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akm September 2, 2013 11:49

WALE model definition !?

I'm trying to implement WALE model in LES code using Fortran for channel flow.
Now the model definition for sgs viscosity is:

\nu_{\tau} = \Delta^2_{s} \frac{\left(S^d_{ij}S^d_{ij}\right)^{3/2}}{\left(\bar{S}_{ij}\bar{S}_{ij}\right)^{3/2}+\left(S^d_{ij}S^d_{ij}\right)^{5/4}}

where, S^d_{ij}S^d_{ij} = \frac{1}{6}\left(S^{2}S^{2} + \Omega^{2}\Omega^{2}\right)+\frac{2}{3}S^{2}\Omega^{2} + 2(IV)_{S\Omega}

S^2 = \bar{S}_{ij}\bar{S}_{ij}
\Omega^2 = \bar{\Omega}_{ij}\bar{\Omega}_{ij}
(IV)_{S\Omega} =  \bar{S}_{ik}\bar{S}_{kj}\bar{\Omega}_{jl}\bar{\Omega}_{li}

I'm having problems understanding the last term in the above equation \left(\bar{S}_{ik}\bar{S}_{kj}\bar{\Omega}_{jl}\bar{\Omega}_{li}\right).
Can anyone please explain how to expand the above term?


sbaffini September 2, 2013 14:26

It seems that there is some algebra which can be exploited:

However, if you define S2=S^2 and W2=W^2 (S_ik S_kj & W_ik W_kj rispectively), you will notice that the term is just S2_mn W2_nm... i guess

Rami October 3, 2013 06:18

It seems that tensor notation is used here, implying summation of repeated indices. If this is the case, the last equation is merely a sum over the all the indices i,j,k,l (each in the proper range, e.g. from 1 to 3 in 3D).

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