Navier Stokes Solvers. Length (Log meters) Particle Methods. Density (Log n/n 0. Knudsen number =1/10 2. Dense Gas

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1 Mesh and Algorithm Reænement Alejandro L. Garcia Center for Comp. Sciences and Engineering Lawrence Berkeley National Laboratory and Dept. Physics, San Jose State University Collaborators: Berni Alder èllnlè, Frank Alexander èbuè, John Bell èlbnlè, Bill Crutchæeld èlbnlè 1

2 Outline Numerical simulation of gas æows combining a particle method and a CFD solver. æ Why use particle methods? æ Direct simulation Monte Carlo èè æ Mesh and Algorithm Reænement èmarè æ Numerical examples æ Future directions 2

3 Particle Methods Hierarchy æ Molecular dynamics æ Boltzmann methods æ Lattice gases æ Pseudo-particle methods 3

4 Dense Gas Continuum vs. Particle When is the continuum description of a gas not accurate? Mean free path Knudsen number ç Characteristic length = ç L 2 Navier Stokes Solvers 0 Particle Methods Knudsen number =1/ Length (Log meters) Density (Log n/n 0 ) 10 4

5 High Kn scenarios æ Aerospace æows æ Micromechanical devices æ Thermal æuctuations and light scattering æ Shock waves and interfaces 5

6 Direct Simulation Monte Carlo is a particle-based algorithm for simulating a dilute gas. Particle collisions are evaluated as a stochastic process. History æ developed by G.A. Bird èlate 60'sè æ Popular in aerospace engineering è70'sè æ Variants & improvements èearly 80'sè æ Applications in physics & chemistry èlate 80'sè æ Used for microscopic æows èearly 90'sè æ Extended to dense gases & liquids èmid 90'sè 6

7 Algorithm æ Initialize system with particles æ Loop over desired number of time steps - Create particles at open boundaries - Move all the particles - Process particleèboundary interactions - Select and execute random collisions 7

8 Collisions æ Sort particles into spatial collision cells æ Loop over collision cells - Compute collision frequency in a cell - Select random collision partners within cell - Process each collision Probability that a pair collides only depends on their relative velocity. Post-collision velocities è6 variablesè given by: æ Conservation of momentum è3 constraintsè æ Conservation of energy è1 constraintsè æ Random collision solid angle è2 choicesè 8

9 vs CFD Advantages: æ Correct at high Kn æ Unconditionally stable æ Boundary conditions are simple æ Correct æuctuation spectrum æ Contains microscopic physics Disadvantage: EXPENSIVE Solution: Only use where it is needed 9

10 Continuum Mesh Reænement Solve equations of the t A =,r æ F èaè using an explicit PDE solver èe.g., Godunovè. CoarseèFine Grid Coupling æ Advance coarse grid æ Fill æneècoarse boundary data - Advance æne grid - Record æuxes at coarseèæne interface - Repeat æne grid calculation æ ëreæux" boundary coarse cells æ Backæll overlying coarse cells 10

11 Mesh Reænement Illustration 11

12 Mesh and Algorithm Reænement Coarseè Coupling æ Advance coarse grid æ Fill boundary data - Create particles in buæer cells - Move all particles - Record particles crossing interface - Discard particles left in buæer region - Collide particles within region - Repeat calculation æ ëreæux" boundary coarse cells æ Implicit correction èviscous solver onlyè æ Backæll overlying coarse cells 12

13 MAR Illustration 13

14 Numerical Examples Parameters æ Hard sphere gas èargonè æ STP conditions upwind æ Mean free path ç = 62:5 nm æ 100 particles per ç 3 æ All computations in full 3D æ Grid size = 2ç æ CFL number = 1=4 æ Explicit Godunov Euler or æ Exèimplicit Godunov NavieríStokes 14

15 Impulsive Piston Euler Rigid wall Rigid wall æ Piston's reference frame æ Mach 2 piston æ Time-dependent æow æ 100 æ 16 æ 16 grid æ Run times: 5 min èeulerè, 13 hr èmarè æ 2 to 7 million particles æ Nearly 200 million collisions 15

16 Piston Density & Temperature Density position (nm) Temperature position (nm) 16

17 Riemann Problem Periodic boundary Euler Euler Periodic boundary æ Mach 2 shock wave jump conditions æ Time-dependent æow æ Initial discontinuity at x = 5000 nm æ 100 æ 16 æ è8 or 16è grid æ Run times: 3 min èeulerè, 2 to 8 hr èeuler MARè, 16 min ènsè, 12 1 to 41 2 hr èns MARè æ 2 to 8 million particles in system æ Over 100 million collisions 17

18 Riemann Problem: Euler Density Initial Discontinuity position (nm) Temperature Initial Discontinuity position (nm) 18

19 Riemann Problem: NavieríStokes Density Initial Discontinuity position (nm) Temperature Initial Discontinuity position (nm) 19

20 Rayleigh Problem Thermal wall Navier-Stokes Reflection wall æ Moving wall's reference frame æ Mach 2 wall; constant temperature æ Time-dependent æow æ 100 æ 16 æ 16 grid æ Run times: 45 min ènav.íst.è, 3 to 6 hr èns MARè æ 2 to 3 million particles æ 40 to 70 million collisions 20

21 Density & Temperature Density position (nm) Temperature position (nm) 21

22 Tangent & Normal Momentum Y-momentum Density position (nm) 6 5 X-momentum density position (nm) 22

23 Flow past a Cylinder Periodic Inflow Euler Periodic Outflow æ Cylinder's reference frame æ Mach 2 inæow æ Steady state æow æ 125 æ 125 æ 4 grid æ region is é 3è of domain æ Run times: 20 hr per 1000 steps èmarè æ 1.5 million particles 23

24 Density past Cylinder 24

25 Particles near Cylinder Sample of particles è1 in 75è 25

26 Particles near Cylinder ècont.è Particles that struck cylinder è1 in 75è 26

27 Future Work æ Parallel version æ Fully adaptive mesh reænement æ Dense gases & liquids ècubaè æ Molecular dynamics MAR æ Applications 27