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where Δ is the filter width and G is a function (Kernel) whose value approaches zero if u_i occurs on a length scale smaller than Δ. A typical example of convolution kernel is the Gaussian filter:

(6) G_Δ(x) = ⎡⎣(6)/(πΔ²)⎤⎦^3 ⁄ 2exp⎛⎝ − (6∥ x∥²)/(Δ²)⎞⎠

A wide range of filters are available in the literature. For a detailed discussion of their properties, the reader is referred to Pope (2000) [5]. The over-line symbol in Eq.(5↑) is used to describe a filtered variable. The next step is to apply this filter to the incompressible Navier-Stokes equations. According to Leibniz’ rule, the partial differentiation with respect to time can be commuted with the filtering operation, or mathematically

(7) (∂u_i)/(∂t) = (∂u_i)/(∂t)

Assuming that the spatial differentiation also commutes with the filtering operation, the convective, pressure gradient and viscous terms can be filtered.

(8) (∂u_iu_j)/(∂x_j) = (∂u_iu_j)/(∂x_j) − (1)/(ρ)(∂p)/(∂x_i) = − (1)/(ρ)(∂p)/(∂x_i) ν∇²u_i = ν∇²u_i

which leads to the filtered incompressible Navier-Stokes equation.

(9) (∂u_i)/(∂t) + (∂u_iu_j)/(∂x_j) = − (1)/(ρ)(∂p)/(∂x_i) + ν∇²u_i

In the spirit of Reynolds decomposition, the velocity can be decomposed into a large scale component as well as a sub-grid component.

(10) u_i = u_i + u_i’ so that u_iu_j = u_i u_j + τ^Δ_ij

where τ^Δ_ij is the subgrid-scale (SGS) stress tensor that includes all unclosed terms. Eventually, the incompressible filtered Navier-Stokes equations become

(11) (∂u_j)/(∂x_j) = 0

and

(12) (∂u_i)/(∂t) + (∂u_i u_j)/(∂x_j) + (∂τ^Δ_ij)/(∂x_j) = − (1)/(ρ)(∂p)/(∂x_i) + ν∇²u_i

Eq.(12↑) is very similar to the Reynolds averaged Navier-Stokes equation. In order to close the system, the SGS stress tensor τ^Δ_ij will need to be modeled in terms of the filtered velocity field.

Smagorinsky model:

Smakorinsky (1963) [7] proposed a SGS model which has been used extensively in the LES literature. Similarly to the Boussinesq assumption, the deviatoric part of the stress tensor is defined by

(13) τ^Δ, smag_ij = τ^Δ_ij − (1)/(3)τ^Δ_kkδ_ij = − 2ν_tS_i, j

where S_i, j is the resolved (filtered) strain rate tensor. The eddy viscosity definition also has much in common with Prandtl’s mixing length hypothesis (Eq.(↓)). It is given by

(14) ν_t = (C^Δ_sΔ)²|S|

with |S| = (2S_i, j S_i, j)^1 ⁄ 2, Δ is used as a mixing length and C^Δ_s is the Smagorinsky coefficient. The SGS model can be validated without running a full Large Eddy Simulation. If the flow field is already known with great detail (DNS), the data can be used to compute the SGS stress tensor using Eq.(13↑) and Eq.(14↑). Such comparison is typically called “a-priori testing". This approach can be used to calibrate the Smagorinsky coefficient C^Δ_s. In the case of statistically stationary isotropic turbulence with Δ≪η, Lilly (1967) showed that C^Δ_s ~ 0.16 [3]. This coefficient is considered to be scale-invariant within the inertial range. Several other SGS models are available in the literature, including the kinetic energy model (Schumann (1975) [6]) and the dynamic model (Germano (1991) [2]). For an exhaustive review of LES modeling, the reader is referred to Meneveau (2000) [4].