Good old source term in a new transport equation

Sun · December 30, 2014, 10:44

Hi, just cross-checking with you guys:
Say I have this equation:

$\frac{\partial {X}}{\partial {t}} + \nabla \cdot (XU) = A X^2$

"A" is a temperature dependent variable being computed, separately, via a header file.
Wondering if the implementation of source term is correct?

Code:

solve(fvm::ddt(X)+fvm::div(phi,X)-fvm::SuSp(AX,X));

Thanks and happy new year (almost!)

Sun · January 5, 2015, 02:22

Any suggestions would be really appreciated. I am impatiently waiting...!
thanks.

Sun · January 9, 2015, 04:21

For future reference:

Quote:

Code:

solve(fvm::ddt(X)+fvm::div(phi,X)-fvm::SuSp(AX,X));

This is totally wrong!

The way to deal with higher order source terms is to linearize them into Sc and Sp parts. I have applied Picard's method which is explained in detailed on page 57 of this document.

For the problem mentioned above, this is how I have implemented it in OF and note that the subscript "o" is for the old value of a field, retrieved by oldTime() function.

Step 1. Source term:

Step 2. Taylor's expansion around the old value of X

$S = S_o + (\frac{\partial S}{ \partial X})_o (X-X_o)$

Step 3. Sc and Sp

$S_c = S_o - (\frac{\partial S}{\partial X})_o X_o = -AX_o^2$

$S_p = (\frac{\partial S}{\partial X})_o = 2AX_o$

Step 4. Create two volScalarField(s) for Sc and Sp and using a forAll loop over the cells

Code:

forAll(mesh.C(), celli)
	{
            X_valOld = X.oldTime()[celli]
            Su[celli] = -1.0*A*Foam::pow(X_valOld,2);
            Sp[celli] = 2.0*A*X_valOld;
        }

Step 5. solving the PDE with the linearized source term

Code:

fvScalarMatrix XEqn
    (
	fvm::ddt(X)
      + fvm::div(phi,X)
     == Su + fvm::Sp(Sp,X)
    );
   XEqn.solve();

Step 6. Compile and do your victory dance!

I haven't run any test cases though! but for now there is no compilation errors.

lbossle · January 13, 2024, 15:05

I landed on this topic because I had stability issues caused by a very large source term. I couldn't perform a taylor linearization to solve my problem but this discussion more or less directed me to 'Numerical heat transfer and fluid flow' (Patankar 1980). Chapter 4.2-5 and especially 7.2 helped me enormously. So perhaps this can be of use to some for future reference.

December 30, 2014, 10:44	Good old source term in a new transport equation	#1
Sun Senior Member Join Date: Nov 2010 Posts: 103 Rep Power: 15	Hi, just cross-checking with you guys: Say I have this equation: $\frac{\partial {X}}{\partial {t}} + \nabla \cdot (XU) = A X^2$ "A" is a temperature dependent variable being computed, separately, via a header file. Wondering if the implementation of source term is correct? Code: solve(fvm::ddt(X)+fvm::div(phi,X)-fvm::SuSp(AX,X)); Thanks and happy new year (almost!)

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January 5, 2015, 02:22		#2
Sun Senior Member Join Date: Nov 2010 Posts: 103 Rep Power: 15	Any suggestions would be really appreciated. I am impatiently waiting...! thanks.

January 13, 2024, 15:05		#4
lbossle New Member Ludwig Bossle Join Date: Jan 2024 Location: Karlsruhe, Germany Posts: 1 Rep Power: 0	I landed on this topic because I had stability issues caused by a very large source term. I couldn't perform a taylor linearization to solve my problem but this discussion more or less directed me to 'Numerical heat transfer and fluid flow' (Patankar 1980). Chapter 4.2-5 and especially 7.2 helped me enormously. So perhaps this can be of use to some for future reference.