34 Turbulence Models Based On Rans

A turbulence model is a set of equations which express relations between unknown terms appearing in Reynolds-averaged governing equations with known quantities. Examination of Reynolds-averaged equations (Eqs (3.10) and (3.11)) reveals that there are four equations (one continuity and three momentum conservation equations) and thirteen unknowns (three mean velocities, mean pressure and nine Reynolds stresses). Similarly, for a general scalar variable, 0, there is one conservation equation (Eq. (3.13)) and four unknowns (mean value of general variable, (j>, and three turbulent fluxes u4>'). The desired turbulence model has to develop a relationship between these extra unknown fluxes and known mean variables. RANS-based turbulence models can be grouped into two classes: one which uses the concept of turbulent or eddy viscosity and another which does not. Models pertaining to these two classes are discussed below. It is not the purpose of this section to present all models in sufficient detail that they can be used without consulting the original references. Instead, the most widely used two-equation model, namely the k-e model will be described in sufficient detail to enable the reader to formulate a 'baseline' model and to appreciate major issues involved therein. Other models are briefly reviewed and key references are cited to assist user in selecting a turbulence model appropriate to the problem (objective) under consideration.

3.4.1. Eddy Viscosity Models

A large proportion of the models of Reynolds stress use an eddy viscosity hypothesis based on an analogy between molecular and turbulent motions. Accordingly, turbulent eddies are visualized as molecules, colliding and exchanging momentum and obeying laws similar to the kinetic theory of gases. This allows the description of Reynolds stresses:

Here, /T is referred to as turbulent or eddy viscosity, which, in contrast to molecular viscosity, is not a fluid property but depends on the local state of flow or turbulence. It is assumed to be a scalar and may vary significantly within the flow domain. k is the turbulent kinetic energy (normal turbulent stresses) and can be expressed as k = 1 uiui (3.15)

Substitution of Eq. (3.14) in the Reynolds-averaged momentum conservation equations (Eqs (3.11)) leads to a closed set, provided the turbulent viscosity is known. The form of the Reynolds-averaged momentum equations remain identical to the form of the laminar momentum equations (Chapter 2 and Table 2.2) except that molecular viscosity is replaced by an effective viscosity, / eff:

By analogy with the kinetic theory of gases, turbulent viscosity may be related to the characteristic velocity and length scales of turbulence (uT and lT respectively):

The turbulence models then attempt to devise suitable methods/equations to estimate these characteristic length and velocity scales to close the set of equations.

Several different models have been developed. Excellent reviews describing the relative merits and demerits of models pertaining to this class are available (Launder and Spalding, 1972; Rodi, 1984; Markatos, 1986; Nallaswamy, 1987). Some salient features, which will provide basic information and guidelines, are discussed here. Most simple models, called zero equation models, estimate characteristic length and velocity scales by algebraic equations. Prandtl (1925) proposed a mixing length hypothesis for two-dimensional boundary layer flows which relates turbulent viscosity to velocity gradient:

This hypothesis works surprisingly well for many boundary layer flows. Prandtl suggested the estimation of characteristic length (mixing length) of turbulence (l) by postulating it to be proportional to the distance from the nearest wall. Several variations of this model and several empirical corrections have been proposed over the years to account for the effect of low Reynolds number, transitional regime, mass transfer, pressure gradient, transverse curvature and three-dimensional flows (for example, see Cebeci and Abbott, 1975). The spreading rates and profiles of velocity, temperature and concentrations of a variety of boundary layer flows can be predicted satisfactorily. However, it is necessary to change the values of model parameters for different flows. This lack of universality indicates that these types of model fail to describe some important features of real flows. The mixing length hypothesis implies that the generation and dissipation of turbulence energy are in equilibrium everywhere. The role of convection and diffusion is ignored in these models. For most internal flows which are of interest to reactor engineers, it may be difficult to obtain satisfactory results using this class of models.

