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. 

The unknown Reynolds stress term can be calculated using the eddy viscosity hypothesis (1.380), or, alternatively, the term in the bracket on the RHS of this relation can be recognized as being approximately equal to the Reynolds stress ... 

On the LHS, the production term is modeled in accordance with the eddy viscosity hypothesis (1.380), hence... 

The corresponding dissipation rate term is approximated using (1.405), the Prandtl-Kolmogorov relation (1.403), the eddy viscosity hypothesis (1.380), and the non-equilibrium boundary layer shear stress approximation (1.441) ... 

For LES performed in physical space, the basic sub-grid stress model is the eddy-viscosity model proposed by Smagorinsky . The Smagorinsky model is based on the gradient transport hypothesis and the sub-grid viscosity concept, just as the Reynolds stress models based on the Boussinesq eddy viscosity hypothesis, and expressed as ... 

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 ... 

All the average model formulations, except the Reynolds Stress Model, are based on the eddy viscosity hypothesis in what is often known as the generalizedBoussinesq hypothesis [19, 20] ... 

By use of the eddy viscosity hypothesis (1.386) and the Prandtl mixing length model (1.362), a similar expression for the turbulent viscosity can be deduced, and given by ... 

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