## Jd

Several filters, such as Gaussian or piecewise continuous, are used. The filter function, G becomes a Dirac delta function in the limit of A tending to zero. Thus, direct simulations are recovered in the fine mesh limit. Such a filtering operation substantially reduces the amplitude of high wave number components of flow variable 0. Similar decomposition can also be made using the volume and surface averaging concepts. With this approach, the velocity associated with the face of a control volume is decomposed into a surface-averaged value and the fluctuation around the surface averaged value, similar to Eq. (3.3). It must be noted that unlike the traditional Reynolds averages (discussed in the next sub-section), in general, the time averages of these fluctuating components around filtered or surface-averaged quantities are not zero:

Fluctuations around the surface- or volume-averaged quantities generate additional terms in the governing transport equations which require further modeling to close the set of equations. These models (representing the effect of scales smaller than the characteristic filter scale) are called 'sub-grid scale' models (SGS). In principle, the characteristic length scale of a filter is not directly related to the grid size (grid size, obviously, cannot be larger than the filter scale). However, the name SGS has stuck and is used to denote these additional models. The additional terms appearing in the governing equations are generally classified into three groups, namely Leonard stresses, sub-grid scale cross-stresses and sub-grid scale Reynolds stresses (Leonard, 1974). Smagorinsky (1963) proposed the first sub-grid scale stress model. Several models have been proposed since then (Ferziger, 1976; Rogallo and Moin, 1984;

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