Numerical modelling of gas flow with kinetic theory Mark Jermy, Lim Chin Wai 林清維, Hadley Cave...

Numerical modelling of gas flow with kinetic theoryMark Jermy, Lim Chin Wai林清維 ,

Hadley Cave 山洞瓦片University of Canterbury

New ZealandProf. Wu J-S 吳宗信 , NCTU and Matt

Smith李文修 NCHC

The thinking, talking animal

Humans are bipedal, which frees our hands. Those hands have opposable thumbs coupled with the most flexible hand of all the great apes which makes us excellent tool-users. We have language which allows us to retain knowledge even when individuals die and organise for efficient food gathering and protection.

ChimpanzeeBonobo Orangutan Gorilla

There are five species of great ape:

With these skills we have built up a continuous material culture of tools, organisations and knowledge which can be passed to new individuals

Human

Great apes are not the only tool usersBut we are the best

Kinetic theory is founded on a few, elegantly simple concepts

MOLECULES HAVE:• MASS• MOMENTUM• ENERGYAND THEY CAN EXCHANGE THESE PROPERITES ONLY VIA COLLISIONS!

Kinetic theory and the Boltzmann equationA gas can be represented by a set of molecules exerting forces on each other, and their surroundings, via the intermolecular potential and other forces

collisionst

f

u

f

m

F

x

fu

t

f

Interactions between molecules can be represented as the effect of collisions: the result of short range forces caused by the intermolecular potentials

x

uThe entire state of the gas can be described with a probability distribution function f

dttt

duuu

dxxx

dxdudttuxf

at time

speedwith

position at

molecule a finding ofy probabilit,,

Which describes the conservation of molecules (mass), momentum and energy under the forces F and the effects of collisions

The evolution in time and space of this p.d.f. is described by the Boltzmann equation:

Probability f

High

Low

The distribution function is in reality discontinuous, but may be treated as continuous when a large number of molecules are present Probability per unit variable

u

Probability per unit variable

u

The molecular probability distribution function

Wait for me!

The Maxwell-Boltzmann distribution is one solution to the Boltzmann equation and is the distribution function of a stationary gas which has come into equilibrium as the result of a large number of collisions

kT

mu

kT

mnf

2

0

21

exp2

A moving gas in local equilibrium may have the same distribution shifted by the bulk velocity

),,( tuxff

The distribution may be skewed by the effect of velocity gradients (shear), temperature gradients, density gradients, or external body or surface forces

Probability (per unit variable)

u

)(

)()(21

exp)(2

)(

22

0 xkT

xuxum

xkT

mxnf

Moments of the molecular pdf

In engineering modelling we wish to predict the macroscopic fluid propertiesDensity == massVelocity == momentumTemperature == energy(all three together determine pressure)

kT

mu

kT

mnf

2

0

21

exp2

The macroscopic properties are moments of the molecular p.d.f.Density or mass == integral of amplitudeMomentum == integral of value (mean)Kinetic energy == integral of value2 (variance)

Probability (per unit variable)

u

dufmu

dufmu

dumf

02

0

0

2

1densityenergy kinetic

density momentum

density mass

Relevant nondimensional numbers

LKn

a

UMa

gas VHS mequilibriuin 2

12nd

c3

1

(3D) 2

(3D) 8

veloc.)molecular rms (3D, 33

RTcc

RTcc

RTm

kTcc

lemostprobab

meanspeed

rms

ν

ULRe

speed of choice on the depends which 1order offactor a is

.

k

kcRTa

These are not independent:

Knudsen- rarefaction

Reynolds- ratio of inertial to viscous forces

Mach- compressibility

x

Q

QKnGL

Local or gradient-length Knudsen

Kn

Ma

2Re

Free molecular flow

Euler equations

Regions of validity

Gradient Kn

Navier-Stokes equations

0.010 0.1 100

Boltzmann equation Collisionless Boltzmann eqn.

