K

where k is the van Karmann constant (0.41), B is an empirical constant related to the thickness of the viscous sub-layer (B « 5.2 in a flat plate boundary layer) and u+ and y+ are defined as follows:

where nP is the normal distance of the considered node at point P from the wall (Fig. 3.5(b)). In addition, the flow is assumed to be in local equilibrium, which means that the production and dissipation terms are nearly equal. These assumptions then allow the use of coarse resolution at the wall. In fact, the wall function approach requires that the dimensionless distance of the adjacent grid node from the wall should be greater than 30 (y+ > 30). For such a case, wall shear stress can be related to the tangential component of the velocity at the grid, as:

For turbulent kinetic energy, k, the normal gradient at the wall is usually set to zero. By assuming the length scale near the wall, L to be given by:

k nP

and by assuming equilibrium between production and dissipation, the turbulent energy dissipation rate at the node adjacent to the wall (denoted by subscript P, located at a normal distance nP from the wall) can be computed without solving the transport equation for e as:

C3/4k3/2

k np

It should be noted that the wall function approach is valid only when the first grid point adjacent to the wall (node P) is within the logarithmic region. For separated flows within the recirculation region and separation and reattachment regions, this condition may not be valid. To rectify this to some extent, several alternative wall functions have been proposed (see, for example, Amano, 1984). Generally, to simulate complex industrial reactors, wall functions are applied everywhere and the regions for which these may not be valid are assumed to be small. When the wall function approach is not applicable over a large portion of the wall boundaries, low Reynolds number models of turbulence should be used to resolve the finer details of near-wall flows.

In addition to representing the influence of walls on turbulence, adequate boundary conditions need to be specified to solve modeled equations of k and e. At computational boundaries far from the wall, the following boundary conditions can be used (Ferziger and Peric, 1995):

• if the surrounding flow is turbulent:

dx dx k

For the inlet boundary conditions, it will be necessary to specify values of k and e. If available, experimental values should be used to set the inlet boundary conditions. If k is not known, it is generally estimated from a suitable guess of turbulence intensity (say 5% ) at the inlet. The value of e is usually estimated from a knowledge of k and assuming a characteristic length scale, L:

k3/2

The characteristic length used in the above equation may be taken as 0.07 times the equivalent pipe radius, in the absence of more information. If the Reynolds stress and mean velocities at the inlet are measured, e can be estimated using the assumption of local equilibrium. The numerical implementation of these boundary conditions and numerical solution of two-equation turbulence models is discussed in Chapter 6.

3.4.2. Reynolds Stress Models

The two-equation models (especially, the k-e model) discussed above have been used to simulate a wide range of complex turbulent flows with adequate accuracy, for many engineering applications. However, the k- model employs an isotropic description of turbulence and therefore may not be well suited to flows in which the anisotropy of turbulence significantly affects the mean flow. It is possible to encounter a boundary layer flow in which shear stress may vanish where the mean velocity gradient is nonzero and vice versa. This phenomenon cannot be predicted by the turbulent viscosity concept employed by the k-e model. In order to rectify this and some other limitations of eddy viscosity models, several models have been proposed to predict the turbulent or Reynolds stresses directly from their governing equations, without using the eddy viscosity concept.

The exact transport equations for turbulent stresses can be derived from the Navier-Stokes equations, following similar procedures to those employed to derive the transport equations for turbulent kinetic energy. These transport equations contain several unknown correlations including the triple correlations of fluctuating velocities. It is, in turn, possible to derive transport equations for these triple correlations, which will, however, contain fourth-order correlations and so on. In general, however, triple and higher correlations are small in practical flows and a reactor engineer may not need to simulate them by transport equations. Therefore, for most reactor engineering flows, second-order closure, that is closing the transport equations for turbulent stresses, will be sufficient. Second-order closure models can be further classified into algebraic stress models (ASM) and differential Reynolds stress models (RSM). The starting point for both of these is the exact transport equation of turbulent stresses (these models can also be developed by a relaxation time approximation around an equilibrium model, Saffman, 1977). In order to understand the physical significance of the various terms appearing in these transport equations, the case of 2D boundary layers is considered here to avoid unnecessary complications. Multiplying each momentum transport equation by a fluctuating velocity component in the other direction and performing time averaging leads to the following equation:

D u'v' —rd U d /-- p'u' \ p' id u' d v' \ I id u'd v' \

Analogous to the transport equation of turbulence kinetic energy, k, the first term on the right-hand side represents 'production or generation', the second represents 'diffusion' and the final term represents 'dissipation'. The third term of the right-hand side, which has no counterpart in the k equation, represents 'redistribution'. It is the correlation between fluctuations in pressure and velocity gradients, which results in enhancement of velocity fluctuations in one direction at the expense of those in the other directions. It is necessary to model these terms in order to close the set of equations.

Production terms do not need any modeling since all the terms appearing there are calculated from the corresponding transport equations. The diffusional transport is assumed to be proportional to the spatial gradient of the stress component. Dissipation is usually assumed to take place isotropically in each of the three normal stress components, and is assumed to be zero in shear stress components. The redistributive action of pressure fluctuations can be represented by two groups of terms, one involving products of Reynolds stress and (e/k), and the other containing products of stresses and mean velocity gradients (more discussion on precise details of these modeled terms can be found in Launder et al., 1975; Launder, 1989). It must be noted that the approximations of dissipative or redistributive terms do not contain gradients of stress terms. Developers of algebraic stress models use this fact and attempt to eliminate convective and diffusive terms from the transport equations of turbulent stresses so as to derive a set of algebraic equations among the Reynolds stresses, the turbulence kinetic energy, the energy dissipation rate and mean velocity gradients. These algebraic equations can be expressed symbolically as

The precise form of the function varies depending on the approximated forms of the dissipative and redistributive terms and how the convective and diffusive transport terms are eliminated. Launder (1971) neglected the latter terms entirely while Rodi (1984) assumed that the convective transport is proportional to the transport of k with an equivalent assumption for the diffusion term. Rodi's algebraic stress model can be written:

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