LES Fundamentals — Theory and Governing Equations

Good question. Turbulence contains eddies of various scales. LES uses spatial filtering to directly calculate the large-scale eddies, while the eddies smaller than the filter width (subgrid-scale, SGS) are approximated by a model.

DNS resolves all eddies down to the Kolmogorov scale $\eta_K$, requiring grid points on the order of $N \sim Re^{9/4}$. This is completely infeasible for industrial problems at practical Reynolds numbers. LES significantly reduces computational cost by taking the filter width $\Delta$ larger than the Kolmogorov scale.

For any physical quantity $\phi(\mathbf{x}, t)$, the filter operation is defined by a convolution integral.

Here, $G$ is the filter kernel, and $\Delta$ is the filter width. Typical filters include the Box (top-hat) filter, Gaussian filter, and Sharp spectral (cut-off) filter.

In finite volume method-based codes (Fluent, STAR-CCM+, OpenFOAM, etc.), implicit filtering based on cell volume is common. The filter width is often defined as the cell characteristic length, such as $\Delta = V_{cell}^{1/3}$ or $\Delta = (\Delta x \cdot \Delta y \cdot \Delta z)^{1/3}$.

For incompressible flow, the filtered continuity and momentum equations become:

The key point here is that the nonlinear term $\overline{u_i u_j}$ cannot be approximated by the product of filtered velocities $\bar{u}_i \bar{u}_j$, giving rise to the SGS stress tensor.

Exactly. The models used to close the SGS stress tensor are called SGS models (subgrid-scale models). Various models have been proposed, such as the Smagorinsky model, dynamic Smagorinsky model, and WALE model.

It applies the Boussinesq hypothesis to the small-scale eddies, making the deviatoric component of the SGS stress tensor proportional to the strain rate tensor.

Here, $\bar{S}_{ij} = \frac{1}{2}\left(\frac{\partial \bar{u}_i}{\partial x_j} + \frac{\partial \bar{u}_j}{\partial x_i}\right)$ is the filtered strain rate tensor, and $\nu_{sgs}$ is the SGS eddy viscosity coefficient. The difference between SGS models lies in how they evaluate this $\nu_{sgs}$.

Exactly. RANS uses a time-averaged eddy viscosity, while LES evaluates a local eddy viscosity for the instantaneous filtered field. This difference is physically very significant.

LES requires accurately resolving eddy structures at scales larger than the filter width, so schemes with low numerical dissipation are needed. Second-order central differencing is the minimum baseline; ideally, higher-order central differencing or compact differencing should be used.

First-order upwind differencing must absolutely not be used. Its numerical dissipation far exceeds the dissipation from the SGS model, artificially wiping out eddy structures. Even second-order upwind requires caution; when using blending schemes (e.g., a mix of central and upwind differencing), the proportion of central differencing should be sufficiently high. OpenFOAM's linearUpwind or Fluent's Bounded Central Differencing are practical compromises, but if the central differencing proportion is too low, LES loses its meaning.

Since LES is inherently an unsteady calculation, time integration accuracy is also important. At least second-order accuracy is required. Typical choices are as follows:

Manage it using the CFL number (Courant-Friedrichs-Lewy number). For explicit schemes, $CFL < 1$ is the stability condition, but even for implicit schemes, it's desirable to keep $CFL \sim 1$ to ensure LES accuracy. The CFL number is defined as $CFL = \frac{u \Delta t}{\Delta x}$.

The PISO (Pressure Implicit with Splitting of Operators) method is the most common. Within each time step, it performs one predictor step and two or more corrector steps. In OpenFOAM, the pimpleFoam solver adopts the PIMPLE (PISO+SIMPLE) algorithm and is widely used for LES. In Fluent, the combination of Non-Iterative Time Advancement (NITA) + fractional step method is recommended.

For wall-resolved LES (wall-resolved LES, WRLES), the grid resolution near the wall has very stringent requirements.

Here, wall units are $y^+ = y u_\tau / \nu$, $u_\tau = \sqrt{\tau_w/\rho}$.

Exactly. The number of grid points for wall-resolved LES increases on the order of $N \sim Re^{13/7}$, making it extremely costly for real-world problems at high Reynolds numbers. That's why hybrid methods like wall-modeled LES (WMLES) and DES/IDDES become important.