Burnett equations…

After Bird, 1994

0 1000 108

Viscous laminar flow Turbulent flow Pseudo inviscid flowRe

Ma

1

0.3

5

Inco

mpr

essi

ble

Com

pres

sibl

esu

bson

icS

uper

soni

cH

yper

soni

c Compressible N-S solvers

Incompressible N-S solvers

Euler solvers

The Chapman-Enskog expansion and the continuum equations

is a is a Taylor series in the gradient lengthscale Knudsen number

Chapman and Enskog showed that for most gas flows, the distribution may be treated as a perturbation from the Maxwell distribution

tuxtuxftuxf ,,,,,, 0

Qquantity in gradient) (of elengthscal

path freemean molecular

x

Q

QKnQ

...1,, 22

1 GLGL KnKntxu

If only the first term is used =1, the distribution function is Maxwellian, the gas is in local equilibrium and the Euler equations of continuum hydrodynamics can be recovered

If the first and second terms are used, the distribution function is perturbed a little from local equilibrium and the Navier-Stokes equations can be recovered

If the higher order terms are used, the distribution function is perturbed further from local equilibrium and the Burnett, super-Burnett, and more complex equations can be recovered.

0

puut

u

0

Tpuut

u

...,42

531,,

22

0

0

2

0

0

2

y

u

ux

T

TO

y

u

usu

x

T

TcvCtxu

Kinetic theory CFDConventional continuum CFD finds approximate solutions to the Euler or Navier Stokes (or, rarely, Burnett) equations directly using some numerical finite volume, finite element or finite difference method. The molecular probability distribution function is never constructed.

Kinetic particle schemesDSMC

Kinetic nonparticle schemesLBMEPSMEFM, TDEFMMBEQDS

Problems with conventional continuum CFD:Non true directionComplex to set upResource hungryUncertain convergence

Kinetic theory CFD schemes construct the molecular p.d.f. at each mesh point and timestep, and use the information contained in it together with conservation laws and some approximate numerical method to predict the state of the molecular p.d.f. at the next timestep.

In both cases, at every timestep the solution is an approximate solution of the Boltzmann equation, though this equation may never be coded into the scheme directly.

God might not play dice but Professor Bird does

The biggest single limitation of conventional CFD schemes?

• Non true-direction fluxes

Solution dependent on grid → reduced usability, convergence issues and dodgy results

Complexity with unstructured grids

Kinetic-Based Schemes

• Take into account the particle based nature of the flow.

• Microscopic ensembles → macroscopic properties

DSMC: Particle based & stochasticEPSM: Particle based at equilibriumEFM: Flux based at equilibrium but NTDTDEFM: Flux based at equilibrium and TDQDS: Flux based at equilibrium, TD and

approximate

Lattice Boltzmann Method

No Poisson problem to solveGood for problems with mesoscopic physics

Weakly incompressibleSome schemes are athermal

Boundary conditions complex- or merely unfamiliar?Early versions limited to low Re, solved by improvements to the scattering matrix

Many similarities with QDSExcept- QDS fully compressible

The D2Q9 solution has some similarities with a 3-particle 2D QDS scheme

Rudiments of QDSIn Quiet Direct Simulation (QDS) the molecular p.d.f. is represented by a small number (usually 3 or 4) sets of velocities (often termed bins).

Probability (per unit velocity)

v

Each bin has a characteristic velocityjj q

m

kTuv

2 and all the molecules in it are assumed to be travelling at this speed.

Each bin has a weight wj. For historical reasons we use the statisticians’ weights which sum to

J

jjw

1necessitating a normalising factor of 1/ when physical quantities are calculated.

Each bin carries a quantity of mass, momentum and energy which are related to the weight and characteristic velocity

1

2 and

2 where

2

1 2

m

kT

vdVxwE

dVvxwvm

dVxwm

j

jjjj

jjjj

jj

molecule one of masssubscript) (no

dimensions of no. and heats specific of ratio

binin gas ofenergy

element volume

gas ofdensity

binin gas of mass

m

E

dV

x

m

j

j

v

Number density

Gauss-Hermite quadrature• A continuous distribution may be represented by a finite number of discrete points. Each point

has an abscissa and a weight (amplitude)• The most accurate discrete representation of the normal distribution is given by the abscissas

and weights generated from the Gauss-Hermite quadrature• ‘Most accurate’ means in effect, the most accurate reconstruction of the moments• For the G-H quadrature, the moments are exact i.e. identical to the moments of the

corresponding normal distribution.

