Lattice-Boltzmann Simulations of Particle Suspensionsjanke/CompPhys07/Folien/lorenz.pdf · Lattice-Boltzmann Simulations of Particle Suspensions in Sheared Flow Eric Lorenz, Alfons

Lattice-Boltzmann Simulations of ParticleSuspensions in Sheared Flow

Eric Lorenz, Alfons G. Hoekstra

Section Computational Science, Faculty of ScienceUniversity of Amsterdam

[email protected]

28/09/2007

Outline

• Lattice-Boltzmann Method (LBM)• Suspension Modeling with LBM• Lees-Edwards Boundary Conditions (LEBC) for LBM Suspensions• Rheology of Suspensions (Validation of LEBC)• Some Cluster Properties

1

Lattice-Boltzmann Method (LBM)

2

LBM’s Predecessor: Lattice-Gas Automaton

• identical particles on a lattice (space, time AND velocities discrete)• propagation• collision: particles reshuffled, mass and momentum conserved

• Frisch, Hasslacher, Pommeau (1986): LGA on hexagonal lattice→ Navier-Stokes

3

Lattice-Boltzmann method

• particles replaced by their ensemble average (reduction of noise):distribution function fi(r, t), giving the probability of finding a particle at site r attime t flying with velocity ci

• evolution follows the Boltzmann-equation for a dilute gas,

∂tfi(x, v, t) + v∇xf(x, v, t) +F

mf(x, v, t) = coll(f(x, v, t)),

in its discretized form (here, the simplest LBGK scheme, single-relaxation time)

fi(r + ei, t+ 1)− fi(r, t) + Fi =1

τ(f

eqi (r, t)− fi(r, t))

where:r lattice citeei velocity, pointing to adjacent nodeτ relaxation time.

• kinematic viscosity ν = (2τ − 1)/6

4

Lattice-Boltzmann method

• moments of fi: density, momentum, momentum flux density

ρ(r, t) =

QXi

fi(r, t)

ρ(r, t) · u(r, t) =

QXi

eifi(r, t)

Π =

QXi

eieifi(r, t)

• equilibrium function (for a weakly compressible fluid) for Ma� 1

feqi (r, t) = ρ(r, t)

“Ai + Bi(ei · u) + Ci(ei · u)

2+Diu

2”

agrees with Maxwell-Boltzmann distribution up to O(u2)

• direction dependent coefficients determined by conservation of mass, momentumand kinetic energy (total energy ρθ + ρuu for thermal LBM)

5

D2Q9 LBM

• isotropy requires at least 9 velocities in 2dimensions

e0 = ( 0, 0),e1 = ( 1, 0), e2 = ( 0, 1),e3 = (−1, 0), e4 = ( 0,−1),e5 = ( 1, 1), e6 = (−1, 1),e7 = (−1,−1), e8 = ( 1,−1)

847

3

6 2 5

1

• coefficients in f eqi

Ai Bi Ci Di

i = 0 4/9 0 0 -2/3i = 1, 2, 3, 4 1/9 1/3 1/2 -1/6i = 5, 6, 7, 8 1/36 1/12 1/8 -1/24

6

Modelling of Suspensions with LBM

7

The ALD method for suspended particles

(Aidun, Lu, Ding, 2003)

• particles mapped to lattice→ broken links

• virtual fluid inside shell• arbitrary surfaces possible

• solid-fluid interaction: (bounce-back at the links with moving wall)

fi(x, t+ 1) =

f−i(x, t

+) + 2ρBiub · ei if BL

fi(x + e−i, t+) else

• momentum exchange, force, torque:

δpi = 2ei[fi(x, t+ 1)− ρBiub · ei]

Fi(x, t0 +1/2) = δpi/∆t, Ti(x, t0 +1/2) = (x−X(t0)×Fi(x, t0 +1/2))

8

Lees-Edwards Boundary Conditions for LBMSuspensions

9

Lees-Edwards Boundaries for Suspensions

wu

f5

Procedure for boundary-crossing densities:- Mapping to off-grid copy (distribute f over destination nodes according to sub-gridshift ratio):

f5(x, 1) = (1−mod1[ut]) · f5(x+ int[x− ut], ymax)

+ mod1[ut] · f5(x+ int[x− ut− 1], ymax)

10

Lees-Edwards Boundaries for Suspensions

wu

f5

Galilei-Transformation of boundary-crossing densities to new reference frame

• difference in fi to get momentum transfer right (all macroscopic variables and theirderivatives will stay the same, image is moving as block at const velocity)

∆f5 = f5(u + uw)− f5(u)

≈ feq5 (u + uw)− f eq

5 (u)

