Turbulent flow mean velocity field closure

Equations (16.56) and (16.57) provide a solution to the closure problem inasmuch as the turbulent fluxes have been related directly to the mean velocity and potential temperature. However, we have essentially exchanged our lack of knowledge of u u2 and u 2B for Km and Kt, respectively. In general, both Km and Kj are functions of location in the flow field and are different for transport in different coordinate directions. The variation of these coefficients, as we will see later, is determined by a combination of scaling arguments and experimental data. 

The shear stresses over the flow boundaries can be rigorously derived as an integral part of the solution of the flow field only in laminar flows. The need for closure laws arise already in single-phase, steady turbulent flows. The closure problem is resolved by resorting to semi-empirical models, which relate the characteristics of the turbulent flow field to the local mean velocity profile. These models are confronted with experiments, and the model parameters are determined from best fit procedure. For instance, the parameters of the well-known Blasius relations for the wall shear stresses in turbulent flows through conduits are obtained from correlating experimental data of pressure drop. Once established, these closure laws permit formal solution to the problem to be found without any additional information. 

G.Gouesbet, A.Berlemont. Prediction of turbulent fields. Including fluctuating velocity correlations and approximate spectra, by means of a simplified second order closure scheme the round free Jet and developped pipe flow. Second Int.Conf. on Numerical Methods In Laminar and Turbulent flow, Venice, Italy, pp 205-216, Plnerldge, Swansea, 1981. 

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