dsmc collision frequency traditional & sophisticated
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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.