Aero Acoustic Field & its Modeling @ Zeus Numerix

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Aero-Acoustic Field & its
Modeling
Advanced topic on Aeroacoustics
Prof GR Shevare

2
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Contents
Aero-Acoustic Field & its Modeling: Zeus Numerix
 Preliminaries
 Acoustic stealth
 Computational aeroacoustics (CAA) vs. Computational fluid dynamics (CFD)
 Quantifying pure tone and tones
 The human hearing
 Analysis of Random noise
 Methods of solution
 Derivation of governing equations
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Contents
 Linearization: from fluid dynamics to aeroacoustics
 CAA for practical applications-Hybrid methods
 Two practical applications of CAA
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Preliminaries
Aero-Acoustic Field & its Modeling: Zeus Numerix21 Oct 2020
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Acoustics – Science & its Applications
 Physics of acoustics can be considered as dynamics (time varying motion) of
continua.
 Study of acoustics includes fields such as building/ room acoustics (sound because of
vibration and reflections), musical instruments, ..., to aeroacoustics (sound induced
by flow of fluids)
 Topics in aeroacoustics
 Generation & amplification for music, communication, etc.
 Physiological acoustics – science of sound reception by ear
 Reducing noise - defence and aerospace
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Aeroacoustics is...
 The origin of noise could be:
 Aero-elastic structural vibration producing and radiating sound. We note aero-elasticity
itself is complex
 Unsteady fluid motion bounded by rigid surfaces
 Unsteady fluid motion in infinite domain (no walls)
 The last two are important and we are interested in the noise produced by aircraft
air-frame, aero-engines, propellers, vibrating machinery, resonance in ducts, free
jets, cavities, air-frame noise, etc.
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 Aeronautics
 Air-frame noise,
 Stator/rotor, jet noise, combustion
noise, propellers
 Cavities, high lift devices
 Naval Architecture
 Machinery, equipment and crew
movements
 Propellers
 Ship hull
 Automobiles
 Mirrors, cavities
 HVAC systems
 Fans, duct acoustics
 wind turbines
 Rotors
Acoustics in Engineering ...
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Acoustic Signals & Humans
 30 Whisper
 50 Rainfall, quiet office, refrigerator
 60 Dishwasher, normal conversation
 70 Traffic, vacuum cleaner, restaurant
 80 Alarm clock, subway, factory noise
 90 Electric razor, lawnmower, heavy truck, road drill at 7 m
 100 Garbage truck, chain saw
 110 Rock concert, power saw
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Acoustic Signals & Humans
 120 Jet take-off, nightclub, thunder
 130 Jack hammer
 140 Shotgun, air raid system
 180 Rocket-launching pad
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Types of Acoustic Signals
 Pure tone
 Superscripts c & s stand for amplitude of cosine and sine
 Tone
 Complex tone : superposition of tones (here p’(t) is generally non-periodic)
 Pure noise : p’(t) has all frequencies. Noise is permanent and stochastic
𝑝′(𝑡) = 𝑝 𝑐
cos( 𝜔𝑡) + 𝑝 𝑠
sin( 𝜔𝑡)
𝑝′(𝑡) =
𝑛=0
∞
𝑝 𝑛
𝑐
cos( 𝑛𝜔𝑡) + 𝑝 𝑛
𝑠
sin( 𝑛𝜔𝑡)
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Types of Acoustic signals
 Noise : superposition of complex tones and pure noise
 Impulse : short duration event (rms-value can not be defined)
 Bang : alternating impulse with zero time integral (sonic boom, N-wave)
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Acoustic stealth
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Noise in Aerospace & its Position Stealth
 In simple words, stealth is reducing the amplitude of signals emanating from
aerospace vehicles.
 AEIO signals:
 Acoustic (unsteady pressure wave)
 Electromagnetic (unsteady electromagnetic wave)
 Infrared (radiation and receipt of thermal energy)
 Optical (receipt of light from illuminated objects)
 Acoustic signals are slower than most aerospace vehicles except in the case of
airships and helicopters
 However, acoustic signals are an order of magnitude faster in marine applications
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Acoustic Signals from Helicopters Rotor
 The rotor generates different types of noise:
 Thickness noise - blade periodically displaces air. This sound propagates mostly in
the plane of the rotor.
