kEqn in kEpsilon.C

fisch · July 20, 2011, 13:35

Hello,

i try to understand the equations for the k-epsilon turb model.
the equation for k is:
tmp<fvScalarMatrix> kEqn
(
fvm::ddt(k_)
+ fvm::div(phi_, k_)
- fvm::Sp(fvc::div(phi_), k_)
- fvm::laplacian(DkEff(), k_)
==
G
- fvm::Sp(epsilon_/k_, k_)
);

i understand all terms except the second and the third...
As far as i get it these terms should represent div(k*phi)
But how can
+ fvm::div(phi_, k_) - fvm::Sp(fvc::div(phi_), k_) == div(k*phi) ???

Thanks for any advice,
rupert

Bernhard · July 21, 2011, 06:09

Be aware, that the term in the original equation is equal to
$\vec{U}\cdot\nabla k$
which is equal to
$\nabla\cdot (k\vec{U}) - k \nabla \cdot \vec{U}$
These are the two terms you see in the kEqn. For a converged solution, the second term vanishes (continuity), but it is numerically favorable to keep it while solving the equation.

By the way:
http://www.cfd-online.com/Forums/ope...silon-eqn.html

fisch · July 22, 2011, 02:21

Thanks Bernhard,
the mentioned origin term (U * grad(k)) was wrong in some of my books ( it was just div(k * U).
I had right now a look into the "Turbulent Flows" book of Pope and you're right with the origin of the term... So the question is answered.

Thanks again.
rupert

Bernhard · July 22, 2011, 03:53

No, it was nog wrong in the other books, the second term is just zero, since the div(U) = 0

fisch · July 22, 2011, 05:30

thats right.

thanks

fisch · July 22, 2011, 10:24

I had a look on this k-epsilon calculation and wanted to figure out what happens if wall functions are active.
Maybe you know answers to the three following questions, too!?

If i use kqR and epsilon wall functions in my calculations he "updates" this G and forces the values for epsilon on the wall region. He's not touching k, right?

And as far as i get it this "modified" G is used in the k equation at the wall region, too, right?

Is there any documentation about OpenFOAM which formulas were used from the theory (like original equations for kEqn, the wall functions, and so on)???

thanks a lot
rupert

July 20, 2011, 13:35	kEqn in kEpsilon.C	#1
fisch Member fisch Join Date: Feb 2010 Posts: 97 Rep Power: 16	Hello, i try to understand the equations for the k-epsilon turb model. the equation for k is: tmp<fvScalarMatrix> kEqn ( fvm::ddt(k_) + fvm::div(phi_, k_) - fvm::Sp(fvc::div(phi_), k_) - fvm::laplacian(DkEff(), k_) == G - fvm::Sp(epsilon_/k_, k_) ); i understand all terms except the second and the third... As far as i get it these terms should represent div(k*phi) But how can + fvm::div(phi_, k_) - fvm::Sp(fvc::div(phi_), k_) == div(k*phi) ??? Thanks for any advice, rupert

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July 21, 2011, 06:09		#2
Bernhard Senior Member Bernhard Join Date: Sep 2009 Location: Delft Posts: 790 Rep Power: 22	Be aware, that the term in the original equation is equal to $\vec{U}\cdot\nabla k$ which is equal to $\nabla\cdot (k\vec{U}) - k \nabla \cdot \vec{U}$ These are the two terms you see in the kEqn. For a converged solution, the second term vanishes (continuity), but it is numerically favorable to keep it while solving the equation. By the way: http://www.cfd-online.com/Forums/ope...silon-eqn.html

July 22, 2011, 02:21		#3
fisch Member fisch Join Date: Feb 2010 Posts: 97 Rep Power: 16	Thanks Bernhard, the mentioned origin term (U * grad(k)) was wrong in some of my books ( it was just div(k * U). I had right now a look into the "Turbulent Flows" book of Pope and you're right with the origin of the term... So the question is answered. Thanks again. rupert

July 22, 2011, 03:53		#4
Bernhard Senior Member Bernhard Join Date: Sep 2009 Location: Delft Posts: 790 Rep Power: 22	No, it was nog wrong in the other books, the second term is just zero, since the div(U) = 0

July 22, 2011, 05:30		#5
fisch Member fisch Join Date: Feb 2010 Posts: 97 Rep Power: 16	thats right. thanks

July 22, 2011, 10:24		#6
fisch Member fisch Join Date: Feb 2010 Posts: 97 Rep Power: 16	I had a look on this k-epsilon calculation and wanted to figure out what happens if wall functions are active. Maybe you know answers to the three following questions, too!? If i use kqR and epsilon wall functions in my calculations he "updates" this G and forces the values for epsilon on the wall region. He's not touching k, right? And as far as i get it this "modified" G is used in the k equation at the wall region, too, right? Is there any documentation about OpenFOAM which formulas were used from the theory (like original equations for kEqn, the wall functions, and so on)??? thanks a lot rupert