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March 21, 2005, 14:47 
LES equations

#1 
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In the LES equations, Favre averaged, all the non linear term give origin to sub grid terms which should be modelled. My code works in the following way: 1) Update the state vector ( ro, ro*u, ro*v, ro*E )
2) Derive the pressure from the Energy and velocity using the the state equation: p=(gamma1)*( ro*E0.5*ro*V^2 ) 3) Derive T form the other state equarion p=ro*R*T Now, my question is: Should the subgrid models appear also in the two state equations? I think so, but I ve not found references mentioning this points. Any help? 

March 22, 2005, 07:21 
Re: LES equations

#2 
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I think if you use Favre averaged there is no need.
\hat{P} = \hat{rho} R \tilde{T} and in the derivation of the LES equations, the Pressure appears normally filtered (\hat{P}) not Favre (\tilde{P}). I do not know in the other equation, but the SGS terms araising are probably small compare with rho V^2 ?? 

March 22, 2005, 10:11 
Re: LES equations

#3 
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Salvador, many thanks for your message. As you mentioned I don't need any model for the first equation. THe problem is the II: not filterd it is:
p=(gamma1)*( ro*E0.5*ro*V^2 ) when you filter you get: \hat{p}=(gamma1)*( \hat{ro}*\tilde{E} 0.5*\hat{ro}*\tilde{V^2} ) where the quantities known are: \hat{ro} \tilde{E} \tilde{V} thus I need to correlate \tilde{V^2} with ( \tilde{V} )^2 At a first look, this 2 quantities seem very different !!! 

March 22, 2005, 12:42 
Re: LES equations

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One classical approache to this problem is the one from Vreman thesis (Twente University, 1995) where an exact equation for the computable energy [rhoE]:
[rhoE]=\hat{p}/(\gamma1)+0.5*\hat{\rho}*\tilde{u_i}*\tilde{u_i} is derived. Formally you have a conservation equation similar to the unfiltered rhoE one but with 7 added subgrid terms B_1 to B_7. Vreman, Geurts & Kuerten (J. Eng. Math 29:299327, 1995) have tested a priori the relative magnitude of these terms : terms B_5 to B_7 can be hold as neglectable, so only 4 terms remain. In my computations, I use a subgrid Prandtl number to model the sum B_1+B_2, the model for B_3 naturally arises from the subrid model of the momentum equation (added subgrid viscosity) and B_4 is ignored for numerical stability reason though it can be explicitly computed from the momentum equation model. If you can't find Vreeman papers, you can have a look at, among others, Larcheveque et al., Ph. Fluids 15(1):193210, 2003 for the formulation of B_1 to B_7 (sorry for the selfpromotion). Note that in your original message you mention [rho, rhoU,rhoV,rhoE] as your conservative variables. I assume that the missing rhoW is a typing error because in most cases 2D LES is completely meaningless. Hope this helps. 

March 22, 2005, 16:38 
Re: LES equations

#5 
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Thank you Lionel, I ll have a look at Vreman's work.
Actually my code is 3D...but why do you say that 2D calculations are meaningless?. 

March 22, 2005, 16:56 
Re: LES equations

#6 
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Hi Lionel, I am at Southampton UK, are u there? Dario


March 23, 2005, 06:46 
Re: LES equations

#7 
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Hi,
LES should be 3D because turbulent flow are 3D with a few exceptions where fluctuations in the third direction are inhibited : stratificated flows or MHD for instance. Basically in LES you need to resolve some of the turbulent scales, including the energy transfer from the largest to the smallest. Since this transfer occurs through vortex streching which is fundamentally a 3D process, the computation has to be 3D. the energy transfer in 2D turbulence is very different, from small scales to the larger ones, eventually resulting in very large vortices. Therefore for 2D LES specific models are required. There was a couple of discussions on that topic in the past years on the forum. Lionel 

March 23, 2005, 06:47 
Re: LES equations

#8 
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Sorry,
I was at Southampton a few months ago but now I'm back in France. Lionel 

March 23, 2005, 07:44 
Re: LES equations

#9 
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well, there's non doubt that what you say it's true. But at the end of the day, and unfortunately, you are just solving a set of equations and for them the LES models appear nothing else than a sink of energy (expecially because in most of the cases the back scattering is avoided by making the average on homegeneous directions if you use the Germano procedure, while if you don 't use is you just model the Reynolds term and neglect Leonard (which accounts for the aliasing error) and the interaction tensor C, (which accounts for back scattering)...thus, the dynamics that you mentioned is invisible to your computations. So, I still believe that like RANS, you can make 2D LES calculations. Regards


March 23, 2005, 09:33 
Re: LES equations

#10 
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Interesting. Where did you get the information you mention above? Which text?


