## Dns

Hussaini et al., 1990). Extensive research efforts have been made to evaluate these SGS models. Two of the main problems identified with different SGS models are (Speziale, 1998):

(1) the inability of SGS models to respond to changes in the local state of the flow, resulting in the need to make ad hoc adjustments in the model parameters. Turbulent channel flow, isotropic turbulence and more general homogeneously strained turbulent flows require values of the Smagorinsky constant that can differ by more than a factor of two;

(2) the generally poor correlation of SGS models with DNS at lower turbulence Reynolds numbers. Even for the simple case of isotropic turbulence, the Smagorinsky model correlates only at about the 50% level. (Note, for example, that the correlation between functions y = x and y = e-x over the interval zero to one is more than 50% despite the fact that they are qualitatively different functions.)

Several attempts have been made to reduce or eliminate these problems. Development of SGS models is an active field of research and further details may be obtained by referring to Germano et al. (1991), Orszag et al. (1993) and Speziale (1998) among others.

The LES approach is capable of capturing time-dependent motions of large scale, which are averaged out in conventional turbulence models based on Reynolds-averaged equations. Small-scale motions exhibit more universal characteristics and, therefore, there is more hope of developing a generalized SGS model. Rigorous DNS simulations may assist such development. The LES approach has been used extensively to simulate turbulent flows with moderately high Reynolds numbers. More details of the LES approach and applications may be found in Reynolds (1990) and Ferziger (1995). More recently, attempts have also been made to apply LES to simulate more complex flows (for example, flow in stirred reactors, by Derksen and van den Akker, 1999). Despite these successes, LES is still computation intensive and suffers some of the disadvantages of DNS, such as difficulties in specifying boundary conditions and generating a huge amount of information, which may not be necessary for a reactor engineer. For steady state flows in particular, LES methods are still much more computation intensive than models based on Reynolds-averaged equations, which are discussed in the following sub-sections.

3.3.3. Reynolds-averaged Navier-Stokes Equations (RANS)

In this approach, an instantaneous value of any variable is decomposed into a mean, obtained by averaging over an appropriate time interval, and a fluctuating component:

The overbar denotes time averaging. The time averaged quantity is defined as:

The Reynolds averaging obeys the following property:

Equation (3.6) is substituted in the basic governing equations for 0 and these are then time averaged to yield the governing equations for mean quantities (using Eq. (3.8)). For compressible turbulent flows, terms appearing in the Reynolds-averaged momentum equations are difficult to categorize according to the acceleration of the mean motion and apparent stresses. In such cases, the use of mass weighted averaging leads to compact expressions. Mass weighted averaging, also called Favre averaging, is defined as:

ft+At p0 dt

For flows in which density fluctuations are negligible, the formulations become identical. Favre-averaged quantities are not easily comparable with experimentally measured quantities, which are normally non-weighted time averages. For most chemical reactor engineering applications (except maybe combustion processes), classical Reynolds averaging is suitable.

The Reynolds-averaged form of the conservation equations of mass (overall) and momentum for an incompressible (constant density) fluid can be written as:

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