Probability per unit velocity

u

Each bin or particle represents the integral over some range of the continuous distribution.This idea is essential to some viscous schemes described later

• The Maxwell (normal) distribution has three nonzero moments• Three independent variables are required to represent three independent nonzero moments• Basic scheme uses a minimum of three particles with fixed weights and variable abscissas• Nonequilibrium distributions have more nonzero moments

1

2 and

2 where

2

1 2

m

kT

vdVxwE

dVvxwvm

dVxwm

j

jjjj

jjjj

jj

molecule one of masssubscript) (no

dimensions of no. and heats specific of ratio

binin gas ofenergy

element volume

gas ofdensity

binin gas of mass

m

E

dV

x

m

j

j

J

jj

j qfw

def1

2 22

1 2

Note the change of axes

Probability

AlgorithmIn Quiet Direct Simulation (QDS) the transport of mass, momentum and energy is calculated by considering the motion (streaming) of bins of gas at their characteristic velocity from one mesh cell to neighbouring cells.After streaming the net mass, momentum and energy in each cell is calculated by summation.After summation, the p.d.f. of the gas in each cell (represented by the small number of bins) is recalculated.

Probability

u

Usually a Cartesian grid is used

1. Gas in one bin in source cell

2. Streams in it’s true direction a distance vit

vi

vit

3. Some of the gas remains in the source cell

4. The rest deposits in a number of neighbouring (destination) cells

5. Repeat with the remaining bins (shown here in blue) of source cell gas

Yes: Move to next cell

All bins in source cell streamed?

AdvantagesSimple explicit scheme arithmetical operations using local informationTrue directionFast to computeEasy to parallelise, with good speedupIntuitiveStableNo pressure coupling (Poisson) equation to solveVersatile

All cells streamed?

Sum mass, momentum, energy in each cell

Calculate state of gas, and hence weights and characteristic velocities

Increment timestep and repear

No: Move to next bin

No: Move to next cell

Courant-Friedrichs-Levy (CFL) criterionPrevents gas passing through a cell without interacting

1max,

x

tvCFL i

x

Calculating fluxes by overlap areas

2nd order flux limited finite volume schemeFluxed quantities are calculated from a second order reconstruction i.e. using the linear gradients of mass, velocity and energy between cells

The overlap areas used to calculate the fluxed quantities also differ from the 1st order schemeThe derivation of these was not trivial in the case of the axisymmetric code

jL

j

xwxdx

d

m

jv

vLx

xj qdx

dx

dx

duuv

2

2

Mass of gas in bin j

Characteristic velocity of bin j

2

2

L

vv

j

xdx

dEnergy per unit volume of gas in bin j

Lin Ya-Ju has studied the effect of using higher orders. The greatest improvement is seen moving from 1st to 2nd order.

ykxjcLL

jk

wwVdy

dy

dx

dx

m

2

2

dy

dy

dx

dx v

Lv

Lv

jk

Computational cost

• Mach 2.0 shock propagation through a pipe• Second-order axisymmetric code• Demonstration case: 11,008 flow field cells• Test case: 33,400 flow field cells

– QDS Simulation Time: 29 seconds (3.3GHz NB)– Godunov solver: 2.75 minutes (3.3GHz NB)– Direct Simulation: 34.2 hours (12 PC cluster)

Boundary conditions

Specular wall

Diffuse wall (not used)

Inlet

Absorbing outlet (1st gradient)

Boundaries are implemented with ghost cells: cells which lie outside the boundary and which serve to allow simple manipulation of the fluxes at the boundaries

Specified ρ, T, vnorm, vparallel

Ghost cell properties match neighbouring fluid cellBut wall normal velocity reversed

Ghost cell properties determined by properties of neighbour and next neighbour flow cell

Wall normal flux

Wall parallel flux

Andersen condition? outlet(2nd gradient)

Bounceback no-slip wall (may be moving)

Fluid domain Ghost cells

Both components of velocity reversedWall normal set to moving wall speed?

Errors in the basic QDS schemeBin velocities and weights are calculated from the Gauss-Hermite quadrature which is the most accurate known discretisation of the normal distribution.Flight (streaming) and collision are separated. i.e. streaming is collisionlessThe Maxwell distribution is forced at the end of each timestep. This is equivalent to assuming an infinite collision rate.

