LES and DNS

Large-eddy simulation (LES) resolves motions larger than the effective filter scale. Direct numerical simulation (DNS) resolves the dynamically relevant scales without a turbulence closure.

Apply a spatial filter

Let $\bar u_i$ denote a filtered velocity. For incompressible, constant-property flow and a filter that commutes with differentiation, the resolved momentum equation is

$$\partial_t\bar u_i+\bar u_j\partial_j\bar u_i =-\frac{1}{\rho}\partial_i\bar p+\nu\partial_{jj}\bar u_i-\partial_j\tau_{ij}^{\mathrm{sgs}}, \qquad \tau_{ij}^{\mathrm{sgs}}=\overline{u_i u_j}-\bar u_i\bar u_j.$$

The filter separates scales by size. An explicit LES filter can be specified directly. In many practical LES calculations, effective resolution and filtering also depend on the grid and numerics. The subgrid-scale stress accounts for interactions between unresolved and resolved scales. Closures include Smagorinsky, dynamic variants and WALE. Some implicit LES approaches rely on the scheme’s dissipation.

A simple example of subgrid closure

For a static Smagorinsky eddy-viscosity model the SGS viscosity is

$$\nu_{\mathrm{sgs}}=(C_s\Delta)^2|\bar S|, \qquad |\bar S|=\sqrt{2\bar S_{ij}\bar S_{ij}}.$$

Here $C_s$ is a model coefficient, $\Delta$ is the filter length and $\bar S_{ij}$ is the resolved strain. The deviatoric subgrid stress is modeled as $-2\nu_{\mathrm{sgs}}\bar S_{ij}$. Dynamic and wall-adapting approaches change how the closure responds to the local flow. The same nominal cell count can yield different effective resolution on different meshes or with different discretizations.

Wall-resolved and wall-modeled LES

Wall-resolved LES aims to resolve near-wall turbulence structures, requiring fine spacing in the wall-parallel directions as well as normal to the wall. Wall-modeled LES (WMLES) uses a model to estimate wall stress and can reduce this cost at high Reynolds number. It still needs enough space and time resolution in the outer flow and wake.

DNS

Direct numerical simulation advances the unaveraged equations and resolves the dynamically relevant scales without a turbulence closure. The mesh and time step must capture the smallest relevant motions. At full aircraft Reynolds numbers this is generally impractical for design, but DNS of simpler flows is valuable for understanding physics and benchmarking models.

Sources

• Bae and Lozano-Durán · explicitly filtered LES and subgrid stresses
• Smagorinsky (1963) · original eddy-viscosity formulation
• NASA · Vision 2030 CFD study
• NASA LAVA · WMLES of a high-lift aircraft configuration