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MULTIPHYSICS 2009

9-11 December 2009

Lille, 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

potential

Lattice Gas automata Lattice Boltzmann

EquationFluid Particule

Probabilty Fluid dynamics Navier-Stokes

EquationsContinum Fluid

Fig.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 r

f 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 ff f

t x

1 eqi ii i i

f fc f f

t x

, , , ,eq

i 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 model

Instead 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 n

tr ,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 1

4 1 1 1 1 2 2 2 2

4 2 2 2 2 1 1 1 1

0 1 0 1 0 1 1 1 1

0 2 0 2 0 1 1 1 1

0 0 1 0 1 1 1 1 1

0 0 2 0 2 1 1 1 1

0 1 1 1 1 0 0 0 0

0 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

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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 i

i

f

As for the pressure, it was can be computed simply by

2 ; 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

xp

xb

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