an introduction to twophaseeulerfoam with addition of an ...hani/kurser/os_cfd_2014/alessandro...

Introduction

An introduction to twoPhaseEulerFoam with additionof an heat exchange model in 23x version

Alessandro Manni

Dept of Industrial EngineeringUniversity of Rome ”Tor Vergata”

Italy

1/2-12-2014

Alessandro Manni OpenFoam Course Presentation 1/2-12-2014 1 / 29

Introduction

Outline

Introduction

How the solver works

Governing equationsKinetic Theory and Frictional modelsNumerical treatment and details in implementationVolume fractions solutionsMomentum equationsPressure correctorMultiphase turbulence modelingPimple algorithm

Heat exchange models

How to add an heat exchange model


Introduction

Introduction

Why twoPhaseEulerFoam?

twoPhaseEulerFoam is a ”solver for a system of 2 compressible fluidphases with one phase dispersed (e.g. gas bubbles in a liquid)including heat-transfer” since the last version (the capability ofmanage compressible fluids has been added since the version 2.1.xwith the analogue solver compressibleTwoPhaseEulerFoam).

The Eulerian approach,considers a solid phase with fluid likebehaviour without particle tracking reducing the numerical effortrequired with a large number of particles.

As a consequence,will be necessary introduce a physical definition forthe particle stress tensor to provide a mathematical closure.


Introduction

Introduction

Why twoPhaseEulerFoam?

Figure : Example of Fluidized Bed (with Gunn Model added in the tutorial)


Introduction

Governing Equations

The Eu-Eu solvers, are based on the assumpion of treating both phases asinterpenetrating continua. In this case with two phases, the followingassumption must be verified:

αs + αg = α1 + α2 = 1 (1)

The mass continuity equation for both the phases are(not exactly but forthe moment it’s ok (fvOptions)):

∂(α1ρ1)

∂t+∇ · (α1ρ1~u1) = R12 = 0 (2)

∂(α2ρ2)

∂t+∇ · (α2ρ2~u2) = R21 = 0 (3)


Introduction

Governing Equations

The momentum equations for both the phases are:

∂(α1ρ1~u1)

∂t+∇ · (α1ρ1~u1~u1) = ∇ · S1 + (α1ρ1~g) + ~I1 (4)

∂(α2ρ2~u2)

∂t+∇ · (α2ρ2~u2~u2) = ∇ · S2 + (α2ρ2~g)− ~I2 (5)

where(NB ∇p1..granularpressure):

S1 = ∇·(α1τ1)−α1∇p−∇p1 with τ1 = µ1[∇~u1+∇~uT1 ]+(λ1−2

3µ1)(∇·~u1)I

(6)

S2 = ∇·(α2τ2)−α2∇p with τ2 = µ2[∇~u2+∇~uT2 ]−2

3µ2(∇·~u2)I (7)

~I1,2 = drag + vme+ lift+ walllubri+ turbulentdispersion (8)


Introduction

Governing Equations

While the energy equation, on which we will ace in the last part of thepresentation,could be written in the following forms:

∂[αiρi(hi +Ki)]

∂t+∇· [αiρi(hi+Ki)~ui] = −∇·αi~qi+αi

∂p

∂t+Kht∆T (9)

and, remembering that h=e+pv

∂[αiρi(ei +Ki)]

∂t+∇·[αiρi(ei+Ki)~ui] = −∇·αi~qi−[

∂α1

∂tp+∇·α1~u1p]+Kht∆T

(10)

where ~qi is the effective energy-flux vector


Introduction

Kinetic Theory and Frictional Models

As said before, we need a mathematical model to close the global solidstress tensor. For this purpose, different flow regimes should be taken intoaccount:

Kinetic Regime → particles present in a ”diluite” state

Collisional Regime→ particles present in a higher conc. with collisions

Frictional Regime → particles present in a very high conc.with friction

For the first two regimes , the Kinetic Theory of Granular Flows (byGidaspow and Lun)gives a closure introducing the concept of granulartemperature, defined as the kinetic energy due to the free motion of theparticles. This gt is then solved (with the aid of other submodels) in abalance equation.


Introduction

Kinetic Theory and Frictional Models

In this equation, are introduced the concepts of radial distribution ,restitution coefficient , transfert rate of granular energy, that aretreated deeply in the report.The only important thing to know at themoment, is that with this equation we can have a solution for thegranular stress tensor (µsolid) and for the granular pressure ps.

