## Large Eddy Simulation

Resolving eddies of all sizes in a turbulent flow simulation is prohibitively expensive. Instead, we could just resolve the large eddies, but model the effects of the small eddies so that the overall computational cost can be dramatically reduced. This is the fundamental idea behind large eddy simulation. To model the small eddies, we need to perform a spatial filtering on the original flow equations. The spatial filtering on any variable $f$ can be written in the following form, $$\bar{f}(x) = \int_{\Omega}G(x-x')f(x')dx',$$ where $\bar{f}$ is the filtered variable, $G(x)$ is the kernel of the filter, $\Omega$ is the size of the filer (which is usually taken as grid size). The following Favre filtering is usually used together to make the filtered equations simpler, $$\widetilde{f} = \frac{\overline{\rho f}}{\bar{\rho}},$$ where $\widetilde{f}$ is a Favre-filtered variable, and $\rho$ is density. Applying these filters to the flow equations, we will have \begin{aligned} &\frac{\partial\bar{\rho}}{\partial t} + \frac{\partial\bar{\rho}\widetilde{u}_j}{\partial x_j} = 0, \\[1em] &\frac{\partial\bar{\rho}\widetilde{u}_i}{\partial t} + \frac{\partial}{\partial x_j}(\bar{\rho}\widetilde{u}_i\widetilde{u}_j+\bar{p}\delta_{ij}) = \frac{\partial\widetilde{\tau}_{ij}}{\partial x_j} - \frac{\partial\tau^{SGS}_{ij}}{\partial x_j},\\[1em] &\frac{\partial\bar{\rho}\widetilde{E}}{\partial t} + \frac{\partial}{\partial x_j}\Big((\bar{\rho}\widetilde{E}+\bar{p})\widetilde{u}_j\Big) = \frac{\partial}{\partial x_j}(\widetilde{u}_i\widetilde{\tau}_{ij}) - \frac{\partial\bar{q}_j}{\partial x_j} - \frac{\partial q^{SGS}_j}{\partial x_j}, \end{aligned} where $\tau_{ij}^{SGS}$ and $q_j^{SGS}$ are the subgrid-scale (SGS) stress tensor and heat flux vector with the following expressions: \begin{aligned} \tau_{ij}^{SGS} &= \bar{\rho}(\widetilde{u_i u_j} - \widetilde{u}_i\widetilde{u}_j), \\[1.0em] q_j^{SGS} &= \bar{\rho}(\widetilde{u_jT} - \widetilde{u}_j\widetilde{T}). \end{aligned} These two terms are modeled using SGS models. For example, the following Smagorinsky model (assume equilibrium energy production and dissipation in small scales), \begin{aligned} \tau_{ij}^{SGS} - \dfrac{1}{3} \tau_{kk}^{SGS} \delta_{ij} &= -2 \bar{\rho} \nu_t (\widetilde{S}_{ij} - \dfrac{1}{3} \widetilde{S}_{kk} \delta_{ij} ) , \\[1.0em] q_j^{SGS} &= -\dfrac{\bar{\rho} \nu_t}{Pr_t} \dfrac{\partial \widetilde{T}}{\partial x_j}, \end{aligned} where $$\widetilde{S}_{ij}= \dfrac{1}{2} \left(\dfrac{\partial \widetilde{u}_i}{\partial x_j} + \dfrac{\partial \widetilde{u}_j}{\partial x_i} \right), \\[1em] \nu_t = (C_s \Delta)^2 \sqrt{2 \widetilde{S}_{ij} \widetilde{S}_{ij}} = (C_s \Delta)^2 \vert\widetilde{S}\vert,$$ are the strain-rate tensor and the turbulent eddy viscosity, respectively; $Pr_t$ is the turbulent Prandtl number whose value is in the range $[0.3,0.9]$; $C_s$ is an empirical constant; $\Delta$ represents grid size.

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