intro lbm fundamentals srt-bgk mrt clbm/fcmkbc … · intro lbm fundamentals srt-bgk mrt...

Intro LBM Fundamentals SRT-BGK MRT CLBM/FCM KBC Turbulence HM Transfer CUDA Show Conclude Hydrodynamic Limit of CTLBM

Introduction

• Lattice Gas Automata was described in 1986 by Frisch,Hasslacher and Pomeau.

• In 1988 McNamara & Zanetti introduced Lattice BoltzmannMethod as an improved method compared to LGA.

• LBM treats fluid as particles that stream along given directions(lattice links) and collide at lattice sites

• It’s strongly based on kinetic theory - discrete approach - wesolve discrete kind of the Boltzmann Transport Equation (BTE).

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LBM in few steps

• Select velocity model(D1Q3,D1Q5,D2Q5,D2Q9,D3Q15,D3Q19,D3Q27), fi,ti,cs

• Divide domain into lattice sites (mark solid and fluid sites,compute links intersections for curved boundary, setup inlet,outlet, symmetry, periodic sites)

• Compute relaxation parameter/parameters• Solve discrete Boltzmann transport equation

1. Apply BC2. Compute moments - macro variables3. Compute equilibrium distribution functions (collision)4. Send fi along its characteristic velocity vector (stream)

Do all of this using CUDA in multiGPU environment to obtainextremely powerfull CFD solver!

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Boltzmann Transport Equation

Equation for the evolution of number of molecules.∂f

∂t+∂f

∂r· c +

F

m· ∂f∂c

= Ω

BTE describes statistics of the system by distribution functions(DF) f(r, c, t) i.e. number of molecules at time t which haveposition and velocity between r + dr and c + dc. Equation for theevolution of number of molecules.

∂f

∂t+∂f

∂r· c +

F

m· ∂f∂c

= Ω

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Moments of distribution function

Relation between BTE and macroscopic quantities are obtainedfrom appropriate moments:

ρ(r, t) = m

∫f(r, c, t)dc

ρ(r, t)u(r, t) = m

∫cf(r, c, t)dc

ρε(r, t) =1

2m

∫|ξ|2f(r, c, t)dc, ε =

DkBT

2m

Pij = m

∫ξiξjf(r, c, t)dξ (stress tensor)

Qijk = m

∫ξiξjξkf(r, c, t)dξ (heat flux tensor)

ξ = ci − u(r, t) (peculiar velocity)5 of 69


Collisional integral & BGK approximation

Collision integral for 2-body collision is of the form:

Ω =

∫(f1′2′ − f12)gσ(g, ω)dωdp2

σ - differential cross section (expressing molecules with relativespeed g = gω around the solid angle ω), Boltzmann’s closureassumption (molecular chaos , Stosszahlansatz)

f12 = f1f2

Generally, Ω is uncloseable (BBGKY hierarchy), in 1954Bhatnagar, Groos and Krook (BGK) introduced simplified collisionoperator:

Ω = ω(f eq − f) =1

τ(f eq − f)

ω collision frequency, τ relaxation factor, f eq the local equilibriumdistribution function6 of 69


Collisional integral properties

Collisional integral Ω has exactly five collisional invariants ψk(c)∫Ωψk(c)dc = 0

The elementary collision invariants read

ψ0 = 1, (ψ1, ψ2, ψ3) = c, ψ4 = c2

general invariants φk(c) can be written as linear combination of ψk

φ(c) = A+ B · c + Cc2

There exist positive functions f of form exp(φ) that give vanishingcollisional integral

Ω(f, f) = 0

Maxwell distribution function is special case of them7 of 69


Discrete Boltzmann Equation

Introducing BGK approximation and neglecting external forces wehave

∂f

∂t+ c · ∇f =

1

τ(f eq − f)

Now we discretize velocities along finite set of specific directions ci

∂fi∂t

+ ci · ∇fi =1

τ(f eqi − fi)

linear PDEs of advection type with source term

next we discretize time and spatial derivatives

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BTE discretization in 1D

fi(x, t+ δt)− fi(x, t)δt

+ cifi(x+ δx, t+ δt)− fi(x, t+ δt)