AdvantagesVery fast to computeSimple to program

Disadvantages• Collisionless streaming overdiffusion of

momentumtoo viscous• Forcing of local equilibrium distribution further

errors in shear stress and heat flux & inability to correctly model nonequilibrium (rarefied or high spatial or temporal gradients)

These errors are not serious in dense hypersonic flows

Solutions?• “Collision en route” or artifical coolingEarly LBM schemes had collisionless streaming and high inherent viscosity• Partial relaxation of molecular pdfChapman Enskog distribution

Application: PP-CVD

High Pressure Gas Source Vessel

Vacuum Pump

Solenoid Valve

Reactor Vacuum Vessel

Substrate Heater

Substrate

Processing Time [s]

Reacto

r P

ress

ure [P

a]

0 ti tp

Pmax

Pmin

Pump-down Phase

Pressure Regulated Precursor Gas Source

Injection Phase

Inlet Orifice

PP-CVD gas injection

• Flow field simulation for a new generation of CVD deposition systems

• Second-order axisymmetric code• 312,744 flow field cells• Simulation time: 560 minutes (standard desktop)• Code is being developed precisely for this

application

Validity of the Maxwell distribution 345mm

325mm

37.5

mm

59m

m Inlet

Orifice 1 mm

substrate

Vacuum pump exhaust

Axis of symmetry

345mm

325mm

37.5

mm

59m

m Inlet

Orifice 1 mm

substrate

Vacuum pump exhaust

Axis of symmetry

Case IIdeal HeliumChoked inlet flow10kPa, 293K supplyReactor initial condition stagnant, 1Pa, 293KCartesian mesh of square cells 312,744 at 0,25mm side11h to simulate 4.0ms flow time on 3GHz Intel Core 2 Duo with 4GB RAMSteady state reached in 4.0ms flow time100Pa af end of injection (experimental)End of injection at 0.1s

Case IIIdeal HeliumChoked inlet flow400kPa, 293K supplyReactor initial condition stagnant, 1,000Pa, 293KCartesian mesh of square cells 312,744 at 0,25mm side12h to simulate 4.0ms flow time on 3GHz Intel Core 2 Duo with 4GB RAMSteady state reached in 4.0ms flow timexxxPa af end of injection (experimental)End of injection at 0.1s

0.01 ms

0.2 ms

0.1 ms

0.05 ms

1.0 ms

0.5 ms

0.3 ms

Case I 0.25mm mesh

Ln(density)

0.1 ms

0.2 ms

0.5 ms

Case I 0.125mm mesh

Convergence of mesh

0.01 ms

0.2 ms

0.1 ms

0.05 ms

2.0 ms

1.0 ms

0.5 ms

0.3 ms

4.0 ms

3.0 ms

0.01 ms

0.05 ms

4.0 ms

0.1 ms

0.2 ms

0.3 ms

0.5 ms

1.0 ms

2.0 ms

3.0 ms

Case I 0.25mm mesh

0. 1s

0.10005ss

0.10001ss

0.101s

0.1005s

0.1001s

Pump down

0.101s

0. 1s

0.10001ss

0.10005ss

0.1001s

0.1005s

Ln(density) Ln(pressure)

Injection

Ln(pressure)Ln(density)

Validity of the Maxwell distribution

3ln

05.0

GLL

GLL

Kn

QQ

Kn

Case I 0.5ms

Case I 4.0ms

Case I


Case I 0.5ms

Case I 4.0ms

Case I

69.0ln

5.0

,

,

t

tt

t

avgcol

avgcol

avgthavgcol v

t,

,

m

kTv avgth

3,

Titov and Levin: collision limited DSMC: Maxwell distribution met to

within a few percent after 2 collisions per molecule

Case I

Case II 0.25mm mesh0.01

ms

0.2 ms

0.1 ms

0.05 ms

2.0 ms

1.0 ms

0.5 ms

0.3 ms

4.0 ms

3.0 ms

0.01 ms

0.05 ms

4.0 ms

0.1 ms

0.2 ms

0.3 ms

0.5 ms

1.0 ms

2.0 ms

3.0 ms

0. 1s

0.10005

ss

0.10001

ss

0.101s

0.1005s

0.1001s

0.101s

0. 1s

0.10001s

s

0.10005s

s

0.1001s

0.1005s

Ln(density)

