Turbulence Boussinesq turbulent viscosity

It is then assumed that due to this separation in scales, the so-called subgrid scale (SGS) modeling is largely geometry independent because of the universal behavior of turbulence at the small scales. The SGS eddies are therefore more close to the ideal concept of isotropy (according to which the intensity of the fluctuations and their length scale are independent of direction) and, hence, are more susceptible to the application of Boussinesq s concept of turbulent viscosity (see page 163). 

Usually, however, the stresses are modeled with the help of a single turbulent viscosity coefficient that presumes isotropic turbulent transport. In the RANS-approach, a turbulent or eddy viscosity coefficient, vt, covers the momentum transport by the full spectrum of turbulent scales (eddies). Frisch (1995) recollects that as early as 1870 Boussinesq stressed turbulence greatly increases viscosity and proposed an expression for the eddy viscosity. The eventual set of equations runs as... 

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. 

To render Eq. (5.62) solvable, it is necessary to provide an expression for the turbulent Reynolds stress. For isotropic turbulent flows, similar to the transport processes in laminar flows, a scalar turbulent viscosity /xT is defined using the Boussinesq formulation... 

As the Reynolds stress has to disappear approaching the wall, the turbulent viscosity cannot be constant. Flows adjacent to the wall cannot be described by Boussinesq s rule, when et = const is presumed. However for flows like those which occur in turbulent free jets, the assumption of constant turbulent viscosity is highly suitable. Corresponding to the Boussinesq rule a turbulent thermal diffusivitf at (SI units m2/s) is introduced, through... 

Prandtl s model derivation can then be briefly sketched, introducing the Boussinesq [19] [20] approximation for the turbulent viscosity. Starting out with the simple kinetic theory relation that the molecular viscosity equals the molecular velocity times the mean free path, an analogous relation can be formulated for the turbulent viscosity in terms of the turbulent mixing length and a suitable velocity scale, Ut Iv. ... 

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. 

In the context of reactor modeling, it is important to notice that this model rely on the Boussinesq eddy-viscosity concept which is based on the assump)-tion that turbulence is isotropic. This means that the normal Re3molds stresses are considered equal and that the eddy viscosity is approximately isotropic. Therefore, the k-e model cannot reproduce secondary flows which arise due to unequal normal Re3molds stresses. Unfortunately, the non-isotropic effects... 

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. 

Two traditional approaches to the closure of the Reynolds equation are outlined below. These approaches are based on Boussinesq s model of turbulent viscosity completed by Prandtl s or von Karman s hypotheses [276, 427]. For simplicity, we confine our consideration to the case of simple shear flow, where the transverse coordinate Y = Xi is measured from the wall (the results are also applicable to turbulent boundary layers). According to Boussinesq s model, the only nonzero component of the Reynolds turbulent shear stress tensor and the divergence of this tensor are defined as... 

The turbulent stresses, Xk, in the momentum equations for the A -phase might be calculated by using the Boussinesq turbulent-viscosity model [8] for both phases or by applying a model of a Newtonian fluid for the gas phase and a granular shear stress for the solid phase [19]. 

The basic idea of RANS models is to account for the change in the fluid transport properties by introducing an eddy viscosity, I, also called turbulence viscosity, which relates the Reynolds stress tensor R to the fluid deformation. Such a relationship was first proposed by Boussinesq in the nineteenth century. More formally, this Boussinesq assumption can be written as... 

In order to close the system of equations, hypotheses or models for the turbulent viscosity are required. Boussinesq [9] was one of the first to develop a model for the Reynolds tensor, introducing the concept of turbulent viscosity. 

Boussinesq assumed that the Reynolds tensor has a behavior similar to the viscous stress, which is proportional to the gradient of velocity multiplied by a coefficient of proportionality, called the turbulent viscosity,... 

In Boussinesq s original concept, the eddy viscosity was assumed to be spatially constant. However, this assumption is valid only if the turbulent flow field is homogeneous, which is not frequent. For general cases, it is necessary to consider the variation of turbulent viscosity along the flow. 

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. 

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. 

Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simphfied here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity... 

In some way, introducing an increased particle drag by means of Eq. (17) resembles the earlier proposal raised by Bakker and Van den Akker (1994b) to increase viscosity in the particle Reynolds number due to turbulence (in agreement with the very old conclusion due to Boussinesq, see Frisch, 1995) with the view of increasing the particle drag coefficient and eventually the bubble holdup in the vessel. Lane et al. (2000) compared the two approaches for an aerated stirred vessel and found neither proposal to yield a correct spatial gas distribution. 

A variety of statistical models are available for predictions of multiphase turbulent flows [85]. A large number of the application oriented investigations are based on the Eulerian description utilizing turbulence closures for both the dispersed and the carrier phases. The closure schemes for the carrier phase are mostly limited to Boussinesq type approximations in conjunction with modified forms of the conventional k-e model [87]. The models for the dispersed phase are typically via the Hinze-Tchen algebraic relation [88] which relates the eddy viscosity of the dispersed phase to that of the carrier phase. While the simplicity of this model has promoted its use, its nonuniversality has been widely recognized [88]. 

Axial dispersion. An axial (longitudinal) dispersion coefficient may be defined by analogy with Boussinesq s concept of eddy viscosity ". Thus both molecular diffusion and eddy diffusion due to local turbulence contribute to the overall dispersion coefficient or effective diffusivity in the direction of flow for the bed of solid. The moles of fluid per unit area and unit time an element of length 8z entering by longitudinal diffusion will be - D L (dY/dz)t, where D L is now the dispersion coefficient in the axial direction and has units ML T- (since the concentration gradient has units NM L ). The amount leaving the element will be -D l (dY/dz)2 + S2. The material balance equation will therefore be ... 

All of these models require some form of empirical input information, which implies that they are not general applicable to any type of turbulent flow problem. However, in general it can be stated that the most complex models such as the ASM and RSM models offer the greatest predictive power. Many of the older turbulence models are based on Boussinesq s (1877) eddy-viscosity concept, which assumes that, in analogy with the viscous stresses in laminar flows, the Reynolds stresses are proportional to the gradients of the time-averaged velocity components ... 

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

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... 

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