Turbulent dissipation rate model

As mentioned before in Eq. (3), the most common source of SGS phenomena is turbulence due to the Reynolds number of the flow. It is thus important to understand what the principal length and time scales in turbulent flow are, and how they depend on Reynolds number. In a CFD code, a turbulence model will provide the local values of the turbulent kinetic energy k and the turbulent dissipation rate s. These quantities, combined with the kinematic viscosity of the fluid v, define the length and time scales given in Table I. Moreover, they define the local turbulent Reynolds number ReL also given in the table. 

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. 

Thus, like the turbulence dissipation rate, the scalar dissipation rate of an inert scalar is primarily determined by the rate at which spectral energy enters the scalar dissipation range. Most engineering models for the scalar dissipation rate attempt to describe (kd, t) in terms of one-point turbulence statistics. We look at some of these models in Chapter 4. 

At the next level of complexity, a second transport equation is introduced, which effectively removes the need to fix the mixing length. The most widely used two-equation model is the k-e model wherein a transport equation for the turbulent dissipation rate is... 

One common difficulty when applying the E-model is the need to know the turbulent dissipation rate e for the flow. Moreover, because e will have an inhomogeneous distribution in most chemical reactors, the problem of finding e a priori is non-trivial. In most... 

Models conditioned on the turbulent dissipation rate attempt to describe non-stationary effects due to the fluctuating strain-rate field, and thus should be adequate for flamelet applications which require a model for the mixture-fraction dissipation rate at the stoichiometric surface. 

Sanders, J. P. H. and I. Gokalp (1998). Scalar dissipation rate modelling in variable density turbulent axisymmetric jets and diffusion flames. Physics of Fluids 10, 938-948. 

Studies on thermodynamic restrictions on turbulence modeling show that the kinetic energy equation in a turbulent flow is a direct consequence of the first law of thermodynamics, and the turbulent dissipation rate is a thermodynamic internal variable. The principle of entropy generation, expressed in terms of the Clausius-Duhem and the Clausius-Planck inequalities, imposes restrictions on turbulence modeling. On the other hand, the turbulent dissipation rate as a thermodynamic internal variable ensmes that the mean internal dissipation will be positive and the thermodynamic modeling will be meaningful. 

These relationships are valid for isolated bubbles moving under laminar flow conditions. In the case of turbulent flow, the effect of turbulent eddies impinging on the bubble surface is to increase the drag forces. This is typically accounted for by introducing an effective fluid viscosity (rather than the molecular viscosity of the continuous phase, yUf) defined as pi.eff = Pi + C pts, where ef is the turbulence-dissipation rate in the fluid phase and Cl is a constant that is usually taken equal to 0.02. This effective viscosity, which is used for the calculation of the bubble/particle Reynolds number (Bakker van den Akker, 1994), accounts for the turbulent reduction of slip due to the increased momentum transport around the bubble, which is in turn related to the ratio of bubble size and turbulence length scale. However, the reader is reminded that the mesoscale model does not include macroscale turbulence and, hence, using an effective viscosity that is based on the macroscale turbulence is not appropriate. 

The primary task of modelling two-phase reaction systems is the estimation of the average diameter of droplets (bubbles and so on) of a dispersed phase and their size distribution in fast interface processes in diffuser-confusor devices. According to Kholmogorov s theory of isotropic turbulence, the specific kinetic energy of turbulence dissipation rates e are limiting in this case. 

The description is based on the previously defined single-particle (Lagrangian) or one-point (Eulerian) joint velocity-composition (micro-)PDF, /(r,yr). As mentioned in Section 12.4.1, in the one-point description no information on the local velocity and scalar (species concentrations, temperature,. ..) gradients and on the frequency or length scale of the fluctuations is included and the related terms require closure models. The scalar dissipation rate model has to relate the micro-mixing time to the turbulence field (see (12.2-3)), either directly or via a transport equation for the turbulence dissipation rate e. A major advantage is that the reaction rate is a point value and its behavior and mean are described exactly by a one-point PDF, even for arbitrarily complex and nonlinear reaction kinetics. 

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. 

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 ... 

Applying a turbulent energy cascade model, Ertesvag and Magnussen [2000] derived expressions for nY and ft in terms of the turbulent kinetic energy k and the turbulence dissipation rate s ... 

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