What happen in the Frictional Regime?? (when α ' 0.6 )When we are close to the critical α, known as αlimit we will’have aswitch in the stress tensor enabling other models. In particular theviscosity of the frictional regime will depend on the I2D on thefrictional angle φ and onps. The ps itself will depend only on the solidfraction with similar correlations:

µs,fric = 0.5p?ssin(φ)√I2D

ps,fric = αp?s,fric = αA(αlimit − α)n (11)


Introduction

Numerical treatment and details in implementation

An issue of the resolution in the mass and momentum equations, isthat the α is present everywhere, having a null equation when we arein a monophase zone(only fluid phase).

Henry Weller, in the first release of the solver, solved that problemwith a non conservative form of the momentum equation expandingthe derivate where α1 is present,to come out the mass conservationequation (=0!) and dividing all for α1.

In this way , the solution could be not so accurate with steepgradients of the solid phase → in the last release (23x 7 months ago)has been implemented the fully conservative formulation of themomentum equation solving the problem and increasing thecomplessive robustness (Passalacqua, Fox 2011)


Introduction

.. let’s start looking at the code!Main Code

cd $FOAM_SOLVERS/multiphase/twoPhaseEulerFoam/

vi twoPhaseEulerFoam.C

// --- Pressure-velocity PIMPLE corrector loopwhile (pimple.loop()){

fluid.solve(); //starts solving the void fractionfluid.correct();volScalarField contErr1(

fvc::ddt(alpha1, rho1) + fvc::div(alphaRhoPhi1)- (fvOptions(alpha1, rho1)&rho1)

);volScalarField contErr2(

fvc::ddt(alpha2, rho2) + fvc::div(alphaRhoPhi2)- (fvOptions(alpha2, rho2)&rho2)

);#include "UEqns.H" //include the momentum equation without solve it#include "EEqns.H" //solves the energy equation// --- Pressure corrector loopwhile (pimple.correct()){

#include "pEqn.H" //construct a combined continuity eq. and correct the velocities}#include "DDtU.H"if (pimple.turbCorr()){

fluid.correctTurbulence(); //calculate the turbulent viscosity}

}


Introduction

.. let’s start looking at the code! Volume fraction Solution.

How is it solved? → the core is the MULES ”MultidimensionalUniversal Limiter with Explicit Solution”

What is MULES?? → is an iterative implementation of the FluxCorrected Transport technique (FCT) , used to guaranteeboundedness in the solution of hyperbolic problems

How does it work? → Compute a corrected Flux between an high anda low order scheme solution with a weighting factor λ

FC = FL+λA where A is the anti− diffusive flux = (FH −FL) (12)

to determine the weighting factor, Zalesak (1979) underlined that thevalue of the net flux in a cell , must be neither greater than a localmaximum nor lesser than a local minimum, correcting the flux in order tonot create nwe maximum or minimum.


Introduction

.. let’s start looking at the code!

The question is: why it’s widely used with multiphase flow? Looking atthe first slides, we find the solution. We need to set global extrema for theproblem! In our case α1 + α2 = 1 and other limits imposed by the users.Parameters are the variable to solve, the normal convective flux and theactual explicit flux of the variable which is also used to return limited fluxused in the bounded-solution.

src\finiteVolume\fvMatrices\solvers\MULES MULESTemplates.C

template<class RhoType, class SpType, class SuType>void Foam::MULES::explicitSolve(

const RhoType& rho,volScalarField& psi,const surfaceScalarField& phi,surfaceScalarField& phiPsi,const SpType& Sp,const SuType& Su,const scalar psiMax,const scalar psiMin

){

const fvMesh& mesh = psi.mesh();const scalar rDeltaT = 1.0/mesh.time().deltaTValue();psi.correctBoundaryConditions();limit(rDeltaT, rho, psi, phi, phiPsi, Sp, Su, psiMax, psiMin, 3, false);explicitSolve(rDeltaT, rho, psi, phiPsi, Sp, Su);

}


Introduction



template<class RdeltaTType, class RhoType, class SpType, class SuType>void Foam::MULES::limit.....

surfaceScalarField phiBD(upwind<scalar>(psi.mesh(), phi).flux(psi));

surfaceScalarField& phiCorr = phiPsi;phiCorr -= phiBD;

.....limiter(

allLambda,rDeltaT,rho,psi,

.

