1. Hybrid Particle-Continuum Computation of
Nonequilibrium Multi-Scale Gas Flows
Iain D. Boyd Department of Aerospace Engineering University of Michigan Ann Arbor, MI Support Provided By: MSI, AFOSR, NASA

2. Overview Problems with DSMC for low-speed MEMS flows.
The Information Preservation (IP) method:
basic ideas and algorithms;
validation studies.
Use of IP in a hybrid CFD-DSMC method:
data communication and interface location;
hypersonic flow applications.
Future directions.

3. Gas Flows at the Micro–Scale
Examples:
accelerometers (air-bags);
micro-turbines (e.g. Karlsruhe);
Knudsen pumps;
Micro-Air-Vehicles (MAV’s).
Physical characteristics:
atmospheric pressure;
characteristic length ~ 10-6 m;
Knudsen number ~ 0.01;
non-continuum effects.

4. DSMC: Established Particle Method for Gas Flows
Direct simulation Monte Carlo (DSMC):
developed by Bird (1960’s);
particles move in physical space;
particles possess microscopic properties, e.g. u’ (thermal velocity);
cell size ~ l, time step ~ 1/n;
collisions handled statistically;
ideal for supersonic/hypersonic flows;
statistical scatter for subsonic flow. {
u’, v’, w’
x, y, z
erot, evib

5. Subsonic Flow Over a 5% Flat Plate
Total sample size:760,000 particles/cell
Flat Plate
Density field
Pressure field

6. Information Preservation (IP)
Relatively new method:
first proposed by Fan and Shen (1998);
further developed by Sun, Boyd, et al.;
evolves alongside DSMC;
particles and cells possess preserved information, e.g. n, <u>, T;
Dn from mass conservation;
D<u> from momentum conservation;
DT from energy conservation;
aims to reduce statistical fluctuations;
may provide connection to CFD code. } Dn
D<u> DT {
u’, v’, w’
n, <u>, T
x, y, z
erot, evib

7. Outline of IP Algorithm
select collision pairs
perform binary collisions
inject new particles Dt
move particles, compute boundary interactions
update information
sample cell information

8. Information Preservation Method
Statistical scatter (from a flow over a flat plate):
Time step: 1
Sampling size: 20
Time step: 1000
Sampling size: 20,000
Time step: 100,000
Sampling size: 2,000,000

9. Information Preservation Method: Computational Performance
Numerically efficient implementation:
only 20% overhead above DSMC;
parallelized using domain decomposition.

10. Thermal Couette Flow
T=373K 1m
argon gas y
T=173K

11. High Speed Couette Flow
V=300m/s, T=273K y 1m
argon gas
V=0m/s, T=273K

12. Rayleigh Flow V=0 T=273K y argon gas t=0: Vx=0, Tw=273K x
t>0: Vx=10m/s, Tw=373K Vx

13. Flow over flat plate of zero thickness
M=0.2
Free-molecular theory: Cd=1.35/M
Compressible experiment:
Schaaf and Sherman (1954)
Slip Oseen-Stokes solution:
Tamada and Miura (1978)
Incompressible N-S:
Dennis and Dunwoody (1966)
Incompressible experiment:
Janour (1951)

14. Flow over a NACA-0012 Airfoil
Air: Ma = 0.8, Re = 73, Kn = 0.014, Lchord= 0.04m
DSMC IP

15. Subsonic Flow Over a 5% Flat Plate
Air: Ma = 0.1, Re = 100, Kn = 0.01, L= 100 m
Density field
Pressure field

16. Flow Over a 5% Flat Plate Pressure contours (angle of attack 10o) IPNS

17. Flow Over a 5% Flat Plate Surface properties IP NS pressure
skin friction

18. Hybrid Method for Hypersonic Flows
Hypersonic vehicles encounter a variety of flow regimes:
continuum: modeled accurately and efficiently with CFD;
rarefied: modeled accurately and efficiently with DSMC.
A hybrid DSMC-CFD method is attractive for mixed flows:
CFD: Navier-Stokes finite-volume algorithm;
DSMC: MONACO+ Information Preservation (DSMC/IP).
Rarefied DSMC approach:
based on kinetic theory
high altitude
sharp edges
Continuum CFD approach:
solve NS equations
low altitude
long length scales

