# LES equations

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 March 21, 2005, 14:47 LES equations #1 Daniel Guest   Posts: n/a 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=(gamma-1)*( ro*E-0.5*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 Salvador Guest   Posts: n/a 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 Daniel Guest   Posts: n/a 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=(gamma-1)*( ro*E-0.5*ro*V^2 ) when you filter you get: \hat{p}=(gamma-1)*( \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 #4 Lionel Larcheveque Guest   Posts: n/a 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}/(\gamma-1)+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 29:299-327, 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):193-210, 2003 for the formulation of B_1 to B_7 (sorry for the self-promotion). 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 daniel Guest   Posts: n/a 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 daniel Guest   Posts: n/a Hi Lionel, I am at Southampton UK, are u there? Dario

 March 23, 2005, 06:46 Re: LES equations #7 Lionel Larcheveque Guest   Posts: n/a Hi, LES should be 3-D because turbulent flow are 3-D 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 3-D process, the computation has to be 3-D. the energy transfer in 2-D turbulence is very different, from small scales to the larger ones, eventually resulting in very large vortices. Therefore for 2-D 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 Lionel Larcheveque Guest   Posts: n/a 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 Daniel Guest   Posts: n/a 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 Interested. Guest   Posts: n/a Interesting. Where did you get the information you mention above? Which text?

 March 23, 2005, 10:42 Re: LES equations #12 Tom Guest   Posts: n/a 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 Daniel Guest   Posts: n/a 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 Srini Guest   Posts: n/a 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 1-eqn 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 Srini Guest   Posts: n/a 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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