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April 29, 2009, 13:53 
Help with einstein notation for a turbulence model

#1 
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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 29, 2009, 13:56 

#2 
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Oh, and the summation is in 3D.


April 30, 2009, 05:13 

#3 
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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 matrixvector 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. 

April 30, 2009, 10:53 

#4 
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That's exactly the explanation I was looking for. Thank you a ton!


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