δx=

= −1

τ(fi(x, t)− f eqi (x, t))

note that δx = ciδt, then

fi(x+ ciδt, t+ δt) = fi(x, t) +δt

τ[f eqi (x, t)− fi(x, t)]

for simplicity we can assume that

δt = δx = 1

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Equilibrium distribution function

Take the normalized Maxwell’s DF in form

f =3ρ

2πe−

32c2e

3(c·u−u2

2

)/2

and expand second exponential around stationary state

f =3ρ

2πe−

32c2[1 + 3(c · u)− 3

2u2 + · · ·

]General form of discrete equilibrium DF is

f eqi = Φti[A+Bci · u + C(ci · u)2 +Du2

]For conserved quantities we have:

Φ =

n∑i=0

f eqi =

n∑i=0

fi, Φui =

n∑i=0

f eqi ci =

n∑i=0

fici

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Chapman-Enskog Expansion in 1D – gateway tomacro scale

Relation between relaxation time τ and macroscopic transportproperty of simulated matter can be obtained by performingChapman-Enskog expansion (small perturbance analysis).First we expand fi in terms of small ε

fi(x, t) = f0i + εf1i + ε2f2i + · · · ,where f0i = f eqi

Updated DF is expanded using Taylor series

fi(x+ ciδt, t+ δt) = fi(x, t) +∂fi∂tδt+

∂fi∂x

ciδt+

+1

2δt2(∂2fi∂t2

+ 2∂2fi∂t∂x

ci +∂2fi∂x2

cici

)+ O(δt)3

Then appropriate scaling is introduced to establish relationshipbetween meso and macro scale parameters.

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LBM D1Q3 Model

1D model with 3 discrete velocities

C E Wci 0 1 -1ti 4/6 1/6 1/6

lattice speed of sound cs = 1√2

(needed for equilibrium DF)

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LBM D2Q9 Model


C E N W S NE NW SW SEci (0,0)(1,0)(0,1)(-1,0)(0,-1)(1,1)(-1,1)(-1,-1)(1,-1)ti 4/9 1/9 1/9 1/9 1/9 1/36 1/36 1/36 1/36

lattice speed of sound cs = 1√3

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3D model with 6 discrete velocities (heat& mass transfer)

cs =1√2

tc =1

4t =

1

8

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LBM D3Q19 Model


weights 1/3, 1/18, 1/36, lattice speed of sound cs = 1√3

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LBM D3Q27 Model

weights 8/27, 2/27, 1/54, 1/216, lattice speed of sound cs = 1√3

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Computing Algorithm

Algorithm for LBM-BGK model• Initialize fi in lattice sites with f eqi (ρini,uini)• Setup τ according to problem solved• Repeat until steady state or desired time is achieved

1. Compute macroscopic quantities from fi2. Compute equilibrium DF and collide3. Stream fi along lattice links4. Apply BC

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Boundary conditions

Imposing BC in LBM is nontrivial except periodic & symmetry BC,in other cases bounce-back is usually usedPossible method for straight boundary velocity/pressureinlet/outlet are

• Inamuro BC• Zhou-He BC• Regularized BC• D’Orazzio

BC for temperature• Yu

BC for curved geometries• and several others18 of 69


LBM models

Presented LBM model is called Single Relaxation Time (SRT) dueto one parameter of relaxation τ . This can (and will for high Re)cause numerical instabilities. To overcome this other approacheswere presented• Multiple Relaxation Time (MRT/TRT) model of d’Humieres et al.• Entropic LBM (ELBM) by Karlin et al.• Hybrid methods like LBM-FD, LBM-FEM and LBM-FVM• Other LBMs with improved Galilean

invariance(CascadedLBM,FCM-LBM), KBC, Cumulant Method

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Heat Diffusion in 2D (D2Q9)

