Download Direct Simulation Monte Carlo - Lecture Notes | ASTR 596 and more Study notes Astronomy in PDF only on Docsity!
Direct Simulation Monte Carlo (DSMC)
Developed by G. Bird (1960s) Stochastic method for solving gasdynamics flows with Kn ~ 1 (mean free path ๎ ~ size of system L ); works well into continuum regime (Kn โช 1) Used frequently in aerospace applications โ also planetary atmospheres, etc. Basic ingredients: โ (^) Particles โ Monte Carlo sampling of real particle velocity field โ (^) Trajectories โ integration of particle motion between collisions โ (^) Collisions โ fast method for including effects of collisions with appropriate correlations โ (^) Boundary conditions โ interactions with surfaces
Basic DSMC algorithm
- Initialize particles using known velocity distribution (Maxwellian)
- Loop:
- Advance particles along free trajectories through time interval ๎ญ t
- Apply boundary conditions to those particles that intersect boundaries
- Choose random pairs of particles ( i , j ) to scatter
- Apply scattering rule to these pairs; modifies their velocities going into the next step
- Compute average quantities โฉ v 2 โช, etc. Choice of ๎ญ t : smaller than average collision time
Scattering
To handle scattering we use a โmeshโ similar to the chaining mesh we use for P 3 M. Particles are associated with cells of size R > ๎ (e.g. by storing in linked lists). For each cell: loop until collisions have occurred: Choose collision pair ( i , j ). Decide if the pair will collide: Choose a random number ๎ ij
๎
[0,1].
If | v i
ij v max , they collide. v max is the largest particle speed in the cell. If they collide: Choose new random directions for both particles. Conserve momentum and energy in the process.
n
2
๎ v rms R
3
๎ญ t
R
Scattering
Probability of scattering depends only on relative particle velocities, not on their positions (except that they must be in same scattering cell). In particular, detailed dynamics of scattering process are not followed. If scattering depends on impact parameter, use โmolecular chaosโ assumption (implicit in the basic procedure) to choose a value at random. Conservation of energy and momentum (elastic hard spheres):
v
cm
๎ v
1
๎ v
2
E =
๎โฃ v 1 โ v cmโฃ
2
๎โฃ v
2
โ v
cm
2
cos ๎พ = random number in [โ1,1]
๎ = random number in [0, 2 ๎]
v 1 ' = v cm๎๎ E ๎sin ๎พcos ๎ , sin ๎พ sin ๎ , cos ๎พ๎
v 2 ' = v cmโ๎ E ๎sin ๎พcos ๎ , sin ๎พsin ๎ , cos ๎พ๎
v
1
' โ v
cm
v
2
' โ v
cm
v
1
โ v
cm
v
2
โ v
cm
Applications
Planetary rings Frezzotti
Applications
Mars Reconnaissance Orbiter aerobraking โ Hanna Prince & Striepe
