How to derive this simple stress tensor calculation?

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 July 4, 2010, 12:33 How to derive this simple stress tensor calculation? #1 Member   bearcat Join Date: Jul 2009 Location: Ohio Posts: 35 Rep Power: 10 Most turbulence models are written in tensor expressions so we have to write out the full formulation before we can implement them. I am not very experienced in difficult tensor analysis. So can anyone explain a little about writing this term in the standard k-e model? From other books, the LHS should equals RHS, which reads: How to get from LHS to RHS? In my understanding, for 3D domain LHS should be a 3x3 matrix as it's a product of two rank-two tensors. But RHS is a scalar. How can this happen? Thank you very much.

 July 4, 2010, 13:46 #2 Administrator     Jonas Larsson Join Date: Jan 2009 Location: Gothenburg, Sweden Posts: 575 Rep Power: 10 Check out: http://www.cfd-online.com/Wiki/Einst...ion_convention The LHS you have written is also a scalar and uses the Einstein summation convention

July 4, 2010, 14:29
#3
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bearcat
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Quote:
 Originally Posted by jola Check out: http://www.cfd-online.com/Wiki/Einst...ion_convention The LHS you have written is also a scalar and uses the Einstein summation convention

The difficulty is that the index does not repeat in any individual term, we don't have any summation.

http://topex.ucsd.edu/geodynamics/shearer.pdf

this is about solid mechanics, but those concepts such as strain tensor and rotation tensor also apply to fluid mechanics.

 July 4, 2010, 17:03 #4 New Member   Guillaume Fournier Join Date: Jun 2010 Posts: 3 Rep Power: 9 Actually, (dui/dxj + duj/dxi)*dui/dxj = (dui/dxj)² + duj/dxi*dui/dxj Now, you have two terms in which the indexes repeat themselves. Therefore, there is a summation on both i and j for each terms After a few step, you will get the RHS from the LHS in your first post

 July 4, 2010, 18:27 #5 Member   bearcat Join Date: Jul 2009 Location: Ohio Posts: 35 Rep Power: 10 yes, Guillaume, you're right from your direction. On the other hand, isn't that (not fully written because of Latex limit here) and They are all 3x3 matrices, how can their product be a scalar? Anything wrong with the above expressions? Last edited by bearcat; July 4, 2010 at 19:00.

 July 5, 2010, 03:45 #6 New Member   Guillaume Fournier Join Date: Jun 2010 Posts: 3 Rep Power: 9 I think you're right in the way you wrote the expressions of and However, you're mistaken in the fact that is not only a product of matrices when using Einstein's convention: you have to sum over i and j, which corresponds to computing the trace of the matrix. Let me show you. I won't develop the Stress tensor, and I will call each of its components S11, S12.... Let's start with the Einstein's summation convention That will eventually lead to the RHS term you wrote in your first message Without the convention, it would be a classical matrix product between two 3x3 matrices, yielding a 3x3 matrix as well. If we call it M, we have and so on for the next lines. We can notice that the RHS term only contains the sum of the diagonal terms of M. Therefore, if you prefer to use the matrix product rather than the Einstein convention to compute and develop your term, you just have to remember that this convention is equivalent to the computation of the trace of the resultant matrix Here we are. I hope it will help a little Cheers Guillaume Last edited by Guillaume_Fournier; July 5, 2010 at 04:42.

July 5, 2010, 13:34
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bearcat
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Thank you for your explanation. This is very helpful discussion.

Quote:
 Originally Posted by Guillaume_Fournier I think you're right in the way you wrote the expressions of and However, you're mistaken in the fact that is not only a product of matrices when using Einstein's convention: you have to sum over i and j, which corresponds to computing the trace of the matrix. Let me show you. I won't develop the Stress tensor, and I will call each of its components S11, S12.... Let's start with the Einstein's summation convention That will eventually lead to the RHS term you wrote in your first message Without the convention, it would be a classical matrix product between two 3x3 matrices, yielding a 3x3 matrix as well. If we call it M, we have and so on for the next lines. We can notice that the RHS term only contains the sum of the diagonal terms of M. Therefore, if you prefer to use the matrix product rather than the Einstein convention to compute and develop your term, you just have to remember that this convention is equivalent to the computation of the trace of the resultant matrix Here we are. I hope it will help a little Cheers Guillaume

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