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3 Lattice Boltzmann Method (LBM) A n alternative to classic CFD Method Big progress made over the last decade Becomes popular Advantages of LBM Ease of BC implementation Well adapted for parallel computations No Poisson equation for pressure Choice Why use the LBM? Motivation
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MULTIPHYSICS 2009
9-11 December 2009Lille, FRANCE
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Computation and control of the near-wake flow over a square cylinder with an upstream rod using an MRT lattice Boltzmann model
by H. Naji a, A. Mezrhab b, M. Bouzidi c
a Université Lille 1 - Sciences et Technologies/ Polytech'Lille/ LML UMR 8107 CNRS, F-59655 Villeneuve d'Ascq cedex, Franceb Laboratoire de Mécanique & Energétique, Département de Physique, Faculté des sciences, Université Mohamed 1, Oujda, Marocc Université Blaise Pascal – Clermont II/ IUT/ LaMI EA3867 – FR TIMS 2856 CNRS, Av. A. Briand, F-03101 Montluçon cedex, France
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Lattice Boltzmann Method (LBM) An alternative to classic CFD Method Big progress made over the last decade Becomes popular
Advantages of LBM Ease of BC implementation Well adapted for parallel computations No Poisson equation for pressure
Choice
Why use the LBM?Motivation
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What is the origin of this method?
LBM has been derived from the Lattice gas Automata (LGA)also It can be obtained using a first order explicit upwind FD discretisatin of the discrete Boltzmann equation.
See next slide
From a historic point view
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Modelling of the fluid process Three levels
Macroscopic, mesoscopic and microscopic
Molecular dynamicsIntermolecular
potentialLattice Gas automata
Lattice Boltzmann Equation
Fluid Particule Probabilty Fluid dynamics
Navier-Stokes Equations
Continum FluidFig.1. Three levels of modeling
Microscopic
Mesoscopic
Macroscopic
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, , f f r t PPD distribution function PPDF
Vector position
Particle velocity
t r e collisionf f f f
External force (EF)
eq
t rf ff f
If we assume: eqe ef f
d
dt
e Complex form !!!!
and neglect EF
feq denotes the equilibrium function; is the relaxation time
small Knudsen numbersmall Mach number
Chapman-Enskog-Expansion
Bhatnagar-Gross-Krook-Approximation (BGK)
discretisation in velocity space
discretisation in space and time
Chapman-Enskog-Expansion
small Knudsen number
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1
, eqf f
f ft x
1 eqf f f ft x
1 eqi ii i i
f fc f ft x
, , , ,eqi i i i i
tf t t x e t f t x f t x f t x
1
vv v p v
t
0vt
Boltzmann equation
discrete Boltzmann equation
Navier Stokes equations
continuity equation
Boltzmann equation
Lattice Boltzmann equation (LBGK)
Overview of the
Modelling
The post-collision state
propagationcollision
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Splitting process
,f t x ,f t x , if t t x e t
Fig. 2. Collision and propagation steps
, , , ,
, ,
eqi i ii
i i i
tf t t x f t x f t x f t x
f t t x e t f t t x
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As computational tool, LBE method differs from NS equations based method in various aspects:
NS eqs are 2d order PDEs; LBE is 1st order PDE NS solvers need to treat the NL convective term; the LBE avoids this term; CFD solvers need to solve the Poisson equation; the LBE method is always local: the pressure is obtained from an equation of state; In the LBE, the CFL number is ~ to Δ t/Δx; Since The LBE is kinetic-based, the physics associated with the molecular level interaction can be incorporated more easily in the LBE model. Hence, the LBE model can be fruitfully applied to micro-scale fluid flow problems; The coupling between discretized velocity space and configuration space leads to regular square grids. This is a limitation of the LBE, especially for Aerodynamics where both the far field boundary condition and the near wall boundary layer need to be carefully implemented.
