Link between eddy viscosity and subgrid-scale turbulent viscosity in LES
I have been working recently with Large Eddy Simulation. There is a point that I would like to discuss here, it concerns the link between the "real" turbulent (or eddy) viscosity and the subgrid-scale turbulent viscosity.
The former comes from the Boussinesq approximation; it links the Reynolds stresses to the gradients of mean velocity in a turbulent flow.
The latter is used for instance in the Smagorinsky subgrid-scale model. It links the subgrid scale stresses to the gradients of filtered velocity. In the dynamic version, it can be time-dependent.
How are these two quantities linked to each other? Is the only difference the level of resolution (subgrid or resolved)? What about the orders of magnitudes? How to access the eddy viscosity in a LES computation?
Any comment, reaction, answer or correction is very welcome. Thanks!
Very roughly speaking, in most of the LES applications (which are based on the implicit filtering approach) the classic Smagorinsky model is nothing very different from a 3D mixing length model in which the length is specified by the grid spacing or, in the general case, by the cubic root of the cell volume (for finite volume methods). The main difference, in this case, is that to talk about LES your resolution should be such that this length is in the inertial range of the turbulent spectrum. Thus, yes the only difference is the level of resolution and the fact that in a URANS mixing length simulation the length is obviously not connected to the resolution of the grid but is somehow larger as it has to "represent" a different range of scales. This also means that the subgrid viscosity in LES is smaller than in URANS (independently from the specific models).
However, this is just the answer to your question. Some people would correctly say that this is not LES. The numerical method employed and the use of an implicit or explicit filtering approach lead to very different LES approaches and none of them is widely recognized by the other researchers in the field. I'd like to explain this better but it would really require too much space (and time). You should check some good reference to get a closer view (probably the Pope's book "Turbulent Flows" is the better choice to begin with as it is well contained, clear enough and probably the most objective).
Thank you for your reply, Paolo.
I will have a look at the reference you proposed me.
Yet, something is still unclear for me: does it make sense to talk about "real" turbulent viscosity in a turbulent flow simulated with LES?
By "real", I mean non-SGS. This quantity would be the proportionality constant between the Reynolds stresses and the gradients of time-averaged velocity and would represent the momentum diffusivity.
In my opinion, it doesn't make sense to define the turbulent Prandtl number or other turbulent dimensionless number by using the SGS turbulent viscosity.
Well, you should be aware of the fact that the turbulent viscosity is just a "position", it is not more "real" in RANS/URANS than in LES. The underlying reasoning is, more or less (under specific hypotheses which i'll not mention):
The application of a statistical operator (RANS/URANS) or a filter (LES) to the nonlinear term of the Navier-Stokes equations leads to an unclosed term: the Reynolds stress tensor in RANS/URANS and the more complex SGS stress tensor in LES. In both RANS/URANS and LES, the turbulent (or SGS) viscosity arises from the Boussinesque hypothesis:
Unknown Reynolds/SGS tensor = -2 Vt S_ij_resolved
where Vt is the turbulent/SGS viscosity and S_ij_resolved is the symmetric part of the gradient tensor evaluated by the statistically-averaged/filtered velocity field, that is by the velocity field which is available in the specific computation.
The basic reasoning for this position is that the action of the non resolved scales (the whole turbulent spectrum in RANS/URANS) on the resolved ones can be assimilated to the action of the molecular motion which draws energy by the flow trough the action of the molecular viscosity. Also, the form of the residual stresses (Reynolds/SGS) is similar to the one which is assumed in the kinetic theory of gases by the actual viscous stresses.
That's all, there is no more than this. However, in LES it can be shown that this form for the SGS stresses can lead (with some additional positions) to the correct energy dissipation when the filter cutoff is in the inertial range and the Kolmogorov hypotheses hold. Hence in LES it is justified by the fact that, by definition, the model has to represent only the universal equilibrium range (whose action is mostly dissipative). This is also why it is usually stated that in LES simpler model (than RANS/URANS) could work properly (in this case the Smagorinsky model is similar to the mixing layer model in the RANS approach).
In contrast, the model in RANS/URANS have to represents all the turbulent scales, even those where there is the turbulent kinetic energy production and the boundary conditions dependent ones. This automatically questions the Boussinesque hypothesis in RANS/URANS as the scales modeled are usually anisotropic and not simply dissipative.
Hence i'd conclude that the Boussinesque hypothesis is more appropriate in LES than RANS/URANS.
Of course (and this is probably about your question), in LES you could compute the approximate Reynolds stresses by a time-average of the resolved velocity field and than compare them with a turbulent viscosity model (again computed with the time-averaged LES velocity field) but this is just a post-processing step which has no sense in LES by itself (at most it can be justified as a benchmark for RANS models when DNS or experiments are not feasible) and, due to the fact that the Boussineque hypothesis is an approximation, you can't obtain directly Vt as a proportionality constant between the Reynolds stresses and -2S_ij_resolved (i.e., comparing them term by term). You can only compare the model as a whole (hence including -2S_ij_resolved in it) or you'll have 6 different values for Vt (5 for incompressible flows).
P.S. In the Pope's book, the specific chapter on LES is the 13th. I don't suggest you to buy it only for the LES. In this case the Sagaut's book is more appropriate.
Thanks again for this very instructive reply.
The cause of my confusion might come from the fact that I was considering the eddy viscosity as a physical quantity (quantifying the diffusion of momentum due to turbulent motions).
Indeed, in the derivation of the law of the wall for instance (for the turbulent flow on a solid boundary), the Boussinesq hypothesis is used together with Prandtl's mixing length hypothesis and leads to a sublime theoretical result verified experimentally and thus reflecting "reality"...
Another reason is that it appears in the definition of "turbulent" dimensionless numbers, Pr_t for instance.
But finally, as you explained, the turbulent viscosity is nothing more than an artefact to close the problem. And when introducing it, one still has to model it. For what I understood, this is the essence of subgrid-scale models.
And, coming back to the law of the wall, this is also what has been done, by Prandtl and von Karman.
That's exactly the point...actually the mixing length is also used by van Driest in the derivation of the wall damping functions. However this result, as well as the one by Prandtl and von Karman, is still based on the basic assumptions of the Boussinesque hypothesis and the mixing length model (indeed their results are quite accurate only for attached boundary layers with no pressure gradients). You can see them as analytical RANS solutions (however the law of the wall can also be derived by scaling arguments only) which, nonetheless, are not exact.
You can think about this: if the boussinesque hypothesis was not an approximation then the 2nd order RANS models (with an additional equation for each Reynolds stress) would not have reason to exist (and would not have been called "Hypothesis" ;) ).
I have borrowed the book by Pope and I think that the chapter about LES is indeed a great source of information.
By the way, to continue a little bit the discussion, it is mentioned in the book that in the case of homogeneous turbulence and with a very large filter width (filter width/integral length scale -> infinity) , the residual (SGS) eddy viscosity becomes the same than the turbulent viscosity...
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