shan-chen multiphase model - lbm...

Multiphase models Shan-Chen model Shan-Chen model in numbers Maxwell reconstruction rule Planar case Summary

SHAN-CHEN MULTIPHASE MODEL

A. Kuzmin J. Derksen

Department of Chemical and Materials EngineeringUniversity of Alberta

Canada

August 22,2011 / LBM Workshop


OUTLINE

1 MULTIPHASE MODELS

2 SHAN-CHEN MODEL

3 SHAN-CHEN MODEL IN NUMBERS

4 MAXWELL RECONSTRUCTION RULE

5 PLANAR CASE

6 SUMMARY


MULTIPHASE VARIATIONS

The multiphase is a really broad term which includes quite differentphenomena. For example one can distinguish the following phenomenabelonging to multiphase phenomena:

One-component different phases systems, i.e. water and water vapour.

Many components one phase systems, i.e. oil and water.

Many components many phases systems, i.e. oil, gas and water withwater vapour.

In this lecture we will proceed with the one-component different phasessystems.


MULTIPHASE MODELS

What does the multiphase model implies? Multiphase model implies certainthings:

The equation of state allowing the separation of phases. For examplethe famous van der Waals equation:

p(ρ) =ρRT

1− bρ + aρ2

The surface tension between phases acting along the interface. Forexample pressure buildup for the Laplace law is because of the surfacetension. The surface tension acts along the curvature and can berepresented through the following equation:

nk(P(1)ik − P

(2)ik ) = σ

( 1

R1+

1

R2

)ni,

where Pik is the momentum flux tensor in phases 1 and 2, ni is thenormal to the interface.


PHYSICAL UNDERSTANDING

How to deal with the equation of state - one can think about it as theattractive and repulsion force introduced on the microlevel. The surfacetension acts along the interface and usually the reconstruction of the interfaceto determine a curvature is required.Therefore, the one-component multiphase models can be described by twostages:

The equation of state can be introduced to the momentum flux tensorPαβ or through the acting force (we will discuss it later).

The surface tension inclusion which is a tricky part and usually is donethrough the interface reconstruction or through the acting force.


CFD MODELS

Basically the surface tension dictates as what the model will be used tosimulate a multiphase system. A few approaches are incorporated in CFD:

The front tracking technique, which impose additional tracking points onthe interface and reconstructs interface with their help.

Volume-of-Fluid (VOF) introduces the phase field which distinguish twophases. The transition between different phases is smooth. VOFreconstructs the interface and then impose boundary conditions.

Diffuse interface models impose the smooth transition between phasesand only integrally fulfill the boundary conditions on the interface.

Level Set method introduces the distance function which implicitly givesthe interface and its curvature.

The Shan-Chen model introduced here is the diffuse interface model whichhelps not to worry about boundary conditions. However, the interface isdiffused over a few lattice units which are not physical (in nature, the interfaceoccupies nm-mm scale).


DIFFUSE INTERFACE MODELS

For diffuse interface models (see review of Andersen) one needs to restorethe following momentum flux tensor:

Pαβ =(p0 −

k

2(∇rho)2 − kρ∆ρ

)δαβ + k∂αρ∂βρ,

where p0 is the bulk pressure responsible for different components, k is thesurface tension coefficient which is involved in the surface tension calculationthrough the interface, as:

σ = k

∫interface

( ∂ρ∂n

)2.

If the model satisfies these conditions then one has a one-componentmultiphase model.


THE SHAN-CHEN FORCE. NAIVE ASSUMPTION

The Shan-Chen force accounts for the attraction force calculated overnearest neighbours of the pseudopotential function ψ:

Fα(x) = −Gψ(x)∑i

wiψ(x+ ci)ci,

where coefficient G controls the strength of the attraction. The function ψ isso-called pseudopotential ψ = ψ(ρ), where ρ depends on time and location.

Therefore, we introduced a force toinclude the attraction. Let us analyze it throughthe Taylor expansion to see contributionsto the equation of state and the surface tension.


TAYLOR EXPANSION

One can perform Taylor expansion of the force:

Fα((x)) = −Gψ(x)(1

3∂αψ +

1

18∂α∆ψ

)+O(∂5).

One can reconstruct the momentum flux tensor from the force by “reverseingeneering”:

∂βPαβ = Fα

Pαβ =(ρ

3+G

6ψ2 +

G

36(∇ψ)2 +

G

18ψ∆ψ

)δαβ −

G

18∂αψ∂βψ

Compare the momentum flux tensor with the diffuse interface method:

Pαβ =(p0 −

k

2(∇rho)2 − kρ∆ρ

)δαβ + k∂αρ∂βρ

They look identical if we assume relationships ψ(ρ) = ρ, k = −G/18,p0 = ρ

3+ G

6ψ(ρ)2.

