Lattice Boltzmann Equation Method in Electrohydrodynamic

Problems

Alexander Kupershtokh, Dmitry Medvedev

Lavrentyev Institute of Hydrodynamics,Siberian Branch of Russian Academy of

Sciences,Novosibirsk, Russia

Equations of EHD

iiiii

iiii

i rwnbqq

nDntn

)(div)( Eu

Hydrodynamic equations:

0)(

ut

uuFu

divgrad)3

()( 2)0(

t

Here uuP )0(

Continuity equation

Navier-Stokes equation

is the main part ofmomentum flux tensor

Concentrations of charge carriers:

Poisson’s equation and definitions:

EFE qnqqq ii ,,,4)(div

Method of splitting in physical processes

1. Modeling of hydrodynamic flows. Lattice Boltzmann equation method (LBE).

2. Simulation of convective transport and diffusion of charge carriers. Additional LBE components (considered

as passive scalars).

3. Calculation of electric potential and charge transfer due to mobility of charge carriers.

4. Calculation of electrostatic forces acting on space charges in liquid and incorporation these forces into LBE.

5. Simulation of phase transition or interaction between immiscible liquids using LBE method.

The whole time step is divided into several stages implemented sequentially:

Development of discrete models of medium

Molecular dynamics (Alder, 1960)

Kinetic Boltzmann equation

(1872)

Lattice Gas Automata Boltzmann equations with discrete set of velocities

Lattice Boltzmann Equation

Macroscopic equations of hydrodynamics

(Navier – Stokes equations)

Chapman – Enskog expansion

1988 1997

1964

Boltzmann equations with discrete velocities

k

kN k

kkNcu k

kkNDu 22

21

)(2

cHydrodynamic variables

The discrete finite set of vectors ck of particle velocitiescould be used for Boltzmann equation at hydrodynamicstage

.kkkkk fftf

c

Usually the populations

Nk are used for eachgroup of particles

)()( kkk Nf cξ

For 1D

Lattice Boltzmann equation method (LBE)

Two-dimensional variants

The main idea is that time step must be so that .tkk ce

One-dimensional isothermal variant (D1Q3)

(D2Q13)

(D2Q9)

Lattice Boltzmann equation method (LBE)

The discrete single-particle distribution functions Nk are used as variables

Evolution equations of LBE method

Hydrodynamic variables

k

kN k

kkNcu k

kkNDu 22

21

)(2

c

.)),((),(),( kkkkk NtNtNtttN xxcx

is the collision operator /)),(),(( ux eqkkk NtN

in BGK form (relaxation to the equilibrium state with

relaxation time ).

.

221),(

2

2

2

uucucu kk

keqk wNExpansion in u

Viscosity ./)2/1( 2 th

kN is the body force term.

New general method of incorporatinga body force term into LBE

Kinetic Boltzmann equation for single particle distributionfunction f(r,,t)

fft

fξaξ

Perturbation method .neqeq fff For any equilibrium distribution function

).( uξ eqeq ff

Hence eqeq ff uξ

From the other hand, the full derivative along theLagrange coordinate at a constant density is equal

to .

d)(d eq

eqf

tf

uau

Thus, we obtained theBoltzmann equationin form t

ff

tf eq

dd

ξtf eq

dd

Exact difference method for lattice Boltzmann equation

After discretization of Boltzmann equation in velocityspace we have

Here the changes of the distribution functions Nk due to

the force F are equal to the exact differences ofequilibrium distribution functions at constant density

.),()),(,(

),(),( kk

eqk

kkk NtNtN

tNtttN

xxuxcx

),(),( uuu eqk

eqkk NNN

The commutative property of body force term and thecollision operator indicates the second order accuracy

intime. The distribution function that is equilibriumin local region of space, is simply shifting under theaction of body force by the value

./ t Fu

Convective transport and diffusion of charge carriers

k kii Qn

iiiiii rwnDnt

n

)( u

),( uiki nQ

Equations for concentrations of charge carriers:

Method of additional LBE components with zero mass(passive scalars that not influence in momentum)

.),(),(

),(),(i

kiieqki

kikkitQnQ

tQtttQ

xuxcx

Equilibrium distribution functions depend onconcentrations of corresponding type of charge carriers and on fluid velocity .u

Diffusivities can be adjusted independently by changingthe relaxation time

./)2/1(3 2 thD ii

2/1i

Calculation of electric field potential and charge transport due to mobility of

charge carriers (conductivity)

The time-implicit finite-difference equations forconcentrations of charge carriers

were solved together with the Poisson’s equation

111 div nn

iii

ini

ni nb

qq

nn

.4)(div 11 nii

n nq

Action of electrostatic forces on space charges in liquid

.4 q The total charge density in the node was calculated from

Hence, we have the finite-difference expressions for electrostatic force

This equation takes into account both free space chargeand charge density due to polarization. Electric field acted on this charge was calculated asnumerical derivative of electric potential.

.2/)(

,2/)(

1,1,,

,1,1,

hqF

hqF

jijijiy

jijijix

Phase transition in 1D

k

kkG eexxxF ))(())(()( 0

To simulate the phase transition, the attractive part of intermolecular potential should be introduced.

For this purpose, the attractive forces between particlesin neighbor nodes was introduced (Shan – Chen, 1993).

Phase transition in 2D

The attraction between particles in neighbor nodes

k

kkkG eexxxF ))(())(()(

These attractive forces ensure also a surface tension.