Ln(density)

Ln(pressure)

Ln(pressure)


3ln

05.0

GLL

GLL

Kn

QQ

Kn

Case II 0.5ms

Case II 4.0ms

Case II


Case II 0.5ms

Case II 4.0ms

Case II

69.0ln

5.0

,

,

t

tt

t

avgcol

avgcol

avgthavgcol v

t,

,

m

kTv avgth

3,

Titov and Levin: collision limited DSMC: Maxwell distribution met to

within a few percent after 2 collisions per molecule

Case II

PP-CVD gas injection

Conclusions:Local equilibrium reasonable in Case II

But not in the more rarefied Case I

• Flow field simulation for a new generation of CVD deposition systems• Second-order axisymmetric code• 312,744 flow field cells• Simulation time: 560 minutes (standard desktop)• Code is being developed precisely for this application

Inviscid and viscous flow• QDS has been described as an Euler solver i.e.

solves inviscid flows

• Inviscid fluid: molecules collide with walls (exert pressure) but not with each other

• This is a reasonable model in highly rarefied flows• Intermolecular collisions give rise to further stress

terms

0

puut

u

0

Tpuut

u

0

puut

u

0

Tpuut

u

Viscous structures in QDS

Viscosity in QDS• QDS would be an Euler solver if it did not

model the intermolecular collisions which give rise to shear stresses

• (Normal stresses i.e. pressure would still appear)

• However tangential momentum is transferred, by collision, from source to destination cell

• Therefore there is viscosity inherent in the scheme

Viscosity depends upon gridsize and timestep.Where the grid spacing is greater than the physical mean free path and the timestep is limited by a reasonable CFL condition, the effective viscosity is usually greater than the real viscosity of the gas simulated.

How viscous is the basic scheme?

uuu

xy

uu

xtvuw

wx

tvuwx

wwvvx

tvuwmw

x

tvwqRTuwmw

x

tvqRTuuwmwp

ii

ii

iiiiii

33

3

1

33

2

3131

3

1

33

3

1

11

33

and as

22

celllower tofluxed momentumNet

Consider:• Simple 2D shear

flow• Uniform density• Uniform

temperature• Bulk velocity

aligned with x axis• 3 particle scheme• Square Cartesian

grid

x

21

cellupper tofluxed Mometum

x

xtvmu

2

3

celllower tofluxed Mometum

x

xtvuum

usually

72.2

3

1

886.0

3

3

1

3

1

stressshear Physical

2

2

12;886.0

1

schemein stressshear Effective

celllower tofluxed momentumNet

physical

333

33

33

3333scheme

33

x

x

mkT

xmkT

m

kTc

y

u

m

kTq

m

kTvw

xvw

xvw

x

u

y

ux

xvuw

tx

xtvuw

t

p

A

xtvuw

physical

scheme

physical

physical

scheme

scheme

schemescheme

How viscous is QDS?

• There is an inherent scheme viscosity which is related to cell size and particle velocity

• The cell size is usually orders of magnitude greater than the physical mean free path

• The scheme viscosity is thus usually orders of magnitude greater than the physical viscosity

t

x

t

x

x

tvCFL

xvw

scheme

scheme

22

3

33

443.02

886.0

5.0

(for a 3 particle scheme)

Timestep and gridspacing are related (via temperature) by CFL criterionInherent viscosity can be manipulated by changing gridspacing and temperature

Another argument for inherent viscosity

t

x

Δxλt

xc

c

eff

effeff

2

3

1

1940) (Jeans 3

1

tvi

ux,vy,

physicalphysicaleff Δx ~ when ~

m

kTc

8

speed almean therm

22

3499.0path freemean

d

M

t

yvc ieff

~condition CFL

m

kTc

8

speed almean therm

22

3499.0path freemean

d

M

Viscosity in PP-CVD simulations1kPa initial pressure underexpanded jet flowReynolds number of real physical flow ~300,000Reynolds number of simulation ~200 to 600 due to inherent viscosity

Test case: incompressible lid driven cavity flowGhia, Ghia and Shin, J. Comp. Phys. 48 387-411 (1982)

