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Estimation of TKE from RANS analysis of Compressible flow with Spalart-Allmaras model

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Old   January 30, 2024, 14:15
Default Estimation of TKE from RANS analysis of Compressible flow with Spalart-Allmaras model
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I'm trying to estimate the TKE from RANS results of a compressible flow with the Spalart-Allmaras turbulence model. I am familiar with the estimation used in OpenFoam from Bourgoin, but this estimation is based on relationships from P Bradshaw, DH Ferriss, and NP Atwell. Calculation of boundary-layer development using the turbulent energy equation. Journal of Fluid Mechanics, which are for incompressible flow.

I am trying to come up with an estimate based on the Boussinesq hypothesis but I'm confused by the notation of the final term in the brackets. for say the xx Reynolds stress, the first term in the brackets will become 2 * par U / par X. The other term in brackets has the subscript k, so I'm not sure if it becomes the sum of the Y and Z normal strains (the ones that aren't i or j), or if it's the sum of all three components (X,Y and Z)
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Old   January 30, 2024, 17:24
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The second term inside brackets is the divergence of U. The repeated index k implies summation (a.k.a Einstein notation):

\frac{2}{3}\frac{\partial U_k}{\partial x_k}\delta_{ij} = \frac{2}{3}\left(\frac{\partial U_1}{\partial x_1} + \frac{\partial U_2}{\partial x_2} + \frac{\partial U_3}{\partial x_3} \right)\delta_{ij} = \frac{2}{3}\mathrm{div}(\vec U) \delta_{ij}
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boussinesq, compressible, reynolds stresses, turbulence, turbulent kinetic energy

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