 Rotating blade at non-zero angle of attack imposes rotating forces onto the
surrounding air This sound generally propagates in a direction perpendicular to the
plane of the rotor
 These two types of noise always occur, even in a hover condition
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Acoustic Signals from Helicopter Rotor
 In level flight, the advancing blade may produce shocks periodically, which result in
high speed impulsive noise (HSI noise)
 On the retreating blade maximum angle of attack causes flow separation. The
separated flow causes broad band noise.
 Tip vortices are shed by blades. While descending or at moderate speeds they may
impact other blades, causing blade-vortex interaction (BVI) also called “blade slap”.
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Acoustic Signals from Helicopters - Anti-
Torque Noise
 The anti-torque refers to tail rotor mechanism. This mechanism is the same as that of
main rotor
 The anti-torque (tail rotor) is subject to non-uniform wake from main rotors, which
produces additional unsteady flow and hence noise
 Ducted tail rotors will have different noise characteristic as the flow entering them is
made somewhat uniform by the nacelle (shielding effect)
 Some helicopters use blower which is completely inside the tail boom instead of
rotor to minimize this noise
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Acoustic Signals from Helicopters- Engine
Noise (Turbo-Shaft Engine)
 This consists of rotational noise by compressor(s) and turbine(s) and broadband
noise from combustion chamber
 The compressor fan produces a high frequency tone emanating from the engine
inlet. But it attenuates quickly in the atmosphere
 Turbo-prop engines exhaust produces broadband noise, though much less than
exhaust of turbo-jet engines. It is prominent when helicopter is overhead (when
rotor noise is less dominant)
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Acoustic Signals from Helicopters- Engine
Noise (Piston Engine)
 Piston engines (normally used on smaller helicopters) produces exhaust noise.
 The exhaust noise is broadband noise dominated by low frequencies
 The exhaust noise has tones associated with the cylinder firings
 Engine exhaust noise can be controlled successfully by relatively easily through
mufflers
 Noise can be reduced using of upturned exhausts, mufflers and resonators
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Acoustic Signals from Helicopters- Flight
Condition Dependent Noise
 Noise depends on flight condition and position of observer
 During the take-off, the main rotor produces maximum thrust. In classical open tail
rotor the tail rotor noise is dominant as it requires to produce maximum anti-torque,
vertical tail being inactive
 Exhaust pipe noise can be appreciable, especially for an observer positioned behind
the helicopter
 For ducted fans, observers directly under the flight path are shielded and they hear
more exhaust noise
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 In level cruise, the power requirement is less than in take-off. The anti-torque
system is augmented by the vertical fin, thus tail rotor provides lower thrust
 Forward speed generates higher velocities on the advancing blades of the main and
tail rotors. This produces high speed impulsive (HSI) noise, especially at low ambient
temperatures. Modern helicopters may therefore operate at lower rpm and may
have thin air foil, especially at the tip. In which case, interaction of open tail rotor
with the main rotors wake can be actually more than that of main rotor noise
 Tail rotor noise is typically the more predominant in light helicopters than in heavy
helicopters.
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 Loudest flight condition is the approach. Even though power requirements is the
lowest, blade vortex interaction (BVI) produces “blade slap” noise
 Blade vortex interactions is difficult to model and is the most difficult source of noise
to predict and/or mitigate in the helicopter design
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Noise in Marine Environment
 In the marine environment, the attenuation of EM waves is higher compared sound
waves. Acoustics in more important in ships, submarines
 Acoustics considerations affect many systems, including sensors and detection,
communication, and stealth operations
 Knowledge of the noise a ship creates during operation is therefore critical for the
successful operation of many vessels.