Non-dimensionalized heat diffusion equation reads

∂φ

∂t= α

(∂2φ

∂x2+∂2φ

∂y2

)with α = λ/ρC. From Chapman-Enskog expansion we have

αlb =δx

3δt

(τ

δt− 1

2

), f eqi = tiφ(x, t)

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Advection-Diffusion in 2D (D2Q9)

Non-dimensionalized advection-diffusion equation reads

∂φ

∂t+ u

∂φ

∂x+ v

∂φ

∂y= α

(∂2φ

∂x2+∂2φ

∂y2

)with α = ρC/λ. From Chapman-Enskog expansion we have

αlb =δx

3δt

(τ

δt− 1

2


(1 +

ci · uc2s

)

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Navier-Stokes in 2D (D2Q9)

Non-dimensionalized NS equations read

∂u

∂x+∂v

∂y= 0

∂u

∂t+∂u2

∂x+∂uv

∂y= −∂p

∂x+

1

Re

[∂

∂x

(∂u

∂x

)+

∂

∂y

(∂u

∂y

)]∂v

∂t+∂vu

∂x+∂v2

∂y= −∂p

∂y+

1

Re

[∂

∂x

(∂v

∂x

)+

∂

∂y

(∂v

∂y

)]with Re = l20/(t0ν). From Chapman-Enskog expansion we have

νlb =δx2

3δt

(τ

δt− 1

2


(1 +

ci · uc2s

+(ci · u)2

2c4s− u2

2c2s

)22 of 69


Navier-Stokes in 3D

Macroscopic equations for 3D Navier-Stokes for naturalconvection flows:

∇ · u = 0

∂u

∂t+ (u · ∇) u = −∇p+ ν∆u− gβ(T − T0)

∂T

∂t+∇ · (uT ) = α∆T

Extra term is Boussinesq forcing term:

FB = −gβ(T − T0)

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A note on units

LBM solves DBTE in terms of lattice units, but the real problem isin physical units, we have to compute the LBM parameters basedon physical units, solve the DBTE and then recompute the unitsback. The procedure is as follows:• Compute Reynolds number of the physical flow, setup the

lattice viscosityu0,pl0,pνp

=u0,lbNlb

νlb

• Solve DBTE, calculate lattice macroscopic variables, andrecalculate units

up =ulbu0,pu0,lb

, tp = tlbu0,lbl0,pNlbu0,p

The only constraint here is ulb < cs = 1√3(D2Q9)

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Multiple Relaxation Time LBM

Stability of SRT can be improved by performing collision inmoment space,

fi(x+ ciδt, t+ δt)− fi(x, t) = −M−1S[m(x, t)−mreq(x, t)]

m = (m0,m1, . . . ,mn)T are vectors of moments, S is diagonalmatrix of relaxation times for each moment and mapping betweendistribution and moment spaces is given by

m = Mf , f = M−1m

for D2Q9 model the vector of moments reads

m = (ρ, e, ε, jx, qx, jy, qy, pxx, pxy)T

and the equilibrium moment vector reads

meq = (ρ,−2ρ+3(j2x+j2y), ρ−3(j2x+j2y), jx,−jx, jy,−jy, (j2x−j2y), jxjy)T

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MRT Matrix

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MRT Projection

Set all moments except density,momentum and momentum flux toequilibrium (Ladd 94’)Compute postcollisional momentum flux

Π∗ = Π− 1

τ

(Π−Π(0)

)and reconstruct postcollisional DF (no matrix operations in orderto transform DF to momentum space and then back!)

f∗i = ti

[ρ

(5

2− 3

2‖ci‖2 + 3u · ci

)+

9

2Π∗ : cc− 3

2TrΠ∗

]+ δtFB,i

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MRT Algorithm - Flow field part

• Compute Πneq = Π−Πeq

• Compute τt = .5(√

τ2 + 18C2∆2x|S| − τ

), where τ = 3νlb + 0.5

• Compute Π∗ = Π + ωΠneq, ω = 1τ+τt

• Updatef∗i = ti

[ρ(52 −

32‖ci‖

2 + 3u · ci)

