LES

Filtered Navier-Stokes Equations

Filter process The spatial filter of a function $\phi$ is defined as its convolution integral with a filter function $G$, according to:

$$\overline{\phi} = \int_V \phi(\mathbf{x}',t) G(\mathbf{x}-\mathbf{x}',\Delta) dV'$$

where $\Delta$ is the characteristic filter width in each respective direction and $\overline{\phi}$ is the filtered quantity.

The actual form of the filter is usually implicit and is not needed while solving the LES equations. Is common to use a box/top-hat filter of the form below, where the cut-off scale/filter width is taken proportional to the mesh size $\Delta \propto h$

Other filter kernels are possible (see Pope's book for more detail on LES filters). In the finite volume method, when the filter width matches the local cell size , i.e., $\Delta= h$ and the filter used is the box filter; the cell-averaged value of a variable is equivalent to its filtered value.

For variable density flows it is convenient to introduce the mass-weighted Favre filtering operation :

$$\widetilde{\phi } = \frac{\overline{\rho \phi} }{\bar{\rho}}$$

It is common to assume commutability of the filtering and derivative operators, that is to say

$$\overline{ \frac{\partial \phi }{\partial x_j} } = \frac{\partial \overline{\phi}}{\partial x_j}$$

This relation is only true under several assumptions, including the restrictive condition of constant filter width throughout the domain. Is common to neglect this error and assume the effects are incorporated into the sub-grid model.

Continuity

Applying the filtering operator to the continuity equation

$$\frac{\partial \overline{\rho} }{\partial t} + \frac{\partial \overline{\rho u_j} }{\partial x_j} = 0$$

and using Favre weigthing

$$\frac{\partial \bar{\rho} }{\partial t} + \frac{\partial \bar{\rho} \tilde{u}_j }{\partial x_j} = 0$$

Filtered Momentum

$$\frac{\partial \overline{\rho u_i} }{\partial t} + \frac{\partial \overline{\rho u_i u_j}}{\partial x_j} = -\frac{\partial \bar{p}}{\partial x_i} + \frac{\partial \overline{\tau}_{ij}}{\partial x_j}$$

Using Favre weigthing

$$\frac{\partial \bar{\rho} \widetilde{u}_i }{\partial t} + \frac{\partial \bar{\rho} \widetilde{u_i u_j}}{\partial x_j} = -\frac{\partial \bar{p}}{\partial x_i} + \frac{\partial \overline{\tau}_{ij}}{\partial x_j}$$

Filtered Species transport

$$\frac{\partial \overline{\rho_k} }{\partial t} + \frac{\partial \overline{ \rho_k u_j }}{\partial x_j} = \frac{\partial \overline{J}_j^k}{\partial x_j} + \overline{\rho \dot{\omega}_k}$$

Using Favre weigthing

$$\frac{\partial \bar{\rho} \bar{Y}_k }{\partial t} + \frac{\partial \bar{\rho} \widetilde{ u_j Y_k}}{\partial x_j} = \frac{\partial \overline{J}_j^k}{\partial x_j} + \bar{\rho} \widetilde{\dot{\omega}_k}$$

Filtered Energy

$$\frac{\partial \overline{E_t} }{\partial t} + \frac{\partial \overline{(E_t + p) u_j}}{\partial x_j} = \frac{\overline{\partial u_i \tau_{ij}}}{\partial x_j} - \frac{\partial \overline{q}_j }{\partial x_j} + \sum_k \frac{ \overline{\partial h_k J_j^k}}{\partial x_j}$$

Sub-grid Closures

The filtered momentum equation requires modelling of the following terms

$$\overline{\rho u_i u_j} =\bar{\rho} \widetilde{u_i u_j}  = \bar{\rho} \tilde{u}_i \tilde{u}_j  - \tau^{sgs}_{ij}$$

where a sub-gris stress tensor is introduced that need modelling

$$\tau^{sgs}_{ij} =   \bar{\rho} \tilde{u}_i \tilde{u}_j  - \bar{\rho} \widetilde{u_i u_j}$$

A common strategy found in many LES studies is the use of an eddy viscosity-like assumption founded on Boussinesq’s hypthothesis

$$\tau_{ij}^{sgs} - \frac{1}{3} \tau_{kk} = \mu_{sgs} \left( \tilde{S}_{ij} - \frac{1}{3} \tilde{S}_{kk} \right)$$

A sub-grid viscosity is introduced, similar to turbulent viscosity in RANS-type models that requires modelling. Implemented models include:

Smagorinsky Model

The Smagorinsky model [3] assumes that small, unresolved turbulent eddies behave like an eddy viscosity, enhancing momentum diffusion.

$$\mu_{sgs} = \bar{\rho} (C_S \Delta) ^2 || \tilde{S}_{ij} ||$$

with $\tilde{S}_{ij}$represents the filtered strain tensor and $C_S$ the Smagorinsky constant, with values between 0.1-0.2. $||\tilde{S}_{ij}|| = \sqrt{2 \tilde{S}_{ij} \tilde{S}_{ij} }$ is the Frobenius norm of the filtered strain tensor. The length scale $l_{sgs}= C_S \Delta$ is a sub-grid length scale, which can be consider proportional to the integral lenght-scale $\ell$. The model assumes that small scales are isotropic

