Turbulence eddy viscosity hypothesis

A large proportion of the models of Reynolds stress use an eddy viscosity hypothesis based on an analogy between molecular and turbulent motions. Accordingly, turbulent eddies are visualized as molecules, colliding and exchanging momentum and obeying laws similar to the kinetic theory of gases. This allows the description of Reynolds stresses ... 

By formulating a proper turbulence model, the total number of equations involved equals the number of unknown variables, and a closed set of model equations is obtained. That is, adopting the eddy viscosity hypothesis we basically need to find proper expressions estimating the eddy viscosity variable. If the term containing the turbulent normal stresses is not neglected in equation (1.380), we need to determine the k variable too. The complexity of such model formulations may vary a lot, thus only a few of the most popular ones will be described below. 

The first-order closure models are all based on the Boussinesq hypothesis [19, 20] parameterizing the Reynolds stresses. Therefore, for fully developed turbulent bulk flow, i.e., flows far away from any solid boundaries, the turbulent kinetic energy production term is modeled based on the generalized eddy viscosity hypothesis , defined by (1.380). The modeled fc-equation is... 

This equation possesses production and dissipation terms that are similar to those in the kinetic energy transport equation, except that they are divided by the turbulence time scale of the energy containing eddies, Tf = K As for the fc-equation, the Rejmolds stresses are parameterized based on the eddy viscosity hypothesis. 

By use of the eddy viscosity hypothesis (1.380) and the Prandtl mixing length model (1.356), a similar expression for the turbulent viscosity can be deduced, and given by... 

A non-isotropic version of the eddy viscosity hypothesis is sometimes used in situations where such effects are essential and the particular flow in question is expected to be far from isotropic [93]. That is, when the turbulent kinetic energy is calculated, the Reynolds stresses can be approximated by ... 

The production of turbulent kinetic energy is modeled with the aid of the eddy viscosity hypothesis ... 

When the concentration profile is fully developed, the mass-transfer rate becomes independent of the transfer length. Spalding (S20a) has given a theory of turbulent convective transfer based on the hypothesis that profiles of velocity, total (molecular plus eddy) viscosity, and total diffusivity possess a universal character. In that case the transfer rate k + can be written in terms of a single universal function of the transfer length L and fluid properties (expressed as a molecular and a turbulent Schmidt number) ... 

According to the ideas of Boussinesq [19] [20], the first turbulent closures was based on the gradient hypothesis and the coefficient of eddy viscosity. 

According to the ideas of Boussinesq [19, 20], the first turbulent closures was based on the gradient hypothesis and the coeflflcient of eddy viscosity. The eddy concept was thus introduced. An eddy still eludes precise definition, but in one interpretation it is conceived to be a turbulent motion, localized within a region of a certain size, that is at least moderately coherent over this region. The region of a large eddy can also contain smaller eddies. 

Within the equilibrium range (also referred to as the inertia subrange) in turbulent flow, eddy size is related to the kinematic viscosity, and local energy dissipation by the Kohnogoroff hypothesis is given in... 

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