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Noise in Marine Applications
 The ship’s acoustic signature includes noise from;
 ship-board machinery
 the flexing of hull plates/members,
 crew operations
 cavitation from propellers and
 hydrodynamic hull noise
 The hydrodynamic hull noise is created the rigid ship hull especially stern (aft-most
part of the hull)
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Noise in Marine Applications
 Though its contribution is not large compared machinery and propeller noise, hull
noise is important in ship’s sonar signature
 Sonar is especially sensitive to hull noise as it is located on the bow of the ship
(ahead of the machinery and propeller)
 The sonar dome hears the noise generated in the bow of the ship and hence limits
the sonar domes detection envelope
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Computational aeroacoustics
(CAA) vs.
Computational fluid dynamics
(CFD)
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Computational Fluid Dynamics (CFD)
 The computational fluid dynamics (CFD) is used for solving spatial differentials equations
governing fluid flow and heat transfer.
 Using CFD, fairly accurate analysis of aerospace vehicles, turbo-machines, nuclear reactors,
etc. can be carried out
 CFD is a mature technology. Many consider CFD as a digital wind tunnel, thanks to
availability of cheap computing power.
 Large eddy simulations (LES), an advanced technique in CFD is considered as the state-of-art
in this area
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Computational Aeroacoustics (CAA)
 Any computational method which simulates the sound associated with a fluid flow can be
called as computational acoustics or CAA
 Though acoustics1 is old as fluid dynamics, Hardin and Lamkin2 are credited with introducing
the terminology computational aeroacoustics and the abbreviation CAA.
 The CAA methods are specially tailored computational fluid dynamics (CFD) methods for
resolving production and propagation of sound waves accurately
 Now CAA is being used for many applications
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CAA & CFD
 In the past, CFD and CAA have evolved independently, despite of the fact that
computational aero acoustics and the large-eddy simulations (LES) in CFD are derived
from the same fundamental equations of fluid dynamics
 In principle, they should be be studied together.
 But acoustics is a multiscale problem.
 The small acoustic perturbations are drowned in the numerical errors of the much
larger aerodynamic quantities
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CAA vs CFD
 The space and time resolution of CAA combined with the large size flow domains
require ridiculously large no of cells and time steps.
 Further, even though necessary computing power is available, the numerical
schemes well known in CFD become useless in CAA as dispersion and diffusion errors
become large
 In real life applications, flow field generating the sound is matched to a acoustic field
in homogeneous flow field
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CAA vs CFD
 Thus in such applications, the sound-generation and sound-propagation are
considered separately with the underlying assumption that the acoustic waves do
not affect the flow field even though flow field is responsible for acoustic field.
 We note that CAA is a newer research field and there are no recommended methods
for reliable methods for predicting sound. Many techniques can be found. Each
seems to work in one area but unsatisfactory in other
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Quantifying pure tone and
tones
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Defining Acoustic Signal
 Sound is unsteady or fluctuating pressure (SI units Pascals)
 In practical applications, averaging happens with some weight W(t’, T) and T is finite,
in which case p0 contains some infra-sonic sound
𝑝′(𝑡) = 𝑝(𝑡) − 𝑝0
𝑤ℎ𝑒𝑟𝑒𝑝0
𝑖𝑠𝑔𝑖𝑣𝑒𝑛𝑏𝑦
𝑝0
= 𝑝 = lim 𝑇→∞
1
𝑇
−𝑇/2
𝑇/2
𝑝(𝑡 + 𝑡′)𝑑𝑡′
𝑝0
≈ 𝑝 𝑇
=
1
𝑇
−𝑇/2
𝑇/2
𝑝(𝑡 + 𝑡′)𝑊(𝑡′, 𝑇)𝑑𝑡′
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Defining Acoustic Signal
 If T=0.5 s and W=1, all frequency components higher than 16Hz will be suppressed
by more than 96% in p0
 Alternately, the error in p’ due to suppression of lower than 16Hz frequencies is less
than 4 %
 If weighing function is W = 1 + cos(2πt′/T), the contribution of higher frequencies to
p0 can be increased to as high as 99.9%
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Acoustic Quantities
 Strength of acoustic signal (the unsteady pressure) is given by its root mean square
or ”rms” value
 The least strength perceivable acoustic signal (by human ear) is 10-5 Pa and
maximum strength of acoustic signal (which does not cause damage to human ear)
is 102 Pa
 This is a large range (approx. ~107) and hence a logarithmic scale is commonly used.
