On the discretization of cross-diffusion like terms in turbulence models

sbaffini · January 27, 2023, 11:31

With a misuse of nomenclature, I define as "cross-diffusion" the terms of the following form:

$CD_f = h \frac{\partial g}{\partial x_j} \frac{\partial f}{\partial x_j}$

which appear in the RHS of a transport equation for the independent variable

and where

and

are non constant and might themselves involve

.

The two most known examples of these terms are for the Spalart-Allmaras model (where really the term "cross" is a misuse):

$CD_{\hat{\nu}} = \rho\frac{c_{b2}}{\sigma} \frac{\partial \hat{\nu}}{\partial x_j} \frac{\partial \hat{\nu}}{\partial x_j}$

and the latest iterations of the Wilcox and Menter $k-\omega$ models, whose $\omega$ equations contain terms of the form:

$CD_\omega = F \frac{\rho}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$

where F is non constant and also discontinuous.

A common way to discretize

in the SA model is by exploiting the following equality:

$CD_f = h \frac{\partial g}{\partial x_j} \frac{\partial f}{\partial x_j} = \frac{\partial}{\partial x_j} \left(h g \frac{\partial f}{\partial x_j}\right) - g \frac{\partial}{\partial x_j} \left(h \frac{\partial f}{\partial x_j}\right)$

which allows an implicit treatment and enhances stability. However, the same is not formally valid for the $k-\omega$ because of the discontinuity in

. Still, for shocks, even the SA rearrangement would be invalid, because of the discontinuity in $\rho$ .

My question then is, have you ever tried this sort of discretization with some turbulence model different from the SA, despite it not being formally valid? Was it effective nonetheless in enhancing stability with only negligible differences in results?

arjun · January 29, 2023, 02:53

Quote:

Originally Posted by sbaffini

With a misuse of nomenclature, I define as "cross-diffusion" the terms of the following form:

and the latest iterations of the Wilcox and Menter $k-\omega$ models, whose $\omega$ equations contain terms of the form:

$CD_\omega = F \frac{\rho}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$

where F is non constant and also discontinuous.

I do this way in Wildkatze and it was observed to produce some instabilities so i under-relaxed this term a bit.

However this is not the biggest source of instability in my opinion, still the term is nasty over-all.

I should add that over-all implementation in Wildkatze is very stable but it is hard to pin point what exactly creates this stability. I implemented based on what i think would be stable method based on over all experiences with numerical method. So under-relaxing this term was not the only thing done. Better gradients i believe also plays a big role in all this.

sbaffini · January 30, 2023, 07:27

Quote:

Originally Posted by arjun

I do this way in Wildkatze and it was observed to produce some instabilities so i under-relaxed this term a bit.

However this is not the biggest source of instability in my opinion, still the term is nasty over-all.

I should add that over-all implementation in Wildkatze is very stable but it is hard to pin point what exactly creates this stability. I implemented based on what i think would be stable method based on over all experiences with numerical method. So under-relaxing this term was not the only thing done. Better gradients i believe also plays a big role in all this.

So, let me ask again more clearly, just to be sure. You have implemented the cross-diffusion term in the k-omega model(s) using the rearrangement shown in my previous post, where the resulting two terms can be implemented like traditional diffusive terms (yet, with the second negative one using different non conservative cell center values for g/k when putting the flux in the two neighbor cells)?

And, while you need to use some underrelaxation factor specific to it, you have found that this way of implementing it is still superior to, say, the one suggested by Menter, where the whole $\omega$ source term implicit part gets the following coefficient (where P, D and C stand for production, destruction and cross-diffusion)?

$\frac{\partial \left(P_\omega + C_\omega - D_\omega\right)}{\partial \omega} = -\frac{\left|C_\omega\right|+2D_\omega}{\omega}$

And finally, that you didn't find notable differences in the results with the two formulations?

Thanks for the clarification

arjun · January 30, 2023, 14:51

I meant I use this one only. I directly compute it from the gradients coming from Turbulence Model.

$CD_\omega = F \frac{\rho}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$

I use it in an explicit manner so no terms go to the Ap. Though now that you have put this thought into me, I MIGHT consider implicit formulation.

I do under-relax it because it was found to be bit unstable.

Nowadays the current implementation is very stable in Wildkatze. Perhaps the reason is that flow model is stable so may be turbulence model is adjusting to it.

Usually turbulence models make flow unstable but since flow model is very robust it all seem to work out.

sbaffini · January 31, 2023, 03:35

Ok, I see, thanks.

Then I suggest you to try the Menter recipe. I never had any specific problem with it.

My alternative formulation is kind of still a random idea at the moment, but it works great for SA. My concern is more around the resulting solution, because what I called F in the CD for k-w is something with a discontinuous derivative and I don't think that the rearrangement is valid there. But it would be even worst for SA with shocks, as the density itself is then discontinuous.

I would have tried myself, but I'm currently validating a rewrite of the original formulation, so this can only happen somehow later.

To be honest, there are also slight inconsistencies in the reformulation above when seen from a FV context. That is, you can't implement it correctly if you just volume integrate the two terms. You need to approach them from a point source perspective.

FMDenaro · January 31, 2023, 03:47

Quote:

Originally Posted by sbaffini

Ok, I see, thanks.

Then I suggest you to try the Menter recipe. I never had any specific problem with it.

My alternative formulation is kind of still a random idea at the moment, but it works great for SA. My concern is more around the resulting solution, because what I called F in the CD for k-w is something with a discontinuous derivative and I don't think that the rearrangement is valid there. But it would be even worst for SA with shocks, as the density itself is then discontinuous.

I would have tried myself, but I'm currently validating a rewrite of the original formulation, so this can only happen somehow later.

