aeroacoustics

name	aeroacoustics
description	Aeroacoustics — Lighthill's equation, Ffowcs Williams-Hawkings (FW-H), trailing-edge noise, tonal vs. broadband noise, Curle's analogy, propeller/fan noise, OASPL prediction, NASA SP-8018.
metadata	{"priority":7,"promptSignals":{"phrases":["aeroacoustics","Lighthill equation","FW-H equation","trailing edge noise","propeller noise","aerodynamic noise","Ffowcs Williams Hawkings"],"minScore":3}}

Aeroacoustics — Complete Skill

Lighthill's Acoustic Analogy

Lighthill equation (exact rearrangement of Navier-Stokes): ∂²ρ'/∂t² - c₀² ∇²ρ' = ∂²T_ij/∂x_i∂x_j

ρ' = ρ - ρ₀ = density perturbation [kg/m³] c₀ = ambient speed of sound [m/s] T_ij = Lighthill stress tensor = ρu_i u_j - e_ij + (p' - c₀²ρ')δ_ij e_ij = viscous stress tensor; p' = pressure perturbation

Physical interpretation: T_ij ≈ ρu_i u_j (for low M, isentropic flow) = quadrupole source Generated by turbulence; scales with U^8 (Lighthill's 8th power law for free-field jet)

Acoustic power from jet: P_acoustic ∝ ρ₀ U^8 L² / c₀⁵ [L = jet diameter; U = jet velocity] Confirmed by experiments; basis for noise reduction via velocity reduction (U↓10% → sound ↓ 8 dB)

Curle's Analogy (Solid Boundaries)

Adds surface forces to Lighthill's analogy: p'(x,t) = [1/(4π c₀)] ∂²/∂t² ∫[T_ij/r] dV - [1/(4πc₀)] ∂/∂t ∫[f_i/r] dS

f_i = surface force per unit area (aerodynamic loading fluctuation) Dipole scaling: P_acoustic ∝ ρ₀ U^6 / c₀³ [6th power law; surface-force dipoles dominate over quadrupoles at low M]

Ffowcs Williams-Hawkings (FW-H) Equation

Extends Lighthill to moving bodies: ∂²p'/∂t² - c₀²∇²p' = ∂²(T_ij H)/∂x_i∂x_j - ∂(f_i δ)/∂x_i + ∂(ρ₀ v_n δ)/∂t

Three source terms:

1. Volume sources (quadrupoles): T_ij in volume (turbulence); U^8 scaling; only at high M
2. Surface loading (dipoles): f_i on body surface; U^6 scaling; dominant at M < 0.5
3. Thickness (monopoles): volume displacement by moving body; U^4 scaling; dominant at low M for rotating blades

FW-H solution (frequency domain, far field): p'(x,ω) = p'_T + p'_L + p'_Q [T = thickness; L = loading; Q = quadrupole]

Far-field approximation (negligible quadrupoles): p'(x,ω) ≈ [jω ρ₀ / (4π c₀ r)] × [∫ v_n e^{jω(r/c₀ - x_i n_i / c₀²)} dS] (thickness)

• [-jω / (4πc₀r)] × [∫ f_i x_i / r × e^{...} dS] (loading)

Trailing-Edge Noise

Dominant broadband noise source for airfoils: Turbulent boundary layer scatters at sharp trailing edge → noise radiated

Brooks-Pope-Marcolini (BPM) model (NASA TM-87328): SPL_TBL_TE = 10 log₁₀(δ_p* M⁵ D_h L / r²) + A(St) + ΔK₁ δ_p* = displacement thickness of pressure side BL [m] M = Mach number; D_h = directivity; L = span [m]; r = observer distance [m] St = Strouhal number = f δ_p* / U

Frequency of peak noise: f_peak ≈ 0.02 U / δ_p* [Hz; Strouhal 0.02 is approximate for TBL-TE]

Leading edge noise (laminar BL instability): Tonal noise at single frequency; f = U / λ_instability Eliminated by: trip wire, roughness elements on suction side LE

Fan/Propeller Noise

Tonal Noise (Blade Passage Frequency)

BPF = N_blades × RPM / 60 [Hz] Harmonics at 2×BPF, 3×BPF, etc.

Sources:

• Loading noise: steady and unsteady lift fluctuations (dominant)
• Thickness noise: volume displacement (significant at M_tip > 0.5)
• Interaction noise: blade passing through wake of upstream stator

Sound power level (propeller, FW-H loading term): L_w = 20 log₁₀(T) + 40 log₁₀(V_tip/c₀) + C [T = thrust; V_tip = tip speed]

Broadband Fan Noise (OASPL Estimation — NASA SP-8018)

Fan total sound power level: L_w = 10 log₁₀(Q × ΔP²) + C_s - 20 log₁₀(η_total) Q = flow rate [m³/s]; ΔP = pressure rise [Pa]; η = efficiency; C_s = specific noise constant

Specific sound power level K_w: L_w = K_w + 10 log₁₀(Q) + 20 log₁₀(ΔP) [K_w = fan specific noise rating; typically 30–50 dB]

Tip speed effect: L_w ∝ 40–50 log₁₀(V_tip) → reduce tip speed to reduce noise

OASPL Prediction Procedures

Far-field OASPL from SPL spectrum: OASPL = 10 log₁₀[Σ_f 10^(SPL(f)/10)] [sum over 1/3-octave bands]

Directivity (jet noise): D(θ) = (1 - M cos θ)^(-5) [relative SPL in direction θ from jet axis; convection amplification]

A-weighting for annoyance: SPL_A = SPL + W_A(f) [W_A: -20 dB at 100 Hz; 0 dB at 1 kHz; +1.2 dB at 4 kHz] OASPL_A [dBA] = 10 log₁₀[Σ 10^((SPL(f)+W_A(f))/10)]

Numerical Aeroacoustics

Direct Noise Computation (DNC): DNS/LES + propagation; expensive; only for short range Hybrid approach: CFD (LES/URANS) for source region → FW-H or Kirchhoff for propagation Software: STAR-CCM+ (FW-H); OpenFOAM + libAcoustics; Ansys Fluent FW-H module; ACTRAN

Standards

Standard	Scope
NASA SP-8018	Propeller/fan noise prediction
NASA TM-87328	BPM airfoil self-noise model
ICAO Annex 16	Aircraft noise certification
ISO 3745	Sound power measurement (anechoic)
SAE AIR1407	Prediction of aircraft external noise

Output

Provide: dominant noise mechanism (trailing-edge/thickness/loading/jet mixing), scaling law (U^n; n = 4, 6, or 8), BPF tones [Hz] for rotating machines, predicted OASPL [dBSPL] at observer distance r [m], 1/3-octave band spectrum (500 Hz – 8 kHz), directivity pattern (angular dependence), noise reduction options (tip speed reduction → ΔL [dB], trailing-edge serrations, sweep angle), A-weighted OASPL [dBA], and applicable standard (NASA SP-8018, NASA TM-87328).