|April 29, 2009, 13:53||
Help with einstein notation for a turbulence model
Join Date: Apr 2009
Posts: 78Rep Power: 9
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!
|April 30, 2009, 05:13||
Join Date: Mar 2009
Blog Entries: 17Rep Power: 20
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,
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.
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