— (pÜ) + V ■ (pÜÜ + pUU) = -Vp -V ■ Ü + pg + F (3.11) d t where the overbar indicates a time-averaged value. u is the fluctuating velocity. The terms appearing in Eq. (3.11) resemble those in Eq. (2.5) except for an additional term appearing on the left-hand side. These extra terms act as apparent stresses due to turbulent motions and are called Reynolds stresses or turbulent stresses and defined as:

The Reynolds-averaged form of conservation equation for a general variable <p can be written as:

d t where the additional term appearing on the left-hand side represents turbulent transport of 0.

In the Reynolds averaging approach, it is not necessary to resolve all the small-scale (spatial and temporal) phenomena since the variation of time-averaged quantities occurs at much larger scales (Fig. 3.4). This approach, therefore, requires much less computing resources than the LES or DNS approaches. However, not resolving the small-scale phenomena comes with an inherent problem, the so-called 'closure' problem. Time averaging of the basic governing equations of flow processes leads to the appearance of new terms in the governing equations, which can be interpreted as 'apparent' stress gradients and heat and mass fluxes associated with the turbulent motion. In principle, governing equations for these new terms can be derived, however, these equations contain further new unknown terms. It becomes, therefore, necessary to introduce a 'turbulence model', which relates the new unknown terms to known terms in order to close the set of governing equations. The process of closing the set of equations through a 'turbulence model' introduces some approximations and assumptions, which are discussed in the following subsection. Generally, for most engineering applications, averaged equations are 'closed' by employing first-order or second-order closures. The cost-to-benefit ratio for employing higher than second-order closures is generally not favorable.

Before discussing RANS-based models, which are commonly used for engineering simulations, it will be worthwhile here to examine the relationship between the three main approaches (DNS, LES and RANS). DNS results can be used to test and develop better sub-grid scale models, which can be used with LES simulations. Implications of the assumptions and approximations employed in RANS modeling can be evaluated using large eddy simulations. With advances in the development of massively parallel computing platforms and efficient computational schemes (for example, lattice Boltzmann methods introduced by Frisch et al., 1986), more and more attempts are being made to employ LES or DNS to flow processes relevant to engineers. In recent years, attempts have been made to develop a consistent modeling framework which can switch over from RANS to LES and then to DNS continuously with increases in scale resolution employed in the computational model (refer to Speziale (1998) for a more detailed discussion). These studies should be used to understand the potential and the limits of RANS-based models. Judicious analysis and engineering creativity is essential to construct computational models to simulate complex industrial engineering flow processes. Some of the key models used to simulate turbulence with RANS equations are discussed below.

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