+ 92Π∗ : cc− 3

2TrΠ∗]

+ FB,i

Sij =3τ

2ρΠneqij Πeq

ij =ρ

3δij+ρuiuj |S| =

(S2ii + 2(S2xy + S2xz + S2yz)

) 12

Πxx =∑i

fic2i,x Πyy =

∑i

fic2i,y Πzz =

∑i

fic2i,z

Πxy =∑i

fici,xci,y Πxz =∑i

fici,xci,z Πyz =∑i

fici,yci,z

ρ =∑i

fi ρu =∑i

fici+FB

2FB,i = c2sti

(1− 1

2τf

)(ci − u)·(FB)

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Cascaded and Factorized Central Moment LBMs

Recent models of LBM uses collisional terms with improvedGalilean invariancy - Cascaded Lattice Boltzmann Models, theypossess enhanced stability and very small numerical diffusion.Based on CLBM Factorized Central Moment LBM was presentedby M. Geier in 2009, ultra stable model with small Mach numberbeing the only one limiting factor. CLBM is multiple relaxation timeLBM and use central moments

κxmyn =∑i

fi(cix − ux)m(ciy − uy)n

to compute post-collision states of fi instead of raw moments

πxmyn =∑i

ficmixc

niy

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Cascaded LBM

Now we take orthogonal transformation matrix K

K =

1 0 0 −4 0 0 0 0 41 −1 1 2 0 1 −1 1 11 −1 0 −1 1 0 0 −2 −21 −1 −1 2 0 −1 1 1 11 0 −1 −1 −1 0 −2 0 −21 1 −1 2 0 1 1 −1 11 1 0 −1 1 0 0 2 −21 1 1 2 0 −1 −1 −1 11 0 1 −1 −1 0 2 0 −2

and assume that post-collision state f∗ is in equlibrium

feq,∗ = f + KT · knext we compute central moments of both sides of the equationabove...30 of 69


Cascaded LBM cnt’d.

So we obtain system of linear equations for k to be solved

6 2 0 0 0 06 −2 0 0 0 00 0 −4 0 0 0−6uy −2uy 8ux −4 0 0−6ux −2ux 8uy 0 −4 0

8 + 6(u2x + u2y) 2(u2y − u2x) −16uxuy 8uy 8ux 4

·

k3k4k5k6k7k8

=

=

κeqxx − κxxκeqyy − κyyκeqxy − κxyκeqxxy − κxxyκeqxyy − κxyyκeqxxyy − κxxyy

=

ρc2s − κxxρc2s − κyy0− κxy0− κxxy0− κxyy

ρc4s − κxxyy

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Cascaded LBM cnt’d.

Solving the system for ki and relaxing them gives us post-collisionstates in momentum space, which is then transformed back byK−1, k3, k4 and k5 are given by

k3 =1

12τ3(ρ(u2x+u2y)−fE−fN−fS−fW−2(fSE+fSW+fNE+fNW−ρ/3)

k4 =1

4τ4(fN + fS − fE − fW + ρ(u2x − u2y)

k5 =1

4τ5(fNE + fSW − fNW − fSE − uxuyρ)

for isotropic viscosity we set τ4 = τ5 = τ and compute τ from

ν+ =1

cs

(τ − 1

2

)This method is stable even for τ4 = τ5 = .5 and all other τi equal to1 (zero viscosity).32 of 69


Factorized Central Moment LBM

To improve Galilean invariancy, and thus stability and accuracy ofthe CLBM, the Factorized Central Moment LBM was proposed byGeier in 2009.For D2Q9 model, the only difference is in k8.

k8 =1

4

[κatxxyy − κxxyy − 8k3 − 6k4(u

2x + u2y)− 2k4(u

2y − u2x)+

+16k5uxuy − 8k6uy − 8k7ux]

where κatxxyy is defined as• κatxxyy = κeqxxyy = ρc4s for CLBM• κatxxyy = κ∗xxκ

∗yy for FCM

and postcollision states κ∗xx, κ∗yy are given by:

