Turbulent flow mean Reynolds-stress closure

The k,E-model is based on a first order turbulence model closure according to Boussinesq. In analogy to laminar flows, the Reynolds stresses are assumed to be proportional to the gradients of the mean velocities. Transport equations for the turbulent kinetic energy and the turbulent dissipation are developed from the Navier-Stokes equations assuming an isotropic turbulence. The implementation of this model and the parameters used can be found in [10],... 

The averaging process creates a new set of variables, the so-called Reynolds stresses, which are dependent on the averages of products of the velocity fluctuations UjUj (which for i = j simply represent the standard deviations of the velocity components). This creates a closure problem, which is one of the fundamental issues that has to be addressed in the modeling of turbulent flows. Importantly, Equation 3.2.12 also indicates that the Reynolds stress terms, which in line with Taylor s fundamental result should be related to the dispersion parameters, are coupled to the gradients of the mean flow velocity. 

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