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Jonathan July 23, 2013 19:12

Extra terms in turbulence model transport equations
 
Hi,

I have been working on re-writing some parts of the kOmegaSST turbulence model, and recently realised there are some 'extra' terms in the transport equations for k and omega, notably -

for omega:

Code:

    tmp<fvScalarMatrix> omegaEqn
    (
        fvm::ddt(omega_)
      + fvm::div(phi_, omega_)
      - fvm::Sp(fvc::div(phi_), omega_)
      - fvm::laplacian(DomegaEff(F1), omega_)
    ==
        gamma(F1)/nut_*min(G, c1_*betaStar()*k_*omega_)
      - fvm::Sp(beta(F1)*omega_, omega_)
      - fvm::SuSp
        (
            (F1 - scalar(1))*CDkOmega/omega_,
            omega_
        )
    );

and for k:

Code:

    tmp<fvScalarMatrix> kEqn
    (
        fvm::ddt(k_)
      + fvm::div(phi_, k_)
      - fvm::Sp(fvc::div(phi_), k_)
      - fvm::laplacian(DkEff(F1), k_)
    ==
        min(G, c1_*betaStar()*k_*omega_)
      - fvm::Sp(betaStar()*omega_, k_)
    );

(relevant lines in the code: kOmegaSST.C lines 408 - 441.)

I have been using a combination of references, but these do not appear in any of them, so they must be an openFoam programming peculiarity.

Does anyone know / could anyone explain where these terms come from / why we have them in the eqns?

thanks very much in advance
much appreciated
jonathan

chegdan July 23, 2013 20:59

This has been discussed in several places

But it is one of my favorite things in fluid mechanics for some reason :D. The definition of the advection operator is

\nabla\cdot(\vec{U}k)=\vec{U}\cdot\nabla k + k(\nabla\cdot\vec{U})

where the left side is the conservative form and the right side is the primitive form ( according to anishtain4 from gas dynamics lingo). The use if this term

Code:

- fvm::Sp(fvc::div(phi_), k_)
in the turbulence model is to subtract off errors associated with incomplete convergence of momentum (last term in right of first equation), since ideally for incompressible flow the velocity field should be solenoidal. So, its an error correction for reducing errors from incompletely converged velocity fields. One more thing, is that I have used other forms of this, mainly
Code:

fvm::SuSp(-fvc::div(phi_),k_)
that will switch between an implicit and explicit source term depending on the sign of
Code:

-fvc::div(phi_)
. Hope this helps!

EDIT: There is actually a little bit of a discussion in "Computational Methods for Fluid Dynamics" by Ferziger and Peric' (3rd edition) on page 162 referring to the conservation of the kinetic energy equation.

Jonathan August 14, 2013 09:24

Hi Daniel,

apologies for the slow reply here - i saw your post and it helped a lot, however, things have been quite hectic this side, hence the late thanks.

again, thanks for your help - much appreciated
jonathan

Quote:

Originally Posted by chegdan (Post 441594)
This has been discussed in several places

But it is one of my favorite things in fluid mechanics for some reason :D. The definition of the advection operator is

\nabla\cdot(\vec{U}k)=\vec{U}\cdot\nabla k + k(\nabla\vec{U})

where the left side is the conservative form and the right side is the primitive form ( according to anishtain4 from gas dynamics lingo). The use if this term

Code:

- fvm::Sp(fvc::div(phi_), k_)
in the turbulence model is to subtract off errors associated with incomplete convergence of momentum (last term in right of first equation), since ideally for incompressible flow the velocity field should be solenoidal. So, its an error correction for reducing errors from incompletely converged velocity fields. One more thing, is that I have used other forms of this, mainly
Code:

fvm::SuSp(-fvc::div(phi_),k_)
that will switch between an implicit and explicit source term depending on the sign of
Code:

-fvc::div(phi_)
. Hope this helps!

EDIT: There is actually a little bit of a discussion in "Computational Methods for Fluid Dynamics" by Ferziger and Peric' (3rd edition) on page 162 referring to the conservation of the kinetic energy equation.


chegdan August 19, 2013 11:01

No worries, I enjoy discussing this topic. If anyone else has some thoughts that could expand this information I would love to hear about it. I can always learn more on the topic!


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