κ∗xx = 6k3 + 2k4 + κxx = 6k3 + 2k4 + πxx − ρu2xκ∗yy = 6k3 − 2k4 + κyy = 6k3 − 2k4 + πyy − ρu2y

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KBC Model

It is LBM with Entropic correction inside

fi(x+ ∆x, t+ 1) = (1− β)fi(x, t) + βfmirri (x, t)

for SRT BGK:

fmirri = 2feqi − fi, ν = c2s

(1

2β− 1

2

)in KBC (Karlin, Bosch, Chikatamarla) one should use:

fmirri = ki + [2seqi − si] + [(1− γ)hi + γheqi ]

wherefi = ki + si + hi

and γ is entropic stabilizer computed by

γ =1

β−(

2− 1

β

)< ∆s|∆h >< ∆h|∆h >

, < x|y >=

b∑i=1

xiyifeqi34 of 69


Smagorinsky-Lilly SGS LES LBM

Stability of LBM can be improved by using turbulence model i.e.alter the viscosity with the turbulent one.

ν = νlaminar + νturbulent

Large Eddy Simulation concept is very convenient for LBM. UsingSmagorinsky approach, the eddy viscosity is given by

νt = (CSM∆)2|S|, |S| =√

2Sij Sij

we need to compute S, but from C-E expansion we directly have

Π(1)ij ≈ −2τc2sρδtSij

but what the Π is?35 of 69


Π(1) approximation

Again from C-E expansion we have

Π(1)ij =

∑α

cαicαjf(1)α ≈

∑α

cαicαj (fα − f eqα )

so we can define

Π =

√Π

(1)ij Π

(1)ij =

√2τc2sδt|S| =

√Q

and then

τ =1

2

τ0 +

√τ20 +

(CSM∆

c2sδt

)2 √8Qρ

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SRT+LES - Temperature field

Simple SRT BGK• Compute T =

∑i gi

• Compute τg,t = .5(√

τ2g + 18C2∆2x|S| − τg

), where

τg = 3αlb + 0.5

• Update g∗i = gi + 1τg+τg,t

(geqi (T,u)− gi)

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MRT - Temperature field

MRT D2Q5 LBM is used for temperature field (passive scalar)approximation. The second population of DF gi is introduced andMRT is applied:

gi(x+ ciδt, t+ δt)− gi(x, t) = −M−1S[m(x, t)−meq(x, t)]

where equilibrium moments and matrix S are defined as

meq = (T, uxT, uyT, aT, 0) S = diag(0,1

τα,

1

τα,

1

τe,

1

τν)

The thermal diffusivity is obtained from

α+ =4 + a

10

(τα −

1

2

)We can use a = −2/3 together with

(τν − .5)(τα − .5) = (τe − .5)(τα − .5) = 1/6

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CTLBM Algorithm - Temperature field part

Cascaded scheme applied to one conservation law (joint workwith Keerti Sharma).

KT =

1 1 1 1 10 −1 0 1 00 0 −1 0 14 −1 −1 −1 −10 −1 1 −1 1

.with a collision step

~gc = ~g + K · ~k(~g,~geq, ω1, . . . , ω5),

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CTLBM - details

Collision here is rather complicated now

k1k2k3k4k5

=

0

ω22

(κeqx − κx)

ω32

(κeqy − κy)

−ω44

(κeqxx+yy − κxx+yy + κ

eqyy−xx − κyy−xx)− uxω2

2(κeqx − κx)− uyω3

2(κeq

y − κy)ω54

(κeqxx+yy − κxx+yy − κ

eqyy−xx + κyy−xx) +

uxω22

(κeqx − κx)− uyω3

2(κeq

y − κy)

,

and gives us following solution

∂T

∂t+∂Tux∂x

+∂Tuy∂y

=a

2

(1

ω2− 1

2

)∂2T

∂x2+a

2

(1

ω3− 1

2

)∂2T

∂y2+O(∆t3).