WALE Model

The eddy viscosity in the WALE model [1] is computed

$$\mu_{sgs} = \bar{\rho} (C_w \Delta) ^2 \frac{(\mathcal{S}_{ij}^d\mathcal{S}_{ij}^d)^{3/2}}{(S_{ij} S_{ij})^{5/2} - (\mathcal{S}_{ij}^d \mathcal{S}_{ij}^d)^{5/4}}$$

where $\mathcal{S}_{ij}^d = \mathcal{S}_{ij} - 1/3 \mathcal{S}_{kk}$ is the traceless, symmetric tensor of the square of the velocity gradient

$$\mathcal{S}_{ij} = \frac{1}{2} \left(\frac{\partial u_i}{\partial x_k} \frac{\partial u_k}{\partial x_j} + \frac{\partial u_j}{\partial x_k} \frac{\partial u_k}{\partial x_i} \right)$$

Model coefficient is in the range $C_w = \sqrt{10.6} \cdot C_S = 0.325 - 0.5$

Diffusivity and heat

The sub-grid transport of a scalar is splitted in

$$\widetilde{u_j Y_k} = \tilde{u}_j \tilde{Y}_k - D_{sgs} \frac{\partial \tilde{Y}_k}{\partial x_j}$$

where $D_{sgs}$ is the sub-grid diffusivity, which taken proportional to sub-grid viscosity

$$\bar{\rho} D_{sgs} = \frac{\mu_{sgs}}{\text{Sc}_{sgs}}$$

with $\text{Sc}_{sgs}$ is a constant often taken as 0.4-1. All species diffuse at the smallest scales due to turbulence at the same speed.

$$\overline{\rho u_j (e_t + P/\rho )}  = \bar{\rho} \widetilde{ u_j h_t} = \bar{\rho}\tilde{u}_j \tilde{h}_t - \lambda_{sgs} \frac{\partial \tilde{T}}{\partial x_j}$$

where $h_t \equiv  e_t + P/\rho$is the specific total enthalpy and $\lambda_{sgs}$ a sub-grid conductivity, that can be related to the sub-grid viscosity through:

$$\lambda_{sgs} = \frac{\mu_{sgs} C_p}{\text{Pr}_{sgs}}$$

Where the sub-grid Prandtl number is introduced, which is a constant taken in the range 0.4-1. Cerisse works with the ratio to molecular Prandtl number and redefines the sub-grid conductivity as

$$\frac{\lambda_{sgs}}{\lambda} = \frac{\mu_{sgs}}{\mu} \frac{\text{Pr}}{\text{Pr}_{sgs}}$$

To avoid computing the specific heat. The ratio $\text{Pr}/\text{Pr}_{sgs}$ is often less than 1 in gases, with a common choice of 0.5/0.7 $\approx$0.7 . In general, if ${\text{Pr}}/{\text{Pr}_{sgs}} > 1$, sub-grid turbulent eddies transport momentum more efficiently than heat (vs. molecular case). If ${\text{Pr}}/{\text{Pr}_{sgs}} < 1$, sub-grid turbulent eddies transport momentum more efficiently than heat (vs. molecular case). The isotropic part of the sub-grid stress $⅓ \tau_{kk}$ is neglected in incompressible flows (absorved in the pressure) and is often modelled using Yoshizawa model [2] in compressible flows

$$\tau_{kk} = \bar{\rho} C_I \Delta^2 ||\tilde{S}_{ij}||^2$$

where $C_I$ is a model constant taken often as 0.008. This expression can be used to estimate the sub-grid kinetic energy

$$k_{sgs} =\frac{3}{2} \tau_{kk}  = C_Y \Delta^2 ||\tilde{S}_{ij}||^2$$

with $C_Y$ taken as 0.0066

Other unknowns

In conventional LES, fluctuations of transport properties are assumed to be small within the filter width and therefore, molecular fluxes can be approximated by

$$\overline{q}_j \approx - \lambda (\tilde{T}) \frac{\partial \tilde{T}}{\partial x_j}$$

Molecular fluxes scale with the inverse of Reynolds number, $\text{Re}^{-1}$, making them relatively small in turbulent flows. Consequently, errors associated with molecular transport properties often (but not always) have a small impact in the solution.

References

[1] Nicoud, F., Ducros, F. Subgrid-Scale Stress Modelling Based on the Square of the Velocity Gradient Tensor. Flow, Turbulence and Combustion 62, 183–200 (1999). [2] Yoshizawa, A. Horiuti, K A Statistically-Derived Subgrid-Scale Kinetic Energy Model for the Large-Eddy Simulation of Turbulent Flows. Journal of the Physical Society of Japan, 54, 2834-2839 (1985) [3] Smagorinsky, J. (1963). General circulation experiments with the primitive equations. Monthly Weather Review, 91(3):99–164.