lattice boltzmann equation method in electrohydrodynamic problems
DESCRIPTION
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. Hydrodynamic equations:. Continuity equation. - PowerPoint PPT PresentationTRANSCRIPT
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
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.