11.2 Acoustic analogy

Lighthill's acoustic analogy

Lighthill's acoustic analogy was the foundational breakthrough in aeroacoustics. It reframes the problem of aerodynamic noise by rearranging the exact equations of fluid motion into the form of a wave equation with source terms. Instead of solving the full compressible Navier-Stokes equations to find the sound field directly, you treat the turbulent flow as a known source that drives acoustic waves through a uniform medium at rest.

This separation is what makes the approach so useful: you can analyze the fluid dynamics and the acoustics somewhat independently.

Derivation of the wave equation

The derivation starts from two fundamental conservation laws for a compressible fluid: the continuity equation (conservation of mass) and the momentum equation (conservation of momentum).

1. Take the time derivative of the continuity equation.
2. Take the spatial divergence of the momentum equation.
3. Subtract one from the other to eliminate the mixed term $\frac{\partial^2 (\rho v_i)}{\partial t \, \partial x_i}$.
4. Rearrange the result into the form of an inhomogeneous wave equation:

$\frac{\partial^2 \rho'}{\partial t^2} - c_0^2 \nabla^2 \rho' = \frac{\partial^2 T_{ij}}{\partial x_i \partial x_j}$

Here $\rho' = \rho - \rho_0$ is the density perturbation, $c_0$ is the ambient speed of sound, and $T_{ij}$ is the Lighthill stress tensor. The left side is the standard wave operator; the right side is the source term that represents how the turbulent flow generates sound.

Lighthill stress tensor

The Lighthill stress tensor is defined as:

$T_{ij} = \rho v_i v_j + (p - c_0^2 \rho)\delta_{ij} - \tau_{ij}$

Each term captures a different physical mechanism:

• $\rho v_i v_j$: The Reynolds stress term, representing momentum flux due to turbulent velocity fluctuations. This is the dominant contributor in most high-speed flows.
• $(p - c_0^2 \rho)\delta_{ij}$: The excess pressure over what a purely acoustic disturbance would produce. This term accounts for entropy fluctuations (e.g., from hot jets) and nonlinear compressibility effects.
• $\tau_{ij}$: The viscous stress tensor. In most aeroacoustic problems at high Reynolds numbers, this term is negligible compared to the others.

The tensor $T_{ij}$ represents the difference between the actual stresses in the real fluid and the stresses that would exist in the uniform acoustic medium assumed on the left side of the wave equation.

Interpretation of source terms

The source in Lighthill's equation is the double divergence $\frac{\partial^2 T_{ij}}{\partial x_i \partial x_j}$, which has the character of a quadrupole. This is significant because quadrupole sources are relatively inefficient radiators at low Mach numbers. Lighthill showed that the acoustic power from a free turbulent flow scales as:

$P \propto \rho_0 \frac{v^8}{c_0^5}$

This is the famous $v^8$ law for jet noise. The steep velocity dependence explains why even modest reductions in jet exhaust velocity produce large noise reductions, and it's the theoretical basis for the shift to high-bypass turbofan engines.

Curle's analogy for stationary surfaces

Lighthill's original formulation applies to unbounded turbulent flows with no solid surfaces. Curle extended this in 1955 to account for stationary rigid boundaries, which is essential for problems like airframe noise, flow over landing gear, or wind noise around buildings.

Extension of Lighthill's analogy

Curle used an integral solution of Lighthill's equation combined with a Green's function that satisfies the boundary conditions on the solid surface. The result adds surface integral terms to Lighthill's volume integral:

• The volume integral over $T_{ij}$ still captures the quadrupole sources from turbulence in the flow.
• A surface integral of the unsteady pressure and viscous stress on the body introduces new source terms.

This means the solid boundary doesn't just reflect sound; it actively generates sound through its interaction with the unsteady flow.

Surface pressure fluctuations

Unsteady pressure fluctuations on a solid surface act as dipole sources. At low Mach numbers, dipoles radiate far more efficiently than quadrupoles. Specifically:

• Quadrupole power scales as $M^8$ (from Lighthill).
• Dipole power from surfaces scales as $M^6$.

So at low to moderate Mach numbers (typical of many airframe and automotive applications), the surface dipole sources dominate the far-field noise. This is why surface pressure fluctuations on landing gear, wing trailing edges, and side mirrors are such important noise sources.

Dipole sources at solid boundaries

The physical origin of the dipole is the fluctuating force that the surface exerts on the fluid (and by Newton's third law, the fluid exerts on the surface). The dipole strength is proportional to the unsteady force integrated over the surface area.

Directivity matters here: a compact dipole radiates most strongly along the axis of the force and has a null in the perpendicular plane. Understanding this directivity pattern helps engineers orient components or add shielding to reduce noise in specific directions.

Ffowcs Williams-Hawkings (FW-H) equation

The FW-H equation (1969) is the most general acoustic analogy. It extends Lighthill's and Curle's work to handle arbitrarily moving surfaces, making it the standard tool for rotating machinery noise: helicopter rotors, propellers, fans, and open rotors.

