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)

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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

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)


Outline

Outline

  • Motivation

  • Formulation and properties

  • Numerical results

  • Summary


Discrete unified gas kinetic scheme for compressible flows

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


Discrete unified gas kinetic scheme for compressible flows

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


Lockerby s test 2005 phys fluid

= 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


Discrete unified gas kinetic scheme for compressible flows

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


Discrete unified gas kinetic scheme for compressible flows

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)


Discrete unified gas kinetic scheme for compressible flows

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)


Outline1

Outline

  • Motivation

  • Formulation and properties

  • Numerical results

  • Summary


Discrete unified gas kinetic scheme for compressible flows

# Kinetic model (BGK-type)

Distribution function

Particel velocity

Equilibrium:

Flux

Conserved variables

Maxwell (standard BGK)

Example:

Shakhov model

ES model


Discrete unified gas kinetic scheme for compressible flows

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


Discrete unified gas kinetic scheme for compressible flows

# 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


Discrete unified gas kinetic scheme for compressible flows

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:


Discrete unified gas kinetic scheme for compressible flows

# Boundary condition

n

n

Bounce-back

Diffuse Scatting


Discrete unified gas kinetic scheme for compressible flows

# 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


Discrete unified gas kinetic scheme for compressible flows

# 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


Discrete unified gas kinetic scheme for compressible flows

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


Discrete unified gas kinetic scheme for compressible flows

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


Discrete unified gas kinetic scheme for compressible flows

# 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


Discrete unified gas kinetic scheme for compressible 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


Outline2

Outline

  • Motivation

  • Formulation and properties

  • Numerical results

  • Summary


Discrete unified gas kinetic scheme for compressible flows

Test cases

  • 1D shock wave structure

  • 1D shock tube

  • 2D cavity flow

Collision model:

Shakhov model


1d shock wave structure

1D shock wave structure

Parameters: Pr=2/3,  = 5/3, Tw

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


Discrete unified gas kinetic scheme for compressible flows

DUGKS agree with UGKS excellently


Discrete unified gas kinetic scheme for compressible flows

Again, DUGKS agree with UGKS excellently


Discrete unified gas kinetic scheme for compressible flows

DUGKS as a shock capturing scheme

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


1d shock tube problem

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


Discrete unified gas kinetic scheme for compressible flows

=10: Free-molecular flow


Discrete unified gas kinetic scheme for compressible flows

=1: transition flow


Discrete unified gas kinetic scheme for compressible flows

=0.1: low transition flow


Discrete unified gas kinetic scheme for compressible flows

=0.001: slip flow


Discrete unified gas kinetic scheme for compressible flows

=1.0e-5: continuum flow


2d cavity flow

2D Cavity 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


Discrete unified gas kinetic scheme for compressible flows

Kn=0.075

Heat Flux


Discrete unified gas kinetic scheme for compressible flows

Kn=0.075

Velocity


Discrete unified gas kinetic scheme for compressible flows

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

Present DUGKS

Temperature and Heat Flux

Kn=1.44e-3; Re=100


Discrete unified gas kinetic scheme for compressible flows

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)


Discrete unified gas kinetic scheme for compressible flows

Velocity

DUGKS

LBM


Discrete unified gas kinetic scheme for compressible flows

DUGKS

LBM

Pressure fields


Discrete unified gas kinetic scheme for compressible flows

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!


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