𝑝 = (𝑝′)2
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Acoustic Quantities
 The logarithmic scale is called sound pressure level Lp or SPL and is defined as
follows:
 The reference pressure pref is the threshold of hearing for sound at 2 kHz.
 Since 20 log (2)  6, decibel value is increased by approx. 6, if the pressure
magnitude is doubled
𝐿 𝑝(𝑑𝑒𝑐𝑖𝑏𝑒𝑙) = 10 log
𝑝
𝑝 𝑟𝑒𝑓
2
= 20 log
𝑝
𝑝 𝑟𝑒𝑓
𝑝 𝑟𝑒𝑓 = 2𝑥10−5 𝑃𝑎
the pressure corresponding to the Brownian motion
of air molecules is ≈ 0.5 · 10−5 Pa
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Acoustic Quantities
 Acoustic signal also has unsteady fluid velocity (popularly referred to as acoustic
particle velocity) v’.
 The sound pressure with the acoustic particle velocity represents the acoustic signal
 This velocity is completely different from the speed of sound. The speed of sound is
the propagation speed of the signal through the fluid
 The acoustic particle velocity refers to signal amplitude similar to sound pressure
level
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Acoustic Quantities
 The sound intensity I and sound intensity level LI are defined in terms of sound
pressure and acoustic particle velocity
𝐿 𝑣(𝑑𝑒𝑐𝑖𝑏𝑒𝑙) = 10 log
𝐯
𝑣 𝑟𝑒𝑓
2
= 20 log
𝐯
𝑣 𝑟𝑒𝑓
𝑣 𝑟𝑒𝑓 = 5𝑥10−8
𝑚/𝑠
𝐈(𝐱) = 𝑝′𝐯′𝑎𝑛𝑑𝐿𝐼(𝑑𝑒𝑐𝑖𝑏𝑒𝑙) = 10 log
𝐈
𝐼𝑟𝑒𝑓
,
𝐼𝑟𝑒𝑓 = 10−12
𝑤/𝑚2
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Acoustic Quantities
 The sound power P emitted in a quiescent fluid is given by integration of I over a
closed surface
 If we neglect dissipation, P is the sound power of the all the sources located inside
the surface
 However, I is affected by all sources irrespective of whether they are inside or
outside the surface. The sound power level LW is defined as
𝑃 =
𝑐𝑙𝑜𝑠𝑒𝑑𝑠𝑢𝑟𝑓𝑎𝑐𝑒
(𝐈 • 𝐧 ⥂)𝑑𝐴
𝐿 𝑤(𝑑𝑒𝑐𝑖𝑏𝑒𝑙) = 10 log
𝑃
𝑃𝑟𝑒𝑓
, 𝐼𝑟𝑒𝑓 = 10−12 𝑤
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The Human Hearing
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Loudness and Loudness Level
 The human ear does not perceive sound of all frequencies equally. Two pure tones
of different frequencies but identical sound pressures are heard with different
loudness
 The “loudness” takes care of this differential sensitivity of ear when sound is pure
tone and when it is heard from the front
 The loudness level LN of a pure tone of any frequency f and sound pressure level L is
the same if it is perceived equally loudly as that sound of frequency f =1000
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 ISO226 is based on data generated from a large no of test persons
 The loudness level LN of a pure tone of any fixed frequency and sound pressure level
of the pure at 1000Hz tone is perceived equally loudly
 Loudness is measured in Phons.
Equal loudness level contours (ELLC) as ISO226
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 The units of loudness level are phons
 By definition the phon-values are equal to the sound pressure levels in dB when f =
1000Hz
 Since ear is less sensitive for low frequencies, (i.e. when f < 1000 Hz), Lp (f < 1000Hz)
> LN(f). For frequencies (1000Hz < f < 4000Hz), Lp < LN. The ear is most sensitive at
frequencies close to 4000Hz
 After decrease in sensitivity beyond 6000Hz, the sensitivity of ear improves roughly
around 10000-12000Hz.