Paolo, just my curiosity, why do you talk of discontinuity, this is for the unresolved shock layer? Formally, in your viscous flow problem there is no singularity. And the action of the turbulence model should smooth the gradients. In conclusion, you could consider to limit the numerical oscillations around the spikes of the functions.

sbaffini · January 31, 2023, 04:07

For SA it was my speculation, and while in practical terms I have a discontinuity, I see what you mean that I formally don't have one. Indeed, I never had any practical problem with the SA reformulation shown above, not even for "practical" discontinuities like shocks.

But as the discontinuity will practically be there for any practical resolution (I mean in case of shocks), the question arises about if the given reformulation will practically work as the original one. I wasn't the one that devised the reformulation for SA, so I never really cared about it.

For the k-w case it is a different matter, it's really just the F definition that involves functions with a discontinuous derivative. Obviously, a much less severe case with respect to a shock, but it is also always there around the boundary layer.

Finally, in both models, a FV implementation also needs the following approximation of the last term:

$\int_V{g \frac{\partial}{\partial x_j} \left(h \frac{\partial f}{\partial x_j}\right) dV} \approx \int_V{g dV} \cdot \int_V{\frac{\partial}{\partial x_j} \left(h \frac{\partial f}{\partial x_j}\right) dV}$

FMDenaro · January 31, 2023, 07:45

Have a look to the second mean value theorem, it could be useful for your FV implementation.

January 27, 2023, 11:31	On the discretization of cross-diffusion like terms in turbulence models	#1
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,156 Blog Entries: 29 Rep Power: 39	With a misuse of nomenclature, I define as "cross-diffusion" the terms of the following form: $CD_f = h \frac{\partial g}{\partial x_j} \frac{\partial f}{\partial x_j}$ which appear in the RHS of a transport equation for the independent variable and where and are non constant and might themselves involve . The two most known examples of these terms are for the Spalart-Allmaras model (where really the term "cross" is a misuse): $CD_{\hat{\nu}} = \rho\frac{c_{b2}}{\sigma} \frac{\partial \hat{\nu}}{\partial x_j} \frac{\partial \hat{\nu}}{\partial x_j}$ and the latest iterations of the Wilcox and Menter $k-\omega$ models, whose $\omega$ equations contain terms of the form: $CD_\omega = F \frac{\rho}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$ where F is non constant and also discontinuous. A common way to discretize in the SA model is by exploiting the following equality: $CD_f = h \frac{\partial g}{\partial x_j} \frac{\partial f}{\partial x_j} = \frac{\partial}{\partial x_j} \left(h g \frac{\partial f}{\partial x_j}\right) - g \frac{\partial}{\partial x_j} \left(h \frac{\partial f}{\partial x_j}\right)$ which allows an implicit treatment and enhances stability. However, the same is not formally valid for the $k-\omega$ because of the discontinuity in . Still, for shocks, even the SA rearrangement would be invalid, because of the discontinuity in $\rho$ . My question then is, have you ever tried this sort of discretization with some turbulence model different from the SA, despite it not being formally valid? Was it effective nonetheless in enhancing stability with only negligible differences in results?

January 31, 2023, 04:07		#7
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,156 Blog Entries: 29 Rep Power: 39	For SA it was my speculation, and while in practical terms I have a discontinuity, I see what you mean that I formally don't have one. Indeed, I never had any practical problem with the SA reformulation shown above, not even for "practical" discontinuities like shocks. But as the discontinuity will practically be there for any practical resolution (I mean in case of shocks), the question arises about if the given reformulation will practically work as the original one. I wasn't the one that devised the reformulation for SA, so I never really cared about it. For the k-w case it is a different matter, it's really just the F definition that involves functions with a discontinuous derivative. Obviously, a much less severe case with respect to a shock, but it is also always there around the boundary layer. Finally, in both models, a FV implementation also needs the following approximation of the last term: $\int_V{g \frac{\partial}{\partial x_j} \left(h \frac{\partial f}{\partial x_j}\right) dV} \approx \int_V{g dV} \cdot \int_V{\frac{\partial}{\partial x_j} \left(h \frac{\partial f}{\partial x_j}\right) dV}$ Last edited by sbaffini; January 31, 2023 at 06:40.

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January 30, 2023, 14:51		#4
arjun Senior Member Arjun Join Date: Mar 2009 Location: Nurenberg, Germany Posts: 1,275 Rep Power: 34	I meant I use this one only. I directly compute it from the gradients coming from Turbulence Model. $CD_\omega = F \frac{\rho}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$ I use it in an explicit manner so no terms go to the Ap. Though now that you have put this thought into me, I MIGHT consider implicit formulation. I do under-relax it because it was found to be bit unstable. Nowadays the current implementation is very stable in Wildkatze. Perhaps the reason is that flow model is stable so may be turbulence model is adjusting to it. Usually turbulence models make flow unstable but since flow model is very robust it all seem to work out.

January 31, 2023, 03:35		#5
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,156 Blog Entries: 29 Rep Power: 39	Ok, I see, thanks. Then I suggest you to try the Menter recipe. I never had any specific problem with it. My alternative formulation is kind of still a random idea at the moment, but it works great for SA. My concern is more around the resulting solution, because what I called F in the CD for k-w is something with a discontinuous derivative and I don't think that the rearrangement is valid there. But it would be even worst for SA with shocks, as the density itself is then discontinuous. I would have tried myself, but I'm currently validating a rewrite of the original formulation, so this can only happen somehow later. To be honest, there are also slight inconsistencies in the reformulation above when seen from a FV context. That is, you can't implement it correctly if you just volume integrate the two terms. You need to approach them from a point source perspective.

January 31, 2023, 07:45		#8
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,781 Rep Power: 71	Have a look to the second mean value theorem, it could be useful for your FV implementation.