numerical modelling of gas flow with kinetic theory mark jermy, lim chin wai 林清維, hadley cave...
TRANSCRIPT
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
Kn
Case I 0.5ms
Case I 4.0ms
Case I
Validity of the Maxwell distribution
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)
Validity of the Maxwell distribution
3ln
05.0
GLL
GLL
Kn
Kn
Case II 0.5ms
Case II 4.0ms
Case II
Validity of the Maxwell distribution
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
PP-CVD Diamond Deposition Reactor
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
22222 ˆˆ vjuiwwx 3232
2 ˆˆ vjuiwwx
23232 ˆˆ vjuiwwx 3333
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
jkflux mA
Am ,
xjjkS
xjflux vmA
Ap ,
22
(max)22
yx
tRTquuCFL
ijyx
yjRR
rj
wrrC
rrB
xm
21
231
3
23 txrPp
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
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 Diamond Deposition Reactor
• 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
J
vw
c
J
jjj
J
jjvj
J
jjvj
J
jjj
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
ni
jjstartijii
jiii
jstartii
jstartistartii
j
jji
ji
jstarti
j
starti
vv
vv
v
vvvvvv
vvmmvmv
vvv
vvmmvmv
v
pvvm
vvm
vv
v
v
2
11
22
1...
4
1
2
1
2
:collisionsn After 24422
2
:collisions After two22
2
:collision oneAfter
:unchanged remains velocity gas blue Assuming
2p
collisioneach after halved is momentumin Difference
is momentumin Difference
is speed Relative
onalunidirecti
travelled distanceevery occurscollision elastican
mass equal of molecules :Assume
speed with begins gas Blue
speed with begins gas Red
2,
2,
,
,1,2,
1,1,2,
,1,
,,1,
i
,
,
:collisionsn After
4
1
4
1
2
1
4
11
424422
2
:collisions After two
2
2
22
2
:collision oneAfter
:momentum) ofion (conservat2
by changes velocity gas blue Assuming
2p
collisioneach after halved is momentumin Difference
is momentumin Difference
is speed Relative
onalunidirecti
travelled distanceevery occurscollision elastican
mass equal of molecules :Assume
speed with begins gas Blue
speed with begins gas Red
,,
,,,,,1,1,2,
1,1,1,2,
,,,1,
,,,1,
,,1,
,,,1,
i
,
,
j
istartj
j
istarti
startjstarti
j
istartjstartjstartijii
jiii
startjstarti
j
istartjj
startjstartiistartjjjj
startjstartii
startjstartistartii
ji
j
ij
j
ij
jji
ji
jstarti
j
starti
w
wv
w
wv
vv
w
wvvvvvv
vvmmvmv
vv
w
wvv
vvmwvmwvmw
vvv
vvmmvmv
vvm
w
wp
w
wv
pvvm
vvm
vv
v
v
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
2lnln
2ln
1
collisions by these unchanged remains Assuming
2
11
2
2int
1
1
speed average of molecules with collisions elastic ofresult a As
collisions ofNumber
ij bins of molecules with collides ibin of molecule a dl distance a streamingIn
,,
,,
,
,
,,
x
0
0
1
1
,
,
L,
,
,
,
Lij
Loldinewi
Loldiijijnewi
L
oldiij
newiij
Loldiijnewiij
v
vv
v
viiji
iij
ij
v
vv
x
x
i
iij
iijiii
iijii
J
jijj
J
jijjj
ij
xv
xvv
xvvvv
x
vv
vv
xvvvv
xxvvdv
xvv
v
dxdvxvxv
dxxvxv
xvdxxvdv
dxxvxvxvdxxvxSpeedatpo
denceinspeegthediffereachhavlin
w
vw
v
dl
newi
oldii
newi
oldi
newi
oldii
L
Distribution narrowing model
v
Number density
at
m
a
tu
mumuma
a
Fm
FmaF
maFF
maF
enroutegstreaenroutecollisions
enroutecollisionsutionestdistribcollisions
enroutecollisionsutionestdistribcollisions
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
ityBulk veloc
ykxk ujui ,,ˆˆ
fluxArbitrary
tu xk ,
tu yk ,
tux yk ,
2
2
2
2
2
1
11ˆ
1
1ˆˆ
itybulk veloc thelar toperpendicur Unit vecto
x
x
x
x
UU
j
UU
iU22
ˆˆˆ
itybulk veloc the toparallelr Unit vecto
yx
yxII
UU
UjUiU
??
11
is scheme by the generated stressshear actual The
is stressshear correct physically The
ˆˆˆ
ˆˆˆˆ
fluxArbitrary
12
,,,
1
,,
11
,,
,,
K
k
ykxkIIkk
K
k
ykxkK
kk
K
k
k
ykxkk
kkIIykxk
x
tuxtuumw
t
x
u
x
uu
x
u
ujuiuU
uUuUujui
yx UjUi ˆˆ
ityBulk veloc
ykxk ujui ,,ˆˆ
fluxArbitrary
tv j
tu yk ,
tux yk ,
2
2
2
2
2
1
11ˆ
1
1ˆˆ
itybulk veloc thelar toperpendicur Unit vecto
x
x
x
x
UU
j
UU
iU22
ˆˆˆ
itybulk veloc the toparallelr Unit vecto
yx
yxII
UU
UjUiU
u
vx
x
22c ji
jiij
qRTvjqRTui
vjuic
2ˆ2ˆ
ˆˆ
fluxes Nine
332313
322212
312111
ccc
ccc
ccc
11c
33c
32c
23c
12c
21c
13c31c
tui
• 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