The Filtered Navier-Stokes Equations

In order to separate out the large eddies from the small-scale motions filtering operations are used. A filtered variable, denoted by an overbar, is defined as

(33.1)

where, D is the entire domain and G is the filter function. The band-pass filter function determines the size and structure of small scales that are eliminated from f(x ) and hence require independent modeling. The most commonly used filter functions are defined in wave space as the sharp Fourier cut off filter

(33.2)

the Gaussian filter

(33.3)

and the tophat filter in real space

(33.4)

Here the large eddies include all wave numbers up to the cutoff wave number k that is included in the numerical approximation. All higher modes are subgrid scales, to which modeling assumptions have to be applied. If the Navier-Stokes equations are approximated by a finite difference scheme, the integrals given above introduce an "approximation filter" which filters out all subgrid scales smaller than Δ = Δx , where Δ is the mesh spacing. The filtering functions are applied to governing equations, the filtered Navier-Stokes and the continuity equations for an incompressible flow will assume the form

(33.5)

(33.6)

The above equations describe large-scale motion. The effect of small scales appears through a subgrid scale (SGS) stress tensor as

(33.7)

If energy is fed into a certain band of scales, then turbulence has the tendency to distribute energy in wave number space from the given mode to all other possible modes. In the absence of sources or sinks of energy, this process will continue to complete redistribution. However in reality, geometry limits the lower end wave numbers and the viscosity damps out energy at high wave numbers. In order to balance the dissipation, more energy has to be transported from the large to small length scales. This is the reason for having a one-way transfer of energy from small to high wave numbers. Thus, the subgrid scale model has to mimic the drainage of energy from the large scales.