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where i is the unit tensor and G is the generation of turbulent kinetic energy (given by Eq. (3.23)). The values of two additional model parameters, Cia and C2a, are reported as 2.2 and 0.45, respectively. The k and e appearing in these equations can be obtained by solving the modeled transport equations of k and e (Eqs (3.21) and (3.22)). Alternatively, one may use Reynolds stresses calculated by the above equations to calculate the generation term, G in the transport equations of k and e. This practice will be more consistent and accurate. Note that Eq. (3.36) is an implicit equation for calculation of Reynolds stress since it appears on both sides of the equation. These implicit models can give rise to multiple solutions or singularities when solved iteratively. Therefore, several attempts have been made to develop explicit algebraic stress models (ASM) in recent years. For example, Gatski and Speziale (1993) developed a regularized, explicit algebraic stress model, which reduces to the k-e model in the limit of homogeneous turbulence in equilibrium. These explicit ASM avoid multiple solutions and can, therefore, be recommended for simulations of complex turbulent flows. Gatski and Speziale (1993) and Speziale (1998) may be referred to for details of model equations and further analysis.

Wherever the convective and diffusive transport of Reynolds stresses are important (flows that are far from equilibrium), algebraic stress models may prove to be inadequate, and solution of the full transport equations of the Reynolds stresses may become necessary. Extensive efforts have been made to model the terms appearing in the exact transport equations of Reynolds stresses. The redistributive term has been the subject of most controversy and experimentation. Several different models have been proposed (Daly and Harlow, 1970; Hanjalic and Launder, 1972; Launder et al., 1975; Reynolds and Cebeci, 1978; Launder, 1989). The current generation second-order closure models do not perform well when there are significant departures from equilibrium. It is a general feeling that it may require quite some time before these models are sufficiently well developed to perform better than simpler two-equation models (such as the k-e model) for engineering problems. Recently attempts have been made to develop second-order closures that are suitable for non-equilibrium flows based on a relaxation time approximation around the non-equilibrium extension of the explicit ASM (Speziale, 1998). Speziale (1998) described this approach to formulate a consistent framework to integrate RANS (two equations to full second-order closure), LES and DNS based on such models. Such an approach looks promising and deserves further research. The details of Reynolds stress models are not included here, since most of the engineering flows may be adequately analyzed using modifications of two-equation turbulence models. Details of RSM can be found in above cited papers and references cited therein. A brief summary of the advantages and disadvantages of two-equation models (k-e and RNG k-e) and Reynolds stress models (algebraic and differential) is given in Table 3.2.

Apart from the models discussed and mentioned here, there are some more miscellaneous attempts to describe specific aspects of turbulence by phenomenological modeling. For example, Professor Spalding and his group have proposed the use of 'two-fluid' models of turbulence to describe the 'fragmentaryness' and 'intermit-tancy' of turbulent flows. The approach has shown some successes in simulating key features of jets and other flows (Spalding, 1983; Malin and Spalding, 1984), however, it is not adequately developed to use as a general approach. For some situations encountered in chemical reactors, such as combustion, this approach may be useful (see a review by Markatos, 1986).

It is important for a reactor engineer to select an appropriate turbulence model for the application at hand from the available turbulence models. It is also important that implications of assumptions underlying these models are adequately understood. It should be kept in mind that a more complex model does not necessarily mean a better model. The reactor engineer has to constantly evaluate implications of the underlying assumptions and performance of the model in light of whatever direct and indirect validation one can carry out and in light of the fulfillment of the simulation objectives. Generally, the two-equation, k-e model can be recommended as a baseline model. When the interest is in simulating unsteady vortex shedding, the renormalization group version of the k-e model may be used. In any case, be it the RNG model or the Reynolds stress models, it is always useful to first carry out simulations using the standard k-e model. While doing this, a reactor engineer must, however, be careful in making appropriate corrections to the standard model to compensate for the known deficiencies of the standard model (such as for modelling axis-symmetric round jets). The predicted results of the two-equation model can then be used as an initial guess to start the simulations using more complex turbulence models. It must be noted that simulated results of many complex, industrial flow processes are more influenced by the employed grid resolution and discretization schemes than the underlying turbulence model. It is, therefore, necessary to clearly identify, monitor and control the influence of numerical issues on predicted results (these aspects are discussed in Chapters 6 and 7).

TABLE 3.2 Summary of Two-equation and Reynolds Stress Models

Model

Advantages

Disadvantages

Standard k-e • Simplest model to represent variation of turbulence length and velocity scales

• Robust and economical

• Excellent performance for many industrial flows

• The most widely validated model

RNG k-e

• Performs better than standard model for some:

— separated flows

— swirling flows

Assumes isotropic eddy viscosity

Not sufficiently validated so far

Algebraic Stress Models (ASM)

Accounts for anisotropy Combines generality of approach with the economy of the k-E model Good performance for isothermal and buoyant thin shear layers

Restricted to flows where convection and diffusion terms are negligible

Performs as poorly as k-e in some flows due to problems with e equation Not widely validated

Reynolds Stress Models (RSM)

Most general model of all classical turbulence models Performs well for many complex flows including non-circular ducts and curved flows

Computationally expensive (seven extra PDEs) Performs as poorly as k-e in some flows due to problems with e equation Not widely validated

### 3.4.3. Scalar Transport Models

Turbulence is often employed to enhance the rates of mixing and transport processes. A reactor engineer is therefore interested in finding out the extent of turbulence generated in a reactor and its influence on other transport processes. The discussion of turbulence modeling so far has been restricted to the modeling of momentum conservation. It is necessary to account for turbulence, while modeling the species and enthalpy conservation equations. Species and enthalpy conservation, for most reactor engineering flows, are modeled using eddy viscosity type models. The Reynolds-averaged equation for a general scalar variable <p is Eq. (3.13). The third term on the left-hand side containing the correlation of fluctuating velocity and fluctuating scalar variable requires further modeling. In writing the time-averaged equations, density fluctuations were assumed to be insignificant. The correlation appearing in this equation is usually modeled using the gradient transport assumption:

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