# Smagorinsky model, length scale

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 September 26, 2022, 23:19 Smagorinsky model, length scale #1 New Member   Michael Join Date: Jan 2022 Location: Birmingham, UK Posts: 8 Rep Power: 2 Dear all, I have some questions about the Smagorinsky model in LES: (1) In a 2D case, when it is in a corner near walls on both x&y directions, what should be the length scale? (I think it should be the smaller one, but I might have seen some codes in which it multiplies x&y length scales.) (2) If it is near a slip boundary, rather than a non-slip wall, what should the length scale be? The Van Driest damping is 1-exp(-yplus/Aplus) I always regard yplus as a concept near non-slip wall as it involves shear stress. Is there still yplus near slip boundary? What should be Van Driest damping if it is near a slip boundary? Thank you

September 27, 2022, 04:36
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Filippo Maria Denaro
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 Originally Posted by MichaelBham Dear all, I have some questions about the Smagorinsky model in LES: (1) In a 2D case, when it is in a corner near walls on both x&y directions, what should be the length scale? (I think it should be the smaller one, but I might have seen some codes in which it multiplies x&y length scales.) (2) If it is near a slip boundary, rather than a non-slip wall, what should the length scale be? The Van Driest damping is 1-exp(-yplus/Aplus) I always regard yplus as a concept near non-slip wall as it involves shear stress. Is there still yplus near slip boundary? What should be Van Driest damping if it is near a slip boundary? Thank you

LES is for 3d, not for 2D flows.
However, you are considering wall bounded turbulence and you have to know that the concept of filtering applies for homogeneous flow directions.
Along the normal to wall direction you cannot thing about a unique characteristic lenght scale, that is the basic topic in turbulence.
The Smagorinsky lenght scale is a function of the distance from the wall.
But be aware that if you want to set physical BCs at the walls, you have to use a DNS-like grid in the BL. That is no filtering in the vertical direction.

 September 27, 2022, 06:30 #3 Senior Member     Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 1,948 Blog Entries: 29 Rep Power: 36 It's always the nearest wall. By definition, a slip boundary is not a wall. However, we are talking here about a model, how it was formulated and later refined. Can we esclude that a slip wall would, inviscidly, block eddies of a certain size and thus be relevant for a damping in the Smagorinsky model? No, of course, but the original damping has a very specific origin: fix the wrong near wall behavior of the resulting eddy viscosity and recover the correct one. It is in this sense, and only this one, that you need to look at the Van Driest correction. Which, again, doesn't mean that the VD correction can't be improved for more general flows, but that's another story and I don't know of any serious attempt at it.

September 27, 2022, 11:36
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Michael
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 Originally Posted by sbaffini It's always the nearest wall. By definition, a slip boundary is not a wall. However, we are talking here about a model, how it was formulated and later refined. Can we esclude that a slip wall would, inviscidly, block eddies of a certain size and thus be relevant for a damping in the Smagorinsky model? No, of course, but the original damping has a very specific origin: fix the wrong near wall behavior of the resulting eddy viscosity and recover the correct one. It is in this sense, and only this one, that you need to look at the Van Driest correction. Which, again, doesn't mean that the VD correction can't be improved for more general flows, but that's another story and I don't know of any serious attempt at it.

Sorry I still don't get it. what is ? Why is it cubic? In the VD damping it's just .

September 27, 2022, 14:12
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Paolo Lampitella
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 Originally Posted by MichaelBham Sorry I still don't get it. what is ? Why is it cubic? In the VD damping it's just .
It is then long to treat here. You might want to have a look at Exercise 13.31 in "Turbulent Flows" by Stephen Pope and the underlying theory spread in that same chapter and previous ones.

Roughly speaking, the Smagorinsky model is there as a surrogate of a turbulent term that, near walls, behaves in a certain way (i.e., ). But it fails to do so (it is constant instead). That's why it is multiplied by a damping term, to recover that very near wall behavior.

Unfortunately, not all the damping terms have been formulated correctly or, at least, there is a lot of confusion on their use and the respective models where they are suitable. So, you might have a damping term that goes to 0 at walls (which is better than nothing), yet it fails to do so with the correct slope. Also, sometimes the correct near wall behavior is not really useful, while a different one works better in practice.

But the underlying theory is still one, the term that one seeks to model actually goes to 0 at walls as . This, in turn, follows from the fact that the term is u'v', with u' going to 0 as and v' going to 0 as (because of continuity and wall parallel velocity gradients being 0 at walls, the wall normal velocity gradient must be 0 as well, meaning that the v' taylor expansion at walls starts with 2nd order terms)

 Tags smagorinsky model, van driest damping