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 The hearing sensitivity is also nonlinear when it comes to intensity.
 A doubling of the intensity of a tone is not perceived as ”double as loud”. In fact
typical increase of 10 phons is perceived only ”double as loud”.
 In order to account for this characteristic of hearing the so called loudness sone is
introduced
 son (units s) is a measure of perception for intensity
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 Loudness defined in terms of s is given by:
 The loudness ratio of two sounds with a loudness level difference of ∆LN is s2/s1 is
given by
 Thus an increasing a signal by 10 phons in loudness level (~10dB in sound intensity
level) corresponds to doubling of the loudness and an increase of 20 phons
corresponds to a fourfold loudness value
𝑠(𝑠𝑜𝑛𝑒) = 2(𝐿 𝑁−40)/10
𝑠2/𝑠1 = 2(𝐿 𝑁2−40)/10−(𝐿 𝑁1−40)/10 = 2Δ𝐿 𝑁/10
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 Loudness defined in terms of s can be converted in terms of LN
𝐿 𝑁 = 40 + 33.2𝐿𝑜𝑔(𝑠)
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Noise Weighting
 It is clear that sound pressure (or sound intensity levels) can not measure perceived
sound most appropriately. At the same time, the hearing sensitivity is too complex
to arrive at one simple formula for this purpose
 Noise weighting is used to simply the procedure, even though some accuracy is lost.
 In noise weighting, very high frequency and very low frequency signals are given low
weights and mid-frequency signals are given higher weights
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Noise Weighting
 Four noise weightings have been agreed (called A-, B-, C- and D-weighting)
internationally. They are designated as LpA, LpB, LpC and LpD and their respective units
are dB(A), dB(C), dB(C), and dB(D)
 A-weighting is most popular and widely. LpA can be obtained from LA
 Similarly LpB, LpC and LpD can be obtained from LB, LC and LD
𝐿 𝑝𝐴 = 𝐿 𝑝 + Δ𝐿 𝐴
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Noise Weighting
Noise weighting (DIN-IEC 651)
The A weighting
corresponds to
negative loudness
level curve LN = 40
phons
The B and C
weighting
corresponds approx.
negative loudness
level curve LN =70
(or 100) phons.
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Noise Weighting
 ∆LA = ∆LA(f) is recommended for low / moderately intense SPL (below 55dB). It
roughly corresponds to the negative loudness level curve LN = 40 phons
 B and C weighting is recommended when the noise is of very high intensity (B for
SPL=55-85dB, C for SPL > 85dB)
 The D-weighting was designed especially for aviation noise. However, it is not widely
used.
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Analysis of Random Noise
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Narrow, Third Octave and Octave Band
Analysis
 For a random variation, sound as fluctuations on top of mean value not be defined.
This is because mean itself can not be defined.
a general random variation of pressure
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Analysis
 But for random processes statistical or ensemble average can be defined using pn(t),
where pn(t) is nth realisation (or the nth measurement) of the process p(t)
 The fluctuating pressure p′ = p−⟨p⟩ can now be defined which gives p′2⟩= 0 & while
p′2⟩ 0
 Luckily, often some even random physical processes show statistical behaviour
which does not change with time
𝑝 (𝑡) = lim 𝑁→∞
1
𝑁
𝑛=0
𝑁
𝑝 𝑛(𝑡)
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Analysis
 Such processes are called statistically stationary processes. For these processes, the
auto correlation P of p′(t) for some values of ∆t and  does not depend on time, i.e.
 For such random functions P() decays to zero for large .
 In other words, in these processes, the process does not depend on its initial values
or forgets it’s past for sufficiently large value of  > ∆t
𝑃 = 𝑝′(𝑡)𝑝′(𝑡 + 𝜏) = 𝑝′(𝑡)𝑝′(𝑡 + Δ𝑡 + 𝜏) = 𝑃(𝜏) ≠ 𝑃(𝑡)
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Analysis
 In this case, the process can be seen as several independent events each with a
time history of the signal p′(t) into several pieces with p′n(t) = p′(t + n∆T) and
consider them as statistically stationary process.
