Turbulence Boussinesq turbulent viscosity hypothesis

To proceed we need to put up dimensionless relations for the heat and mass transfer fluxes in the turbulent boundary layer using a procedure analoguous to the one applied for the momentum flux (5.249) in which the Boussinesq s turbulent viscosity hypothesis is involved. 

Provided that this hypothesis holds the heat and mass transfer rates can be estimated from the rate of momentum transport. It is noted that Sideman and Pinezewski [111], among others, have examined this hypothesis in further details and concluded that there are numerous requirements that need to be fulfilled to achieve similarity between the momentum, heat and mass transfer fluxes. On the other hand, there are apparently fewer restrictions necessary to obtain similarity between heat and low-flux mass transfer. This observation has lead to the suggestion that empirical parameterizations developed for mass transfer could be applied to heat transfer studies simply by replacing the Schmidt number Sct = ) by the Prandtl number (Pr, = and visa versa. To proceed we need to put up dimensionless relations for the heat and mass transfer fluxes in the turbulent boundary layer using a procedure analoguous to the one applied for the momentum flux (5.229) in which the Boussinesq s turbulent viscosity hypothesis is involved. 

The Reynolds-averaged approach is widely used for engineering calculations, and typically includes models such as Spalart-Allmaras, k-e and its variants, k-co, and the Reynolds stress model (RSM). The Boussinesq hypothesis, which assumes pt to be an isotropic scalar quantity, is used in the Spalart-Allmaras model, the k-s models, and the k-co models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, fit. For the Spalart-Allmaras model, one additional transport equation representing turbulent viscosity is solved. In the case of the k-e and k-co models, two additional transport equations for the turbulence kinetic energy, k, and either the turbulence dissipation rate, s, or the specific dissipation rate, co, are solved, and pt is computed as a function of k and either e or co. Alternatively, in the RSM approach, transport equations can be solved for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (usually for s) is also required. This means that seven additional transport equations must be solved in 3D flows. 

Note that the turbulent viscosity parameter has an empirical origin. In connection with a qualitative analysis of pressure drop measurements Boussinesq [19] introduced some apparent internal friction forces, which were assumed to be proportional to the strain rate ([20], p 8), to fit the data. To explain these observations Boussinesq proceeded to derive the same basic equations of motion as had others before him, but he specifically considered the molecular viscosity coefficient to be a function of the state of flow and not only on the system properties [135]. It follows that the turbulent viscosity concept (frequently referred to as the Boussinesq hypothesis in the CFD literature) represents an empirical first attempt to account for turbulence effects by increasing the viscosity coefficient in an empirical manner fitting experimental data. Moreover, at the time Boussinesq [19] [20] was apparently not aware of the Reynolds averaging procedure that was published 18 years after the first report by Boussinesq [19] on the apparent viscosity parameter. 

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... 

Turbulent viscosity based models start from the Boussinesq hypothesis [1877] relating the Reynolds stresses to the mean velocity gradients, the turbulent kinetic energy and the turbulent viscosity ix. ... 

To summarize the solution process for the k-e model, transport equations are solved for the turbulent kinetic energy and dissipation rate. The solutions for k and 8 are used to compute the turbulent viscosity, Xf Using the results for Xt and k, the Reynolds stresses can be computed from the Boussinesq hypothesis for substitution into the momentum equations. Once the momentum equations have been solved, the new velocity components are used to update the turbulence generation term, Gk, and the process is repeated. 

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. 

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