Help with einstein notation for a turbulence model
 

I need some help translating a summation from Einstein notation to matrix notation. Can anyone help?

The summation is equation 4.2 in this paper: http://www.stanford.edu/group/ctr/Re.../duraisamy.pdf

Equation 4.2 is actually 2 equations, I'm interested in the one on the right.

Thanks in advance!

Oh, and the summation is in 3D.

The second equation is composed of 2 main terms.

1) a tensor of the form:

A_ij = k * (a * d_ij + b * S_ij + c * (S*S)_ij)

2) another tensor of the form:

B_pq = (S*DS/Dt)_pq

where the products S*S and S*DS/Dt are tensor products (they're like matrix products), and k,a,b,c are simple scalar constants.

The final point is the evaluation of:

e_pqj * A_ij * B_pq

this is evaluated by first contracting the following:

e_pqj * B_pq = e_jpq * B_pq = v_j

i don't actually remember if this is any known form in vector notation
but it is just a vector and each component is obtained summing on p and q with fixed j. In example:

v_1 = e_111*B_11 + e_112 * B_12 + e_113 * B_13 + e_121 * B_21 + e_122 * B_22 + e_123 * B_23 + ... = e_123 * B_23 + e_132 * B_32 = B_23 - B_32

And for the the others:

v_2 = e_213 * B_13 + e_231 * B_31 = B_31 - B_13

v_3 = e_312 * B_12 + e_321 * B_21 = B_12 - B_21

Which, as expected, is identically zero when B is symmetric. The final point is just a matrix-vector multiplication:

A_ij * v_j

So, if you're able to identify the proper vector notation for v_j,
you're done.

If it can help, the form e_jpq * B_pq is what usually
comes out when taking the moment of the momentum equations in Fluid Dynamics with B playing the role of the stress tensor. When there is no applied volume moment then it is identically zero and this explain why the stress tensor, in this case, is symmetric.

That's exactly the explanation I was looking for. Thank you a ton!

You're welcome.