Finite difference implicit multigrid Navier Stokes solver

NxN cells where N=129, 257, 1024

Re=100, 400, 1000, 3200, 5000, 7500, 10,000

L

H*

*

*

* Stationary no-slip walls

*** ** **

*

**

No-slip wall moving at U0

*

**

ResultsBasic QDS Rescheme = 90.5 for N=128

t

xscheme

2

3

1

tN

xNLxN

ULU

scheme

scheme

2

00

3Re

;1

;Re

yConclusions• QDS is not inviscid• If the effective kinematic viscosity=∆x2/(3∆t),

results are similar to Ghia et al’s• QDS scheme does have viscosity of order

gridspacing2/ (3*timestep)• Some discrepancy with Ghia et al. remains:

wall treatment, imperfect estimate of viscosirty

Rescheme 420 for N=512

Rescheme= 739 for N=1024

How to restore physically correct viscosity?

• Reduce gridsize to order of mean free path

• Reduce weight- doesn’t work: moments do not all scale properly: instead reduce velocities

• Reduce velocity of extreme bins• Equivalent to momentum exchange within source

cell via in cell collisions during streaming• Equivalent to reducing gas temperature• Equivalent to increasing effective mass of the

particles (retaining the same kinetic energy)

m

kT

qm

kTvv

xvw

physical

scheme

3

233

33

Probability per unit velocity

u

m

kT

xwv

3

1

33

Removing inherent viscosity: Modelling collisions en route

Gas in fluxing bin (red) passes through rest of gas in source cell (blue)Inter-molecular collisions occur during flight

iv

2path freemean

travelleddistance tvn i

n collisions occur during flighttvi

Simple collision modelimv

Collisions occur between partners of equal mass m

Moving at

Change to centre of mass frame of reference

2imv

2imv

Assume each collision reduces difference in momentum by half

4imv

4imv

Return to laboratory frame of reference

4

3 imv

4imv

gas ofity bulk veloc

molecules gas fluxing ofvelocity collision -pre

molecules gas fluxing ofvelocity collision -post

2

11

2

,

,

,

,

v

v

v

vv

v

oldi

newi

nn

oldi

newi

After n collisions:

Assumptions: velocity of blue gas doesn’t change

v

It’s a model

It’s not the real thingbut it looks like it

Representing non-equilibrium distributions by manipulating the abscissas

-1500 -1000 -500 0 500 1000 15000.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

Molecular velocity m/s

Pro

bab

ilit

y p

er u

nit

vel

oci

ty

a1

b1 = a2

f(v)

W1

W2

W3

b2 = a3

b3

xxxxxxxxx

yzzyxx

CCCC

dCdCCCCfC

2

1

5

21

2

51

,,

22

Smith and Kuo NCHC

Matt’s test case - Simulation Setup:L = H = 1, number of cells: 200x200R = 1.0, gamma = 1.4Initial conditions: (density, u, v, temp) = (1, 0, 0, 1)Top wall speed: Ux = 0.11832159 m/sViscosity: power law – m = m0*(T/T0)w; m0 = 0.0011832, T0 = 1.0, w = 0.9Boundary conditions: non-slipSimulation order: 2nd order 2N flux, MC limiter, 3-particle, dynamic time step adjustment with CFLmax = 1.0

QDS with CE Abscissas correctionBasic QDS

Ghia’s test case:L = H = 1, number of cells: 128x128R = 208.0, gamma = 1.667 (Argon)Initial conditions: (density, u, v, temp) = (1, 0, 0, 1)Top wall speed: Ux = 1.0 m/sViscosity: power law – m = m0*(T/T0)w; m0 = 2.125e-5, T0 = 273.0, w = 0.81Boundary conditions: non-slipSimulation order: 2nd order 2N flux, MC limiter, 3-particle, dynamic time step adjustment with CFLmax = 0.1

QDS with CE Abscissas correction; dt = 4.29e-5s

Nominal Re=10,000

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Ux

y

Ux on vertical centreline (QDS cells:128x128)

Ghia (Re=100)

Basic scheme

coll-en-route, 2-collisions

CE Abscissas correction

Ghia’s test case:

• Inherent viscosity (=gridspacing2/(3*timestep)) gives Re~10

• Basic QDS some discrepancy with Ghia Re=100• QDS with collision en route smaller discrepancy• Inherent viscosity is reduced with collision en route

