les-simulation

name	les-simulation
description	Large Eddy Simulation (LES) — turbulence modeling theory, Smagorinsky model, dynamic Smagorinsky, wall models (WMLES), numerical schemes (implicit/explicit), filter width, grid requirements, subgrid-scale stress, LES vs. DNS vs. RANS, bluff body flows, combustion LES, acoustic source prediction, OpenFOAM/Fluent setup.
metadata	{"priority":7,"promptSignals":{"phrases":["large eddy simulation","LES turbulence","LES simulation","subgrid scale model","Smagorinsky model","LES vs RANS"],"minScore":3}}

Large Eddy Simulation (LES) — Complete Skill

Turbulence Scale Hierarchy

Energy Cascade

Kolmogorov microscale (η): η = (ν³/ε)^(1/4) [m; ν = kinematic viscosity; ε = turbulent dissipation rate] For air at Re_L = 10⁶: η ≈ 0.1–10 μm (extremely small → DNS computationally prohibitive)

Integral length scale (L₀): L₀ = k^(3/2) / ε [k = turbulent kinetic energy; ε = dissipation rate] Energy-containing eddies; O(1) of characteristic length

Taylor microscale (λ_g): λ_g = √(15 ν k / ε) [intermediate scale between L₀ and η]

LES philosophy: Resolve large eddies (L₀ to ≈ 20–50η) explicitly; model subgrid-scale (SGS) eddies below filter width Δ

DNS requires: resolve all scales down to η → N_DNS ∝ Re_L^(9/4) grid points (impractical above Re ~ 10,000) RANS models: all turbulence → approximate with model → fast but misses unsteady/large-scale effects LES: resolve 80–90% of turbulent kinetic energy explicitly; model only smallest SGS → most accurate computationally feasible approach

Filtered Navier-Stokes Equations

Filtering Operation

Spatial filter: ū_i(x,t) = ∫ G(x - x') × u_i(x',t) dx' [G = filter function (box, Gaussian, spectral cutoff)]

Box filter (most common in FVM): G(x) = 1/Δ if |x| ≤ Δ/2; 0 otherwise [Δ = filter width = grid spacing (implicit filtering)]

Filtered NSE: ∂ū_i/∂t + ∂(ū_i ū_j)/∂x_j = -1/ρ × ∂p̄/∂x_i + ν × ∂²ū_i/∂x_j² - ∂τ_ij^SGS/∂x_j

SGS (subgrid-scale) stress tensor: τ_ij^SGS = u_iu_j̄ - ū_i ū_j [Leonard, cross, and Reynolds stresses; must be modeled]

Subgrid-Scale Models

Smagorinsky Model (Constant)

Eddy viscosity assumption: τ_ij^SGS - (1/3)τ_kk δ_ij = -2 ν_SGS × S̄_ij where S̄_ij = (1/2)(∂ū_i/∂x_j + ∂ū_j/∂x_i) [resolved rate-of-strain tensor]

SGS eddy viscosity: ν_SGS = (C_s × Δ)² × |S̄| [|S̄| = √(2 S̄_ij S̄_ij)]

Smagorinsky constant: C_s ≈ 0.1–0.2; theoretical value for isotropic turbulence: C_s = 0.18 Practical: C_s = 0.1 (near-wall regions); C_s = 0.17 (free shear flows) Problem: C_s fixed → over-dissipation near walls; doesn't account for anisotropy or transitional flows

Filter width (implicit in FVM): Δ = (Δx × Δy × Δz)^(1/3) [geometric mean of cell dimensions] For structured mesh: Δ = Δx (if uniform and cubic cells)

Dynamic Smagorinsky Model (Germano-Lilly)

Key improvement: C_s computed dynamically from resolved field → adapts to local flow state Germano identity: test filter L̂_ij = Ξ_ij - τ̂_ij where L̂_ij = û_iû_j̄ - ū̂_i ū̂_j

C_s computation: C_s² = (L_ij × M_ij) / (M_ij × M_ij) [Lilly least-squares contraction; local field quantity] where M_ij = -2Δ̂² |Ŝ| Ŝ_ij + 2Δ² |S̄| S̄_iĵ [difference between test-filtered and grid-level SGS models]

