Unit 4: LINEARIZED AERODYNAMICS FOR AEROACOUSTIC APPLICATIONS

I. Orientation — Linear Flow Perturbations and Sound

Linearized aeroacoustics describes small unsteady disturbances superposed on a steady mean flow. It connects aerodynamic fluctuations—especially vortices, wakes, blade loading, and boundary interactions—to propagating pressure waves.

• Perturbation convention: Each flow variable is decomposed into mean and fluctuating parts:
  
  TEXT

    ρ = ρ₀ + ρ′,   p = p₀ + p′,   u = U₀ + u′

  
  Here, ρ, p, and u are total density, pressure, and velocity; subscript 0 denotes the mean state; and a prime denotes a small perturbation.
• Linearization assumption: Products of perturbations, such as ρ′u′ and u′·∇u′, are neglected because their magnitudes are second order.
• Acoustic condition: For an isentropic disturbance,
  
  TEXT

    p′ = a₀²ρ′

  
  where a₀ is the mean speed of sound.
• Flow–sound distinction: Vortical disturbances are mainly convected with the flow, whereas acoustic disturbances propagate relative to the fluid at approximately a₀.
• Frequency convention: Harmonic quantities are commonly represented as q′ = Re{q̂e⁻ⁱωt}, where q̂ is complex amplitude, ω is angular frequency in rad/s, t is time, and Re denotes the real part.
• Applicable regime: The theory is strongest for small amplitudes, modest mean-flow gradients, and wavelengths large enough for continuum mechanics to remain valid.

II. Linearized Unsteady Flow — Governing Equations

A. Basic linearized unsteady aerodynamic equations

The linearized conservation equations determine how small velocity, pressure, and density disturbances evolve in a prescribed mean flow.

• Material derivative: For a uniform mean velocity U₀, convection is represented by
  
  TEXT

    D₀/Dt = ∂/∂t + U₀·∇

  
  where ∂/∂t is the local time derivative and ∇ is the spatial-gradient operator.
• Continuity equation: Linearized conservation of mass gives
  
  TEXT

    D₀ρ′/Dt + ρ₀∇·u′ = 0

  
  where ∇·u′ is the perturbation dilatation.
• Momentum equation: For an inviscid uniform mean flow,
  
  TEXT

    ρ₀D₀u′/Dt = −∇p′

  
  Pressure gradients therefore accelerate the perturbation velocity; viscosity is normally retained only near walls or inside liners and apertures.
• Pressure equation: Linearized isentropic energy conservation gives
  
  TEXT

    D₀p′/Dt + γp₀∇·u′ = 0,
    a₀² = γp₀/ρ₀

  
  where γ is the ratio of specific heats.
• Convected wave equation: Eliminating u′ and ρ′ yields
  
  TEXT

    (D₀²/Dt² − a₀²∇²)p′ = 0

  
  where ∇² is the Laplacian. With U₀ = 0, this reduces to the ordinary acoustic wave equation.
• Velocity-potential form: For irrotational perturbations, u′ = ∇φ and
  
  TEXT

    p′ = −ρ₀D₀φ/Dt

  
  where φ is the perturbation velocity potential.
• Boundary condition: On a rigid stationary wall with unit normal n, no penetration requires u′·n = 0; on a vibrating wall with normal velocity vₙ, it becomes u′·n = vₙ.

B. Applicability and limitations

Linear equations efficiently predict wave propagation but exclude several mechanisms that become important at high disturbance amplitudes.

• Useful applications: Modal duct acoustics, blade-response calculations, scattering by edges, and impedance-boundary analysis can all be formulated in the frequency domain.
• Mean-flow effects: Convection changes apparent wavelength and directivity; nonuniform mean flow also introduces refraction and coupling between acoustic and vortical modes.
• Excluded behavior: Shock formation, strongly separated flow, turbulence–turbulence interaction, and nonlinear aperture losses require higher-order models or computational fluid dynamics.
• Singular regions: Leading edges, trailing edges, and perforation rims can generate locally large gradients even when the incident disturbance is small.

III. Vortex-Induced Sound — Aerodynamic Sources

A. Introduction to sound generation induced by vortex flow

Vortex sound is produced when unsteady vorticity interacts with mean-flow gradients, solid boundaries, other vortices, or changes in geometry.