Generalization for arbitrarily moving surfaces

The FW-H approach introduces a mathematical control surface $f = 0$ that can move through the fluid. Using generalized functions (the Heaviside function $H(f)$ and the Dirac delta $\delta(f)$), the equation is written so that it's valid everywhere in space, with the surface effects appearing naturally as source terms.

The control surface can be either:

• Impermeable (coinciding with the physical body surface)
• Permeable (a fictitious surface placed in the flow to enclose all significant sources, sometimes called a Ffowcs Williams-Hawkings surface or integration surface)

The permeable surface formulation is especially useful in computational work because it can capture nonlinear near-field effects that would otherwise require evaluating the expensive volume quadrupole integral.

Monopole, dipole, and quadrupole sources

The FW-H equation decomposes the sound field into three distinct source types:

• Monopole (thickness noise): Caused by the displacement of fluid as the body moves through it. For a helicopter rotor, this is the sound generated simply because the blade occupies volume and pushes air aside. It depends on the blade geometry and its velocity.
• Dipole (loading noise): Caused by the unsteady aerodynamic forces (lift and drag) on the surface. For a rotor, this includes both the steady loading rotating through space and any unsteady loading from gusts, blade-vortex interaction, etc.
• Quadrupole (volume sources): Distributed throughout the flow volume outside the surface. These capture nonlinear propagation effects and become important at high tip Mach numbers (transonic and above), where shocks form near the blade tips.

At subsonic conditions, thickness and loading noise dominate. As the tip Mach number approaches 1.0, quadrupole contributions grow rapidly and can produce the sharp, impulsive "HSI noise" (high-speed impulsive noise) characteristic of helicopter rotors.

Retarded time formulation

Sound takes a finite time to travel from the source to the observer. The retarded time $\tau$ is defined by:

$\tau = t - \frac{|\mathbf{x} - \mathbf{y}(\tau)|}{c_0}$

where $\mathbf{x}$ is the observer position, $\mathbf{y}(\tau)$ is the source position at emission time, and $t$ is the observer time. Notice that $\tau$ appears on both sides, making this an implicit equation that must be solved iteratively for moving sources.

For a source approaching the observer, multiple emission times can map to the same observer time, producing signal compression and amplification (Doppler effects). This is particularly important for high-speed rotors where the advancing blade tip moves toward the observer at near-sonic speeds.

Kirchhoff's theorem and analogy

Kirchhoff's approach provides an alternative integral method for computing far-field noise. Rather than using the fluid mechanics equations directly, it starts from the linear wave equation and expresses the solution as a surface integral over a closed control surface.

Integral formulation for moving surfaces

The Kirchhoff integral relates the far-field sound to three quantities on the control surface:

• The acoustic pressure (or density perturbation) on the surface
• Its normal derivative on the surface
• Its time derivative on the surface

If you can obtain these quantities from a CFD simulation on a suitably placed control surface, you can propagate the sound to any far-field observer location without computing the acoustics in the entire volume between the surface and the observer.

Assumptions and limitations

Kirchhoff's analogy carries stricter assumptions than the FW-H equation:

• The control surface must enclose all nonlinear flow regions and noise sources. If any significant source lies outside the surface, it will be missed.
• The region outside the surface must satisfy the linear wave equation (no mean flow gradients, no nonlinear effects).
• The method is sensitive to the placement of the control surface. If it's too close to the body, nonlinear effects may not be fully enclosed. If it's too far, the CFD grid resolution may be insufficient to capture the acoustic waves accurately.

Comparison with FW-H equation

In practice, the permeable-surface FW-H formulation has largely replaced Kirchhoff's method in modern computational aeroacoustics because it offers similar computational convenience with fewer restrictive assumptions.

Applications of acoustic analogies

These analogies aren't just theoretical constructs. They're the workhorses behind noise prediction and reduction across the aerospace and turbomachinery industries.

Jet noise prediction

Lighthill's analogy remains the foundation for understanding jet noise. Turbulent mixing in the jet shear layer generates quadrupole sources, and the $v^8$ scaling law directly motivated the development of high-bypass turbofan engines, which produce the same thrust at lower exhaust velocities.

Modern jet noise reduction techniques informed by acoustic analogy predictions include:

• Chevrons (serrated nozzle trailing edges) that enhance mixing and reduce peak turbulence levels
• Microjets injected into the shear layer to modify the turbulence structure
• Variable-geometry nozzles that optimize the exhaust profile for different flight conditions

Helicopter rotor noise

The FW-H equation is the standard tool for rotor noise prediction. It captures the key noise mechanisms:

• Blade-vortex interaction (BVI) noise: Occurs during descent when blades pass through the tip vortices shed by preceding blades. This produces sharp, impulsive loading noise and is often the most annoying component.
• Thickness noise: Dominant at high advance ratios when the advancing tip Mach number is high.
• Broadband self-noise: Generated by turbulence in the blade boundary layer passing over the trailing edge.