Advantages:

• C_s → 0 near walls naturally (correct asymptotic behavior)
• C_s → 0 in laminar regions (no over-dissipation)
• Correct in backscatter (negative C_s → negative ν_SGS locally; requires clipping to prevent instability) Disadvantage: computationally expensive (30–50% overhead vs. constant C_s); numerical noise requires spatial averaging

WALE Model (Wall-Adapting Local Eddy-viscosity)

Better near-wall behavior than Smagorinsky without dynamic procedure: ν_SGS = (C_w × Δ)² × (S_ij^d × S_ij^d)^(3/2) / ((S̄_ij S̄_ij)^(5/2) + (S_ij^d × S_ij^d)^(5/4)) S_ij^d = (1/2)(ḡ_ij² + ḡ_ji²) - (1/3)δ_ij ḡ_kk² [ḡ_ij = ∂ū_i/∂x_j; velocity gradient tensor] C_w = 0.5 (standard); naturally gives ν_SGS ∝ y³ near walls (correct wall damping without explicit damping function)

Coherent Structure Model and Sigma Model

Sigma model: ν_SGS ∝ σ₃(σ₁ - σ₂)(σ₂ - σ₃)/σ₁² [σ_n = singular values of ∇ū] Correct wall behavior; accounts for pure rotation (no SGS viscosity in pure rotation)

VMS (Variational Multiscale) LES: Separate resolved field into large (resolved large) and small (resolved small) scales Apply SGS model only to small resolved scales → reduces over-dissipation

Grid Requirements

Resolution Criteria

Channel flow (wall-bounded): Streamwise: Δx⁺ = u_τ Δx / ν ≤ 50–100 [u_τ = friction velocity; ν = kinematic viscosity] Spanwise: Δz⁺ ≤ 15–30 Wall-normal: Δy₁⁺ ≤ 1 (first cell at wall); y⁺ ≤ 10 for 80% resolved TKE

Free shear flows (jets, wakes, mixing layers): Δ / L_I ≤ 0.1–0.2 [L_I = integral length scale] Minimum 8–12 cells per largest eddy

Bluff body (cylinder, square column): D / Δ ≥ 30–50 (across wake region) for well-resolved LES of wake shedding D = cylinder diameter; 20–30 cells across diameter

Total cells: channel LES: 10⁷–10⁹; combustion chamber LES: 10⁷–10⁸; automotive bluff body: 10⁸–10⁹

Wall-Modeled LES (WMLES)

Challenge: wall-resolved LES requires y⁺ ≤ 1 → ≫ 10× more cells for high Re WMLES: coarse grid in outer flow + wall model for near-wall layer Common wall models: ODE wall model: solve simplified 1D boundary layer equations (simplified NSE) at wall Equilibrium wall model: log-law assumption; u = (u_τ/κ) × ln(y/y₀) + B Werner-Wengle: algebraic piecewise wall model; less accurate but fast

WMLES cost: O(Re) vs. O(Re^(9/4)) for wall-resolved LES → enables engineering Re numbers

Numerical Schemes for LES

Temporal Discretization

Explicit time integration (preferred for accuracy): 2nd order Adams-Bashforth or Runge-Kutta; no numerical dissipation from time scheme Time step: CFL = u_max × Δt / Δx ≤ 0.2–0.5 (LES; much more restrictive than RANS)

Semi-implicit (Crank-Nicolson): CFL ≤ 1.0 stable; reduces acoustic time step constraint; slight numerical dissipation at high CFL

Implicit for stiff problems: Large Δt possible → but artificial dissipation contaminates LES solution → avoid for high-fidelity

Spatial Discretization

Central differencing: preferred for LES; non-dissipative (energy conserving) → Risk: numerical instability → use 4th order or symmetric 2nd order with energy conservation Upwind schemes: dissipative → damps resolved turbulence → corrupt solution → AVOID pure upwind for LES Blended (partial upwind): 5% upwind + 95% central → control instability without too much artificial dissipation; common in commercial codes

Spectral methods: exact spatial differentiation → ideal accuracy; but require structured grid, periodic BC → research use

