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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation Viscoelasticity and constitutive relations Amith Balasubramanya January 22, 2016 Amith balasubramanya Viscoelasticity January 22, 2016 1 / 44

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Page 1: Viscoelasticity and constitutive relations - Chalmers · Viscoelasticity and constitutive relations Amith Balasubramanya ... Presentation OutlineIntroductionGoverning ... dimensionedScalar

Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Viscoelasticity and constitutive relations

Amith Balasubramanya

January 22, 2016

Amith balasubramanya Viscoelasticity January 22, 2016 1 / 44

Page 2: Viscoelasticity and constitutive relations - Chalmers · Viscoelasticity and constitutive relations Amith Balasubramanya ... Presentation OutlineIntroductionGoverning ... dimensionedScalar

Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Presentation Outline

Introduction to viscoelasticity

Governing Equations

Constitutive Relations

High We problem and stability

viscoelasticFluidFoam

viscoelasticLaws

Case-Setup

Amith balasubramanya Viscoelasticity January 22, 2016 2 / 44

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Introduction

Materials that exhibit gradual deformation and recovery whensubjected to loading and unloading.

The response of such materials is dependent upon how quickly theload is applied or removed, the extent of deformation being dependentupon the rate at which the deformation-causing loads are applied.This time-dependent material behavior is called viscoelasticity.

viscoelasticity can be explained by a set of springs and dashpot asshown in the figure

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Governing Equations

The continuity equation is given as:

∇.u = 0

The momentum equation is given as:

∂ρu

∂t+∇.(ρuu) = −∇p+∇τ

Stress tensor is split up as:

τ = τs + τp

Where τs is the solvent vicosity and τp is the polymeric part of we can callit newtonian and non-newtonian components

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

τs = 2ηpkD

where;

D =1

2(∇u+∇uT )

τp is calculated from the solution of an appropriate constitutive equation

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Some important relations to know

Upper Convected Derivative is given as:

Oτpk =

DτpkDt

− [∇u.τpk]− [∇u.τpk]T

whereDτpkDt is the material derivative of the extra stress tensor defined as:

DτpkDt

=∂τpk∂t

+ u · ∇τpk

The Gordon-Schowalter derivative of the polymer stress tensor are givenby:

�τpk =

DτpkDt

− [∇uT .τpk]− [τpk.∇u] + ηk(τpk.D +D.τpk)

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Important Definitions to know are the following dimensionless numbers:

Weissenberg Number(We)

We = λ•γc

Deborah Number(De)De =

λtc

The figure represents exactly what causes the High We Problem:

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

DEVSS

The momentum equation is written as:

∂u

∂t+∇.(ρuu)− (ηs + κ)∇.(∇u) = −∇p+∇.τp − κ∇.(∇u) (1)

Here κ is a positive number. Usually ηpk is chosen as κ. This is observedin all constitutive relations.

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Constitutive Relations

Linear maxwell

τpk + λk∂τpk∂t

= 2ηpkD

Oldroyd-B

τpk + λkOτ = 2ηpkD

Both equations are derived from the Johnson-Segalman Equation.

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Giesekus

τpk + λkOτ + αk

λkηpk

(τpk.τpk) = 2ηpkD

Leonov is a special case of Giesekus when αk =12 . the equation is

given as:

τpk + λkOτ +

1

2

λkηpk

(τpk.τpk) = 2ηpkD

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

FENE-Finitely Extensible Non-Linear Elastic

FENE-P

(1 +

31−3/L2

k+ λk

ηpktr(τpk)

L2k

)τpk + λkOτpk = 2(

1

1− 3/L2k

)ηpkD

FENE-CR

(L2kλkηpktr(τpk)

L2k − 3

)τpk + λkOτpk = 2((

L2kλkηpktr(τpk)

L2k − 3

))ηpkD

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

PTT- Phan Thein Tanner

LPTT

(1 +εpkτpkηpk

tr(τpk))τpk + λk�τpk = 2ηpkD

EPTTexp(

εpkτpkηpk

tr(τpk))τpk + λk�τpk = 2ηpkD

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

POM-POM Models

τpk =ηpkλk

(3 ∧2pk Spk − δ)

