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Solving coupled scalar transport equations in openFoam

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Old   December 27, 2019, 04:58
Default Solving coupled scalar transport equations in openFoam
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Pavan Inguva
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Hi, I am trying to solve a series of coupled scalar transport equations i.e. temperature and chemical species transport for reacting flows by modifying icoFoam. I understand from a past forum post (quite a while ago) (How to solver coupled transport equations), that an iterative scheme is needed within openFoam.

If anyone could share what is the best practice for doing so, or if there a better way of solving these coupled equations, that would be immensely helpful. The model equations are as follows:
\frac{\partial T}{\partial t} + \bold{u}\cdot \nabla T = \alpha \nabla^{2}T + \sum_{i}(-\Delta H)r

\frac{\partial C_{A}}{\partial t} + \bold{u}\cdot \nabla C_{A} = D\nabla^{2}C_{A} + r

where r is given by a standard arrhenius expression which couples the two transport equations.:
r = Aexp(\frac{E_{a}}{RT})C_{A}

A, E_{a} are the preexponential factor and activation energy and R is the ideal gas constant.

I intend to solve this part outside the PISO loop as the temperature and chemical species do not strongly affect flow and can be neglected. But I am not too sure how to couple these two equations within openFoam. I understand that the following implementation provides a 1 way coupling of the velocity to the temperature equation:

```
// --- PISO loop
while (piso.correct())
{
volScalarField rAU(1.0/UEqn.A());
volVectorField HbyA(constrainHbyA(rAU*UEqn.H(), U, p));
surfaceScalarField phiHbyA
(
"phiHbyA",
fvc::flux(HbyA)
+ fvc::interpolate(rAU)*fvc::ddtCorr(U, phi)
);

adjustPhi(phiHbyA, U, p);

// Update the pressure BCs to ensure flux consistency
constrainPressure(p, U, phiHbyA, rAU);

// Non-orthogonal pressure corrector loop
while (piso.correctNonOrthogonal())
{
// Pressure corrector

fvScalarMatrix pEqn
(
fvm::laplacian(rAU, p) == fvc::div(phiHbyA)
);

pEqn.setReference(pRefCell, pRefValue);

pEqn.solve();

if (piso.finalNonOrthogonalIter())
{
phi = phiHbyA - pEqn.flux();
}
}

#include "continuityErrs.H"

U = HbyA - rAU*fvc::grad(p);
U.correctBoundaryConditions();
}
// This is where we add the temperature transport equations:
fvScalarMatrix TEqn
(
fvm:: ddt(T)
// The phi bit is a function argument which represents a vol<Type>Field
+ fvm:: div(phi, T)
- fvm:: laplacian(alpha, T)
// We need to add the source term below here:
);

TEqn.solve();
```
Thanks for your help!
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Old   December 28, 2019, 08:28
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Michael Alletto
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There are at least two option you can follow:


1) use a segregated approach: you first solve for T using the values of CA of the old time step and than you solve for CA using the newly computed values of T. I do not how good this will work because the equation for CA is very stiff. But this is implemented quit easily


2) Use the same method as above but apply an operator splitting method for CA: First solve the ordinary differential equation \frac{\partial Ca}{\partial t} = Aexp(\frac{E_{a}}{RT})C_{A}. Use the averaged value over the time step of CA computed by solving the differential equation to compute the reaction rate and solve the partial differential equation for CA. This method is used in the solver of openfaom involving chemical reactions like chemFoam, reactingFoam ore fireFaom.


Best


Michael
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