epsilon as implicit term in kEqn's

agustinvo · March 3, 2021, 11:06

I have remarked that in several turbulence models where the turbulent kinetic energy is solved, the dissipation term appears as:

Code:

 - fvm::Sp(alpha*rho*epsilon_/k_, k_)

and I was wondering the reason behind this piece of code. ¿Does it give some stability? As far as I see, if all the left input is constant, it means that the contribution of epsilon on the equation will be affected by the factor k_calculated/k.

¡Thank you for your answers!

Tobermory · March 3, 2021, 12:02

Yes, stability is the reason I believe. Remember that for the source term linearisation, you must ensure that Su is positive and Sp negative in order to preserve stability (check out Patankar's book, for example). Here we have a $-\epsilon$ term, and so we cannot put it into Su, but instead put it into Sp ... but that means that the Sp coefficient must be $-\epsilon/k$ since the linearised source is

.

agustinvo · March 3, 2021, 12:52

So, do you think this is the best way to introduce the dissipation rates on turbulence models?

Tobermory · March 3, 2021, 13:01

Yes. Indeed, I think it is the only way to do it without risking instability.

Okay, well strictly speaking if you have other Su source terms with positive coefficients you could try implement a scheme where you put part of the dissipation term in Su (making sure that Su stayed positive) and put the rest in Sp ... but I don't see any benefit from doing that, and it's certainly a lot of work.

agustinvo · March 3, 2021, 13:21

What disturbs me in this approach is that in the k equation you should the dissipation term as $\epsilon\frac{k}{k_{init}}$ , where the k on the numerator changes as solving the equation. It means that in certain regions this dissipation term might be larger than the existing epsilon value (as it is multiplied by a factor $k_{new}/k_{init}$ ).

This method helps to ensure stability but I find it not so physical because of this factor.

Tobermory · March 3, 2021, 13:28

Yes, understood and agreed, it is an approximation. However, it will only matter in parts of the flow where $\partial k / \partial t$ is large; for most flows $k^{n+1}/k^n \approx 1$ and a good approximation to the solution is better than no solution!

agustinvo · March 4, 2021, 06:34

Quote:

Originally Posted by Tobermory

Yes, understood and agreed, it is an approximation. However, it will only matter in parts of the flow where $\partial k / \partial t$ is large; for most flows $k^{n+1}/k^n \approx 1$ and a good approximation to the solution is better than no solution!

Indeed, specially when running steady state simulations as me... I had a look on Patankar's book and I found what you meant about the source terms... I will revise my implementations! I actually had some other divergence problems, and they might be caused by this... Thank you!

Tobermory · March 4, 2021, 06:54

My pleasure! Glad to be of help.

March 3, 2021, 11:06	epsilon as implicit term in kEqn's	#1
agustinvo Senior Member Agustín Villa Join Date: Apr 2013 Location: Alcorcón Posts: 313 Rep Power: 15	I have remarked that in several turbulence models where the turbulent kinetic energy is solved, the dissipation term appears as: Code: - fvm::Sp(alpharhoepsilon_/k_, k_) and I was wondering the reason behind this piece of code. ¿Does it give some stability? As far as I see, if all the left input is constant, it means that the contribution of epsilon on the equation will be affected by the factor k_calculated/k. ¡Thank you for your answers!

March 3, 2021, 13:21		#5
agustinvo Senior Member Agustín Villa Join Date: Apr 2013 Location: Alcorcón Posts: 313 Rep Power: 15	What disturbs me in this approach is that in the k equation you should the dissipation term as $\epsilon\frac{k}{k_{init}}$ , where the k on the numerator changes as solving the equation. It means that in certain regions this dissipation term might be larger than the existing epsilon value (as it is multiplied by a factor $k_{new}/k_{init}$ ). This method helps to ensure stability but I find it not so physical because of this factor. Last edited by agustinvo; March 4, 2021 at 06:32. Reason: LaTeX

March 3, 2021, 13:28		#6
Tobermory Senior Member Join Date: Apr 2020 Location: UK Posts: 669 Rep Power: 14	Yes, understood and agreed, it is an approximation. However, it will only matter in parts of the flow where $\partial k / \partial t$ is large; for most flows $k^{n+1}/k^n \approx 1$ and a good approximation to the solution is better than no solution! agustinvo likes this.

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March 3, 2021, 12:02		#2
Tobermory Senior Member Join Date: Apr 2020 Location: UK Posts: 669 Rep Power: 14	Yes, stability is the reason I believe. Remember that for the source term linearisation, you must ensure that Su is positive and Sp negative in order to preserve stability (check out Patankar's book, for example). Here we have a $-\epsilon$ term, and so we cannot put it into Su, but instead put it into Sp ... but that means that the Sp coefficient must be $-\epsilon/k$ since the linearised source is .

March 3, 2021, 12:52		#3
agustinvo Senior Member Agustín Villa Join Date: Apr 2013 Location: Alcorcón Posts: 313 Rep Power: 15	So, do you think this is the best way to introduce the dissipation rates on turbulence models?

March 3, 2021, 13:01		#4
Tobermory Senior Member Join Date: Apr 2020 Location: UK Posts: 669 Rep Power: 14	Yes. Indeed, I think it is the only way to do it without risking instability. Okay, well strictly speaking if you have other Su source terms with positive coefficients you could try implement a scheme where you put part of the dissipation term in Su (making sure that Su stayed positive) and put the rest in Sp ... but I don't see any benefit from doing that, and it's certainly a lot of work.

March 4, 2021, 06:54		#8
Tobermory Senior Member Join Date: Apr 2020 Location: UK Posts: 669 Rep Power: 14	My pleasure! Glad to be of help.