Noise predictions from the FW-H equation guide rotor blade design (tip shape, twist distribution) and operational procedures (approach angles, descent rates) to minimize community noise.

Turbomachinery noise

Acoustic analogies help identify and rank noise sources in fans, compressors, and turbines:

• Rotor-stator interaction noise: Unsteady wakes from rotor blades impinge on downstream stator vanes, generating tonal noise at blade-passing frequency and its harmonics. Increasing the rotor-stator spacing or choosing appropriate blade/vane count ratios can reduce this.
• Blade self-noise: Broadband noise from trailing-edge turbulence, tip leakage flows, and laminar boundary layer instabilities.
• Buzz-saw noise: At supersonic tip speeds, shocks from individual blades propagate forward through the inlet, producing a characteristic harsh tone.

Computational aeroacoustics (CAA)

CAA methods numerically solve the governing equations to predict noise generation and propagation. They turn the acoustic analogies from analytical frameworks into practical engineering tools applicable to real geometries.

Numerical methods for acoustic analogies

Acoustic waves carry very small energy compared to the mean flow, so numerical schemes must be exceptionally accurate to resolve them without excessive dissipation or dispersion. Key requirements:

• High-order spatial schemes (4th order and above) to minimize numerical dispersion, which causes wave speeds to depend on frequency and corrupts the acoustic signal.
• High-order time integration (e.g., low-dissipation Runge-Kutta schemes) to maintain accuracy over the many time steps needed for wave propagation.
• Non-reflecting boundary conditions (e.g., characteristic-based, perfectly matched layers/PML, or buffer zones) to prevent spurious reflections from computational domain boundaries from contaminating the solution.

Hybrid CFD-CAA approaches

Direct computation of both the turbulent flow and the acoustic field in a single simulation (direct noise computation, or DNC) is prohibitively expensive for most practical problems because acoustic wavelengths and energy levels differ enormously from the turbulent scales.

The hybrid approach splits the problem:

1. Near-field CFD: Use LES, DES, or unsteady RANS to compute the turbulent flow in the source region. This captures the noise-generating mechanisms.
2. Acoustic propagation: Extract source data (surface pressures, flow variables on a permeable surface) from the CFD and feed them into an acoustic analogy (FW-H, Kirchhoff) or a linearized Euler solver to propagate the sound to the far field.

Coupling is typically one-way (flow affects acoustics, but acoustics don't feed back into the flow), which is valid when the acoustic energy is much smaller than the flow energy. Two-way coupling is needed only in special cases like combustion instabilities or resonance phenomena.

Challenges and future developments

• Computational cost: High-fidelity LES of the source region remains expensive, especially at high Reynolds numbers. Wall-modeled LES and adaptive mesh refinement are active areas of development.
• Turbulence modeling: The accuracy of noise predictions depends heavily on how well the simulation captures the unsteady turbulent structures. RANS-based methods often underpredict broadband noise.
• Complex geometries: Full aircraft or engine configurations require massive grids and efficient parallel solvers.
• Multidisciplinary integration: Coupling aeroacoustic predictions with structural vibration (vibroacoustics) and optimization algorithms for automated low-noise design.

Experimental validation

No prediction method is trustworthy without experimental validation. Acoustic measurements provide the ground truth against which analytical and computational models are assessed.

Acoustic measurements in anechoic chambers

An anechoic chamber is a room lined with sound-absorbing wedges that eliminate reflections, creating a free-field environment. This allows you to measure only the direct sound from the source without contamination from room acoustics.

Typical uses include measuring noise from:

• Scaled jet rigs
• Airfoil sections in open-jet wind tunnels
• Small-scale propeller and rotor models

Scaling from model to full scale requires careful attention to Reynolds number effects and Mach number matching.

Microphone array techniques

A single microphone tells you how loud the total source is, but not where the noise comes from. Phased microphone arrays solve this by using the time delays between signals at different microphones to localize sources spatially.

• Beamforming is the standard processing algorithm. It steers the array's focus to different locations and maps the source strength distribution.
• Deconvolution methods (e.g., DAMAS, CLEAN-SC) improve the spatial resolution beyond the classical beamforming limit.
• Arrays are routinely used in wind tunnel tests and flyover measurements to identify dominant noise sources on aircraft (e.g., distinguishing landing gear noise from flap side-edge noise).

Comparison with analytical and numerical results

Validation follows a systematic process:

• Compare predicted and measured spectra (sound pressure level vs. frequency) at multiple observer angles.
• Compare directivity patterns (how the noise level varies with angle from the source).
• Compare source maps from microphone arrays with predicted source distributions.
• Quantify agreement using metrics like overall sound pressure level (OASPL) and effective perceived noise level (EPNL).

Discrepancies between predictions and measurements highlight where the models need improvement, whether that's in the turbulence simulation, the acoustic propagation, or the assumptions of the analogy itself.