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k-e model is that it overpredicts turbulence generation in regions where the mean flow is highly accelerated or decelerated. Kato and Launder (1993) proposed a modified k-e model to overcome this problem. It will not be possible to discuss all the proposed modifications of the k-e model here. Launder and Spalding (1972), Rodi (1984), Markatos (1986) and Nallaswamy (1987) discuss these modifications, among others. More discussion on the influence of compressibility and other issues can be found in Wilcox (1993).

In recent years, renormalization group (RNG) methods have been used to formulate two-equation turbulence models. These methods are a general framework for model building in which the complex dynamics is described in terms of so-called 'coarse-grained' equations governing the large-scale, long-time behavior. The basic idea applied to turbulence modeling is the elimination of small-scale eddies by employing RNG methods. As the small-scale eddies are removed, the effective viscosity of the system is increased. Through the scale elimination procedure, RNG theory develops an equation for effective viscosity and the corresponding transport equations of k and e (Yakhot and Orszag, 1986; Yakhot et al., 1992). The overall form of the model closely resembles the standard k-e model except for the values of the model parameters. The values of model parameters derived by RNG methods are also listed in Table 3.1. The main difference between the standard and RNG version lies in the equation of turbulent energy dissipation rate. In large strain rate flows, the RNG model predicts a lower turbulent viscosity (larger e and lower k) than the standard model. Although the RNG model has been shown to perform better than the standard model for flows with high streamline curvature, vortex shedding etc., it has not yet been validated as extensively as the standard k-e model.

The RNG version of the k-e model has been extended to employ a differential form of the equation for calculating effective viscosity from a knowledge of k and e (Fluent User Guide, Vol. 4, 1997):

This form allows extension to low Reynolds number and near wall flows, unlike the standard k-e model, which is valid only for fully turbulent flows. Despite such an extension, the standard and RNG versions of k-e models are normally valid for turbulent flows away from the walls. The presence of a wall alters turbulence in a nontrivial way, by damping turbulence in the region very close to the wall. At the outer part of the near-wall region, turbulence is rapidly generated due to the large gradient in mean velocity. Correct representation of the influence of walls on turbulent flows is an important aspect of simulating wall-bounded flows.

Numerous experiments have shown that the near-wall region can be divided into three layers (Fig. 3.5(a)).

(1) The innermost layer, called the viscous sub-layer in which flow is laminar-like and the molecular viscosity plays a dominant role;

(2) an intermediate buffer layer, where molecular viscosity and turbulence are equally important;

(3) the outer layer, called the fully turbulent layer, where turbulence plays a major role.

inner layer

inner layer

fully turbulent region or log-law region

Upper limit depends on Reynolds no.

lnUT y/v buffer layer or blending viscous sublayer region fully turbulent region or log-law region

Upper limit depends on Reynolds no.

lnUT y/v buffer & sublayer

FIGURE 3.5 Near wall flows. (a) Flow structure near wall, (b) wall function approach, (c) low Re model approach (adapted from FLUENT user guide, Vol. 1).

There are two main approaches to modeling the near-wall region. In one approach, the so-called 'wall function' approach, the viscosity-affected inner regions (viscous and buffer layers) are not modeled. Instead, semi-empirical formulae (wall functions) are used to bridge the viscosity-affected region between the wall and the fully turbulent region. In another approach, special, low Reynolds number turbulence models are developed to simulate the near-wall region flow. These two approaches are shown schematically in Fig. 3.5(b) and 3.5(c).

In most high Reynolds number flows, the wall function approach gives reasonable results without excessive demands on computational resources. It is especially useful for modeling turbulent flows in complex industrial reactors. This approach is, however, inadequate in situations where low Reynolds number effects are pervasive and the hypotheses underlying the wall functions are not valid. Such situations require the application of a low Reynolds number model to resolve near-wall flows. For the low Reynolds number version of k-e models, the following boundary conditions are used at the walls:

where Ut is the velocity component tangential to the wall and n is the co-ordinate normal to the wall. A number of low Reynolds number modifications to the k-e model have been proposed (Chen and Patel, 1988; Wilcox, 1993; Hrenya and Sinclair, 1995). These models are too numerous to discuss here. Instead, the wall function approach, which is commonly used in reactor engineering applications, is briefly discussed below.

In the wall functions approach, a universal velocity profile of the form shown below is assumed to exist near the wall:

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