1 a different view of the lattice boltzmann method for simulating fluid flow jeremy levesley robert...
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A different view of the Lattice Boltzmann method for simulating fluid flow
Jeremy LevesleyRobert BrownleeAlexander Gorban
University of Leicester, UK
Supported by EPSRC
June 27th 2007 Dundee 2007 2
Movie
More Space Balls than Star Wars.
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High Reynold’s Number Flow
Very low viscosity leads to complicated dynamics.
PDE model for such flows is the Navier-Stokes equation with small viscosity, the Euler equation for zero one.
Examples of shock tube, square cylinder, lid-driven cavity.
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Talk
A new framework for looking at LBM. Not the Boltzmann equation.
It is close to the Navier-Stokes’ equations in some sense.
Stabilisation of method via targetted introduction of diffusion.
Filters and entropy limiters.
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Fraud and hypocrite Approximation theorist
talking about “PDEs” Numerical analysis
conference – what is the order of convergence of your method?
Spot the hypocracy.
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Little quiz.
0111
1011
1101
1110
What is the condition number?
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Smoothed Particle Hydrodynamics
velocity
pressure
density
v
P
F)exp()(
2yxc x
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The shock tube simulation
Gas A Gas B
diaphragm
Simulation of pressurewith time.
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Poorly modelled physics
Artificial viscosity Slope limiters Initial smoothing
Continuum equations do not model the physics
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Radial basis approximation
RR : univariate function
A data set Y
Approximate
)()( 1 xpyxxs kYy
yY
Low degree polynomial
h
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Examples
function Wendland
icmultiquadr dgeneralise
spline icpolyharmon
),31()1(
,)(c
),(log
)(3
2/22
2
rr
r
rr
r
dk
Some grow at infinity!!
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Micchelli (1986, CA) Interpolation problem is always
solvable. In all space dimensions For any configuration of points (with
some very mild restrictions). A great challenge to find appropriate
methods for solving real high dimensional problems.
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Another representation
Consider
x=(x+1x(x+
x)=|x|
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More smoothness
Cubic B-splines from iterating twice. Shape to the data – partition of unity.
3)( rr
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A Good Basis Basis functions
which match the shape of the data.
Discrete Laplacians formed using the data points.
2v
v1v
3v
4v
5v
Laplacian - linears sannihilate
1
ii
iii
vv
vv
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Two or more dimensions (Beatson)
Generalised barycentric coordinates Sibson – Stone (boundary over distance) Mean value (Floater et. al.)
rrr log)( 2
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Simulation using B-splines
Still need artificial viscosity!!
Brownlee, Houston, Levesley, Rosswog, Proceedings of A4A5 (2005)
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Approximation in high dimension
0 3.8095 4.1986 4.1797
3.8095 0 4.1214 4.0608
4.1986 4.1214 0 3.9250
4.1797 4.0608 3.9250 0
d Condition number
1 5.1 8
2 1.4 6
4 3.5 4
8 1.3 4
16 4.4 3
32 3.4 3
64 2.8 3
d=100
1000
1,
jijiijA xx
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Lattice Boltzmann Method
is the probability density function on phase space. This is a microscopic description.
Recover macroscopic variables via integration in phase space
),,( tf vx
densityenergy ),,(),(
density momentum),,(),(
density),,(),(
2
vvxx
vvxx
vvxx
dtfvte
dtfvtu
dtft
jj
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Equilibrium distribution Many different microscopic
descriptions lead to the same macroscopic description.
For each macroscopic description M there is a distribution which maximises the entropy.
This is the quasi-equilibrium distribution.
Totality of these distributions is the quasi-equilibrium manifold.
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Microscopic dynamics Boltzmann
equation, the collision operator Q conserves the macroscopic variables.
)( fQft
f
v
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Popular choice of collision is
*1)( fffQ
Bhatnagar-Gross-Krook collision (BGK) (1954, PR)
is a relaxation time and is viscosity parameter.
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Lattice Boltzmann dynamics
Lattice Boltzmann method – break into a finite number of populations each moving with a fixed velocity.
)( iiii fQft
f
v
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Recover macroscopic variables Sum rather than
integrate
Operator form
densityenergy ),(),(
velocity),(1
),(
density ),(),(
2
,
iiii
iijiij
iii
tfWte
tfvWtu
tfWt
xvx
xx
xx
)( fmM
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Example – the shock tube There are three velocities allowed
Excellent exposition on LBM by Karlin et al. (2006, CCP).
Three populations with two conservation laws to satisfy – density and momentum.
We can make trade between populations, conserving the macroscopic dynamics so as to control the introduction of diffusion.
cc ,0,
0c c
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2d
Lattice in computational space – velocities allow us to movefrom one point in the lattice to a neighbouring one.