OpenFOAM schemes for LES: divSchemes: div(phi,U) Gauss linear; (pure central) or Gauss linearUpwind grad(U) (blended) laplacianSchemes: Gauss linear corrected (corrected for non-orthogonal cells)

Applications

Bluff Body (Cylinder) Wake

Problem: resolve unsteady von Kármán vortex shedding + turbulent wake St (Strouhal) = f_shed × D / U_∞ ≈ 0.2 → resolved with LES (RANS gives steady solution) Critical Re = 3.9×10⁵ (transition to turbulent BL on cylinder) → LES captures correctly Drag crisis at Re ≈ 5×10⁵: C_D drops from 1.2 to 0.3 → LES captures; RANS misses

Combustion LES

Premixed flame: resolve flame brush and turbulence-chemistry interaction SGS combustion model: Flamelet model (G-equation); Thickened Flame Model (TFM); PDF transport Reaction progress variable c: 0 (unburned) to 1 (burned); transport with SGS closure Acoustic sources: LES captures flame-acoustic coupling (thermoacoustic instability in gas turbines) Target: pressure fluctuation amplitude and frequency match experiment ±10% for validated LES

Aeroacoustics (Lighthill-Curle)

Noise from resolved LES field: Lighthill analogy: sound power ∝ M^8 for free-field jet noise (Lighthill) Curle's extension: solid boundaries → M^6 dependence → dominant at low M Extract acoustic source terms from LES → propagate to far field with Ffowcs Williams-Hawkings (FWH) formulation

FWH surface integration: P'(x,t) = ∫_S [Ṗ/c₀r + v_n(Ṗ/r + P̈/c₀) + ...] dS [simplified; P' = far-field acoustic pressure]

OpenFOAM LES Setup

Solver: pisoFoam (incompressible LES); reactingFoam (combustion); rhoPimpleFoam (compressible)

turbulenceProperties:

simulationType      LES;
LES
{
    LESModel        dynamicKEqn;  // or Smagorinsky, WALE
    delta           cubeRootVol;  // filter width = (V_cell)^(1/3)
    turbulence      on;
    printCoeffs     on;
}

fvSolution:

PISO
{
    nCorrectors             2;  // 2 pressure corrections per time step
    nNonOrthogonalCorrectors 1;  // for non-orthogonal mesh
}

Typical runtime: 10⁸ cells at CFL=0.3 on 2,000 CPU cores → 10⁵ time steps = several days to weeks

Validation Strategy

Quality indicators: LES_IQ (LES Index of Quality): k_resolved / (k_resolved + k_SGS) > 0.8 (80% energy resolved) k_SGS from SGS model; k_resolved from resolved fluctuations Turbulent kinetic energy spectra: compare to Kolmogorov E(κ) ∝ κ^(-5/3) in inertial range Grid convergence: compare key quantities (C_D, Nu, St) on successively refined grids (factor √2 refinement)

Standards and References

Source	Scope
Smagorinsky (1963) Mon. Wea. Rev.	Original SGS model
Germano et al. (1991) Phys. Fluids	Dynamic Smagorinsky model
Pope (2000) Turbulent Flows (textbook)	Comprehensive LES/DNS/RANS reference
AIAA CFD Validation Guide	Best practices for turbulence model assessment
ISO 13999 (in progress)	CFD simulation quality assurance
NASA CFD Best Practices	NASA/TM reports for LES validation

Output

Provide: flow regime (wall-bounded/free shear/combustion), Re and characteristic velocity/length, LES model chosen (Smagorinsky/dynamic/WALE) with C_s or C_w value, grid resolution (Δx⁺, Δy⁺, Δz⁺ for wall-bounded; Δ/L_I for free shear), total cell count and estimated WMLES vs. wall-resolved approach, time step Δt [s] and CFL, numerical scheme (spatial + temporal), SGS model ν_SGS/ν_molecular ratio at critical location, LES_IQ value (> 0.80 required), key outputs (C_D, Nu, St, p' [Pa] as applicable), turbulent kinetic energy spectra slope verification (−5/3 in inertial range), and applicable reference (Germano 1991, Pope 2000, OpenFOAM user guide).