19. Motivation for Hybrid Method
Flow Regimes:
continuum
slip
transitional
free-molecular Kn
0.01
0.1 10
DSMC
Model Accuracy:
Boltzmann Equation
Navier-Stokes
Collisionless
Boltzmann Eqn
Euler
Burnett
high
Numerical Performance:
DSMC
CFD Kn
0.01
0.1 10
low
Accuracy + }
hybrid approach
Performance

20. Sometimes CFD Works Best

21. Sometimes DSMC Works Best

22. MacCormack and Candler Comp. Fluids, 17, 1989
Hybrid Approach
MacCormack and Candler Comp. Fluids, 17, 1989
Sun and Boyd
J. Comp. Phys., 179, 2002
Navier-Stokes solver
DSMC/IP method
2nd order accurate modified
Steger-Warming flux-vector
splitting approach (FVM)
Macroscopic properties are
preserved as well as microscopic
particle information
interface

23. Domain Coupling

24. Continuum Breakdown Parameters
Interface Location:
Continuum Breakdown Parameters
Local Knudsen number, hypersonic flow (Boyd et al., 1995)
where Q= r, T, V.
Determined through detailed DSMC versus CFD comparisons:
Parameters also under investigation for use inside DSMC:
- failure of breakdown parameters at shock front;
- use DSMC to evaluate continuum onset parameter?

25. Cut-off Value = 0.05

26. Hybrid CFD/DSMC-IP Process
Solution
Interface &
Particle Domain
Determined
Hybrid
Calculation
Final
Solution No
Yes
End

27. Summary of Hybrid Code Numerical methods:
2d/axially symmetric, steady state;
CFD: explicit, finite volume solution of NS Eqs.;
DSMC: based on MONACO;
interface: Information Preservation scheme;
implementation: a single, parallel code.
Physical modeling:
simple, perfect gas (rotation, but no vibration);
walls: slip / incomplete accommodation;
breakdown parameter: local Knudsen number.

28. Normal Shock Waves of Argon
Numerical Example (1)
Normal Shock Waves of Argon
Argon normal shocks investigated:
relatively simple hypersonic flow;
Alsmeyer experimental measurements;
well-known case for testing new algorithms.
Simulations:
modeled in 2D (400 x 5 cells);
initialized by jump conditions;
pure DSMC;
pure CFD (Navier-Stokes equations);
hybrid code initialized by CFD solution.

29. Mach 5 Profiles

30. Reciprocal Shock Thickness

31. Numerical Performance
Hybrid simulation employed 57% particle cells
DSMC time-step could be 20 times larger

32. Numerical Example (2) Blunted Cone 104.4 T (K) 297.2 2680.2 2630.4
5.61810-4
1.310-4
12.6
Tw (K)
Tvib (K)
U (m/s)
r  (kg/m3)
l  (m)
Ma

33. Particle Domain (Knmax > 0.03)

34. Comparisons of Density Contours

35. Comparisons of
Temperature Contours

36. Comparisons of
Surface Properties

37. Along the Stagnation Streamline
Detailed Comparisons
Along the Stagnation Streamline

38. Detailed Comparisons
at x = 2 cm

39. Detailed Comparisons
at x = 4 cm

40. Summary Simulation of subsonic nonequilibrium gas flows:
difficulties caused by low speed and rarefaction;
DSMC not effective due to statistical fluctuations;
IP method developed and applied to several flows;
IP is effective at reducing statistical fluctuations;
IP provides a basis for creating a hybrid method.
Simulation of hypersonic nonequilibrium gas flows:
vehicle analysis involves continuum/rarefied flow;
hybrid CFD-DSMC/IP method looks promising;
physical modeling must be improved;
numerical performance must be improved;
much work still to be done!

41. Future Directions Algorithm development for hybrid method:
CFD: parallel, implicit solver on unstructured mesh;
DSMC: implicit and/or other acceleration schemes.
Physics development for hybrid method:
vibrational relaxation;
chemically reacting gas mixture.
Development of hybrid methodology:
refinement of breakdown parameters;
evaluation against data (measured, computed).