WALE model definition !?

akm · September 2, 2013, 10:49

Hello,

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?

Cheers

sbaffini · September 2, 2013, 13:26

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

http://www.math.univ-montp2.fr/~nico...S_BayaToda.pdf

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, 05:18

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).

September 2, 2013, 10:49	WALE model definition !?	#1
akm New Member Join Date: Jan 2010 Location: Netherlands Posts: 28 Rep Power: 16	Hello, 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? Cheers

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September 2, 2013, 13:26		#2
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,152 Blog Entries: 29 Rep Power: 39	It seems that there is some algebra which can be exploited: http://www.math.univ-montp2.fr/~nico...S_BayaToda.pdf 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

October 3, 2013, 05:18		#3
Rami Senior Member Rami Ben-Zvi Join Date: Mar 2009 Posts: 155 Rep Power: 17	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).