Hi,
In a simple form, the Boussinesq hypothesis reads
R = -2 nu_t * S,
where R and S are tensors (well known).
The previous hypothesis works if the two tensors are aligned; basically if the cosine
( | R : S | ) / ( ||R|| ||S|| ) > 0.86 (approximation).
I would like to have some suggestions for cases of non-aligned tensors. Any method to estimate nu_t?
I thank you in advance and I look for your reply.
BR
Elie
Dear Elie, multiply using scalar product by S the two equation members you have (R and S are modules of the two tensors and x is scalar product):
nu = -1/2 R/S (R x S)/(RS) = -1/2 R/S cos(alpha)
Now use a Taylor expansion around alpha = 0:
nu = -1/2 R/S (1 - 1/2 alpha2 + 1/24 alpha4 - ...)
which is an approximation to viscosity when R and S are not parallel. Gianluca
Dear Gianluca
what R x S means for you? Isn't it still a tensor?
Dear Gianluca,
In fact I can not recognize how you obtained the expression of nu_t after themultiplication.
Second, do you mean by the scalar product nothing but the tensorial contraction operator?
Thanks
Dear Prof. Denaro and Elie, yes, R x S is the "scalar" product in the sense of contraction operator Rij Sji. Gianluca
Dear Gianluca,
Thanks for your reply.
However, I see that the result of nu_t is a tensor, and not a scalar from this formulation!
Elie
I think that Gianluca means that RxS is the same as R:S
However, rather than defining an angle (I don't like a definition of one angle between two second order tensors), I would focus on the anisotropy of the eddy viscosity coefficients.
I mean, we use a diagonal matrix I nu_t but one can write a matrix NU_t
Part of the essence of this, is that one may end up with a vector viscosity, or more accurately a tensor viscosity. It's a matter of what you're wanting to use the model for, and which anisotropies are important (and how much so).
Thanks for the discussion.
Yes Mark, what is obtained directly from the hypothesis is a tensor viscosity.
However, I was looking to calculate a scalar viscosity using the relation (can be derived easily)
nu_t = 0.5* (R_ij : S_ij ) / (S_ij : S_ij ).
In fact, I obtain very high viscosity values. I think the problem is that the tensors are not aligned and basically the hypothesis is not therefore valid....
What I want from this model is to estimate a nu_t, to compare with my nu_SGS obtained by my LES.
Are you considering the tensors filtered from the DNS fields? The resulting eddy viscosity strongly depends on the type and width of filtering you used. Be careful in comparing with the LES, what about the LES filter?
Hi Filippo,
The tensors I use in the post-poscessing are the time averaged fields
< rho. u' . u' >,
obtained from DNS ( I look to the RANS equations ).
To be more precise, I will explain the procedure as follows.
I initially carried out LES (Smagorinsky's model, filter top hat). What I obtain is a small viscosity ratio (nu_SGS / nu < 0.4). This means basically that the mesh is already fine and almost no viscosity is required to me modelled.
To the following result, I perform a DNS on the same grid. I save all the time averaged quantities appearing in the RANS equations (basically FANS: Favre averaged NS). I hope to estimate from those a nu_t to see if also the ratio is small...
Why are thinking to compare the eddy viscosity computed from LES to the RANS-based one obtained from DNS? For me that makes no sense.
Run directly a RANS instead of an LES
Thanks Elie for confirming.
But with significant mis-alignment of R and S, then shouldn't one be concerned with the use of Smagorinsky? It seems that, e.g. for flows highly influenced by geometry(spatial boundary conditions), then use of "Smag" itself may cause non-physical misalignments, which in turn may reduce the meaning of a scalar nu_t. (Recall the lack of backscatter with Smag, e.g. Wyngaard 2004).
So is part of your question, to see how the (tensorial!) diffusivity compares, from "DNS-FANS" versus LES with Smag.-SGS?
Thanks Mark for your remark.
In fact, the geometry I simulate influences significantly the flow (basically the outlet bc's). What I can ensure is that I have performed an LES convergence study (by grid influence). I note that a relative error (referenced to a fine simulation) decreases to 5% for the concentration and 6% for the velocity magnitude.
Regarding if the Smago itself induces a non-physical misalignments in my case, I see this probably require further investigations...
Elie, be careful in analysing your results. You cannot have a real grid-independent LES solution if yoy ar not using an explicit filtering approach. Indeed, if you do not use it, you are simply tending towards the DNS solution, not an LES one. This is congruent to the fact you have a vanishing eddy viscosity. That not happens in RANS where the turbulent viscosity is a result of the model of all the scales and do not vanish for successive refined grids.
Yes Filippo, it is exactly the case where the solution approaches the DNS. I just give the information for completeness... and it is not to say that the LES is well representing the flow (although the results are comparable with the PIV measurements)
I see that performing a RANS simulation is necessary for the comparison.
Yes. You can for sure use the DNS solution to extract all the RANS averaging, you do not need an LES field.
I never worked using hybrid formulation. I suspect that there are few strong theories on what solution one really get in the blended regions...
I used a blended approach to calculate the angular velocity for the inner moon of Mars.
It blended the densities.
The result was such that presumably the theories* were based on those data. *Chandrasekhar potential.
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