thermal diffusivities in x and y directions are defined by

αx =a

2

(1

ω2− 1

2

)αy =

a

2

(1

ω3− 1

2

)40 of 69


Using CUDA for GPU

• parallel computing platform and programming model created byNVIDIA

• CUDA platform is accessible through extensions to C, C++ andFortran

• domain is divided into the grid consisting of blocks of threads• pull algorithm with flattened arrays

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Pull vs. algorithm

Pull algorithm• Stream step (uncoalesced read)• Apply BC• Compute macroscopic variables• Collide & write post-collision state back (coalesced write)Push algorithm• Read DF (coalesced read)• Apply BC• Compute macroscopic variables• Collide & Stream(uncoalesced write)

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CUDA performance D2Q9 FCM-MRT

43 of 69


Validation of the MRT-TLBM - Thermal storageexperiment

We try to simulate experiment1 of rock packed bed heating.

1Meier et al., Solar Energy Materials (24) 199944 of 69


Comparison of results with experimental data

200

300

400

500

600

700

800

900

0 0.2 0.4 0.6 0.8 1 1.2 1.4

tem

pe

ratu

re [

K]

distance [m]

average temp 1200saverage temp 3000saverage temp 4800s

exp 1200sexp 3000sexp 4800s

45 of 69


Shaft furnace

No we consider larger problem of approx 5m x 2.5m of packedbed of solid lumps of different diameters.

46 of 69


Shaft furnace - Results

47 of 69


Forced convection - cylinder with heated core

Several benchmark and other fancy heat-transfer problemsincluding forced and natural convection.

U0

D2D

10D

40D

20D

10D

48 of 69


Cooling of Cylinder with Heated Core

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Cooling of Cylinder with Heated Core

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

(T-T

c)/(

Th-

Tc)

φ

ks/kf=0.5 Refks/kf=0.5 CTLBM

ks/kf=1 Refks/kf=1 CTLBM



50 of 69


Cooling of Cylinders with Heated Core

U0 D

3D20D

4D

2D2.5D

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Cooling of Cylinders with Heated Core Pr=1

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Cooling of Cylinders with Heated Core Pr=7.2

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Double Shear Layer Vortex Test

Ulb(x, y, 0) =

U0 tanh(80(y/(N − 1)− 0.25)) y/(N − 1) ≤ 0.5

U0 tanh(80(0.75− y/(N − 1))) y/(N − 1) > 0.5

Vlb(x, y, 0) = 0.05U0 sin(2π(x/(N − 1) + .25))

Tlb(x, y, 0) =

1 1

4 ≤ y/(N − 1) < 34

0 elsewhere

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Double Shear Layer Vortex Test

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CTLBM for natural convection flows

2D simulation of natural convection around heated cylinder indifferentially heated square cavity at Ra=2.24 · 107

(ongoing work with Keerti Sharma)

56 of 69


Heating furnace

• We want to compute flow in an heating furnace• 3D turbulent flow - D3Q19/D3Q27 lattices are used• CO2 = 7.81% H2O=14.84% N2=73.01% O2=4.88%• vin = 1.808 m/s Tin = 1000C ν = 2.22 · 10−4m2/s

57 of 69


Summary

• ∆t = 8.8 · 10−4 s, ∆x = 0.016 m ≈ 12M LS• 1s - 1130 iterations, 1 it takes approx. 0.01 s of GPU time

58 of 69


Fixed bed

• Fixed bed of different icosahedral particles - generated bysettleDyn

• 3D turbulent flow - D3Q19 lattice is used• Air is flowing across the bed• vin = 0.5 m/s, ν = 1 · 10−5m2/s

59 of 69


Fixed bed

• ∆t = 3.2 · 10−4 s, ∆x = 0.016 m ≈ 5.7M LS• 1s - 3125 iterations, 1 it lasts ≈ 0.007 s of GPU time

60 of 69


Natural convection in a room

• Cold room (Tini = 15C) is heated by two heaters(Theater = 70C)

• Outside the building, the Winter is coming (Tout = −20C)• Heaters are situated under the window and at the opposite wall.61 of 69


Multiphase

Future work? What about multiphase flows?