01 4

2FF

01 41GG

Phase transitions for Chan-Chen models

:10

.3/1

,41

01 GG

and for

20 GP The equation of state:

:))/exp(1()( 00 For specific function

,693.032

0 G

For isothermal models

D1Q3

D2Q9

D3Q19 ,21

01 GG

,0G

2/3

3

1

,03

dd

GP

032

22

2

2

d

ddd

GP

Critical point:

,2ln0

Steady state of 1D phase transition layer

))./exp(1()( 00

3/1

.3/20 G

Critical point

For specific function

,693.0

for isothermalcaseand ,10

20 GP Equation of state:

Phase transition in 2D

202

3 GP Equation of state:

))/exp(1()( 00 And for specific function

9/40 G

isotherms

metastablestates

Simulation of immiscible liquids

s

s

s

kss N

ss

suu

The attraction between particles in neighbor nodes wasintroduced (Shan – Chen, 1993).

k

kkksss G eexxxF ))(())(()(

Momentum of each component is

SHere we denote the components by the indexes and .

The total fluid density at a node depends on densities of allcomponents as

Here

The total momentum at a node

ks

ksss Nu cThe interaction forces change thevelocity of each component as sss t / Fu

Phase transition from unstable state(waves of higher density)

Red – liquid in unstable state. LightRed – liquid. Black – vapour.

1200t 1300tGrid 160x160

Phase transition from metastable state with different nucleuses

Red – liquid in metastable state (G=0.6; 0 = 1.6)

Black – vapour. LightRed - liquid

Grid 160x160

0 = 0.8 0 = 0.67

0 = 0.5 Small

0 = 0.4

Deformation and fragmentation of conductive vapor bubbles in electric field

t = 0 100 200 300 400 500 600 700

t = 100 200 300 400 500 600 700 770 850

= 0.5

= 0.38

Deformation and fragmentation of conductive vapor bubbles in electric field

t = 100 200 300 400 500 600 700 800 850

= 0.2

The droplet of higher permittivity in liquid dielectric under the action of

electric field

E = 0.035 E = 0.1

= 1.41;

Deformation of vapor bubble under the action of electric field due to “electrostriction”

Permittivity 01

Conclusions

A new method for simulating the EHD phenomena usingthe LBE method is developed: Hydrodynamic flows and convective and diffusivetransfer of charge carriers are simulated by LBE scheme,as well as interaction of liquid components and phasetransitions and action of electric forces on a chargedliquid. Evolution of potential distribution and conductivetransport of charge are calculated using the finitedifference method.

The exact difference method (EDM) is not an expansionbut is a new general way to incorporate the body forceterm into any variant of LBE Simulations show great potential of the LBE methodespecially for EHD problems with free boundaries(systems with vapor bubbles and multiple componentswith different electric properties).

Lattice Boltzmann equation methodwith arbitrary equation of state

Zhang, Chen (Phys. Rev. E, 2003)

Idea: to use the isothermal LBE method (T=T0)For mass and momentum conservation laws+Usual energy equation, that can be solved by ordinaryfinite-difference method.

Here energy equation is written in divergent form and canbe solved, for example, by Lax–Vendroff two-step method. The equation of state was introduced by means the bodyforces acted on the liquid in the nodes

Fuu qTpuetue

)2/()2/( 2

2

;N UFU ),( Tp 0T

here the potential is expressed throughequation of state

Liquid boiling with free surface in gravity field

Density distribution. Pr = 10, Re = 3·105

t = 39.5 t = 41.5

t = 43.5 t = 45.5

Stages of evolution of multiparticle system (N. N. Bogolubov)

1. Stage of initial randomizing (t 0).

2. Kinetic stage (0 << t < ).3. Hydrodynamic stage (t > ). The local equilibrium

was settled in small volumes. Even the exact information about single particle

distribution function f(r, , t) is unnecessary. Only several first moments of it are enough to know

zyx dddtf ),,( ξr

zyx dddtf ),,( ξrξu

zyx dddtf ),,()(21 2 ξruξ

1) Kupershtokh A. L., Calculations of the action of electric forces in the lattice Boltzmann equation method using the difference of equilibrium distribution functions // Proc. of the 7th Int. Conf. on Modern Problems of Electrophysics and Electrohydrodynamics of Liquids, St. Petersburg, Russia, pp. 152–155, 2003.

2) Kupershtokh A. L., Medvedev D. A., Simulation of growth dynamics, deformation and fragmentation of vapor microbubbles in high electric field // Proc. of the 7th Int. Conf. on Modern Problems of Electrophysics and Electrohydrodynamics of Liquids, St. Petersburg, Russia, pp. 156–159, 2003.

Publications

Previous forms of body force term in LBE

The terms that are proportional to are absent at all.

Method of modifying the BGK collision operator (MMCO)(Shan, Chen, 1993)

Methods of explicit derivative (MED) of the equilibriumdistribution function (He, Shan, Doolen, 1998)

,)()(

),(),(

uuuxx k

eqk

kkNN

tNttN

where u+ = F/.

,)( eqeq ff

uξa

a ).(

)(u

uuc eqk

kk NN

.2

)1( 22

u

uc

kkk

wR

.2

22

u

uc

kkk

wR

2uIf the first orderexpansion of in u is used we have

eqkN

The deviation from EDM

Previous forms of body force term in LBE

The deviation from EDMfor coefficients that werefound by authors is equal

to

Method of undefined coefficients (MUC)(Ladd, Verberg, 2001)

In method of Guo, Zheng, Shi (2002) the MUC was used incombination with MMCO

Its were found as A=0, B=u, and

.ijjiij uuuuC

.),(),(

),(),( kk

eqk

kkk NtNN

tNtttN

xuu

xcx

.8

22

u

uc

kkk

wR

where u* = u / 2.

.

2

:2

2

1ccCBc kkkkk AwN

This method exactly coincide with the methodof modification of collision operator at = 0.5.