For such flows it is necessary to devise a model which relies on using partial differential equations for estimating both length scale and velocity scales of turbulence (two-equation model). Several such models have been proposed (Launder and Spalding, 1972; Rodi, 1984; Wilcox, 1993). Two-equation turbulence models are the simplest ones that promise success for flows in which length scales cannot be prescribed empirically and are, in general, the recommended first choice for simulating internal turbulent flows. There are several different two-equation models proposed in the literature. All of these models employ a modeled form of turbulent kinetic energy (modeling of the gradient diffusion term may, however, be different). The choice of the second model transport equation, from which the length scale is determined, is the main differentiating factor among these models. Instead of the length scale itself, generally a combination of turbulent kinetic energy, k and length scale, lT, having the form:

is chosen as dependent variable, z. Some of the popular forms of z are:

• turbulence frequency of energy containing eddies = f (m = 1/2; n = -1)

• time averaged square of the vorticity fluctuations = W (m = 1; n = -2)

• turbulent energy dissipation rate = e (m = 3/2; n =-1)

• product of energy and length scales = kl (m = 1; n = 1)

The modeled transport equations for z differ mainly in the diffusion and secondary source term. Launder and Spalding (1972) and Chambers and Wilcox (1977) discuss the differences and similarities in more detail. The variable, z = e is generally preferred since it does not require a secondary source, and a simple gradient diffusion hypothesis is fairly good for the diffusion (Launder and Spalding, 1974; Rodi, 1984). The turbulent Prandtl number for e has a reasonable value of 1.3, which fits the experimental data for the spread of various quantities at locations far from the walls, without modification of any constants. Because of these factors, the k-e model of turbulence has been the most extensively studied and used and is recommended as a baseline model for typical internal flows encountered by reactor engineers.

In the k-e model of turbulence, turbulent viscosity is related to k and e by the following equation:

Cap k2

where CIX is an empirical coefficient. Therefore, in order to close the set of equations, it is necessary to obtain values for k and e. Local values of k and e can be obtained by solving their transport equations. Exact transport equations for k and e can be derived from the Navier-Stokes equations (see, for example, Ranade, 1988 for such a derivation). Without going into details of exact transport equations, various terms appearing in exact equations of k and e are interpreted by classifying them into four groups: convective transport, diffusive transport, generation and dissipation. Diffusive transport comprises a molecular as well as turbulent component. Velocity and pressure fluctuations contribute to the diffusive transport of turbulent kinetic energy as well as energy dissipation rates and are usually modeled using a gradient diffusion approximation. The turbulent diffusivity of k and e are related to turbulent viscosity with additional empirical constants, which are known as turbulent Prandtl numbers for k and e. Turbulent kinetic energy is generated by extracting energy from the mean flow, and the terms representing this are also modeled using the assumption of turbulent viscosity. The generation term in the transport equation for e represents vortex stretching by mean flow and fluctuating flow. The dissipation term in the transport equation for k is simply equal to e. The viscous dissipation term appearing in the equation for e cannot be modeled separately. However, the difference between the generation and dissipation term can be modeled with the help of two additional empirical constants. The modeled form of transport equations for k and e can be written (Launder and Spalding, 1972):

d t d xt d xt \ ae dxi J k where G is the turbulence generation term given by:

These transport equations contain four empirical parameters, which are listed in Table 3.1 along with the parameter appearing in Eq. (3.20). The values of these parameters are obtained with the help of experimental information about simple flows such as decay of turbulence behind the grid (Launder and Spalding, 1972). Before discussing the modifications to the standard k- model and its recent renormalization group version, it will be useful to summarize implicit and explicit assumptions underlying the k- model:

• Turbulence is nearly homogeneous.

• The spectral distributions of turbulent quantities are similar.

• Diffusion is of the gradient type with constant effective Prandtl numbers.

• High Reynolds numbers.

It must be remembered that since all the assumptions may not be valid for flows of practical interest, the model parameters are not truly universal but are functions of characteristic flow parameters. Several attempts have been made to enhance the applicability of the k- model by modifying these empirical parameters to suit the specific requirements of different types of flow. One of the weaknesses of the standard

TABLE 3.1 Parameters of the k-e Model

Sr. No Parameter Standard k-e RNG k-e cd ci

0 0

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