Basic QDS

NCHC 2nd order QDS code with C-E abscissa correction

High speed LDC test case:L = H = 1, number of cells: 200x200R = 1.0, gamma = 1.4Initial conditions: (density, u, v, temp) = (1, 0, 0, 1)Top wall speed: Ux = 0.11832159 m/sViscosity: power law – m = m0*(T/T0)w; m0 =

0.0011832, T0 = 1.0, w = 0.9Boundary conditions: non-slipSimulation order: 2nd order 2N flux, MC limiter, 3-particle, dynamic time step adjustment with CFLmax = 1.0Mach number contours shown

QDS with CE Abscissa correction

8th order WENO solution60x60 cells

(courtesy xxxxx)

Re=100Ma=0.1

Kn=Ma/ (/2)0.5Re=0.0007Kncell=0.13 or greater

Matt’s test case:

-0.1

0.0

0.0

0.0

0.0

0.0

0.1

0.1

0.1

0.1

0.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Ux

y

Ux on vertical centreline (QDS cells:200x200)

Basic scheme

CE Abscissas correction

Note: have not run with CER since molecular data e.g. molecular diameter is unknown

Conclusions• Basic QDS scheme has an inherent viscosity • This is different from (and additional to) the numerical viscosity of any finite order FV scheme• The inherent viscosity is large for practical computations

• The inherent viscosity can be reduced by implementing a collision-en-route model to reduce the transfer of momentum (and mass, and energy) to the destination cell

• Other corrections are possible (cooling, effective mass increase) but all reduce the velocities of the outlying bins

• The speed of computation is acceptable• 128 cells: 20mins for 1s simulation• 512cells: 18hours for 1s simulation• initial T of 273K with 128 cells: 5hours for 1s simulation• (for the Argon coll-en-route simulations)

• There is a separate effect of imperfect relaxation to equilibrium• NCHC CE model can correct this

• Some discrepancies with Ghia’s results remain• Perhaps due to:

• Imperfect collision en route model• Slight difference in Reynolds number• Imperfect convergence• Wall treatment• Viscous heating

• More testing is required- Poisueulle flow

Thank you for listening

I invite your questions

Separate Fluxes of Species

• Fluxing species separately is useful, especially in hypersonic flows (although there is a limit to what we might want to do).

• Standard CFD methods cannot do this (multi-species fluxes just use effective mixture properties).

• Particle-based methods can do this, but have big problems with trace species (sample size).

Species Fluxing in QDS

• Is trivial!• Macroscopic properties are calculated using

effective mixture values.• Trace species are no problem.• Flow-field chemistry models should be easy to

implement.

Hypersonic Flow over a Forward Facing 2D Step

• 50% Helium, 50% Xenon (97.04% Xe by mass)• Mach 20 flow (ρ=1.0 kgm-3, T=1.0K, ux=286.3ms-1)• 100 x 100 grid• Two simulation methods:

Direct Simulation (DS) with ~1.5M particlesFirst order, multi-species QDS

Hypersonic Flow over a Forward Facing 2D Step

0.98

0.97

0.96

0.95

0.94

0.93

0.92

0.91

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

[kgm-3]

Density Mass Fraction of Xe

QDS Simulation• 4 processors• Sim. time: 31s

Direct Simulation• 1 processor• Sim. time: 13hrs

PP-CVD Diamond Deposition Reactor

• Used to deposit carbon nano-fibres and diamond films by Dr. Maxim Lebedev

• Used a hot wire to decompose methane

Mass Fraction of Xenon


Future Work…

• Implement a model for viscosity.• Implement flow-field chemistry.• Improve the code’s versatility (cut cells etc).

