Unit 4: LINEARIZED AERODYNAMICS FOR AEROACOUSTIC APPLICATIONS - Subjective Questions

The principal assumptions are:

• The flow variables are decomposed into a steady mean component and a small perturbation:
  
• Perturbations are small compared with their mean values, such that , , and .
• Products of perturbation quantities, such as , are second order and are neglected.
• The fluid is usually assumed to be inviscid and compressible.
• Acoustic perturbations are commonly treated as isentropic, giving , where is the mean speed of sound.
• The mean flow is assumed to be known. For the simplest equations, it is uniform and steady.

These assumptions transform the nonlinear governing equations into linear equations that permit superposition and frequency-domain analysis.

Let

The nonlinear continuity equation is

Substitution followed by removal of steady terms and products of perturbations gives

For inviscid flow, the momentum equation is

For a steady, uniform mean flow, its linearization yields

Defining the mean-flow material derivative as

the equations become


Together with , these form the basic linearized inviscid aerodynamic system.

For a uniform mean flow, the linearized equations are


with .

Apply to the continuity equation:

Taking the divergence of momentum gives

Combining these equations and using produces

or

Physical meaning:

• The equation describes acoustic disturbances carried by a uniform moving medium.
• Relative to the fluid, waves propagate at .
• Relative to a stationary observer, propagation speed and wavelength depend on direction because the mean flow convects the disturbance.
• When , it reduces to the ordinary acoustic wave equation.

Acoustic mode:

• Contains coupled pressure, density, and velocity fluctuations.
• Is compressible and propagates relative to the mean flow at approximately the sound speed .
• For a plane wave, pressure and particle velocity are related through an acoustic impedance.

Vortical mode:

• Is characterized primarily by velocity and vorticity fluctuations.
• In a uniform inviscid flow, it has negligible pressure and density perturbations to first order.
• It is convected at the mean-flow velocity rather than propagating at the sound speed.

Entropy mode:

• Contains entropy, density, and temperature fluctuations, but has negligible pressure fluctuation in uniform flow.
• It is also convected with the mean flow.

Vortical and entropy disturbances can generate sound when they encounter nonuniform mean flow, solid boundaries, blade rows, trailing edges, or changes in area. Thus, non-acoustic modes may be converted into acoustic modes.

An isolated vortex in uniform, unbounded, low-Mach-number flow mainly produces a hydrodynamic pressure field that decays rapidly and does not efficiently radiate to the far field. Sound is generated when the vortex experiences acceleration, deformation, or interaction.

Important sound-producing interactions include:

• Vortex passage near a rigid edge or body, which creates an unsteady surface force.
• Vortex stretching, pairing, breakdown, or rapid changes in circulation distribution.
• Interaction with mean-flow shear, shocks, or another vortex.
• Impingement on blade leading edges, trailing edges, liners, or perforated plates.

The interaction converts part of the vortical energy into propagating pressure fluctuations. In vortex-sound theory, the effective source is related to the Lamb vector

where . Compact unsteady forces commonly radiate as dipoles, while free turbulent flow is often represented by weaker quadrupole sources at low Mach number.

The Lamb vector is defined as

where is vorticity. Using the identity

the inviscid momentum equation may be expressed in terms of stagnation enthalpy as

Taking the divergence and combining it with the linearized continuity relation gives, under common low-Mach-number and homentropic approximations, an acoustic analogy of the form

Thus, the divergence of the Lamb vector acts as a vortex-sound source. A steady, symmetric vortex distribution can have weak radiation, whereas rapid changes in caused by edges, blades, or vortex deformation produce stronger acoustic waves.

Hydrodynamic pressure fluctuations:

• Are associated with the near field of vortices and turbulent structures.
• Are strongly convected with the flow.
• Commonly have subsonic phase velocity and high spatial wavenumber.
• Decay rapidly with distance from the vortex region.
• Do not necessarily satisfy the homogeneous acoustic wave equation.

Acoustic pressure fluctuations:

• Are compressible disturbances that propagate away from the source.
• Travel at the sound speed relative to the fluid.
• Have a pressure-velocity relationship characteristic of acoustic waves.
• In the far field, their amplitude generally decays algebraically, such as for spherical spreading.
• Carry net acoustic energy to distant observers.

The distinction is essential in experiments and simulations because microphones close to a vortex may measure predominantly hydrodynamic pressure rather than radiated sound.

As a vortex approaches a solid edge, the no-penetration condition alters its velocity and pressure field. The edge scatters the convected hydrodynamic disturbance and produces a rapidly varying aerodynamic force on the surface.

For a compact body, the far-field pressure associated with an unsteady force has the dipole form

where points toward the observer and is the observer distance.