 Suc processes called ergodic process. For ergodic process, the temporal mean and
the ensemble averages are the same
 Most notable example of stationary process is quasi-steady turbulence
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Fourier Analysis of Acoustic Signals
 In acoustics, signal is considered as statistically stationary.
 However, loudness depends on frequency, it is essential to decompose the signal in
to frequency components, i.e. Fourier Analysis of signal is required
 Fourier transform of a signal h’(t) is given by
 here f = ω/2π is the frequency and i = √−1
ℎ(𝜔) =
−∞
∞
ℎ′(𝑡)𝑒−𝑖𝜔𝑡
𝑑𝑡
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 The inverse transformation in time domain is given by
 Unfortunately, Fourier analysis is possible only if h′(t) it is square integrable, i.e.
 And p′(t) violates the condition of being square integrator
ℎ(𝑡) =
1
2𝜋
−∞
∞
ℎ(𝜔)𝑒 𝑖𝜔𝑡
𝑑𝜔
−∞
∞
ℎ′(𝑡) 2 𝑑𝑡 ⥂⥂< ∞
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 To determine the frequency components of the mean square sound pressure, we
take the auto correlation P() = ⟨p′(t)p′(t + )⟩.
 We assume that P() decays to zero sufficiently fast in  to be square integrable so
that Fourier decomposition is possible and which produces power spectral density
given by
 The inverse transformation for P(τ) can now be calculated P(τ=0)
𝑃(𝜔) =
−∞
∞
𝑃(𝜏)𝑒−𝑖𝜔𝜏
𝑑𝜏
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 The above equation shows that for every frequency interval df there is a certain
contribution to p
 Thus once the power spectral density is known each frequency component of the
random signal may be considered can be frequency weighted
 Naturally, to display the spectral content of the rms-value of a random signal the
integral over frequency bands is required
𝑝2
= 𝑃(𝜏 = 0) =
1
2𝜋
−∞
∞
𝑃(𝜔)𝑑𝜔 =
−∞
∞
𝑃(𝑓)𝑑𝑓
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Preferred Frequencies for Third Octave
Bands (In Hz)
Nominal frequency Exact 2000t (ISO 266) 10i/10 103 Third octave 2i/3 103
… … …
400 398.1 396.9
500 501.1 500.0
630 631.0 630.0
800 794.3 793.7
1000 1000.0 1000.0
1250 1258.9 1259.9
1600 1584.9 1587.4
2000 1995.2 2000.0
2500 2511.9 2519.8
3150 3162.3 3174.8
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Methods of Solution
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Methods of Simulations vs Distance from
the Source
Computational
boundary
Full nonlinear
equations
Linear equations
with variable
coefficients
Linear equations
with constant
coefficients
Disturbance
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Direct Numerical Simulation and Large
Eddy Simulation
 Direct numerical simulation (DNS) solves compressible Navier-Stokes equation for
obtaining both the flow field, and the aerodynamically generated acoustic field.
 This requires very high numerical resolution schemes due to the large differences in the
length scales of acoustic field and the flow field.
 DNS is unsuitable for any practical application Even in CFD. Large eddy simulations (LES)
or detached eddy simulation (DES) are useful the sound is from turbulence and domain
size is small
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Euler and Linearised Euler Equations
 Euler and Linearized Euler Equations consider noise as small disturbances
superimposed on a uniform mean flow of density ρ0 , p0 and velocity on x-axis as u0
 Linearized Euler Equations can be used in the simulation of engine noise
 LEE is not a good mathematical model when acoustic propagation is non-linear,
which what happens in high Mach number flows
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64
Euler and Linearised Euler Equations
 Euler and Linearized Euler Equations (LEE)
 These methods are useful when viscous effects are less important
 Splitting methods Based on LEE
 These are applied to specific unsteady fluid-structure interaction problems
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65
Integral or Hybrid Methods
 Here computational domain is split into flow field domain and the acoustic field
domain. The methods require two different numerical solvers
 The flow field solvers RANS, SNGR (Stochastic Noise Generation and Radiation), DNS,
LES, DES, URANS, are used for simulating flow field and finding out acoustic sources.
