98 0.994 0.94 0.955 0.75 0.77

21

N

Quick “Fix” in Simple ExampleSince 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

V

tvNM MAXr

TRY 2

20

V

tvNM r 1

2

20

Results for Quick “Fix”Quick calculation estimates that number of

collisions will be higher by a factor of

<N> Prediction Simulation32 1.03 1.0316 1.06 1.068 1.12 1.134 1.25 1.275 1.55 1.57

11

N

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.