The interaction is efficient because:

• The edge imposes a sharp spatial change in boundary conditions.
• Hydrodynamic pressure is converted into a propagating acoustic wave.
• The resulting fluctuating lift or force produces dipole radiation.
• The source can be coherent over part of the edge span.

The sound has directional lobes and is usually strongest in directions related to the fluctuating-force axis.

A cascade is modeled as a periodic row of blades subjected to small unsteady disturbances.

Typical assumptions are:

• The mean flow through the blade row is steady and known.
• Incoming wakes, vortical gusts, or acoustic waves have small amplitudes.
• Blade loading perturbations can therefore be determined using linearized equations.
• Blades are identical and equally spaced, allowing a periodic or phase-shifted condition.
• The fluid is inviscid outside thin boundary layers and wakes.
• Two-dimensional blade sections may be used when spanwise variations are negligible.

Important parameters include:

• Blade chord and pitch .
• Solidity .
• Mean Mach number and flow angle.
• Reduced frequency .
• Interblade phase angle .
• Blade count, stagger angle, and incident-gust wavenumbers.

These parameters control unsteady blade loading, acoustic-mode generation, and the propagation or decay of pressure modes.

A wake or vortical gust arriving at a compressor blade row contains velocity fluctuations. As the gust reaches each blade, it changes the local incidence angle and relative velocity.

The process is:

1. The incoming gust is convected toward the cascade.
2. Its normal velocity component perturbs blade circulation and surface pressure.
3. The pressure difference between the two blade surfaces creates fluctuating lift.
4. The fluctuating lift radiates dipole sound.
5. Cascade periodicity organizes the response into spatial modes determined by the interblade phase angle.

If the incident gust is represented by

the response has the same frequency but its amplitude and phase are modified by the cascade transfer function. Some generated modes are cut-on and propagate through the duct, while others are cut-off and decay exponentially away from the blade row.

Consider a rotor with blades rotating at revolutions per minute. The rotational frequency is

A stationary observer encounters blade passages during each revolution. Therefore, the blade-passing frequency is

Its harmonics occur at

Rotor blades generate periodic wakes, potential-field disturbances, and sometimes tip vortices. When these disturbances strike downstream stator vanes, they produce periodic incidence changes and fluctuating stator loading. Conversely, the stator potential field can affect the rotor.

The interaction creates discrete tonal sound because the disturbances are phase coherent. Blade and vane counts determine the circumferential mode orders, while duct geometry and mean flow determine whether each mode is cut-on and able to propagate. Broadband noise may also be produced by turbulence within the rotor wakes.

Pressure fluctuations in a uniform circular duct can be decomposed into modes:

where is the circumferential order, is the radial order, and is the axial wavenumber.

Cut-on mode:

• Has a real axial wavenumber .
• Propagates along the duct and transports acoustic energy.
• Can contribute to inlet or exhaust far-field noise.

Cut-off mode:

• Has an imaginary axial wavenumber.
• Decays exponentially with axial distance.
• Remains localized near the blade row or disturbance.

The cut-on condition depends on frequency, duct dimensions, modal eigenvalue, sound speed, and mean Mach number. Cascade interactions may generate many spatial modes, but only cut-on modes radiate efficiently from the duct. This explains why a large unsteady blade loading does not always produce equally large far-field noise.

Trailing-edge noise is produced when boundary-layer turbulence or coherent vortices convect over an airfoil trailing edge. Before reaching the edge, these disturbances mainly create a non-radiating hydrodynamic pressure field.

At the trailing edge:

• The abrupt termination of the solid boundary scatters the hydrodynamic pressure.
• The Kutta condition constrains the fluctuating flow leaving the edge.
• Time-dependent surface pressure creates fluctuating lift near the edge.
• Part of the vortical disturbance is converted into acoustic waves.

In a vortex-sound interpretation, the relevant source is connected with the interaction of vorticity and velocity through . The solid edge modifies this source and introduces an effective dipole. Noise depends on convection velocity, turbulence intensity, boundary-layer thickness, spanwise coherence, edge geometry, Mach number, and frequency. Sharp trailing edges generally scatter disturbances more efficiently than smoothly varying boundaries.

Frequency dependence:

• A turbulent eddy of streamwise scale convected at velocity produces a characteristic frequency of order
  
• Large eddies dominate lower frequencies, while small eddies contribute at higher frequencies.
• Boundary-layer thickness supplies an important length scale, often represented by a Strouhal number such as
  
• Finite trailing-edge thickness or bluntness can introduce vortex-shedding tones.

Directivity:

• The fluctuating aerodynamic force makes trailing-edge noise predominantly dipolar.
• Radiation is weak along the dipole axis and stronger in directions normal to it.
• Mean-flow convection skews and amplifies the downstream radiation pattern.
• Airfoil geometry and observer position modify the lobes through diffraction and interference.

