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Discrete unified gas-kinetic scheme for compressible flows

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### 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

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)

Outline Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

- Motivation
- Formulation and properties
- Numerical results
- Summary

Motivation Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

Non-equilibrium flows covering different flow regimes

Free-molecular

Continuum

Transition

Slip

10-3

10-2

10-1

100

10

Re-Entry Vehicle

Inhalable particles

Chips

Challenges in numerical simulations Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

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

Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China = const

Lockerby’s test (2005, Phys. Fluid)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

MD Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

NS

A popular technique: hybrid method

Limitations

Numerical rather than physical

Artifacts

Time coupling

Dynamic scale changes

Hadjiconstantinou

Int J Multiscale Comput Eng 3 189-202, 2004

Hybrid method is inappropriate for problems with dynamic scale changes

Efforts based on kinetic description of flows Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

# 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, 3781 (2008).

[5] K. Xu and J.-C. Huang, J. Comput. Phys. 229, 7747 (2010)

Efforts based on kinetic description of flows Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

# 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)

Outline Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

- Motivation
- Formulation and properties
- Numerical results
- Summary

# Kinetic model (BGK-type) Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

Distribution function

Particel velocity

Equilibrium:

Flux

Conserved variables

Maxwell (standard BGK)

Example:

Shakhov model

ES model

Conserved variables Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

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

# Formulation: A finite-volume scheme Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

Trapezoidal

Mid-point

j+1/2

j

j+1

Point 1: Updating rule for cell-center distribution function

1. integrating in cell j:

2. transformation:

3. update rule:

Key: distribution function at cell interface

explicit Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

Implicit

j+1/2

So

j

j+1

Point 2: Evolution of the cell-interface distribution function

How to determine

Again

Again

1. integrating along the characteristic line

2. transformation:

Slope

3. original:

# Boundary condition Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

n

n

Bounce-back

Diffuse Scatting

# Properties of DUGKS Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

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

# Comparison with UGKS Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

j+1/2

j

j+1

Unified GKS (Xu & Huang, JCP 2010)

Starting Point:

Macroscopic flux

Updating rule:

If the cell-interface distribution f(t) is known, the update bothfand W can be accomplished

Unified GKS Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China (cont’d)

j+1/2

j

j+1

Key Point:

Integral solution:

Free transport

Equilibrium

After some algebraic, the above solution can be approximated as

Chapman-Enskog expansion

Free-transport

DUGKS vs UGKS Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

- 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

# Comparison with Finite-Volume LBM Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

ci

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

# Comparison with Finite-Volume LBM Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

j+1/2

j

j+1

Finite-volume LBM (Peng et al, PRE 1999; Succi et al, PCFD 2005; )

Micro-flux is reconstructed without considering collision effects

Viscosity:

Numerical dissipation cannot be absorbed

Limitations(Succi, PCFD, 2005):

2. Large numerical dissipation

1. time step is limited by collision time

Difference between DUGKS and FV-LBM:

DUGKS is AP, but FV-LBM not

Outline Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

- Motivation
- Formulation and properties
- Numerical results
- Summary

Test cases Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

- 1D shock wave structure
- 1D shock tube
- 2D cavity flow

Collision model:

Shakhov model

1D shock wave structure Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

Parameters: Pr=2/3, = 5/3, Tw

Left: Density and velocity profiles; Right: heat flux and stress (Ma=1.2)

DUGKS agree with UGKS excellently Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

Again, DUGKS agree with UGKS excellently Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

DUGKS as a Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China shock capturing scheme

Density (Left) and Temperature (Right) profile with different grid resolutions (Ma=1.2, CFL=0.95)

1D shock tube problem Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

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

= Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China 10: Free-molecular flow

= Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China 1: transition flow

= Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China 0.1: low transition flow

= Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China 0.001: slip flow

Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China =1.0e-5: continuum flow

2D Cavity Flow Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

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

Kn=0.075 Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

Heat Flux

Kn=0.075 Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

Velocity

UGKS: Huang, Xu, and Yu, CiCP 12 (2012) Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

Present DUGKS

Temperature and Heat Flux

Kn=1.44e-3; Re=100

Comparison with LBM Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

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)

Velocity Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

DUGKS

LBM

DUGKS Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

LBM

Pressure fields

Summary Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

- 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!

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