62 of 69


Conlusions

PROS• Easy implementation of complex geometry• LBM proved its efficiency in computing various types of fluid

problems• Scheme is local in space and explicit in time i.e. good

scalability for parallel computing (GPGPU, clusters...)• There are still open problems and ongoing research of LBMCONS• Dreadful evaluation of BC’s compared to traditional CFD

methods• Regular square grid, need for multi-block lattices or

interpolation schemes for very accurate curved geometry• Numerical instabilities for high Re flows with BGK (MRT, ELBM,

Fractional step LBM, CLBM, Cumulants,...)63 of 69


CTLBM - CLBM for 1 conservation law

Cascaded scheme applied to one conservation law.

KT =

1 1 1 1 10 −1 0 1 00 0 −1 0 14 −1 −1 −1 −10 −1 1 −1 1

.

K′ =[~M0, ~Mx, ~My, ~Mxx+yy, ~Myy−xx

],

where ~M0 = [1, 1, 1, 1, 1]T, Mx,i = ci,x, My,i = ci,y,Mxx+yy,i = c2i,x + c2i,y, Myy−xx,i = c2i,y − c2i,x, with a collision step

~gc = ~g + K · ~k(~g,~geq, ω1, . . . , ω5),

64 of 69


Collision cascade

Now compute central moments for both sides of the previous eq.

~gc = ~g + K · ~k(~g,~geq, ω1, . . . , ω5) /∑i

(ci,x − ux)m(ci,y − uy)n

to obtain system of eq. with the shift matrix S

S

k1k2k3k4k5

=

0 0 0 0 00 2 0 0 00 0 2 0 00 −4ux 0 −2 −20 0 −4uy −2 2

k1k2k3k4k5

=

0

κeqx − κxκeqy − κyκeqxx − κxxκeqyy − κyy

(1)

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CTLBM Collision Scheme

We solve for ki’s and eventually obtain

k1 = 0

k2 =ω2

2(κeqx − κx)

k3 =ω3

2(κeqy − κy)

k4 = −ω4

4(κeqxx − κxx + κeqyy − κyy)−

uxω2

2(κeqx − κx)− uyω3

2(κeqy − κy)

k5 =ω5

4(κxx − κeqxx + κeqyy − κyy)−

uxω2

2(κeqx − κx) +

uyω3

2(κeqy − κy)

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CTLBM in Raw Moments Representation

We can also reformulate the collision in raw moment to obtain

~mc = ~m+ M ·K · M(~meq − ~m).

with

M ·K · M =

0 0 0 0 00 ω2 0 0 00 0 ω3 0 00 (ω2 − ω4)2ux (ω3 − ω4)2uy ω4 00 (ω5 − ω2)2ux (ω3 − ω5)2uy 0 ω5

where

~m = [m0,mx,my,mxx +myy,myy −mxx]T

~meq = [T, Tux, Tuy, aT, 0]T

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Derivation of an equivalent partial differential equation (EPDE -very useful technique! promoted by F. Dubois) yields2

m(3)0 −m

c(3)0 + ∂tm

eq(1)0 + ∂xm

eq(2)x + ∂ym

eq(2)y =(

1

ω2+

1

ω3− 1

)∂xym

eq(1)xy +

(1

ω2− 1

2

)∂xxm

eq(1)xx +

(1

ω3− 1

2

)∂yym

eq(1)yy .

For D2Q5 lattice model mxy = meqxy = 0 and m0 = meq

0 = T ,meqx = Tux, meq

y = Tuy, meqxx = meq

yy = T a2 the final PDE is

∂T

∂t+∂Tux∂x

+∂Tuy∂y

=a

2

(1

ω2− 1

2

)∂2T

∂x2+a

2

(1

ω3− 1

2

)∂2T

∂y2+O(∆t3).

i.e. the Fourier-Kirchhoff with diffusivities

αx =a

2

(1

ω2− 1

2

)αy =

a

2

(1

ω3− 1

2

)2For details check our last article in International Journal of Thermal Sciences

];)68 of 69

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