[LuckyTran]

Context: I am trying to back-calculate the eddy viscosity from knowledge of the Reynolds stresses and velocity strain that is consistent with the Boussinesq eddy viscosity hypothesis for a 2D axissymmetric case.


I get the general Boussinesq eddy viscosity hypothesis in 3D cartesian coordinates but consider statistically axis-symmetric flow through a circular pipe. The gradients in the azimuthal direction are 0 by symmetry but the Reynolds normal stress in the azimuthal direction are not necessarily zero.

[FMDenaro]

Originally Posted by LuckyTran

Context: I am trying to back-calculate the eddy viscosity from knowledge of the Reynolds stresses and velocity strain that is consistent with the Boussinesq eddy viscosity hypothesis for a 2D axissymmetric case.


I get the general Boussinesq eddy viscosity hypothesis in 3D cartesian coordinates but consider statistically axis-symmetric flow through a circular pipe. The gradients in the azimuthal direction are 0 by symmetry but the Reynolds normal stress in the azimuthal direction are not necessarily zero.

I am not sure about your question...
Boussinesq just postulated that the momentum transfer caused by turbulent eddies can be modeled with an eddy viscosity, that is independent from the reference system.

If you are using RANS and a 2D domain you set the symmetry condition for all the variables.
If you are using the Bousinnesq hypothesis in the LES formulation then the velocity is filtered, not statistically averaged.

My answer can be not focused on the real meaning of your question, could you better explain?

[LuckyTran]

A naive interpretation of the Boussinesq hypothesis is that assuming the eddy viscosity can be somehow prescribed, one can relate the Reynolds streses to the mean flow.

So I am trying to do the reverse. Given that I know the Reynolds streses and mean-flow, how to compute the eddy viscosity?

Originally Posted by FMDenaro

If you are using RANS and a 2D domain you set the symmetry condition for all the variables.

The symmetry condition sets the out of plane velocity and gradients to 0 but this does not prevent Reynolds stresses in that direction.

In the simple 2D case, the Boussinesq hypothesis is not helping? I have no relation between the the out of plane Reynolds stresses and the mean flow? What am I misunderstanding?

[FMDenaro]

Originally Posted by LuckyTran

A naive interpretation of the Boussinesq hypothesis is that assuming the eddy viscosity can be somehow prescribed, one can relate the Reynolds streses to the mean flow.

So I am trying to do the reverse. Given that I know the Reynolds streses and mean-flow, how to compute the eddy viscosity?

The symmetry condition sets the out of plane velocity and gradients to 0 but this does not prevent Reynolds stresses in that direction.

In the simple 2D case, the Boussinesq hypothesis is not helping? I have no relation between the the out of plane Reynolds stresses and the mean flow? What am I misunderstanding?

you can use the definition

http://www.cfd-online.com/Wiki/Bouss...ity_assumption

of course, you have 1 scalar function unknown, therefore you need to do some contraction, for example as I did here

http://link.springer.com/article/10....162-010-0202-x

[LuckyTran]

I forgot to mention that I considering a potentially (and most likely) anisotropic eddy viscosity.

Originally Posted by FMDenaro

you can use the definition

http://www.cfd-online.com/Wiki/Bouss...ity_assumption

I am referring to that as the definition of the eddy viscosity assumption yes. But in 2D, the third (out of plane) mean strain is zero. So what I get from that relation is that the third Reynolds stress is whatever is left of the turbulent kinetic energy that is not part of the 1st or 2nd stress, which is trivial. It still doesn't get me an eddy viscosity without additional assumptions.

[FMDenaro]

but you are someway considering a system with more equations than unknown...
Your problem is: Given all components of the Reynolds and velocity gradient tensors, which is the scalar function that gives a linear relation between them?

Therefore, you need a contraction, this is somehow resembling the problem in the determination of the eddy viscosity in the dynamic modelling....