March 23, 2005, 10:14 
Re: LES equations

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I agree with your view on the purely energy sink nature of most of the models, as in RANS computation.
However look at the momentum equation : even if you use Leonard's triple decomposition with a term L accounting for resolvedresolved interaction (complementary to the longrange interaction term C and the Reynolds tensor), you obtain an equation for the resolved scales which includes a nonlinear term \bar{u_i}\bar{u_j}. This term coupled to the time derivative leads to energy exchanges to be resolveld between resolved scales, at least if the (effective) Reynolds number is high enough. Such turbulent exchanges are inherently 3D. So even if part of the computation, namely the model, can be 2D or even less if the flow has homogeneous direction(s) (that the idea behind the averaging process sometime used to stabilize the dynamic model), the resolveld scales should be computed as 3D to described the vortex stretching Now the comparison with RANS. To my view, RANS of purely turbulent flows is strictly wellposed for steady or coherent unsteady flows (because of the shift from ensemble averaging to time averaging invoking the statisticaly steadyness assumption). For the steady definition the model has to take into account the whole energy transfer process and 3D is not required unless the mean flowfield is statistically 3D itself. But the problem of the U(nsteady) RANS remains. As for LES there is one part of the energy spectrum which is computed and another part which is resolved. Well, I think that unsteady RANS should be use only if the unsteadyness comes from another source than turbulence and is (mostly) decoupled from it. My interpretation is based on the idea of an ensembleaveraging of multiple unsteady but (partly) coherent process. Note however that the model should be adapted (i.e. lowering in some wey the added viscosity) because some unsteady scales do not contribute to the turbulent energy cascade. For instance it is well know that for cavity flows (one of my personnal field of interest) URANS models generally yield poor predictions because of their too disipative characteristics. Eventually, from my personnal experience, 2D LES generally performs (slightly) better thant 2D URANS. However they compare very badly with a 3D (true?) LES. Particularly, the high viscosity damps smaller scales as the size of the large coherent vortices is most often widely overestimated (a 2D dynamics artifact?). See for instance the unsteady wake behind an airfoil with high angle of attack. Hybrid methods such as DES can solve the problem, but once again they should be 3D in their LES subzones. Anyway, I hope this clarifies a little bit my twopenny thoughts despite my poor english. Regards Lionel 

March 23, 2005, 10:42 
Re: LES equations

#12 
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The reason why an LES needs to be 3D, just as in an DNS, is simply that the vortex stretching term is missing from the 2D equations. In order to simulate the dynamics of a turbulent flow you need to include these terms and hence you need to be in 3D.


March 24, 2005, 08:23 
Re: LES equations

#13 
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I am new to LES, but I found a useful and clear approach on a degree thesis written by Marzio Piller, available on line. Unfourtunately, it's written in italian (fortunately for me, as I am italian as well ).
Lionel is clearly much more expert than me on the topic, and we should ask him about a very good reference. 

March 26, 2005, 08:28 
Re: LES equations

#14 
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Hi Daniel and Lionel
\bar{rho * E} = \bar{rho} * \tilde{E} = \bar{rho} * (\tilde{ei} + \tilde{u_iu_i}/2) \tilde{u_iu_i} = \tilde{u_i}\tilde{u_i}  (\tilde{u_iu_i} \tilde{u_i}\tilde{u_i}) The second term is the subgrid TKE and a transport equation can be solved for this (easy enough to derive). Then you dont have any trouble with the equations of state. There are very good dynamic versions of the 1eqn LES model also  check the literature. You can some references by visiting Dr. Menon's web site http://www.ccl.gatech.edu/ I am not too sure about this, but I think the number of modeled terms decreases tremendously. Regards Srini 

March 26, 2005, 08:30 
Re: LES equations (errata)

#15 
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Sorry it should have been :
\tilde{u_iu_i} = \tilde{u_i}\tilde{u_i} + (\tilde{u_iu_i} \tilde{u_i}\tilde{u_i}) Srini 

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