| 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:
- Volume sources (quadrupoles): T_ij in volume (turbulence); U^8 scaling; only at high M
- Surface loading (dipoles): f_i on body surface; U^6 scaling; dominant at M < 0.5
- 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).
