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

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
f 
N vr
V
So the total number of collisions in a time step is
M  12 N f t 
N 2 vr t
2V
DSMC Collisions
DSMC uses this result to determine the number of
attempted collisions in a cell as
N ( N  1) vr MAX t
M TRY 
2V
Attempted collisions are accepted with probability,
PACCEPT 
vr
vr MAX
Traditional DSMC Collisions
In traditional DSMC, the average number of
collisions is
M  M TRY PACCEPT

N ( N  1)  vr MAX t
2V

vr
vr MAX
This gives the correct result since for Poisson,
N ( N  1)  N
2
 N  N
2
Alternative Formulation
In Graeme’s 1994 book he uses
M TRY 
N N e  vr MAX t
2V
This also gives the correct result since,
N N
e
 N N
e
 N
2
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 t and V are also random variables.
Sophisticated DSMC Collisions
In sophisticated DSMC, the average number of
collisions is
M  M TRY PACCEPT
N ( N  1) vr MAX t
vr


vr MAX
2V
N ( N  1) vr t ? N ( N  1)  vr t


2V
2V
If N, V, and t 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
N 0 ( N 0  1) vr t 1
M 
2
V
Two problems:  N 0 ( N 0  1)  N 02  N
1
1


V
V
2
Results Simple Example
Quick calculation estimates that number of
collisions will be lower by a factor of 1  N
<N>
32
16
8
4
2
Prediction
1.00
1.00
0.98
0.94
0.75
Simulation
1.00
1.00
0.99
0.95
0.77
2
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
M TRY
so that
N 02 vr MAX t

2V
N 02 vr t 1
M 
2
V
Results for Quick “Fix”
Quick calculation estimates that number of
collisions will be higher by a factor of 1  N
<N>
32
16
8
4
2
Prediction
1.03
1.06
1.12
1.25
1.55
Simulation
1.03
1.06
1.13
1.27
1.57
1
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.