• Vorticity definition: Vorticity is
  
  TEXT

    ω = ∇ × u

  
  where ω is the vorticity vector and × denotes the vector cross-product. Uniform translation alone does not radiate efficiently because it produces little fluctuating compression.
• Lighthill analogy: Flow-generated sound in a nominally stationary medium may be written as
  
  TEXT

    ∂²ρ′/∂t² − a₀²∇²ρ′ = ∂²Tᵢⱼ/(∂xᵢ∂xⱼ)

  
  where xᵢ are Cartesian coordinates and Tᵢⱼ is Lighthill’s stress tensor. Its dominant high-Reynolds-number contribution is approximately ρuᵢuⱼ.
• Source character: Free turbulence behaves primarily as a quadrupole source, while fluctuating forces exerted on solid surfaces create stronger dipole radiation at low Mach number.
• Vortex-force form: In low-Mach-number homentropic flow, a useful source term is associated with
  
  TEXT

    ∇·(ρ₀ω × u)

  
  The vector ρ₀ω × u is the Lamb-vector force density and identifies regions where vortical motion can exchange energy with sound.
• Boundary interaction: A vortex convecting steadily in an unbounded uniform flow is acoustically inefficient; rapid distortion near an edge or blade produces a time-varying surface force and substantial radiation.
• Scaling: The acoustic efficiency generally increases rapidly with Mach number M = U₀/a₀, so low-speed vortical flows contain much more hydrodynamic energy than radiated acoustic energy.

B. Physical interpretation and model limits

Vortex-source models reveal where sound originates, but source strength alone does not determine the measured field.

• Propagation dependence: Geometry, reflections, duct modes, and refraction determine whether locally generated sound reaches a receiver.
• Compact-source approximation: A source is acoustically compact when its dimension L satisfies kL ≪ 1, where k = ω/a₀ is acoustic wavenumber.
• Energy transfer: Sound is generated when the acoustic particle velocity correlates with the vortex-force field; an unfavorable phase relationship can instead remove acoustic energy.
• Modelling limit: Decomposing a disturbance into “sound” and “flow” becomes difficult inside strongly sheared or turbulent near fields because both may have comparable pressure signatures.

IV. Turbomachinery Cascades — Periodic Blade–Wake Interaction

A. Analysis of fan/compressor cascade

Cascade analysis predicts the unsteady blade loading and duct sound generated when periodic wakes or gusts encounter a row of fan or compressor blades.

• Cascade idealization: An infinite or annular periodic array represents blades of chord c, spacing s, and stagger angle relative to the mean flow. Periodicity reduces the analysis to one blade passage.
• Incident gust: An upstream wake may be represented by
  
  TEXT

    u′g = Re{ûg exp[i(kₓx + kᵧy − ωt)]}

  
  where ûg is gust amplitude, kₓ and kᵧ are streamwise and transverse wavenumbers, and x and y are corresponding coordinates.
• Interblade phase angle: Adjacent blades experience a fixed phase shift β; consequently, the blade response and pressure field satisfy a relation such as q′(y+s) = q′(y)eⁱᵝ.
• Reduced frequency: Unsteadiness is measured by
  
  TEXT

    kᵣ = ωc/(2U₀)

  
  where kᵣ is reduced frequency. Large kᵣ implies that quasi-steady aerodynamic assumptions are unsuitable.
• Blade conditions: No penetration is imposed on each blade surface, while a Kutta condition selects finite velocity and physically smooth circulation shedding at the trailing edge.
• Mode generation: Rotor–stator interaction produces circumferential duct modes. A common indexing relation is
  
  TEXT

    m = nB − qV

  
  where m is circumferential mode order, n is shaft-harmonic order, B is rotor-blade count, V is stator-vane count, and q is any integer.
• Cut-on condition: A mode radiates only if its axial wavenumber is real; otherwise, it decays exponentially and is cut off.

B. Applications and limitations

Cascade solutions connect aerodynamic forcing to tonal noise but rely on idealized geometry and flow.

• Outputs: The analysis supplies unsteady lift, surface-pressure distributions, wake response, modal amplitudes, and acoustic directivity.
• Design use: Blade-count selection, rotor–stator spacing, sweep, lean, and acoustic treatment can reduce the strength or cut-on status of dominant modes.
• Limitations: Real machines introduce finite blade counts, tip clearance, swirl, radial flow variation, turbulence, and nonlinear transonic effects.
• Three-dimensional correction: Annular-duct eigenfunctions and radial mode order are required when spanwise variation cannot be represented by a two-dimensional cascade.

V. Edge Scattering and Acoustic Treatment — Coupled Noise Control

A. Vortex sound model of trailing edge noise and liner impedance

Trailing-edge models describe vortical disturbances scattered into sound, while liner impedance specifies how treated boundaries absorb or reflect that sound.