 The propagation of acoustic field uses a different kind of solver (wave operator)
 These methods are widely practiced
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66
Derivations of governing
equations
66

67
Governing Equations for Compressible
Viscous Flow
 Navier-Stokes equations in conservative form describe motion of compressible
viscous flows3-5
𝜕𝜌
𝜕𝑡
+ ∇ • (𝜌𝐯) = 𝑚. . . . 1
𝜕𝜌𝐯
𝜕𝑡
+ ∇ • (𝜌𝐯𝐯) + ∇𝑝 = ∇ • 𝛕 + 𝐟 + 𝑚. 𝐯 . . . 2
𝜕𝜌𝑒 𝑇
𝜕𝑡
+ ∇ • (𝜌𝑒 𝑇 𝐯) + ∇ • (𝑝𝐯) =
= −∇ • 𝑞 + ∇ • (𝛕𝐯) + 𝜃. + 𝐟 • 𝐯 + 𝑚. 𝑒 𝑇 . . . 3
67

68
Viscous Flow
ρ - density, v - velocity vector, and p – pressure
et = e+0.5 v2 - specific total energy
e – internal energy
µ (T) - dynamic viscosity
 = µ(∇v + t∇v − (2/3)I∇·v) - Stress tensor due to friction
q(T, k) = −k∇T –heat flux tensor and
k(T) – heat conductivity
68

69
Viscous Flow
 While modelling 2D, axisymmetric or 3D domains as 1D domain, it helps in having
some terms representing sources for mass, momentum and energy. Thus
 m –source term for mass,
 f - external forces and
 θ - heat addition
 The above equation have the 7 unknowns ρ, v, et (or equivalently e), p and T but the
no of equations is only 5
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70
Viscous Flow
 The system of equations (1-3) along with the expressions for τ, µ, q and k is not
closed as the no of unknowns (namely ρ, v, et, p & T) is more than the no of
equations by 2
 We use 2 more relations to close the system of equations
 the thermal equilibrium equation:
 = (p, T) …4
 the caloric state equation:
e=e(T, p) …5
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71
Viscous Flow
 The system of equations is in conservative form.
 Other than the 1st term on LHS of the these equations, other terms represent flux of
the conserved quantities. The source terms are on RHS
 The injected mass appears in the momentum equations because the fluid must force
the injected mass to the ambient velocity v. The mass also appears in the energy
equation for the same reason
 The force on RHS contributes to power
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72
Viscous Flow
 In acoustics the above equations are not used as they are
 We prefer to use equations written in terms of primitive variables. Also we would
like to have entropy s as one of the primitive variables. This can be done by
 Multiplying the continuity equation (1) by v and subtracting it
from the momentum equation (2)
 Taking dot product of the momentum balance (2) with v and
subtracting the product from energy equation (3)
 Also, multiplying (1) by e and subtract from (3)
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73
Arriving at Suitable Equations for
Aeroacoustics
 Here D/Dt:= ∂/∂t+v∇ is the material derivative, i.e. the change felt by the fluid
element in motion, as it flows past position x at time t
 In acoustics, we write energy equation (7) in terms of the entropy
𝐷𝜌
𝐷𝑡
= −𝜌∇ • 𝐯 + 𝑚. . . . 5
𝜌
𝐷𝐯
𝐷𝑡
= −∇𝑝 + ∇ • 𝛕 + 𝐟 . . . 6
𝜌
𝐷𝑒
𝐷𝑡
= −𝑝∇ • 𝐯 + 𝛕: ∇𝐯 − ∇ • 𝑞 + 𝜃.