OSpk + 2[D : Spk]Spk +

1

λk[Spk −

1

3δ]

D(∧pk)Dt

= ∧pk[D : Spk] +1

λsk[∧pk − 1];

The pom-pom model however has the following disadvantages: It tends topresent a discontinuous solution at high-shear rates, does not predict thesecond normal stress difference

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

XPP-SE

Because of these disadvantages, new equations are developed as follows:Extended Pom-Pom Model-Single Equation(XPP-SE)

Oτpk + λ(τ)−1.τpk =

2ηpkD

λOBk

λ(τ)−1 =1

λOBk

[αkλOBk

ηpkτpk + f(τ)−1δ +

λOBk

ηpk(f(τ)−1)τ−1pk

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

XPP-DE

Extended Pom-Pom Model-Double Equation(XPP-DE)

OSpk + 2[D : Spk]Spk+

1λOBk

∧2pk [3αk ∧4pk Spk.Spk+(1− αk − 3αk ∧4pk IS.S)Spk −

1−αk3 δ] = 0

D(∧pk)Dt

= ∧pk[D : Spk] +1

λsk[∧pk − 1];

OSpk + 2[D : Spk]Spk +

1

λk[Spk −

1

3δ]

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

viscoelasticFluidFoam

The solvers can be accessed at :

cd $FOAM_SOLVERS\viscoelastic\viscoelasticFluidFoam

vi viscoelasticFLuidFoam.C

Transient solver for incompressible, laminar flow of viscoelastic fluids.

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Algorithm

With the given initial fields of velocity u, stress τ and pressure p , theexplicit calculations of the pressure gradient and stress divergence arecarried out, and, subsequently, the momentum equation is solvedimplicitly for each component of the velocity vector, computing a newvelocity field estimate u∗.With the new velocity values u∗, the new pressure field p∗ isestimated and, subsequently, the correction of velocity is carried out,leading to a new velocity field u∗∗ which satisfies the continuityequation. In this step either SIMPLE or PISO algorithm can be usedto obtain p∗ and u∗∗, with the more accurate PISO being the bestoption for transient flows.With the corrected velocity field u∗∗ the new estimate τ∗ for the stresstensor field is calculated by solving the specified constitutive equation.Steps 1,2 and 3 may be repeated recursively within each time step inorder to generate more accurate solutions in transient flows. For this,u,p and τ are updated with u∗∗, p∗ and τ∗, respectively.

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

viscoelasticFluidFoam.C

We begin from Line 66

// Pressure-velocity SIMPLE corrector loop

for (int corr = 0; corr < nCorr; corr++)

{

// Momentum predictor

tmp<fvVectorMatrix> UEqn

(

fvm::ddt(U)

+ fvm::div(phi, U)

- visco.divTau(U)

);

UEqn().relax();

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

viscoelasticFluidFoam.C from Line 80

solve(UEqn() == -fvc::grad(p));

p.boundaryField().updateCoeffs();

volScalarField rUA = 1.0/UEqn().A();

U = rUA*UEqn().H();

UEqn.clear();

phi = fvc::interpolate(U) & mesh.Sf();

adjustPhi(phi, U, p);

// Store pressure fo runder-relaxation

p.storePrevIter();

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

viscoelasticFluidFoam.C from Line 92

// Non-orthogonal pressure corrector loop

for (int nonOrth=0; nonOrth<=nNonOrthCorr; nonOrth++)

{

fvScalarMatrix pEqn

(

fvm::laplacian(rUA, p) == fvc::div(phi)

);

pEqn.setReference(pRefCell, pRefValue);

pEqn.solve();

if (nonOrth == nNonOrthCorr)

{

phi -= pEqn.flux();

}

}Amith balasubramanya Viscoelasticity January 22, 2016 20 / 44

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

viscoelasticFluidFoam.C from Line 111

// Explicitly relax pressure for momentum corrector

p.relax();