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The lattice Bolzmann Method for the D2Q9 square lattice modelInstead of a complex integro-differential operator Ω we can use two dfferent approximations:
1 eqr,t M S Mf r,t _m r,t
Multi Relaxation Time (MRT) model
Single Relaxation Time (SRT) model
eq
i n i i ntr ,t ,t f ,t
if r r
In this aproach the distribution is transformed into moment space before relaxation The utilisation of several different relaxation rates for the non-conserved moments leads to an increase in stability and thus to more efficient simulations
where Mf=m
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For the D2Q9 square lattice model
the form of the transformation matrix is
is the transformation matrix such that
1 1 1 1 1 1 1 1 14 1 1 1 1 2 2 2 24 2 2 2 2 1 1 1 10 1 0 1 0 1 1 1 10 2 0 2 0 1 1 1 10 0 1 0 1 1 1 1 10 0 2 0 2 1 1 1 10 1 1 1 1 0 0 0 00 0 0 0 0 1 1 1 1
M
the relaxation matrix S in the moment space is
S = diag(0,s1,s2,0,s4,0,s6,s7,s8); si being the collision rates
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( )ac bc eq bcj j j j jm m s m m
where mjac is the moment after collision, mj
bc is the moment before collision (the post- collision value)sj are the relaxation rates which are the diagonal elements of the matrix S and are the corresponding equilibrium momentsj
eqm
The moments (9) are separated into two groups: (ρ, m3, m5) are the conserved moments which are locally conserved in the collision process; (m1, m2, m4, m6, m7, m8) are the non-conserved moments and they are calculated from the relaxation equations:
The macroscopic fluid variables, (ρ , velocity u and pressure P, are obtained from the moments of the distribution functions as follows
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0
ii
f ( , )x y i ii
j j f J u e 2 / 3 sp c
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The D2Q9 model
Fig. 3. A 2-D 9-velocity lattice (D2Q9) model
(0,0), 0
cos ( 1) / 2 ,sin( ( 1) / 2 ) , 1 4
2 cos (2 9) / 4 ,sin (2 9) / 4 , 5 8
i
i
i i c i
c i i i
e
In LB flow simulations, Discrete Particle Distribution Functions (PPDF) are propagated with discrete velocities
e8e4e7
e3 e0 e1
e6 e2 e5
1
2
3
4
56
7 8
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In the discrete velocity space, the density and momentum fluxes can be evaluated as
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0
ii
f
1u ei ii
f
As for the pressure, it was can be computed simply by2; sp c / 3s cc
The corresponding viscosity in the NS equations is
212 s tc
Flow configuration
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Fig. 4. Schematic representation of the configuration and nomenclature
h
w
xpxb
the rod (Bi-partition)
the obstacle
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e8e4e7
e3 e0e1
e6 e2 e5
Fig. 5. Mesh structure around the control partition and square cylinder
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Re
Cd
0 50 100 150 200 250 3001.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Our resultsBreuer et al. FVMBreuer et al.LBA
Re
St
50 100 150 200 250 300
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
Our resultsBreuer et al. FVMBreuer et al. LBA
(a) (b)
Fig. 6. Comparison with previous work for: (a) the drag coefficient, (b) Strouhal number .
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h/d
Cd
0 0.2 0.4 0.6 0.8 1-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
Our resultsZhou et al.
h/d
rmsCl
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.05
0.1
0.15
0.2
0.25
0.3
Our resultsZhou et al.
(a ) (b)Fig. 7. Comparison with previous work for: (a) the drag coefficient, (b) rms lift coefficient at w/d=1.5.
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(a)
(c)
Fig. 8. Streamlines at Re = 250 and h*=0.5: (a) without control,(b) w/d = 1, (c) w/d = 2;
(b)
The bi-partition location effects
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(d)
(e)
Fig. 9. Streamlines at Re = 250 and h*=0.5: (d) w/d = 3, (e) w/d = 4, (f) w/d = 5.
(f)
Effect of the bi-partition location (continued)
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(b)
(d)
(a)
Fig. 10. Streamlines for different bi-partition heights at w/d =5 and Re=250; (a) h*= 0.1, (b) h* = 0.3, (c) h* = 0.5, (d)
h* = 0.6
(c)
The bi-partition height effects
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(f)
(g)
(h)
(e)
Fig. 11. Streamlines for different bi-partition heights at w/d =5 and Re=250; (e) h* = 0.7, (f) h* = 0.8, (g) h* = 0.9,
(h) h* =1.
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The present LBM showed:
1. LBM is a reliable alternative Method to the classical based on the resolution of the NS-equations (at least for incompressibl flows)
2. LBM can be used for research and development
BUT
Need to convince more researchers (e.g. turbulence community) to use LBM (How ?)
Conclusion