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

Acknowledgements: RGC6108/02E, 6116/03E, 6102/04E,6210/05E 6102/04E,6210/05E Collaborators: Changqiu Jin, Meiliang Mao, Huazhong Tang, Chun-lin Tian Huazhong Tang, Chun-lin Tian

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

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

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

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

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

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 Attached shock Detached shock

Double-cone M=9.50 (RUN 28 in experiment) Mesh: 500x100

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

The 2D BGK model under the transformation Particle velocitymacroscopic velocityGrid velocity

The computed paths fluttering tumbling

computed experiment

fluid force as functions of phase

3D cavity flow

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

Integral solution: Gravitational potential

where for x<0 for x>0 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):

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

andhave different. Gas-kinetic scheme for multi-component flow

Gas distribution function at a cell interface:

Shock tube test:

= + Sod test

A M s =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 1D

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) j+1/2 free transport

Flux splitting for MHD equations:

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

Equilibrium flux function: The BGK flux is a combination of non-equilibrium and equilibrium ones: (K. Xu, JCP159)

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

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

Orszag-Tang MHD Turbulence: t=0.5 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy 5th WENO

t=2.0 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy 5th WENO

t=3.0 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy 5th WENO

t=8.0 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy

3D examples:

BGK (100^3)

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

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

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

A time-dependent gas distribution function at a cell interface where Extended Navier-Stokes-type Equations 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: N2 (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