1. Trailing-edge noise

   • Mechanism: Turbulent boundary-layer pressure and vorticity convect toward the trailing edge; termination of the solid surface forces a rapid pressure adjustment that radiates as a dipole-like field.
   • Convection relation: A frozen disturbance approximately satisfies
     
     TEXT

          kₓ = ω/Uc

     
     where kₓ is streamwise hydrodynamic wavenumber and Uc is convection velocity, commonly below the external flow speed.
   • Scattering condition: The edge converts a subsonically convected, non-radiating disturbance into acoustic waves whose wavenumber magnitude is k = ω/a₀.
   • Influencing quantities: Noise depends on boundary-layer thickness, wall-pressure spectrum, spanwise coherence, edge geometry, observer angle, and Mach number.
   • Finite span: Only spanwise-correlated portions radiate coherently; contributions separated by more than the coherence length add mainly in mean-square pressure rather than amplitude.

2. Liner impedance

   • Definition: The locally reacting acoustic impedance is
     
     TEXT

          Z(ω) = p′/vₙ = R + iX

     
     where vₙ is inward normal acoustic velocity, R is resistance, X is reactance, and i² = −1.
   • Normalized form: ζ = Z/(ρ₀a₀) compares liner impedance with the characteristic impedance of the fluid.
   • Physical roles: Positive R dissipates acoustic energy; positive or negative X represents inertive or compliant energy storage, depending on the harmonic convention.
   • Optimum behavior: Strong attenuation requires both adequate resistance and reactance that couples the liner to the targeted duct mode; excessive resistance makes the surface nearly rigid.
   • Mean-flow effect: Grazing flow changes boundary-layer refraction, effective impedance, and stability, so a no-flow impedance measurement may not represent installed performance.

B. Applications and limitations

The combined framework supports quieter blades and treated ducts but requires separate near-field and propagation models.

• Applications: Serrated or porous trailing edges alter scattering, while nacelle liners attenuate the resulting fan tones and broadband noise.
• Frequency dependence: A single-degree-of-freedom liner is strongly effective near resonance but provides limited broadband attenuation.
• Model limits: Sharp-edge theory, frozen turbulence, local reaction, and uniform grazing flow may fail for thick edges, separated flow, or highly nonuniform liners.

VI. Perforated Interfaces — Vorticity, Scattering, and Dissipation

A. Vortex sound interaction of perforated plates

A perforated plate couples acoustic pressure to oscillatory aperture flow, generating rim vorticity that can absorb, scatter, or nonlinearly modify incident sound.

• Geometric parameters: Performance depends on hole diameter d, plate thickness t, open-area ratio σ, hole spacing, and aperture shape.
• Pressure-jump model: For a homogenized plate,
  
  TEXT

    Δp = Zp v̄ₙ

  
  where Δp is pressure difference across the plate, Zp is specific plate impedance, and v̄ₙ is surface-averaged normal velocity.
• Hole velocity: Continuity gives approximately
  
  TEXT

    vh = v̄ₙ/σ

  
  where vh is average velocity inside each hole. Small σ therefore creates large local velocities and strong rim shear.
• Linear impedance: A simple aperture model is
  
  TEXT

    Zp ≈ Rp + iωρ₀le/σ

  
  where Rp is viscous and radiation resistance and le is effective neck length, including end corrections.
• Vortex production: Flow separates at the sharp rim during each half-cycle and forms alternating vortex rings or sheets. Their kinetic energy is supplied by the acoustic field.
• Dissipation mechanism: Viscous wall stress, vortex shedding, and turbulent mixing convert organized acoustic energy into heat and small-scale vortical motion.
• Nonlinear regime: At high acoustic velocity, resistance becomes amplitude-dependent, often increasing roughly with ρ₀|vh|; harmonic generation then invalidates a purely linear impedance.
• Bias and grazing flow: A steady flow through or across the holes changes separation, convection, resistance, and possible feedback between apertures.

B. Applications and limitations

Perforated plates are compact acoustic-control elements, but their performance is sensitive to operating conditions.

• Applications: They serve as facesheets over honeycomb cavities, microperforated absorbers, combustor liners, mufflers, and flow-resistant trailing-edge treatments.
• Resonant absorption: The aperture mass combines with cavity compliance to form a Helmholtz-type resonator; maximum absorption occurs near the frequency where net reactance approaches zero.
• Interaction effects: Closely spaced apertures have overlapping near fields, so isolated-hole end corrections and independent-vortex assumptions become inaccurate.
• Practical limits: Temperature, contamination, manufacturing tolerance, high sound pressure, and mean-flow direction can shift impedance and resonance substantially.