.

.);

if (returnCorr){

phiCorr *= lambda;}else{

phiPsi = phiBD + lambda*phiCorr;}

}


Introduction



void Foam::MULES::explicitSolve( const RdeltaTType& rDeltaT,const RhoType& rho,.......){ Info<< "MULES: Solving for " << psi.name() << endl;

const fvMesh& mesh = psi.mesh();scalarField& psiIf = psi;const scalarField& psi0 = psi.oldTime();

psiIf = 0.0;fvc::surfaceIntegrate(psiIf, phiPsi);

if (mesh.moving()){ }else{

psiIf =(

rho.oldTime().field()*psi0*rDeltaT+ Su.field()- psiIf

)/(rho.field()*rDeltaT - Sp.field());}psi.correctBoundaryConditions();

}

∂(α1ρ1)

∂t+∇ · (α1ρ1~u1) =

αn1 − α01

δt+ psiIf = Spα

n1 + Su → αn1 =

α01δt

+ Su + psiIf1δt− Sp

(13)

where psiIf is the net flux,while Spαn1 + Su is the source term (will beexplined better in the report) NB in the MULES alg the density isconsidered constantAlessandro Manni OpenFoam Course Presentation 1/2-12-2014 15 / 29

Introduction


before executing MULES, for the mass conservation,a trick is done for the solidfraction. The velocities aren’t considered as ~u1and ~u2 but are defined the mixtureand relative velocities ( with the respective fluxes). This approach will be veryusefull in the solution of the pressure equation.

U = α1U1 + α2U2 Ur = U1 − U2 → U1 = U + Ur(1− α1) (14)

U = (U +∇p1λs

)− ∇p1λs

= U∗ − ∇p1λs→ U∗ = U +

α1

λs

∂p1∂α1∇α1 (15)

U∗ = U +α1

λs

∂p1∂α1∇α1 with λs =

1

As +Ksg

ρs

(16)

where As condensates the diagonal coefficients of the matrix of the momentumequation (Passalacqua,Fox 2011)


Introduction

Overview

twoPhasesystem/twoPhaseSystem.C

MULES::explicitSolve(

geometricOneField(),alpha1,phi_,alphaPhic1,Sp,Su,phase1_.alphaMax(),0

);

then the solid phase continuity equation is finally solved:

∂α1

∂t+∇ · (α1~u

∗1)−∇ · [α1

λs

∂p1∂α1∇α1] = Spα1 + Su (17)


Introduction

Volume Fraction Solution

UEqns.H

U1Eqn =(

fvm::ddt(alpha1, rho1, U1) + fvm::div(alphaRhoPhi1, U1)- fvm::Sp(contErr1, U1)+ mrfZones(alpha1*rho1 + virtualMassCoeff, U1)+ phase1.turbulence().divDevRhoReff(U1)

==- liftForce- wallLubricationForce- turbulentDispersionForce- virtualMassCoeff*(

fvm::ddt(U1)+ fvm::div(phi1, U1)- fvm::Sp(fvc::div(phi1), U1)- DDtU2

)+ fvOptions(alpha1, rho1, U1)

);

U2Eqn =..........................

In this file, the momentum equations are constructed but still notsolved!It’s important to notice that are not included the terms related tothe pressure and the solid pressure


Introduction

Pressure equation

Summing the continuity eqns 19,20 we obtain the combined continuityequation 21 :( where Di

Dt = ∂∂t + ui · ∇)

∂α1

∂t+∇ · (α1~u1) = −α1

ρ1

D1ρ1Dt

(18)

∂α2

∂t+∇ · (α2~u2) = −α2

ρ2

D2ρ2Dt

(19)

∇ · ~u = −α1

ρ1

D1ρ1Dt

− α2

ρ2

D2ρ2Dt

(20)

as well known, the momentum equation could be written in short in thesemi discrete form:

AsUsp = Hs(~u1)− α1∇p (21)

AgUgp = Hg(~u2)− α2∇p (22)


Introduction

pressure equation

Combining togheter the equation 21 with 22 and 23 we obtain thepressure equation :

∇·~u = ∇·(α1Hs

As+α2

Hg

Ag)−∇·[(α1

α1

As+α2

α2

Ag)∇p] = −α1

ρ1

D1ρ1Dt−α2

ρ2

D2ρ2Dt(23)

that is solved in pEqn.H


Introduction

Turbulence modeling

Multiphase Turbulence

Issue of compressibility managed for many years using two distinctturbulence modelling frameworks → constant/variable density flows.

what about multiphase turb? the phase-fraction must be includedinto all transport and source terms of the turbulence propertyequations → Code is largely duplicated !