−τ(∂t[feq5 (u + uw)− f eq

5 (u)] + u5∂r[feq5 (u + uw)− f eq

5 (u)])

≈ feq5 (u + uw)− f eq

5 (u)

• last step: only u-dependence kept, skip O(∂) terms because their 0th and 1stmoments neglectible (Wagner, Pagonabarraga, 2006)

11

Lees-Edwards Boundaries for Suspensions

wu

f5

Sub-grid Boundary Reflection (according to sub-grid shift ratio):

• modification of distribution step to allow fluid-solid interaction

fi(x, t+ 1) = (1−mod1[uwt]) ·fi(x− int[x− uwt] + ei, t

+)

f−i(x, t+)− ρBiub · ei if BL

+ mod1[uwt] ·fi(x− int[x− uwt+ 1] + ei, t

+)

f−i(x, t+)− ρBiub · ei if BL

12

Rheology of Suspensions

13

Apparent Viscosity, Dependence on Concentration φ

• for a dilute dispersed suspension (Einstein, 1906): νapp = νf(1 + 2.5φ)

assumptions: no hydrodynamical interaction (∼ φ2) , Brownian motion insignificant• semi-empirical Krieger-Dougherty relation:

νapp = νf

„1−

φ

φmax

«−[η]φmax

(1)

• simulation results for R = 8, Re = 0.001

0

1

2

3

4

5

6

7

0 0.1 0.2 0.3 0.4 0.5

ν app

/νf

φ

Krieger-DoughertyEinstein

Re=0.001

14

App. Viscosity νapp as a Function of Shear Rate γ

• generic behaviour

νlo

g

I II III IV V

log γ

φ

– I - Newtonian plateau: Brownian motion dominates– II - shear-thinning: increasing shear decreases disorder of particle structure– III - Newtonian plateau: particles strongly orientated– IV - shear-thickening: local structures, broken by shear, momentum transfer via

particles dominates– V - unknown: some experimental results show repeated shear-thinning

15

Shear-Thickening

• Rp = 8, Lx,y = 259 ≈ 16 · 2Rp, φ = 0.40, νf = 0.0125, uw < 0.0864

Lees-Edwards vs. planar Couette scheme

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

-2.5 -2 -1.5 -1 -0.5 0 0.5

ν eff/

(1-φ

s/φ m

ax)-[

η]φ m

ax

log(Reshear,p)

φ=0.40 (Kromkamp+)φ=0.40 (Lees-Edwards)

φ=0.40 Couette

0

20

40

60

80

100

-2.5 -2 -1.5 -1 -0.5 0

τ s,f/

τ T

log(Reshear,p)

solidfluid

0.8

0.85

0.9

0.95

1

1.05

1.1

0 50 100 150 200 250

φ(y)

/<φ>

y

Couette, φ=0.40, Re=1.8Lees-Edwards, φ=0.40, Re=1.8

-1

-0.5

0

0.5

1

0 50 100 150 200 250

u x(y

)/uw

y

Couette, φ=0.40, Re=1.8Lees-Edwards, φ=0.40, Re=1.8

• clear: Couette scheme suffers from wall effects at higher shear rates γ– wall induces different particle structures→ depletion zone→ wall slip→ lower

apparent viscosity

16

Particle Clusters

17

Emerging Particle ClustersSnapshot of a sheared suspension

Rpart = 3.75, Reshear,part ≈ 0.1, φ = 0.431, red∼high pressure

18

Cluster Properties

30

20

10

0

10

20

30

25 20 15 10 5 0 5 10 15 20 25

angle distribution of particle in a cluster

0.0001

0.001

0.01

0.1

1

10

100

1 10 100

n(m

)

m

Re=1.0Re=0.1

• Definition: Cluster = agglomeration of particles,connected via links d < dcrit = 2.2Rp (Max in PDF)

• Angle distribution of linked particles shows stronganisotropy→ mostly aligned to a direction diagonal to shear

• Rod-like clusters in diagonal direction are perfect toolsto transport momentum through the system→ possibleexplanation for shear thickening behaviour

• KCM model (Raiskinmaki, 2004)– tube rotates in shear: particles initially uncorrelated,

density of centres of mass Poisson distributed– tube deformes, particles collide

n(m) ∼ m−1.5

exp(−m/m0)

m0 = 1/(λ− log(λ)− 1), λ = φ/φc(γ)

– cutoff cluster size m0 diverges when φ→ φc(γ)

19

Outlook

• integration into the COAST multi-scale environment

φ

γ.

u

Γν

Further objective: blood flow

• implementation of (deformable) RBC-like particles• implementation of walls, LCs and their sticking behaviour (LC rolling)

Thank you!

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