Filtered Navier-Stokes equation

A detailed description of LES filtering is beyond the scope of this book (see, for example, Meneveau and Katz (2000) or Pope (2000)). However, the basic idea can be understood by considering a so-called sharp-spectral filter in wavenumber space. For this filter, a cut-off frequency kc in the inertial range of the turbulent energy spectrum is chosen (see Fig. 4.1), and a low-pass filter is applied to the Navier-Stokes equation to separate the... 

It can be easily shown (Pope 2000) that filtering and space/time derivatives commute in homogeneous flows. The filtered Navier-Stokes equation can thus be expressed as... 

Our intent here is not to suggest a solution method but rather to use the stream-function-vorticity formulation to comment further on the mathematical characteristics of the Navier-Stokes equations. In this form the hyperbolic behavior of the pressure has been lost from the system. For low-speed flow the pressure gradients are so small that they do not measurably affect the net pressure from a thermodynamic point of view. Therefore the pressure of the system can simply be provided as a fixed parameter that enters the equation of state. Thus pressure influences density, still accommodating variations in temperature and composition. Since the pressure or the pressure gradients simply do not appear anywhere else in the system, pressure-wave behavior has been effectively filtered out of the system. Consequently acoustic behavior or high-speed flow cannot be modeled using this approach. 

S. Paolucci. On the Filtering of Sound from the Navier-Stokes Equations. Technical Report SAND82-8257, Sandia National Laboratories, 1982. 

The modelling example of the previous section shows that to simplify the general mathematical model of the studied process, the real flow in the filter unit has been considered in terms of its own simplified model. Indeed, it is difficult to understand why we have used a flow model, when in fact, for the flow characterization, we already have the Navier-Stokes equations and their expression for the computational fluid dynamics. To answer this question some precisions about the general aspects of the computational fluids dynamics have to be given. 

The maximum wave number resolved with the LES approach is chosen to lie in the inertial sub-range of the turbulence energy spectrum. The governing transport equations are derived either by filtering the Navier-Stokes equation or using volume... 

The equations for the evolution of the filtered velocity field are derived from the Navier-Stokes equations. 

The opposite viewpoint, advocated by Boris et al. [16], is that no explicit filtering should be performed and no explicit residual stress model should be used (Oj = 0). Instead, an appropriate numerical method is used to attempt to solve the Navier-Stokes equation for v(r, f). Because the grid is not fine enough to resolve the solution to the Navier-Stokes equation, significant numerical stresses arise. Thus, filtering and residual-stress modeling are performed implicitly by the numerical method. 

Considering the Navier-Stokes equation (1.383), the linear terms are straightly treated in a similar manner but, as in the Re3uiolds averaging procedure, the non-linear terms require further analysis. Applying the filter operator (1.476) on the non-linear terms in (1.383), yields... 

In practice, the Leonard stress is often dominated by the numerical errors inherent in the finite difference (and finite volume) representation and is thus neglected or lumped into the deviatoric stress tensor (e.g., [97] [106] [186], p. 325). Consequently, as the box filter is applied to the Navier-Stokes equation, the residual stresses assume the form of sub-grid scale stresses ... 

When dealing with turbulent flows all the relevant dimensionless numbers are evaluated with the available quantities. For example, in DNS, the fluid and particle instantaneous velocities will be employed, whereas in large-eddy simulation (LES) or in Reynolds-average Navier-Stokes-equations (RANS) simulations the filtered or Reynolds-average values will be used. 

In LES/FMDF calculations of single-phase flows, the resolved hydrodynamic field is obtained by solving the filtered form of the compressible Navier-Stokes equations [3] ... 

Mixing phenomena between cold and hot fluids at the core exit, investigated in simulation experiments with air (AIRJECO) and sodium (NAJECO, were already presented [Ref. 1-2]. Simulations were obtained with Large Eddies Simulations (LES). The LES filtered Navier-Stokes equations were solved using the TRIO-VF code. 

To facilitate the manipulation of the Navier-Stokes equations, it is necessary that the filter satisfies the following [15] ... 

The physics of the problem under study is assumed to be governed by the compressible form of the Favre-filtered Navier-Stokes energy and species equations for an ideal gas mixture with constant specific heats, temperature-dependent transport properties, and equal diffusion coefficients. The molecular Schmidt, Prandtl, and Lewis numbers are set equal to 1.0, 0.7, and 1.43, respectively [17]. 

For the purpose of understanding pressure filtering, attention may be restricted to the single-component, constant-property, nonreacting equations for a perfect gas. Introducing the nondimensional variables into the vector forms of the mass-continuity, constant-viscosity Navier-Stokes, and perfect-gas thermal-energy equations yields the following nondimensional system ... 

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