Therefore, only for twoPhaseEulerFoam in this version, a new single,general templated turbulence modelling framework was created thatcovers all four of the combinations.


Introduction

The framework can be found in FOAMSRC/TurbulenceModel andincludes the incompressible, compressible, phase-incompressible andphase-compressible options including support for laminar, RAS andLES simulations with wall-functions and other specialised boundaryconditions.

This new framework is very powerful and supports all of theturbulence modelling requirements needed so far.


Introduction

A quick re−overview on the solution procedure: the PIMPLE algorithmIt’s a combination of PISO and SIMPLE with the power of quick solution forunsteady flows coming from the PISO , and the possibility to use under−relaxation

Start the mass conservation loop

Solve the eqn 17 (without the particle pressure contribution) using the”now well known” MULEScalculate ∂p1

∂αs, correct the c.e. for a new α1 .For the ps it’s obviously

used the formulation related to value of alpha1 (k g or f regime(eq 11))calculates the alpha2iterations

calculate and update the coefficients needed in the term ~I of the momentumequation ( drag vme lift ecc....)

solve a predicted velocity field with the last calculated pressure

evaluates the energy equation

pressure equation loop

iteration until is reached the convergency criteriaAlessandro Manni OpenFoam Course Presentation 1/2-12-2014 23 / 29

Introduction


In the energy equation 9 or 10, we’d see the term Kht∆T . Add anheat ex model means to add another way to calculate Kht dependingon the conditions (for instance the value of the Re)

in this version is present only the Ranz Marshall correlation. In thisquick guide we will add togheter the Gunn model . In the uploadedfiles you’ll find the Tomiyama and LiMason(2000)models as well.

Kht = k2Nu1d1

where k2 is the thermal fluid condictivity, Nuconv/cond, d is the solid diameter

the gunn correlation is more suitable for granular flows and very highRe105. The LiMason includes the Ranz Marshall for Re < 200 andpresents two other correlations in the range 200/1500 and over1500,while the Tomiyama is similar to the RM.


Introduction


Let’s start copying the following lines on the terminal

OF23x

cd $WM_PROJECT_DIR

cp -r --parents applications/solvers/multiphase/twoPhaseEulerFoam \

$WM_PROJECT_USER_DIR

cd $WM_PROJECT_USER_DIR/applications/solvers/multiphase

mv twoPhaseEulerFoam myTwoPhaseEulerFoam

cd myTwoPhaseEulerFoam

./Allwclean

now you have a clean src code in your project user dir


Introduction


now copy the AMpresentation.tar.gz into the /heatTransferModelsfolder , then

cd interfacialModels/heatTransferModels/

tar xzf AMpresentation.tar.gz

ls

now we have to simply modify the make/files

cd ..

cd Make

vi files

and insert after the line 31 the following lines

heatTransferModels/Gunn/Gunn.C

heatTransferModels/LiMason/LiMason.C

heatTransferModels/Tomiyama/Tomiyama.C


Introduction


before to compile the code we have to modify other few things:

cd ..

cd ..

mv twoPhaseEulerFoam.C myTwoPhaseEulerFoam.C

cd Make

sed -i s/twoPhaseEulerFoam/myTwoPhaseEulerFoam/g files

now we can compile!

cd ..

./Allwmake


Introduction


now we need a fresh tutorial to test

run

cp -r $FOAM_TUTORIALS/multiphase/twoPhaseEulerFoam/RAS/fluidisedBed .

cd fluidisedBed/constant

sed -i s/RanzMarshall/ASD/g phaseProperties

cd ..

blockMesh

myTwoPhaseEulerFoam

now check the error!

--> FOAM FATAL ERROR:

Unknown heatTransferModelType type ASD

Valid heatTransferModel types are :

4

(

Gunn

LiMason

RanzMarshall

Tomiyama

)Alessandro Manni OpenFoam Course Presentation 1/2-12-2014 28 / 29

Introduction

Thanks for your

Patience


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