Gaskinetic schemes for flow computations Kun Xu Mathematics

Gas-kinetic schemes for flow computations Kun Xu Mathematics Department Hong Kong University of Science and Technology

Collaborators: Changqiu Jin, Meiliang Mao, Huazhong Tang, Chun-lin Tian Acknowledgements: RGC 6108/02 E, 6116/03 E, 6102/04 E, 6210/05 E

Contents • Gas-kinetic BGK-NS flow solver • Navier-Stokes equations under gravitational field • Two component flow • MHD • Beyond Navier-Stokes equations

FLUID MODELING Continuum Models Molecular Models Deterministic Liouville MD DSMC Kn Euler Statistical Chapman-Enskog Boltzmann 0. 1 0. 001 Continuum Navier-Stokes Burnett Slip flow 10 Transition Free moleculae

Gas-kinetic BGK scheme for the Navier-Stokes equations fluxes

Gas-kinetic Finite Volume Scheme • Based on the gas-kinetic BGK model, a time dependent gas distribution function is obtained under the following IC, • Update of conservative flow variables,

BGK model: Equilibrium state: Collision time: A single temperature is assumed: To the Navier-Stokes order: in the smooth flow region !!!

• Relation between and macroscopic variables • Conservation constraint

• BGK flow solver Integral solution of the BGK model

• Initial gas distribution function on both sides of a cell interface. The corresponding is where the non-equilibrium states have no contributions to conservative macroscopic variables,

• Equilibrium state

• Equilibrium state is determined by

Where is determined by

• Numerical fluxes: • Update of flow variables:

Double Cones Detached shock Attached shock

Double-cone M=9. 50 (RUN 28 in experiment) Mesh: 500 x 100

Unified moving mesh method physical domain computational domain Unified coordinate system ( W. H. Hui, 1999) geometric conservation law

The 2 D BGK model under the transformation Particle velocity macroscopic velocity Grid velocity

The computed paths - fluttering - - tumbling -

computed experiment

fluid force as functions of phase

fluid force as functions of phase

3 D cavity flow

BGK model under gravitational field: Integral solution: where the trajectory is

Integral solution: Gravitational potential

X=0 where for x<0 for x>0

Initial non-equilibrium state: Equilibrium state

The gas distribution function at a cell interface: Flux with gravitational effect: Flux without gravitational effect (multi-dimensional):

Steady state under gravitational potential N=500000 steps Diamond: with gravitational force term in flux Solid line: without G in flux

Gas-kinetic scheme for multi-component flow and have different .

Gas distribution function at a cell interface:

Shock tube test:

Sod test = +

A Ms=1. 22 shock wave in air hits a helium cylindrical bubble

Shock helium bubble interaction (Y. S. Lian and K. Xu, JCP 2000)

Ideal Magnetohydrodynamics Equations in 1 D

Moments of a gas distribution function: Equilibrium state: The macroscopic flow variables are the moments of g. For example, Then, according to particle velocities, we can split flow variables as:

With the definition of moments: We have Recursive relation:

Therefore,

Kinetic Flux vector splitting scheme (Croisille, Khanfir, and Ghanteur, 1995) free transport j+1/2

Flux splitting for MHD equations:

Construction of equilibrium state: j free transport collision , where j+1/2

Equilibrium flux function: The BGK flux is a combination of non-equilibrium and equilibrium ones: (K. Xu, JCP 159)

1 D Brio-Wu test case: Left state: Right state: density x-component velocity solid lines: current BGK scheme dash-line: Roe-MHD solver

y-component velocity shock By distribution Contact discontinuity +: BGK, o: Roe-MHD, *: KFVS

Orszag-Tang MHD Turbulence: t=0. 5 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy 5 th WENO

t=2. 0 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy 5 th WENO

t=3. 0 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy 5 th WENO

t=8. 0 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy

3 D examples:

BGK (100^3)

FLUID MODELING Continuum Models Molecular Models Deterministic Liouville MD DSMC Kn Euler Statistical Chapman-Enskog new continuum models Boltzmann 0. 1 0. 001 Continuum Navier-Stokes Burnett Slip flow 10 Transition Free moleculae

Generalization of Constitutive Relationship Gas-kinetic BGK model: Compatibility condition: Constitutive relationship:

With the assumption of closed solution of the BGK model: is obtained by substituting the above solution into BGK eqn. The solution becomes

Extended Navier-Stokes-type Equations A time-dependent gas distribution function at a cell interface where Viscosity and heat conduction coefficient

Argon shock structure Observation: Experiment: Alsmeyer (‘ 76), Schmidt (‘ 69), . . . Shock thickness: Mean free path (upstream):

Density distribution in Mach=9 Argon shock front Circles : experimental data (Alsmeyer, ‘ 76); dash-dot line: BGK-NS; solid line: BGK-Xu

Diatomic gas: N 2 (two temperature model: bulk viscosity is replaced by temperature relaxation) ,

BGK Compatibility condition

M=12. 9 nitrogen shock structure

M=11 nitrogen shock structure Efficiency: DSMC: hours Extended BGK: minutes