Populations in phase space each moving in the direction ofone of the arrows.
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Numerical discretisation
)'(12
)()(2
')',()(1
),(),(
3
*
tQt
tQttQt
dttxfxftxftttvxftt
t
High Reynolds number has tending to 0.
3
1)'(
tQ
3
t
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Either
High viscosity and we can approximate the Boltzmann equation.
Or Low viscosity and we cannot let time
step get less than without incurring huge computational cost.
Not approximating Boltzmann!!
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New idea Simulate transport equation by
Free flying for time t Equilibration
Macroscopic variables are transported by free flight.
Microscopic variable redistributed leaving macroscopic variables locally unchanged.
Smallness parameter is t.
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“Nonlinearity is local, non-locality is linear”
(Sauro Succi)Moreover, non-locality is linear,
exact and explicit
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Numerically
iii
iiii
fff
tftftttf
~
),(~
2),(),( xxvx
Numerical scheme is
Free flight Equilibration
(ELBM)entropy equal
mequilibriu toreturned21
(LBGK)limit viscosityzero1
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Stability problem is nontrivial: Entropic LBM does not solve it
ELBM
LBGK
Shock tube 1D test {-c,0,c}.
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Coarse-graining the Ehrenfests’ way
Formal kinetic equation
Microscopic dynamics
)( fJdt
df
))((1 iti ff
ffJdt
df v)(
t
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Macroscopic dynamics
Match the microscopic and macroscopic dynamics to order
),( tMFdt
dM
))(()( *fmtM t
2t
***
)(2
))(( PffJDmt
fJmdt
dMff
Euler Navier-Stokes
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Summarise
Free fly for time and equilibrate populations f*
Integrate to recover macroscopic variables
Navier-Stokes’ equations to order with viscosity
t
2t2/t
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Coupled steps – a scheme of LBM stabilization
QE manifold
Free flight steps t
Overrelaxation step
Complete relaxation(Ehrenfests’ step)
The mirror image
f0
f1
f *-(f1-f*)f *-(2β-1)(f1-f*)
f *
f2
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Decoupled viscosity from timestep
***
)()())(( PffJDmfJmdt
dMff
Controlled viscosity
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Shock tube 1D test {-1,0,1}
LBGK
ELBM
magicsteps
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The Ehrenfests’ Step Potential problem near shocks where we are
too far from the quasi equilibrium.
0f
too big
1f
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Not enough artificial dissipation LBGK and
ELBM Step back from mirror. Not enough dissipation.
Ehrenfest Introduces dissipation in a very precise and targetted way.
0f1f
mirrordissipation
0f
1f dissipation
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Simulation looks good
June 27th 2007 Dundee 2007 42
Simulation of square cylinder
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Relationship for Strouhal number and Reynold’s number
Okajima’s experiment (1982) ….
LBM simulations, Ansumali et. al.(2004)
Ehrenfest’s steps(2006)
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Lid-driven cavity flow with ES(movie of vorticity)
(k,δ)=(32,10-3)
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Flux limiters
S.K. Godunov (1959) we should choose between spurious oscillation in high order non-monotone scheme and additional dissipation in first order scheme.
Flux limiter schemes are invented as the “formulas of compromise” to combine high resolution schemes in areas with smooth fields and first-order schemes in areas with sharp gradients.
The additional dissipation control is difficult.
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Nonequilibrium entropy limiters for LBM
Entropy is a scalar quantity Entropy trimming: we monitor local deviation of f from the
correspondent equilibrium f*, and correct most nonequilibrium states (with highest ΔS(f)=S(f*)-S(f));
0f
too big
1fEhrenfest
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Positivity rule
f *
f
f *+(2β-1)(f*-f)
Positivity fixation
Positivity domain
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Entropy Filtering
)))((( ** fffSffF
Ehrenfest
.,0
,,1)(
dt
dtt
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Median Filter Choose a number of neighbouring points. Arrange the non-equilibrium entropies in order of
size. Choose middle one.
Very robust and gentle in places where signal is smooth.
Preserves edges, but reduces oscillation.
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Lid driven cavity For Re < 7000
steady flow For Re > 8500
periodic flow Bifurcation point
between
Peng, Shiau, Hwang (2003)
(100 by 100 grid)
3007402
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Velocity at monitor point
Reynolds’ number7375
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Locating bifurcation point
Re=7135
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Conclusions Navier-Stokes’ equations arise naturally via
free flight and equilibration in phase space. The viscosity, both actual and artificial can
be controlled precisely. The appropriate notion of smallness is the
free-flight time, which is a computational, not physical number.
Non-locality is exact and computable, non-linearity is local.
Reproduce statistics in some standard tests. Flux limiting can be done via control of a
scalar variable entropy.