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Discrete unified gas-kinetic scheme for compressible flows

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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)

- Motivation
- Formulation and properties
- Numerical results
- Summary

Motivation

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

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

= 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

MD

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

# 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

# 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)

- Motivation
- Formulation and properties
- Numerical results
- Summary

# Kinetic model (BGK-type)

Distribution function

Particel velocity

Equilibrium:

Flux

Conserved variables

Maxwell (standard BGK)

Example:

Shakhov model

ES model

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

# Formulation: A finite-volume scheme

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

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

n

n

Bounce-back

Diffuse Scatting

# 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

# Comparison with UGKS

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 (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

- 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

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

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

- Motivation
- Formulation and properties
- Numerical results
- Summary

Test cases

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

Collision model:

Shakhov model

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

Again, DUGKS agree with UGKS excellently

DUGKS as a shock capturing scheme

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

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

=10: Free-molecular flow

=1: transition flow

=0.1: low transition flow

=0.001: slip flow

=1.0e-5: continuum flow

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

Heat Flux

Kn=0.075

Velocity

UGKS: Huang, Xu, and Yu, CiCP 12 (2012)

Present DUGKS

Temperature and Heat Flux

Kn=1.44e-3; Re=100

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)

Velocity

DUGKS

LBM

DUGKS

LBM

Pressure fields

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!