Large Eddy Simulation for detailed resolution of turbulent structures
Large Eddy Simulation (LES) emerged in the 1970s as a revolutionary approach to turbulence modeling, pioneered by Smagorinsky (1963) and further developed by Deardorff (1970). Unlike RANS methods that model all turbulent scales, LES resolves the large, energy-containing turbulent structures directly while modeling only the smaller, more universal subgrid scales.
The fundamental insight behind LES lies in the recognition that large turbulent eddies are highly anisotropic and flow-dependent, while small eddies exhibit more universal, isotropic characteristics. This scale separation, formalized through spatial filtering, allows LES to capture the dominant turbulent physics directly on the computational grid while employing simpler models for subgrid-scale (SGS) turbulence.
Key developments include the Smagorinsky model (1963), dynamic modeling by Germano et al. (1991), the WALE model by Nicoud and Ducros (1999), and modern machine learning-enhanced SGS models. Each advancement has improved LES accuracy and applicability across different flow regimes.
Contemporary LES benefits from advances in computational power, enabling wall-resolved simulations at moderate Reynolds numbers and wall-modeled approaches for high Reynolds number engineering flows.
LES is based on spatial filtering of the Navier-Stokes equations, decomposing flow variables into resolved (grid-scale) and subgrid-scale components through a filter operation that depends on the computational grid spacing.
| SGS Model | Model Constant | Typical Value | Advantages |
|---|---|---|---|
| Smagorinsky | C_s | 0.17-0.18 | Simple, robust |
| Dynamic Smagorinsky | Dynamic C_s | Variable | Automatic tuning |
| WALE | C_w | 0.5-0.6 | Better near-wall behavior |
| Kinetic Energy | C_k | 0.094 | Transition handling |
LES success critically depends on adequate grid resolution to capture the energy-containing turbulent scales while maintaining computational feasibility through subgrid-scale modeling of the smallest eddies.
The turbulent energy spectrum follows Kolmogorov's -5/3 law in the inertial range. LES grids must resolve scales containing 80-90% of the turbulent kinetic energy, typically requiring grid spacing on the order of the Taylor microscale λ or integral length scale L in homogeneous turbulence.
For wall-bounded flows, different resolution requirements apply in different regions. Near walls, y⁺ < 1 is needed for wall-resolved LES, while wall-modeled LES can operate with y⁺ ≈ 30-100. In the streamwise and spanwise directions, Δx⁺ < 100 and Δz⁺ < 20 are typical requirements for channel flow LES.
The filter width Δ is typically defined as the cube root of the cell volume: Δ = (ΔxΔyΔz)^(1/3). For highly anisotropic grids, alternative definitions such as Δ = max(Δx, Δy, Δz) or more sophisticated formulations accounting for grid anisotropy may be employed.
LES quality can be assessed through several metrics: resolved turbulent kinetic energy should exceed 80% of total TKE, subgrid viscosity should remain below 10% of molecular viscosity in most regions, and the effective Reynolds number based on grid spacing should ensure adequate scale separation.
Subgrid-scale models represent the effect of unresolved turbulent scales on the resolved motion, bridging the gap between the computational grid and the universal dissipation scales.
The most common SGS models employ the eddy viscosity hypothesis, relating subgrid stresses to the resolved strain rate tensor. The Smagorinsky model provides the foundation, with the dynamic procedure offering automatic calibration of the model constant based on resolved flow information.
Scale-similarity models, such as Bardina's model, assume that subgrid scales are similar to the smallest resolved scales. These models often require combination with eddy viscosity models to provide adequate dissipation but offer better correlation with actual subgrid stresses.
Mixed models combine multiple approaches, such as scale-similarity and eddy viscosity components, to capture both the structural and dissipative aspects of subgrid turbulence. These models often provide superior performance in transitional and separated flows.
Modern developments include machine learning-enhanced models trained on high-fidelity data, fractional-step models accounting for non-local effects, and variational multiscale methods that provide rigorous mathematical frameworks for scale separation.
Near-wall treatment represents one of the most challenging aspects of LES, as the smallest turbulent scales concentrate near solid boundaries where viscous effects dominate.
Wall-resolved LES integrates the filtered equations down to the wall with no-slip boundary conditions. This approach requires y⁺ < 1 and sufficient resolution to capture near-wall turbulent structures, making it computationally expensive but highly accurate for moderate Reynolds numbers.
Wall-modeled LES (WMLES) employs wall functions or near-wall RANS models to bridge the gap between the first grid point and the wall. This approach allows coarser near-wall grids (y⁺ ≈ 30-100) while maintaining reasonable accuracy for high Reynolds number flows.
Outlet boundaries require convective conditions to minimize spurious reflections, while far-field boundaries in external flows may employ sponge layers or buffer zones to prevent contamination of the physical domain by artificial boundary effects.
LES places stringent requirements on numerical methods to maintain the accuracy and physical realism of resolved turbulent structures while minimizing numerical errors that can contaminate the solution.
Central difference schemes are preferred for their low numerical dissipation, though they require careful attention to grid quality and potential numerical instabilities. High-order finite difference, finite volume, and spectral methods are commonly employed, with compact finite difference schemes offering good accuracy-to-cost ratios.
Time integration must resolve the convective time scales of the largest eddies, typically requiring CFL numbers of 0.5-1.0. Runge-Kutta schemes and Adams-Bashforth methods are popular choices, with implicit methods used for stiff terms such as viscous contributions.
Incompressible LES requires special treatment for pressure-velocity coupling due to the elliptic nature of the pressure equation. Projection methods, SIMPLE-family algorithms, and pressure correction approaches are adapted for LES with careful attention to temporal accuracy.
Contemporary LES applications span fundamental research, engineering design, and emerging interdisciplinary fields, driven by advances in computational power and modeling sophistication.
LES serves as a bridge between theory and experiments in turbulence research, providing detailed flow field information for understanding energy cascade processes, intermittency phenomena, and coherent structure dynamics. It enables studies of turbulent mixing, scalar transport, and multi-phase flows at scales inaccessible to experiments.
Industrial LES applications include combustion chamber design, aeroacoustics prediction, heat exchanger optimization, and environmental flow modeling. The method excels in capturing unsteady phenomena critical for performance, emissions, and noise prediction in engineering systems.
Future LES developments focus on improving subgrid models through data-driven approaches, developing efficient algorithms for exascale computing, and extending applications to extreme-scale geophysical and astrophysical flows. Integration with artificial intelligence promises adaptive models that optimize themselves for specific flow configurations.
Modern LES practice emphasizes rigorous validation against experimental data and verification through grid convergence studies. Standard test cases include channel flow, backward-facing step, periodic hills, and flow around bluff bodies, with established databases enabling systematic model assessment and improvement.