Extra terms in turbulence model transport equations

Jonathan · July 23, 2013, 19:12

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

. 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

This has been discussed in several places

But it is one of my favorite things in fluid mechanics for some reason

. 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!

July 23, 2013, 19:12	Extra terms in turbulence model transport equations	#1
Jonathan Senior Member Join Date: Mar 2010 Posts: 173 Rep Power: 17	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 HPE likes this.

July 23, 2013, 20:59		#2
chegdan Senior Member Daniel P. Combest Join Date: Mar 2009 Location: St. Louis, USA Posts: 621 Rep Power: 0	This has been discussed in several places http://www.cfd-online.com/Forums/ope...tml#post411753 http://www.cfd-online.com/Forums/ope...silon-eqn.html http://www.cfd-online.com/Forums/ope...tml#post280210 But it is one of my favorite things in fluid mechanics for some reason . 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. Bernhard and cfdonline2mohsen like this. Last edited by chegdan; August 22, 2013 at 16:37. Reason: Added more information and a literature source and forgot a dot product

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August 19, 2013, 11:01		#4
chegdan Senior Member Daniel P. Combest Join Date: Mar 2009 Location: St. Louis, USA Posts: 621 Rep Power: 0	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!