LL Calculation on the Smagorinsky Model

[eduardovital31]

Hello,

I am encountering some confusion regarding the operations on vector and tensor fields in the following code snippet:

dev(filter_(sqr(U)) - (sqr(filter_(U))));

From my understanding, both filter_(sqr(U)) and (sqr(filter_(U))) should return vector fields. However, the dev function is typically used for tensor fields. I am not aware of any function overloading for sqr that computes the outer product, nor for dev that accepts vector fields. Could you please help clarify what I might be missing here?

Thank you for your attention!

[AlbertoPa]

Hi @eduardovital31

if you apply the sqr method to a volVectorField like U, it calcuates the outer product, so it will return a volTensorField, of which you can then compute the deviatoric part with dev.

You can see this happening in the implementation of sqr for GeometricField: here https://cpp.openfoam.org/v12/namespaceFoam.html#aa4b0f76f190f109b3af558a5e95904d4 and at line 205 here https://cpp.openfoam.org/v12/GeometricFieldFunctions_8C_source.html#l00205

I hope this helps.

[eduardovital31]

Thank you for the clarification, @AlbertoPa !

I now understand that applying sqr to a volVectorField like U results in the outer product, which indeed produces a volTensorField. This clears up my confusion regarding the use of dev on the result.

I appreciate the detailed explanation and the links to the relevant implementation in the OpenFOAM codebase—they were very helpful in understanding how the sqr method behaves for GeometricField.

Thanks again for your assistance!

[eduardovital31]

Hello @AlbertoPa, I'd like to ask another question regarding the repository. I've noticed various implementations of both classical and dynamic Smagorinsky models online.

For instance, the classical Smagorinsky model typically uses a Cs coefficient (https://www.sciencedirect.com/topics/engineering/smagorinsky-model), whereas the Openfoam version employs Ck and Ce (https://www.openfoam.com/documentation/guides/latest/doc/guide-turbulence-les-smagorinsky.html).

Similarly, the dynamic Smagorinsky model often omits the term 2/3kI, focusing only on the right side of the equation (https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/large-eddy-simulations-of-stratified-turbulence-the-dynamic-smagorinsky-model/9EF56F397C92F45507E5FED05E1808A3).

Could you explain these differences?

Additionally, I haven't been able to determine if the equations, this repository, and the classical Smagorinsky model from Openfoam are specifically for incompressible flows, or if modifications are necessary for compressible cases.

Finally, I question myself if SGS stress tensor \tau_ij (devReff) generated by both Smagorinsky models are interchangeable, as I see quite a magnitude difference between both. Are they equivalent to the SGS stress tensor \tau_ij from the LES equation: dev(filter(u.u)-filter(u).filter(u))? Or even to the test SGS stress tensor as discribed in https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/large-eddy-simulations-of-stratified-turbulence-the-dynamic-smagorinsky-model/9EF56F397C92F45507E5FED05E1808A3: filter2(u.u)-filter2(u).filter2(u)?

Thanks for any assistance!

[AlbertoPa]

The model here is derived by the dynSmagorisky model in older versions of OpenFOAM. The difference is that their implementation averaged the dynamic coefficient over the whole domain to stabilize the simulation, while I stabilized it by using a local average involving neighbor cells.

The implementation is for incompressible flows (density is assumed constant, and nut is calculated).

Concerning the Ce and Ck coefficients, OpenFOAM has chosen to implement an explicit representation of k, while typically this is not done. That is why you have that term. You could work out the equations to see how their form relates to the more common form, or just replace the implementation with the one using Cs.