1. Discrete unified gas-kinetic scheme for compressible flows
Sino-German Symposium on Advanced Numerical Methods for Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China
Discrete unified gas-kinetic scheme for compressible flows
Zhaoli Guo
(Huazhong University of Science and Technology, Wuhan, China)
Joint work with Kun Xu and Ruijie Wang (Hong Kong University of Science and Technology)

2. Outline Motivation Formulation and properties Numerical results
Summary

3. Motivation Non-equilibrium flows covering different flow regimes Slip
 Re-Entry Vehicle
Chips
Inhalable particles
Slip
Continuum
Transition
Free-molecular 10
10-3
10-1
100
10-2

4. Challenges in numerical simulations
Modern CFD:
Based on Navier-Stokes equations
Efficient for continuum flows
does not work for other regimes
Particle Methods: (MD, DSMC… )
Noise
Small time and cell size
Difficult for continuum flows / low-speed non-equilibrium flows
Method based on extended hydrodynamic models :
Theoretical foundations
Numerical difficulties (Stability, boundary conditions, ……)
Limited to weak-nonequilibrium flows

5. Lockerby’s test (2005, Phys. Fluid)
 = const
the most common high-order continuum equation sets (Grad’s 13 moment, Burnett, and super-Burnett equations ) cannot capture the Knudsen Layer, Variants of these equation families have, however, been proposed and some of them can qualitatively describe the Knudsen layer structure … the quantitative agreement with kinetic theory and DSMC data is only slight

6. A popular technique: hybrid method
Limitations MD NS
Numerical rather than physical
Artifacts
Time coupling
Dynamic scale changes
Hadjiconstantinou
Int J Multiscale Comput Eng , 2004
Hybrid method is inappropriate for problems with dynamic scale changes

7. Efforts based on kinetic description of flows
# Discrete Ordinate Method (DOM) [1,2]:
Time-splitting scheme for kinetic equations (similar with DSMC)
dt (time step) <  (collision time)
dx (cell size) <  (mean-free-path)
numerical dissipation  dt
Works well for highly non-equilibrium flows, but encounters difficult for continuum flows
# Asymptotic preserving (AP) scheme [3,4]:
Consistent with the Chapman-Enskog representation in the continuum limit (Kn  0)
dt (time step) is not restricted by  (collision time)
at least 2nd-order accuracy to reduce numerical dissipation [5]
Aims to solve continuum flows, but may encounter difficulties for free molecular flows
[1] J. Y. Yang and J. C. Huang, J. Comput. Phys. 120, 323 (1995)
[2] A. N. Kudryavtsev and A. A. Shershnev, J. Sci. Comput. 57, 42 (2013).
[3] S. Pieraccini and G. Puppo, J. Sci. Comput. 32, 1 (2007).
[4] M. Bennoune, M. Lemo, and L. Mieussens, J. Comput. Phys. 227, (2008).
[5] K. Xu and J.-C. Huang, J. Comput. Phys. 229, 7747 (2010)

8. Efforts based on kinetic description of flows
# Unified Gas-Kinetic Scheme (UGKS) [1]:
Coupling of collision and transport in the evolution
Dynamicly changes from collision-less to continuum according to the local flow
The nice AP property
A dynamic multi-scale scheme, efficient for multi-regime flows
In this report, we will present an alternative kinetic scheme (Discrete Unified Gas-Kinetic Scheme), sharing many advantages of the UGKS method, but having some special features .
[1] K. Xu and J.-C. Huang, J. Comput. Phys. 229, 7747 (2010)

9. Outline Motivation Formulation and properties Numerical results
Summary

10. # Kinetic model (BGK-type)
Distribution function
Particel velocity
Equilibrium:
Conserved variables
Flux
Maxwell (standard BGK)
Example:
Shakhov model
ES model

11. Conserved variables
Conservation of the collision operator
A property: for any linear combination of f and f eq , i.e.,
The conservation variables can be calculated by

12. # Formulation: A finite-volume schemej
j+1
j+1/2
Point 1: Updating rule for cell-center distribution function
1. integrating in cell j:
Trapezoidal
Mid-point
2. transformation:
3. update rule:
Key: distribution function at cell interface

13. Point 2: Evolution of the cell-interface distribution function
How to determine j
j+1
j+1/2
Again
Again
1. integrating along the characteristic line
explicit
Implicit
2. transformation: So
Slope
3. original:

14. # Boundary condition
Bounce-back n
Diffuse Scatting n

15. # Properties of DUGKS 1. Multi-dimensional
It is not easy to device a wave-based multi-dimensional scheme based on hydrodynamic equations
In the DUGKS, the particle is tracked instead of wave in a natural way (followed by its trajectory)
2. Asymptotic Preserving (AP)
(a) time step (t) is not limited by the particle collision time ():
(b) in the continuum limit (t >> ):
Chapman-Ensokg expansion
in the free-molecule limit: (t << ):
(c) second-order in time; space accuracy can be ensured by choosing linear or
high-order reconstruction methods

16. # Comparison with UGKS Unified GKS (Xu & Huang, JCP 2010)
Starting Point:
Macroscopic flux
Updating rule: j
j+1
j+1/2
If the cell-interface distribution f(t) is known, the update both f and W can be accomplished

17. Unified GKS (cont’d) Key Point: j j+1 j+1/2 Integral solution:
Free transport
Equilibrium
After some algebraic, the above solution can be approximated as
Chapman-Enskog expansion
Free-transport

18. DUGKS vs UGKS Common: Finite-volume formulation; AP property;
collision-transport coupling
(b) Differences: in DUGKS
W are slave variables and are not required to update simultaneously with f
Using a discrete (characteristic) solution instead of integral solution in the construction of cell-interface distribution function

19. # Comparison with Finite-Volume LBMci
Lattice Boltzmann method (LBM)
Standard LBM: time-splitting scheme
Collision
Free transport
Evolution equation:
Viscosity:
Numerical viscosity is absorbed into the physical one
Limitations:
1. Regular lattice
2. Low Mach incompressible flows

20. # Comparison with Finite-Volume LBM
Finite-volume LBM (Peng et al, PRE 1999; Succi et al, PCFD 2005; ) j
j+1
j+1/2
Micro-flux is reconstructed without considering collision effects
Viscosity:
Numerical dissipation cannot be absorbed
Limitations (Succi, PCFD, 2005):
1. time step is limited by collision time
2. Large numerical dissipation
Difference between DUGKS and FV-LBM:
DUGKS is AP, but FV-LBM not

21. Outline Motivation Formulation and properties Numerical results
Summary

22. Test cases 1D shock wave structure 1D shock tube 2D cavity flow
Collision model:
Shakhov model

23. 1D shock wave structure Parameters: Pr=2/3,  = 5/3, Tw
Left: Density and velocity profiles; Right: heat flux and stress (Ma=1.2)

24. DUGKS agree with UGKS excellently

25. Again, DUGKS agree with UGKS excellently

26. DUGKS as a shock capturing scheme
Density (Left) and Temperature (Right) profile with different grid resolutions (Ma=1.2, CFL=0.95)

27. 1D shock tube problem Parameters: Pr=0.72,  = 1.4, T0.5
Domain: 0  x  1;
Mesh: 100 cell, uniform
Discrete velocity : 200 uniform gird in [-10 10]
Reference mean free path
By changing the reference viscosity at left boundary, the flow can changes from continuum to free-molecular flows

28. =10: Free-molecular flow

29. =1: transition flow

30. =0.1: low transition flow

31. =0.001: slip flow

32. =1.0e-5: continuum flow

33. 2D Cavity Flow Parameters: Pr=2/3,  = 5/3, T0.81
Domain: 0  x, y  1;
Mesh: 60x60 cell, uniform
Discrete velocity : 28x28 Gauss-Hermite
Parameters: Pr=2/3,  = 5/3, T0.81
Kn=0.075
Temperature.
White and background: DSMC
Black Dashed: DUGKS

34. Kn=0.075
Heat Flux

35. Kn=0.075
Velocity

36. Temperature and Heat Flux
UGKS: Huang, Xu, and Yu, CiCP 12 (2012)
Present DUGKS
Temperature and Heat Flux
Kn=1.44e-3; Re=100

37. Comparison with LBM Stability: Re=1000
LBM becomes unstable on 64 x 64 uniform mesh
UGKS is still stable on 20 x 20 uniform mesh
80 x 80 uniform mesh
LBM becomes unstable as Re=1195
UGKS is still stable as Re=4000 (CFL=0.95)

38. Velocity
DUGKS
LBM

39. DUGKS
LBM
Pressure fields

40. Thank you for your attention!
Summary
The DUGKS method has the nice AP property
The DUGKS provides a potential tool for compressible flows in different regimes
Thank you for your attention!