Acknowledgements

• This work is funded by the NZ Foundation for Research, Science and Technology under contract UOCX0710

• Slide template « Butterfly Garden » www.templateswise.com

2N and N2 flux schemes

Probability

u

vi

j=1 j=2 j=3i=1i=2i=3

11112 ˆˆ vjuiwwx

12122 ˆˆ vjuiwwx

13132 ˆˆ vjuiwwx

21212 ˆˆ vjuiwwx 3131

2 ˆˆ vjuiwwx


2 ˆˆ vjuiwwx


2 ˆˆ vjuiwwx

N2 fluxes are generated independently

N bins

N fluxes are generated in the i-directionN fluxes are generated in the j-direction

These are combined

True direction

DSMC

150

200

250

300

350

400

450

500

0 0.02 0.04 0.06 0.08 0.1x[m]

Tem

pera

ture

[K]

CFL condition

jkykxjjk

Sjkflux vvm

A

AE 22

, 2

1

jkS

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22

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• Used to deposit carbon nano-fibres and diamond films by Dr. Maxim Lebedev

• Used a hot wire to decompose methane

High Pressure Gas Source Vessel

Vacuum Pump

Solenoid Valve

Reactor Vacuum Vessel

Substrate Heater

Substrate

Processing Time [s]

Reacto

r P

ress

ure [P

a]

0 ti tp

Pmax

Pmin

Pump-down Phase

Pressure Regulated Precursor Gas Source

Injection Phase

Inlet Orifice


• 1vol% CH4 (7.453% by mass). Remainder is H2.

• PS=100kPa Pmin=30Pa• Axisymmetric second order

QDS solver• 1.9M flowfield cells• Simulation used a 32

processor cluster• Sim. time: 15.5hrs

Things to doNeeds a more general analytical treatment

Generalise to arbitrary number of particles and 2 or 3 dimensions

Try correction schemes!

(subsonic)

222 22

J

qw

m

kT

J

qw

J

quw

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vw

c

J

jjj

J

jjvj

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jjvj

J

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eff

the simulation ran longer, the flow pattern gets odd with the centre vortex diminishes towards the upper right corner. Attached powerpoint file is the streamline plot of the solution at every 10s from 10-110s. This solution is obtained from running with 4-particel scheme.

R=1 and gamma=1.4

Collision models

njstarti

njstarti

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Continuum collision model

v

Number density

QDS gas consists of J=no. bins sets of molecules each at one of the discrete speedsThe molecular distribution function i.e. the set of J velocities and J weights is set up by the collision process(complete relaxation to Maxwellian- basic scheme or incomplete relaxation- CE scheme)These sets of molecules pass through each other during streaming, colliding as they go

2exp1

2exp

2exp

2exp

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2

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Distribution narrowing model

v

Number density

at

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21

211

min

Source cell contains set of moleculesCollisions between these molecules during the timestep establishes the velocity distribution

Streaming occurs at the same time (physically)

Direct correction to expected shear stress

yx UjUi ˆˆ

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ykxk ujui ,,ˆˆ

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tu xk ,

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• U0=1, L=H=1• Initial conditions density=1 u=0 v=0 T=1 everywhere• Argon:• m=39.38/1000/avogno• gamma=1.4• R=248 (speed of sound=sqrt(gamma*R*T)=323m/s at 30m/s)• does using R=1 make sense if we use the physical viscosity?• mu_0=2.12E-5 Pa s at T_0=273K with power omega=0.81• • So at T=1, U=1, L=1:• mu=0.0106 so Re=100• speed of sound=308m/s (R=248) or 19.5m/s (R=1) so Ma=0.003 or 0.05• Kn=sqrt(pi*gamma/2)*Ma/Re=4E-5 or 7E-4• Kn(cell)=Kn/128 so many mfp per cell• • To raise Re to 400 reduce the density by a factor of 4• • Ghia's results are with an incompressible solver, effectively Ma=0 as sound speed=infinite whereas our computational gas is compressible. So we can

never have truly identical results. But with this low Mach nuber we expect very low compressibility.

• Re 100, T=1 hence mu=2.25128e-7, rho = 2.25128e-5 • Re=100, T=273 hence mu=2.117e-5, rho=2.117e-3 • Re=400, T=1 hence mu=2.5128e-7, rho=9.005e-5• Re = 100, rho should be 2.25128e-5 for T=1, or rho=2.117e-3 for T=273, and rho=9.005e-5 for T=1, Re=400, are these right

Numerical modelling of gas flow with kinetic theory Mark Jermy, Lim Chin Wai 林清維, Hadley Cave...

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Transcript of Numerical modelling of gas flow with kinetic theory Mark Jermy, Lim Chin Wai 林清維, Hadley Cave...