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How to calculate Reynolds Stress using LES data?

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Old   February 22, 2001, 19:04
Default How to calculate Reynolds Stress using LES data?
  #1
Jongdae Kim
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I have u,v,w velocities on staggered grid. The velocities are from Large Eddy Simulation. I'd like to compare LES results with experimental data (from LDV).

Please let me know how to calculate Reynolds stress with LES data.

************************************************** ******

LES data : space averaged values

Reynolds Stress (from Experimental data) : time averaged values
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Old   March 1, 2001, 09:18
Default Re: How to calculate Reynolds Stress using LES dat
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Lionel Larcheveque
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LES is by nature an unsteady computation. So to calculate components of the Reynolds stress tensor, you need to time average u, v, w, u^2, v^2, w^2, uv, uw, vw as soon as the flow seems establish and for a time period long enough. Then use the formula : <u'v'>=<uv>-<u><v>, and so on, ..., where <> denotes time averaging, with ' defined by u'=u-<u> (Reynolds decomposition, assuming ergodicity, i.e. time average and average on multiple identical experiences are equivalent). I suppose that space averaged values you mentioned correspond to homogeneous direction(s) of the flow. You can use averaged values along this (these) direction(s) at each time step to perform time averaging, in order to accelerate the convergence of statistical values and greatly reduce data size.
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Old   March 1, 2001, 09:24
Default Re: How to calculate Reynolds Stress using LES dat
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Lionel Larcheveque
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It seems that there is a little HTML problem in the formula I've just posted (use of <>). The correct one is : [u'v']=[uv]-[u][v], where [] denotes time averaginf, with ' defined by u'=u-[u]...

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Old   March 1, 2001, 09:43
Default Re: How to calculate Reynolds Stress using LES dat
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andy
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But what about the SGS contribution? (particularly the normal components).

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Old   March 1, 2001, 18:44
Default Re: How to calculate Reynolds Stress using LES dat
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Jongdae Kim
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Notations : { * } space (or volume) averaging [ * ] time averaging

Procedure of calculation of Reynolds stresses

[1] DNS & Experiments

(1) measure u (= [u]+u') for longtime

(2) u'=u-[u]

(3) u'v' = u' * v'

(4) [u'v'] : time averaging of (3)

or [uv]-[u][v] : this is not space averaged

[2] RANS equations with Finite volume method :

tau = [u' * v'] : calculated from model

= {[uv]}-{[u][v]}

[3] LES : implicit box-filter

Governing equation : space averaged-NS

Numerical scheme : Finite volume method

Output of numerical scheme : space averaged velocity

R12={uv}-{u}{v}

[R12]=[{uv}]-[{u}{v}]

Questions : 1. [{*}] = {[*]} .ie. tau=[R12] ?

2. How to compare Reynolds stress of LES output with DNS(Experiment) output ?

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Old   March 2, 2001, 09:45
Default Re: How to calculate Reynolds Stress using LES dat
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Lionel Larcheveque
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First my notations (slightly different from yours):

u=[u]+u' Reynolds decomposition, with[*] as an average on multiple realizations which can become, using ergodicity, space, time or space and time averaging, according to the problem you want to solve (2D, 3D, statistically steady or unsteady).

u={u}+u" scales separation using space filtering, with {*} a space filter.

LES outputs space filtered velocity {u}.

Three remarks : - I haven't understood what you was meaning by space averaging. LES is based on a space filtering : there is from a mathematical point of view a great difference between to average and to filter : the relation {{u}}={u} is false, and then {u"} differs from 0.)

- I don't know much about RANS, but, for a flow with one homogeneous direction, it seems to me that : * if you consider steady 2D RANS, the average is both on time and space (you will iterate until you have a converged solution and the statistical quantities are given by the final results without "explicit" time averaging during the convergence process) :[*] means time and space averaging. * if you consider steady 3D RANS, the average is on time, and once you have converged you can average on space (and then you have the relation from your last message : tau={[uv]}-{[u]}[v]} (using your notations)) * if you consider 2D URANS, at each time step you obtain space (and maybe phase time) averaged values that you will time average during the computation. * if you consider 3D URANS, the quantities at each time steep can be regarded as phase average. There is a kind of link between this king of computations and VLES (Very large eddy simulation).

- At a rough estimate, I will say that time averaging and space filtering commute on discrete level (they're both integration, but the problem is the infinite bounds of the integrals), but even then the relation you mentioned become : [R12]=[{uv}]-[{u}{v}]={[uv]}-[{u}{v}], which is different from {[uv]}-{[u][v]} (product doesn't commute).

Now from the LES point of view :[*] means time averaging, and maybe space averaging if you have homogeneous flow direction(s).

Using both the decompositions mentioned above, formally : u'={u}+u"-[{u}]-[u"]

=> [u'v']=[{u]{v}]-[{u}][{v}] : can be computed +[{u}v"]-[{u}][v"]+[u"{v}]-[u"][{v}] : cross terms +[u"v"]-[u"][v"] : subgrid terms

As far as i know, as general rules the comparisons between DNS or experimental data and LES computations are made using the first line of the formula. The idea is that if you have a mesh finer enough (cut length of the implicit filter far in the inertial range of the kinetic turbulent energy spectrum), subgrid scales energy has a minor impact on the global fluctuations. If you really want to go further, here are some possibilities (not an exhaustive list):

1-Filtering your DNS data : the main problem is you don't know precisely the characteristics of the (implicit) LES filter, which is by the way far different from an academical one (gaussian, cutoff) due to the adding filtering effects of the numerical scheme.

2-Using deconvolution techniques to recalculate u". Problems similar to point 1 occur : what are the characteristics of the filter you want to inverse. To reach the same objective, maybe is it possible to use NLDE (Non Linear Disturbance Equation I think) to regenerate an estimation of the fluctuations of the small scales?

3-using models for the third (and the second?) term(s) of the formula. Models based on the Boussinesq approximation (subgrid viscosity) seems less pertinent than those based on a Bardina's approach (scales similarity) who reproduces more accurately the mathematical structure of the subgrid tensor (Is there a reason to consider turbulence only from a dissipative point of view while modeling the effects of the subgrid terms on the Reynolds stresses : the stability of the scheme is not a problem in an a posteriori computation of the Reynolds stresses). I heard about some attempts to use such approach, but it seems that few were conclusive.

I hope this little text (I apologize for my English) is helping to clarify your problem.

Good luck
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