, // Momentum corrector

U -= rUA*fvc::grad(p);

U.correctBoundaryConditions();

visco.correct();

}

runTime.write();

Info<< "ExecutionTime = "

<< runTime.elapsedCpuTime()

<< " s\n\n" << endl;

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Constitutive Relations OpenFOAM

The constitutive relations are located in:$FOAM_SRC/transportModels/viscoelastic/viscoelasticLaws

The following constitutive relations are found in OpenFOAM:

DCPP FENE-CR Feta-PTT Leonov Maxwell Oldroyd-B

viscoelasticLaw XPP_DE EPTT FENE-P Giesekus LPTT

multiMode S_MDCPP WhiteMetzner XPP_SE

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Line 70 of Oldroyd.C

Foam::tmp<Foam::fvVectorMatrix>

Foam::Oldroyd_B::divTau(volVectorField& U) const

{

dimensionedScalar etaPEff = etaP_;

return

(

fvc::div(tau_/rho_, "div(tau)")

- fvc::laplacian(etaPEff/rho_, U, "laplacian(etaPEff,U)")

+ fvm::laplacian( (etaPEff + etaS_)/rho_, U, "laplacian(etaPEff+etaS,U)")

);

}

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

The equation that is represented using this is:

∇. τpρ− ηp

ρ∇.(∇U) +

ηs + ηpρ∇.(∇U)

This value is calculated and used in viscoelasticFluidFoam.C

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Line 84 of Oldroyd.C

void Foam::Oldroyd_B::correct()

{

// Velocity gradient tensor

volTensorField L = fvc::grad(U());

// Convected derivate term

volTensorField C = tau_ & L;

// Twice the rate of deformation tensor

volSymmTensorField twoD = twoSymm(L);

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

These are represented as:

L = ∇uC = τpk.∇u

twoSymm(C) = C + CT = [τpk.∇u] + [τpk.∇u]T

Mathematical Operator OpenFOAM implementationDτpkDt fvm::ddt(tau_)+fvm::div(phi(),tau_)

2ηpkD etaP_/lambda_*twoD

[τ.∇U ] + [τ.∇U ]T twoSymm(c)1λkτpk fvm::Sp(1/lambda_,tau_)

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Implementing your own constitutive relation

Let us implement the LCM model. The equation is given as:

τpk + λk4τpk = 2ηpkD

where,4τpk is the Lower Convected Derivate given by:

4τpk =

DτpkDt

+ [∇u.τpk] + [τpk.∇uT ]

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Now, we need to copy Oldroyd-B and make a copy of it and create adirectory structure. This is done by:

cd $WM_PROJECT_DIR

cp -r --parents src/transportModels/viscoelastic/

viscoelasticLaws/Oldroyd_B $WM_PROJECT_USER_DIR

cd $WM_PROJECT_USER_DIR/src/transportModels/viscoelastic/

viscoelasticLaws

Now rename Oldroyd-B as LCM

mv Oldroyd_B LCM

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We need to create a Make directory in

cd $WM_PROJECT_USER_DIR/src/transportModels/viscoelastic/

Type mkdir Make

First, we create Make/files, add the following lines:

echo "viscoelasticLaws/LCM/LCM.C

LIB= \$(FOAM_USER_LIBBIN)/libmyviscoelasticModels"

> Make/files

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

First, we create Make/options, add the following lines:

echo "EXE_INC = \\

-I\$(LIB_SRC)/finiteVolume/lnInclude \\

-I\$(LIB_SRC)/transportModels/viscoelastic/lnInclude

LIB_LIBS =" > Make/options

Now we need to rename the .C and .H files in our constitutiverelations

cd viscoelasticLaws/LCM

mv Oldroyd_B.C LCM.C

mv Oldroyd_B.H LCM.H

rm Oldroyd_B.dep

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

In LCM.C and LCM.H, change all occurances of Oldroyd_Bto LCM sothat we have a new class name:

sed -i s/Oldroyd_B/LCM/g LCM.C

sed -i s/Oldroyd_B/LCM/g LCM.H

Now, we edit the .C file of the new constitutive relation according toequation . If we go to LCM.C, from line 96 to 104, replace theequation as follows:

fvSymmTensorMatrix tauEqn

(

fvm::ddt(tau_)