“MAGIC“ OF THE SHAN-CHEN FORCE

The magic of the Shan-Chen force is not only in the attraction force but insuch a form of the attraction potential that it modifies not only the equation ofstate but also gives the surface tension through the higher order of Taylorexpansion (term G

18ψ∇∆ψ).


THE CHOICE OF PSEUDOPOTENTIALS

However, the choice of ψ = ρ(x, t) is not the best in terms of stability. Onecan see that if we choose ψ = ρ, it becomes larger for larger ρ. Thus, theattractive potential contains a “disease” loop - the larger density ρ, the largerψ given usually larger gradients which lead to instabilities. ψ = ρ is good forsmall gas-liquid density ratios.Therefore one wants to bound pseudopotential ψ for large ρ and have itproportional to ρ for smaller ρ. It can be achieved by the following choice ofthe pseudopotential (usual Shan-Chen pseudopotential):

ψ(ρ) = 1− exp(−ρ),

which is for small ρ equals to ψ(ρ) = ρ and for large densities ψ(ρ) = 1. Thischoice of the pseudopotential allows larger separation for gas and liquid,usually no more than 60− 70.


CRITICAL VALUES

We see that the form of the Shan-Chen force contributions resembles themomentum flux tensor of the diffuse interface method. So what?The main question what kind of separation I can expect if the equation ofstate is modified through the force inclusion? Let us look to the equation ofstate:

p0 =ρ

3+G

6(1− exp(−ρ))2.

It is known from the thermodynamic theory that the critical values of pressurep0, density ρcrit and temperature like parameter in the equation of state Gcritwhere the separation occurs cna be obtained from these two equations:

dp0dρ

= 0

d2p0dρ2

= 0,

which becomes after algebraic manipulations as follows:

1

3+Gcrit

3exp(−ρcrit)(1− exp(−ρcrit)) = 0

− Gcrit3

exp(−ρcrit)(1− exp(−ρcrit)) +Gcrit

3exp(−2ρcrit) = 0.


CRITICAL VALUES

One can solve the system for Gcrit and ρcrit to obtain in numbers:

Gcrit = −4

ρcrit = ln 2

That means if the system is initialized with the liquid density more than ln 2and the gas density less than ln 2 and temperature-like parameter G ≤ −4,then one will obtain the stable system with separated gas-liquid phases.


MAXWELL RECONSTRUCTION RULE

Still we don’t know what wil be values of ρgas and ρliq when one gives thevalue of G. To obtain these values one need to use the Maxwellreconstruction rule. The Maxwell reconstruction rule states that the integralaround the transition pressure by the volume should equal to zero or in otherwords - the transion happens only for such pressures where the integralequals to zero which gives the equilibrium values of gas and liquid densities:∫ Vgas

Vliq

(P (V )− P )dV =(∫

adb

+

∫bec

)(P (V )− P )dV


MAXWELL RECONSTRUCTION RULE IN TERMS

As far as the equation of state uses density as the independent variable, oneneeds to translate the Maxwell reconstruction rule in terms of densities:

V ∝ 1

ρ∫ Vgas

Vliq

(P (V )− P )dV =

∫ ρliq

ρgas

(P (ρ)− P )

ρ2dρ = 0

Overall, once parameter G is given one needs to solve the following systemof integral equations to obtain values of ρgas and ρliq:∫ ρliq

ρgas

p0(ρ)− pρ2

dρ = 0

p = p0(ρgas)

p = p0(ρliq)


However, approaches presented before are only for the ideal case of thethermodynamics. The case of the diffuse interface is different - the transitionhappens because of the force and is diffused through a few lattice units.Overall, to see how the separation occurs for diffuse interface models oneneeds to eliminate the surface tension and solve the equation for the planarcase.

Liquid phase, ρliq

Gas phase, ρgasy

The boundary conditions areas follows assuming that function ρ = ρ(y):

ρ(∞) = ρgas

ρ(−∞) = ρliq

As far as the curvature of the interface is infinity, the boundary conditionbecomes Pyy = const.


THE SOLUTION FOR THE PLANAR CASE

For the one-dimensional case which we have here the momentum flux tensoris:

Pyy(x) = p0 =ρ

3+G

6ψ2 − G

18

(dψ

dy

)2+G

18ψ

d2ψ

dy2

Pyy(∞) = p0 =ρ

3+G

6ψ2(ρgas)

Pyy(−∞) = p0 =ρ

3+G

6ψ2(ρliq)

The derivative of the pseudopotential by y can be expressed through thechain rules:

dψ

dy=

dψ

dρ

dρ

dy= exp(−ρ)(1− exp(−ρ))

dρ

dy

In what follows we do notations ˙standing for ddρ

and ′ standing for ddy

. Theequation you come up is:

p0 =ρ

3+G

6ψ2 − G

36(ψρ′)2 +

G

18ψ(ψρ′

2+ ψρ′′)