The final spectrum also depends on spanwise coherence: coherent source regions add constructively, whereas poorly correlated regions contribute largely through their mean-square pressures.

The specific acoustic impedance of a liner is the ratio of acoustic pressure to the wall-normal acoustic particle velocity:

Its normalized form is

Resistance :

• Is the real part of impedance.
• Represents irreversible acoustic-energy dissipation caused by viscosity, vortex shedding, turbulence, and thermal losses.
• A suitable resistance is required for effective absorption.

Reactance :

• Is the imaginary part.
• Represents reversible storage of kinetic or compressive energy.
• Positive reactance is conventionally associated with stiffness-like behavior, while negative reactance may be associated with mass-like behavior, depending on the adopted time convention.

Near a liner resonance, the reactance passes through zero and absorption can become high if the resistance is appropriately matched to the surrounding acoustic field.

An impedance liner replaces a perfectly rigid boundary with a surface across which pressure fluctuations can drive normal velocity. Its boundary condition is

The liner can affect trailing-edge noise through several mechanisms:

• Absorption: Its resistive component converts acoustic energy into viscous and thermal losses.
• Phase modification: Its reactive component changes the phase between pressure and wall-normal velocity.
• Reduced scattering: A compliant or porous region can weaken the abrupt rigid-boundary termination responsible for edge scattering.
• Vortex interaction: Flow through pores can create small vortices, changing near-edge hydrodynamic pressure and coherence.
• Mode control: In a duct or cascade, impedance changes modal attenuation and propagation constants.

Performance is frequency dependent. Too little resistance provides weak dissipation, while excessive resistance makes the surface nearly rigid. Strong grazing flow may also change the effective impedance and produce additional flow noise.

A perforated plate backed by a sealed cavity can be represented by a resistance, a perforation inertance, and a cavity compliance.

Let the plate have open-area ratio , effective hole length , and a resistive loss . The air moving through the holes behaves like an acoustic mass per unit total area:

Its impedance is

For a cavity of depth that is acoustically compact, its compliance per unit area is approximately

and its impedance is

The total surface impedance is therefore

Resonance occurs when the net reactance vanishes:

so

At resonance, . Strong absorption occurs when this resistance is suitably matched to the acoustic field.

When a convected vortex approaches a perforated plate, it produces an unsteady pressure difference across the plate. This pressure drives oscillatory flow through the holes.

The main processes are:

• The incident vortex is distorted by the blockage and redistributed around the holes.
• Fluid accelerates through each aperture, creating localized jets.
• Shear layers form at the hole rims and may roll up into secondary vortices.
• Vorticity may be transmitted, reflected, dissipated, or converted into acoustic waves.
• Viscous losses occur within the holes and near their edges.
• A backing cavity can store compressive energy and create resonance.

The interaction depends on hole diameter, plate thickness, porosity, vortex size and strength, grazing-flow velocity, and excitation frequency. At small amplitudes, a linear impedance model may be adequate. At larger amplitudes, vortex shedding and jet separation make the resistance amplitude dependent and the response nonlinear.

Rigid plate:

• Enforces approximately zero wall-normal velocity.
• Strongly deflects the vortex and supports substantial unsteady surface pressure.
• Sharp edges can efficiently scatter hydrodynamic disturbances into sound.

Porous plate:

• Permits distributed flow through interconnected pores.
• Reduces the pressure difference across the surface.
• Can weaken coherent edge scattering and dissipate energy through viscous resistance.
• Its behavior is often represented by an effective impedance or permeability.

Perforated plate:

• Allows flow through discrete apertures.
• Produces concentrated aperture jets, rim shear layers, and secondary vortex shedding.
• Exhibits inertial, resistive, and sometimes cavity-compliance effects.
• May absorb sound near resonance but can generate extra flow noise at high amplitudes.

Thus, rigid plates mainly reflect and scatter disturbances, whereas porous and perforated plates can transmit and dissipate energy. The acoustic benefit depends on selecting appropriate porosity, hole dimensions, impedance, and operating conditions.

Porosity: Increasing the open-area ratio generally reduces both acoustic resistance and inertance per unit total area because a larger area is available for oscillatory flow.

Hole diameter: Smaller holes increase viscous influence and usually increase resistance. Larger holes reduce linear viscous resistance but can support stronger jetting and nonlinear vortex shedding at high amplitudes.

Plate thickness: Increasing thickness increases the effective air-column length and hence the inertive reactance. A simple estimate is

where includes end corrections.

Grazing flow: Mean flow across the plate modifies the aperture shear layers, increases or alters resistance, changes effective reactance, and may introduce bias-flow or grazing-flow noise. It can also make impedance dependent on flow direction and acoustic amplitude.

These parameters must be optimized together. Low resistance gives poor dissipation, excessive resistance limits aperture motion, and incorrect reactance moves the absorption peak away from the desired frequency.