Turbulence models kinetic energy based

In this model, the velocity disturbance by the particle is from both the wake behind the particle (Rep > 20) and the vortex shedding (Rep > 400). Hence, the changes in the kinetic energy associated with the turbulence production are proportional to the difference between the squares of the two velocities and to the volume where the velocity disturbance originates. It is further assumed that the wake is half of a complete ellipsoid, with base diameter of dp (same as the particle diameter) and wake length of /w. Thus, the total energy production of the gas by the particle wake or vortex shedding is... 

As was the case with the full equations, these contain beside the three mean flow variables u, v, and T (the pressure is, of course, by virtue of Eq. (2.157) again determined by the external in viscid flow) additional terms arising as a result of the turbulence. Therefore, as previously discussed, in order to solve this set of equations, there must be an additional input of information, i.e., a turbulence model must be used. Many turbulence models are based on the turbulence kinetic energy equation that was previously derived. When the boundary layer assumptions are applied to this equation, it becomes ... 

The k,E-model is based on a first order turbulence model closure according to Boussinesq. In analogy to laminar flows, the Reynolds stresses are assumed to be proportional to the gradients of the mean velocities. Transport equations for the turbulent kinetic energy and the turbulent dissipation are developed from the Navier-Stokes equations assuming an isotropic turbulence. The implementation of this model and the parameters used can be found in [10],... 

Pacanowski and Philander, 1981 Peters et al., 1988). More sophisticated methods are based on prognostic equations for the turbulent kinetic energy k and a second quantity, which is either the dissipation rate e or a length scale in the turbulent flow see Burchard (2002) for a recent review and applications of two-equation turbulence closures for onedimensional water column models. A two-equation turbulent closure has been applied by Omstedt et al. (1983) and Svensson and Omstedt (1990) for the Baltic Sea surface boundary layer under special consideration of sea ice, whereas the application in three-dimensional circulation models is described by Burchard and Bolding (2002) and Meier et al. (2003). 

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... 

The two-phase k — e model analyzed was based on the Favre averaged transport equation for turbulent kinetic energy developed by [73, 74]. The resulting transport equation for kinetic energy is similar to the one obtained from the single phase model (5.2), supporting the semi-empirical modification introduced in that model. 

Based on these observations [93] proposed a modified model containing two time constants, one for the liquid shear induced turbulence and a second one for the bubble induced turbulence. The basic assumption made in this model development is that the shear-induced turbulent kinetic energy and the bubble-induced turbulent kinetic energy may be linearly superposed in accordance with the hypothesis of [128, 129]. Note, however, that [82] observed experimentally that this assumption is only valid for void fractions less than 1 %, whereas for higher values there is an amplification in the turbulence attributed to the interactions between the bubbles. The application of this model to the high void fraction flows occurring in operating multiphase chemical reactors like stirred tanks and bubble columns is thus questionable. 

To parameterize the new quantities occurring in these equations a few semi-empirical relations from the literature were adopted. The asymptotic value of bubble induced turbulent kinetic energy, fesia, is estimated based on the work of [3]. By use of the so-called cell model assumed valid for dilute dispersions, an average relation for the pseudo-turbulent stresses around a group of spheres in potential flow has been formulated. Prom this relation an expression for the turbulent normal stresses determining the asymptotic value for bubble Induced turbulent energy was derived ... 

In this section the application of multiphase flow theory to model the performance of fluidized bed reactors is outlined. A number of models for fluidized bed reactor flows have been established based on solving the average fundamental continuity, momentum and turbulent kinetic energy equations. The conventional granular flow theory for dense beds has been reviewed in chap 4. However, the majority of the papers published on this topic still focus on pure gas-particle flows, intending to develop closures that are able to predict the important flow phenomena observed analyzing experimental data. Very few attempts have been made to predict the performance of chemical reactive processes using this type of model. 

The theoretical description of the turbulent mixing of reactants in tubular devices is based on the following model assumptions the medium is a Newtonian incompressible medium, and the flow is axis-symmetrical and nontwisted turbulent flow can be described by the standard model [16], with such parameters as specific kinetic energy of turbulence K and the velocity of its dissipation e and the coefficient of turbulent diffusion is equal to the kinematic coefficient of turbulent viscosity D, = Vj- =... 

Thus, the study of two-phase flows in diffuser-confusor devices can provide us with reliable results, based on the interpenetrating continuum model (the Euler approach). The numerical solution of the partial derivatives of the differential equations in the C-e turbulence model, using the implicit integro-interpolation finite volume method, provides us with the following fields of functions for a diffuser-confusor reactor axial u and radial v rates for each of the phases pressure p volume fractions of continuous and dispersed phases specific kinetic energy of turbulence k and its dissipation s, as well as some other characteristics. 

In fact, the pitch of triangular flow channels is very small compared to the reactor radius, that is, the torsion tends to zero. (Liu, 1994) and (Bolinder, 1996) predicted a much weaker effect of a relatively small torsion on fluid flow in hehcal ducts. Then in this work, a deeply study of turbulent flow in the jacket with triangular hehcal duct based on the simplified physical model is presented. With numerical simulation method, velocity fields and turbulence kinetic energy of fully developed flow for different curvature ratio are obtained. The local coefficient of resistance on outer walls are... 

In the RANS-approach, turbulence or turbulent momentum transport models are required to calculate the Reynolds-stresses. This can be done starting from additional transport equations, the so-called Reynolds-stress models. Alternatively, the Reynolds-stresses can be modeled in terms of the mean values of the variables and the turbulent kinetic energy, the so-called turbulent viscosity based models. In either way, the turbulence dissipation rate has to be calculated also, as it contains essential information on the overall decay time of the velocity fluctuations. In what follows, the more popular models based on the turbulent viscosity are focused on. A detailed description of the Reynolds-stress models is given in Annex 12.5.l.A which can be downloaded from the Wiley web-page. 

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. ... 

The calculation of the six components of the Reynolds stress tensor, that is, six second-order moments of the micro-PDF, f v,yf), is reduced to the calculation of k and the modeling of the turbulent viscosity pf As seen from (12.5.1-2), is a function of a limited number of second-order moments of the micro-PDF. Turbulent viscosity based closure models for the Reynolds-stresses can be used at relatively low computational effort. In the two-equation model approach, the turbulent viscosity is expressed in terms of the turbulent kinetic energy, k, and the turbulence dissipation rate, s, according to ... 

Based on turbulent kinetic energy balance (k-equation) and a model for the rate of dissipation of mrbulent kinetic energy (s-equation) Variations have been developed to model subclasses of flows... 

Using turbulence models, this new system of equations can be closed. The most widely used turbulence model is the k-e model, which is based on an analogy of viscous and Reynolds stresses. Two additional transport equations for the turbulent kinetic energy k and the turbulent energy dissipation e describe the influence of turbulence... 

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