Turbulence production (TP) is calculated by taking a product of Reynolds shear stress and velocity gradient, whereas, turbulent kinetic energy (TKE) is calculated by adding normal stress terms. How are TP and TKE related?
The TKE has is own equation defining the rate in which it changes in time (for statistically unsteady flows) as a consequence of convection, production, dissipation. The form of the equation depends also on the definition of TKE, you can build the equation from the total velocity field or by using the residual field.
When you see the definition of TKE involves the averaged fluctuations therefore their meaning is formally different if referred as to either RANS or LES variables.
Some description of the equations can be found in the book of Sagaut.
No, it can only quantify how much energy is produced by mean shear (i think it's also not the only production mechanism, but likely a dominant one). See Tennekes and Lumley "A first course in turbulence" that explains it very well.
I totally agree, the problem I see in incompressible flows is that the mechanical energy is just a consequente of continuiy and momenti equations, the only ones really solved...
I) Turbulence production term, would illustrate the source term (+ve, usually) in the TKE budget equation, that would contribute to the spatio-temporal (D/Dt) evolution of TKE. the production term, in tensorial form is -\overline{u'_uu_j}S_{ij}, so essentially the TKE evolution has contribution from normal stresses as well as Shear stresses. However, in a wall-bounded turbulence, -\overline{u'_uu_j}S_{ij} ~ -\overline{u'w'} \partial U/\partial y. Hence, the production of tke in wall bounded turbulence has a contribution from its interaction of mean shear and Reynolds stresses.
ii) So, I agree with Alex Liberzon , it can only quantify how much energy is produced by the mean shear, in wall-bounded turbulence for example. But, for grid-decaying anisotropic turbulence, normal stresses can contribute to the production of turbulence too.
Performing a scaling analysis, or analysis of TKE budget in the spectral space rather than the physical space, can give you interesting revelations for example, the region and scales where production approximately balances dissipation, where logarithmic trends in velocity moments are observable.
Finally, apart from the book which Alex suggested, another book with much details might be of interest: "Turbulent Flows" by Stephen B. Pope (specially, the chapter on Reynolds Stress models)
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