Turbulence model Boussinesq hypothesis

The velocities and other solution variables are now represented by Reynolds-averaged values, and the effects of turbulence are represented by the Reynolds stresses, (—pu pTl) that are modeled by the Boussinesq hypothesis ... 

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. 

In the previous section, stability criteria were obtained for gas-hquid bubble columns, gas-solid fluidized beds, liquid-sohd fluidized beds, and three-phase fluidized beds. Before we begin the review of previous work, let us summarize the parameters that are important for the fluid mechanical description of multiphase systems. The first and foremost is the dispersion coefficient. During the derivation of equations of continuity and motion for multiphase turbulent dispersions, correlation terms such as esv appeared [Eqs. (3) and (10)]. These terms were modeled according to the Boussinesq hypothesis [Eq. (4)], and thus the dispersion coefficients for the sohd phase and hquid phase appear in the final forms of equation of continuity and motion [Eqs. (5), (6), (14), and (15)]. However, for the creeping flow regime, the dispersion term is obviously not important. 

Although numerous turbulence models are reported in the literature,1113 by far the most popular is the two-equation k-e model, first proposed by Jones and Launder.14 In this model, the turbulent stresses are recast in a form similar to the molecular stress tensor with mean velocity gradients, an assumption generally known as the Boussinesq hypothesis ... 

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. 

The third term of Eq. (3) contains eji, which is generally modeled in terms of turbulent dispersion in a manner analogous to the well-known gradient hypothesis of Boussinesq, as proportional to the gradient of holdup in the z direction, the constant of proportionality being referred to as the turbulent dispersion coefficient ... 

Although the Boussinesq s hypothesis does not determine a complete model for turbulence, its importance lies in the fact that it provides a relationship between the apriori unknown Reynolds transport terms and the mean flow field variables. To close the model it is necessary to specify the turbulent coefficients in terms of known quantities. 

The covariance is modeled by use of the gradient hypothesis and the Boussinesq turbulent transport coefficient concept ... 

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