How to derive this simple stress tensor calculation? -- CFD Online Discussion Forums

July 4, 2010, 12:33	How to derive this simple stress tensor calculation?	#1
bearcat Member bearcat Join Date: Jul 2009 Location: Ohio Posts: 35 Rep Power: 16	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? $LHS=S_{ij}\frac{\partial U_{i}}{\partial x_{j}}=(\frac{\partial U_{i}}{\partial x_{j}}+\frac{\partial U_{j}}{\partial x_{i}})\frac{\partial U_{i}}{\partial x_{j}}$ From other books, the LHS should equals RHS, which reads: $RHS=2[(\frac{\partial u}{\partial x})^{2}+(\frac{\partial v}{\partial y})^{2}+(\frac{\partial w}{\partial z})^{2}]+(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x})^{2}+( \frac{\partial u}{\partial z}+\frac{\partial w}{\partial x})^{2}+(\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z})^{2}$ 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.

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