+ fvm::div(phi(), tau_)

==

etaP_/lambda_*twoD

- twoSymm(C)

- fvm::Sp(1/lambda_, tau_)

);

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

In case of an unsteady simulation, DEVSS causes additional diffusion.So to remove the stabilizing scheme, we edit line 73 as:dimensionedScalar etaPEff = etaP_;

The final step is to compile

cd ..

cd ..

wmake libso

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Setup

Copy the tutorial Oldroyd-B to your home directory. This is done by:

cp -r $FOAM_TUTORIALS/viscoelastic/viscoelasticFluidFoam/

Oldroyd_B $FOAM_RUN

cd $FOAM_RUN/Oldroyd B

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The Simulation is governed by some important definitions:

Re =2ρuhη0

Where,e is the Reynold’s Number, u is the averagevelocity at the downstream section, η0 = (ηs + ηp)

De =uh Where λ is the relaxation time

the parameters used in the simulation are: ηs = 0.0067, ηp = 0.0017,ρ = 1050 and = 0.008

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Create the mesh using blockMesh

The geometry used is a 4:1 planar contraction which is the standardtest geometry for polymeric fluids. The diameter upstream is2H = 0.0254 m and downstream is 2h = 0.00064m.

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Boundary Conditions

In the 0 directory, u has the following boundary conditions

boundaryField

{

inlet

{

type fixedValue;

value uniform (0.58 0 0);

}

fixedWalls

{

type fixedValue;

value uniform (0 0 0);

}

outlet

{

type zeroGradient;

}

}

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The boundary conditions for p and tau is the same as the tutorial.

N1 and N2 are the normal stress differences that are calculated usingthe stressDifferences utility. This is done after solving for alltime-steps.

Change the properties in the Constant\viscoelasticproperties

rheology

{

type Oldroyd-B;

rho rho [1 -3 0 0 0 0 0] 1050;

etaS etaS [1 -1 -1 0 0 0 0]0.0067;

etaP etaP [1 -1 -1 0 0 0 0]0.0017;

lambda lambda [0 0 1 0 0 0 0] 0.008;

}

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

Check the controldict file in the System directory

application viscoelasticFluidFoam;

startFrom startTime;

startTime 0.0;

stopAt endTime;

endTime 0.1;

deltaT 1e-3;

writeControl adjustableRunTime;

writeInterval 0.01;

purgeWrite 0;

writeFormat ascii;

writePrecision 6;

writeCompression uncompressed;

timeFormat general;

timePrecision 6;

graphFormat raw;

runTimeModifiable yes;

adjustTimeStep on;

maxCo 0.5;

maxDeltaT 0.1;

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Run the solver using viscoelasticFluidFoam >& log &

Calculate N1 and N2 using stressDiffrences

Calculate the components of shear stress usingstressSymmComponents

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Presentation Outline Introduction Governing Equations Constitutive Relations Implementing constitutive relation

2 4 6 80.000e+00 8.975e+00

U Magnitude

Velocity after 1 sec

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5e+4 1e+5 1.5e+5-1.593e-01 2.098e+05

tauxx

τxx after 1 sec

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-75000 -5e+4 -25000-1.038e+05 5.560e+02

tauxy

τxy after 1 sec

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2e+4 3e+4 4e+4 6e+4-1.785e-01 7.314e+04

tauyy

τyy after 1 sec

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Future Work

Simulate with geometries like Y-bifurcations, T-bifurcations, thosethat resemble arterial geometries.

Oldroyd-B though stable doesn’t take into account Shear-thinning. Ageneralized Oldroyd-Model(Anand) should be implemented inOpenFOAM

Once 2D geometries are successful, a 3D model should be simulated.

Introduce a clot and try simulating.

Amith balasubramanya Viscoelasticity January 22, 2016 44 / 44