THE SOLUTION FOR THE PROFILE

The previous equation can be much simplified by the introduction of the newvariable z = ρ′

2. Then the second order derivative becomes:

ρ′′ =d

dy

√z =

1

2√z

dz

dy=

1

2√z

dz

dρ

dρ

dy=

1

2√z

√zz =

z

2

Therefore one comes up with the the following first order differential equationfor z:

G

36ψψz +

G

36(2ψψ − ψ2)z = p0 −

ρ

3− G

6ψ2

The solution of the similar equation y′ + p(x)y = q(x) is known and equals toy(x) = e−

∫p(x)dx

(∫q(x)e

∫p(x′)dx′dx+ C

). In our case the solution

becomes:

z(ρ) =ψ

ψ2

36

G

∫ ρ

ρgas

(p0 −

ρ

3− G

6ψ2) ψ

ψ2dρ∫ ρliq

ρgas

(p0 −

ρ

3− G

6ψ2) ψ

ψ2dρ = 0

The latter equation is to satisfy boundary conditions.


PROFILES

Overall the profiles ρ(y) can be restored by reversing the following equation:

z = ρ′2; y =

∫ ρ

ρgas

dρ′√z(ρ′)

The simulation and theoretical profiles for different temperature-likeparameter G are presented in the figure below.

æææææææææææ

ææ

æææææææææææææ

à à à à à

à

à

à

à à à à

ìììììììììììììì

ììì

ììììì

ììììììììììììììììì

ò ò òò

ò

ò

ò ò ò ò ò ò

ôôôô ôôôôôôôô

ôô

ôô

ôô

ôô

ôô

ôô

ôô

ôô

ôôôôôôôôôôôôôô

ç ç

ç

ç

çç ç ç ç ç ç ç

0 2 4 6 8 100

1

2

3

4

Coordinate

Density

ç Sim,G=-7.5

ô Theor,G=-7.5

ò Sim,G=-6.5

ì Theor,G=-6.5

à Sim,G=-5.5

æ Theor,G=-5.5


GAS AND LIQUID BRANCHES

One can solve the system of equations to obtain not only profiles but also gasand liquid densities depending on temperature-like parameter G:


THERMODYNAMIC INCONSISTENCY

Notice that the Maxwell reconstruction rule is quite different in the case of thereal Shan-Chen force:∫ ρliq

ρgas

(p0 −

ρ

3− G

6ψ2) ψ

ψ2dρ = 0

Usual Maxwell reconstruction rule is as follows:

ψ

ψ2

36

G

∫ ρliq

ρgas

(p0 −

ρ

3− G

6ψ2)dρ

ρ2= 0

It is so-called a thermodynamicinconsistency which leads to different values of gasand liquid densities in comparison with the Maxwellreconstruction. One can see the differencein the figure below (courtesy of Shan andChen). y axis is the temperature like parameterG, x axis is the density. One can see that twocurves (maxwellian and shan-chen) are different.


BOUNDARY CONDITIONS

As far as the Shan-Chen model is one-component model all the boundariesconditions for one phase stay (pressure, velocity, BB conditions). However,for the wall boundary conditions one needs to do something with theShan-Chen force which accounts for all neigbours densities. There are twoapproaches as to handle the solid boundaries:

The first one is a logical choice. As far as the Shan-Chen force accountsfor densities of nearest neighbours, i.e. F = −Gψ(x)

∑i wiψ(x+ ci)ci.

In this case if one wants a function ψ defined on walls, one needs tospecify density ρwall. From simulations one can see that ρwall controlsthe contact angle.

Another approach is to specify (not widely used) another parameterGads, which transforms a force:

F = −Gψ(x)∑i

wiψ(x+ ci)ci

= −Gψ(x)∑fluid

wiψ(x+ ci)ci −Gadsψ(x)∑wall

ci


BE AWARE

There are a few different things which you need to know before startingworking with the Shan-Chen model:

Limited stability, basically for dynamic problems you don’t want tosimulate systems with smaller G ≤ −7. Usually spurrious currentscoming from discretization destroy stability.

Additional condensation and evaporation occuring in the systems

Problems with the initialization and stabilization

While the liquid densities are predicted correctly one needs to be awareabout the vapour branch densities. Because of the discrete effects thevapour density is not predicted correctly and one needs to be accuratewith the gas liquid density ratio prediction.

The original force implementation is due to shift of velocityρueq = ρu+ Fτ , absolutely inconsistent choice. Gives the dependencyof the surface tension on τ

The surface tension is not specified explicitly and needs to be calculatedthrough the Laplace law.


BUT..

But there is always something good:

Really fast to implement and see first results

Good for problems involving macroscopic phenomena, bulk pressures.Not so good for interface shapes.

Good for the simulation of liquids.

Be always in a stable regime. Match your non-dimensional parameters.

The next lecture is about binary liquid model which avoids condensation.

shan-chen multiphase model - lbm...

Documents