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Smagorinsky term in SGS kinetic energy equation

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Old   January 21, 2013, 10:15
Default Smagorinsky term in SGS kinetic energy equation
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Ivan
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Dear all,

I want to model the equation for the SGS turbulent kinetic energy (TKE) in LES context. Without entering the details, I have two terms I am dealing with. The first is the triple correlation

(a) 1/2*(\overline{u_i u_i u_j} - \overline{u_i u_i} \bar{u_j})

which I model as c*rho*delta*sqrt(k)*dk/dt, where k is the SGS TKE and c is a certain constant, and i put together with the diffusion term. All ok until here. There is, however, the term

(b) \frac{\partial}{\partial x_j} (\tau^r_{ij}  u_i)

where \tau^r_{ij} is the residual term of the momentum equation, modelled as

(c) \tau^r_{ij} = -2\mu_t\bar{S}_{ij}

My code follows a finite volume approach. I do not know then how to calculate this term (b). For example, is there a easy way to insert this term into the diffusion term? How this term is usually modelled (ex. implicitely or explicitely)?. Is there some paper to look at?

Hope someone can help!
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Old   January 21, 2013, 12:19
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Filippo Maria Denaro
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Quote:
Originally Posted by Ivan View Post
Dear all,

I want to model the equation for the SGS turbulent kinetic energy (TKE) in LES context. Without entering the details, I have two terms I am dealing with. The first is the triple correlation

(a) 1/2*(\overline{u_i u_i u_j} - \overline{u_i u_i} \bar{u_j})

which I model as c*rho*delta*sqrt(k)*dk/dt, where k is the SGS TKE and c is a certain constant, and i put together with the diffusion term. All ok until here. There is, however, the term

(b) \frac{\partial}{\partial x_j} (\tau^r_{ij}  u_i)

where \tau^r_{ij} is the residual term of the momentum equation, modelled as

(c) \tau^r_{ij} = -2\mu_t\bar{S}_{ij}

My code follows a finite volume approach. I do not know then how to calculate this term (b). For example, is there a easy way to insert this term into the diffusion term? How this term is usually modelled (ex. implicitely or explicitely)?. Is there some paper to look at?

Hope someone can help!

Maybe I have not well understood your question, but for what I see the term b) is in divergence form, thus I don't see particular problems to insert it in a finite volume constest...
(\tau^r_{ij}  u_i)
is the additional flux ...
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Old   January 21, 2013, 13:43
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Originally Posted by FMDenaro View Post
Maybe I have not well understood your question, but for what I see the term b) is in divergence form, thus I don't see particular problems to insert it in a finite volume constest...
(\tau^r_{ij}  u_i)
is the additional flux ...
Well, that's true, but the point is this:
The procedure to calculate the flux requires first the calculation of S_{ij} (which involves also an interpolation from points at the cell center to point at boarder-cell surface centers) and then a second interpolation to the surface centers. When I look at the TKE for RANS, this term is never present, so I was wondering if it is usually neglected for some reason or incorporated in some other term in some way.

Of course, if not, the solution will be to compute the flux! =)
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Old   January 21, 2013, 16:22
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Filippo Maria Denaro
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Originally Posted by Ivan View Post
Well, that's true, but the point is this:
The procedure to calculate the flux requires first the calculation of S_{ij} (which involves also an interpolation from points at the cell center to point at boarder-cell surface centers) and then a second interpolation to the surface centers. When I look at the TKE for RANS, this term is never present, so I was wondering if it is usually neglected for some reason or incorporated in some other term in some way.

Of course, if not, the solution will be to compute the flux! =)
It is not so strange that RANS equations and LES equation differ each other, the meaning of the variables you solve is totally different.
I never worked personally on LES modelling supplied by the kinetic energy equation, I suggest to have a look at the book of Sagaut, you can find the TKE equation and have an idea of the modelling approximations
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