LES idea and overview

properties of turbulent scales

LES core idea: resolve large structures and model influence of smallest structures

How is the turbulence kinetic energy affected?

topics in LES modeling

• filtering in complex domains
• subgrid scale models
• supergrid models
• wall/hybrid models

filtered equations

filtering in 3D, no boundaries

$$ \widetilde{\varphi} (\mathbf{x}) = \int_{\mathbb{R}^3} G(\mathbf{x}, \mathbf{y}, \Delta (\mathbf{x}))\ \varphi (\mathbf{x}) \mathrm{d}\mathbf{y} $$

$$ \int_{\mathbb{R}^3} G(\mathbf{x}, \mathbf{y}, \Delta (\mathbf{x})) \mathrm{d}\mathbf{y} = 1 $$

$\Delta $ - filter width, $G$ - filter function

3D expressed as 1D filtering

$$ G(\mathbf{x}, \mathbf{y}, \Delta (\mathbf{x})) = \prod_{i=x,y,z} G_i(x_i, y_i, \Delta_i (x_i)) $$

typical filters for theoretical analysis

• box filter
• Fourier cut-off
• Gauß

some mathematical properties of filtering

• $\widetilde{\varphi + \psi} = \widetilde{\varphi} + \widetilde{\psi}$
• $\widetilde{\partial_i \varphi} = \partial_i \widetilde{\varphi}$
• $\widetilde{\widetilde{\varphi}} \neq \widetilde{\varphi}$
• $\widetilde{\psi\widetilde{\varphi}} \neq \widetilde{\psi} \widetilde{\varphi}$
• $\widetilde{\varphi^\prime}\neq 0$

derivation of filtered momentum and mass conservation equation

further aspects

• inhomogeneous filters
• finite domains
• filtering by discretization

subgrid scale models

SGS model overview

• eddy viscosity models
  • algebraic (Smagorinsky, WALE)
  • transport equation ($k$, $\widetilde{\nu}$)
• stress models
  • ...

additional stochastic and dynamic variants

brief historical perspective

• 1963: Smagorinsky model
  J. Smagorinsky
• 1991: dynamic Smagorinsky model
  M. Germano et al.
• 1999: WALE model
  F. Nicoud, F. Ducros

eddy viscosity approach

$$\tau_{ij} = \widetilde{u_iu_j} - \widetilde{u_i}\widetilde{u_j}$$

$$\tau_{ij}^{mod} = -2\nu_t\widetilde{S}_{ij} + \frac{1}{3}\delta_{ij}\tau_{kk}$$

$\rightarrow$ similar advantages and shortcomings as in RANS approach but typically less pronounced

Prandtl's mixing length idea for $\nu_t$

$$\nu_t = l_c u_c$$

$l_c := f(\Delta)$ (in contrast to RANS)

typical choice $\Delta = V^{1/3}$

$\rightarrow$ only $u_c$ needed

$\nu_t$-based SGS models

some (rough) guidelines for meshing - wall-resolved LES

• $y^+ < 5$, $x^+\approx z^+ < 50$
• cell expansion ratio less than $10\%$
• $80-90\%$ of $k$ resolved

boundary conditions

How to deal with inflow BCs in reduced domains?

summary

• LES solves filtered transport equations
• SGS stress tensor needs to be modeled
• eddy viscosity models most common
• complex boundary/inflow conditions
• wall-resolved LES mostly used in research