. . . 7
73

74
Aeroacoustics
Entropy (Gibbs) is defined by equation:
T δs = δe−(p/ρ2)δρ = δh − (1/ρ) δp …8
 Here δ is the total variation of one thermodynamic variable in terms of the variation
of two other thermodynamic variables, all taken at the same point in time and space
in sense of equilibrium thermodynamics
 If the fluid is globally in thermodynamic equilibrium, then it does not matter, how
the change δ comes about
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75
Aeroacoustics
 Note that the same kind of variation must be considered for all variables, may it be
due to a separate temporal change (δ = dt · ∂/∂t) or a separate spatial change (δ = dx
· ∂/∂x)
 If the equilibrium is only local, i.e. only within the size of a fluid element, then this
couples the temporal and spatial variations of the variables and δ = dt· D
Dt must be understood as a material change, i.e. according to some D/Dt.
75

76
Aeroacoustics
 It can be seen that heat conduction, the mass and heat sources can increase or
decrease the entropy depending their signs
 However, the dissipation function Φ := τ:∇ v is never negative and hence it always
increases the entropy making the process irreversible
 Since internal energy e is eliminated, to close the equations we replace equation of
state, ρ = ρ(p, T)
by new thermodynamic variables, i. e ρ = ρ(p, s)
𝜌
𝐷𝑠
𝐷𝑡
=
1
𝑇
𝛕: ∇𝐯 − ∇ • 𝑞 + 𝜗
.
− 𝑚
𝑝
𝜌
.
. . . 9
76

77
Arriving at suitable equations for aeroacoustics
𝛿𝑝 =
𝜕𝜌
𝜕𝑝 𝑠
𝛿𝑝 +
𝜕𝜌
𝜕𝑠 𝑝
𝛿𝑠 . . . 10
1/𝑎2 =
𝜕𝜌
𝜕𝑝 𝑠
𝑎𝑛𝑑𝜎 =
1
𝜌
𝜕𝜌
𝜕𝑠 𝑝
. . . 11
 It can be seen that heat conduction, the mass and heat sources can
increase or decrease the entropy depending their signs
77

78
Linearization:
from fluid dynamics to
aeroacoustics
78

79
Linearization of Fluid Dynamic Equations
to get CAA Equations
 We note that acoustic field is the pressure fluctuation p′ over and above from the
mean ambient pressure p0, which is varying with time.
 Thus we need equation for p' or some field from which p' can be obtained
algebraically (i.e. without numerical method which always produce approximate
results)
 Mathematically this means we write p=p0+p'. But this results in perturbations in all
fluid dynamic conditions
79

80
 Thus we write
 We exclude global hydrodynamic instabilities (e.g. von Karman vortex street), which
are self-sustained flow oscillations and assume that the perturbations are small, ϵ ≪
1 as required by definition of aeroacoustics
𝜌
𝑣
𝑝
𝑎2
𝜎
. . .
=
𝜌0
+ 𝜀𝑝′
𝑣0
+ 𝜀𝑣′
𝑝0 + 𝜀𝑝′
(𝑎2
)0
+ 𝜀 ⥂ (𝑎2
)′ ⥂
𝜎0
+ 𝜀 ⥂ 𝜎′
. . .
. . . 19
80

81
𝐯0
• ∇𝑝0
= −𝜌0
∇ • 𝐯0
. . . 50
𝜌0 𝐯0 • ∇𝐯0 − ∇ • 𝛕0 = −∇𝑝0 . . . 51
(𝐯0
• ∇𝑝0
)/(𝑎2
)0
= −𝜌0
∇ • 𝐯0
+ 𝜎0
(𝛕0
: ∇𝐯0
− ∇ • 𝑞0
)/𝑇0
. . . 52
 We also assume that the mean flow is steady without source terms, i.e. sources are
only aero acoustic in nature
 We insert equation 19 into equations (5, 18 and 13) and collect first order associated
with  we aero acoustic equations
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82
Linearised Aero Acoustic Equations
𝐷0 𝜌′
𝐷𝑡
+ 𝜌0∇ • 𝐯′ + 𝐯′ • ∇𝑝0 + 𝜌′∇ • 𝐯0 = 𝑚.′ . . . 13
𝜌0
𝐷0
𝑝′
𝐷𝑡
+ ∇𝑝′ + 𝜌0
𝐯′ • ∇𝐯0
+ 𝑝′𝐯0
• ∇𝐯0
= 𝑓′ . . . 14
1
(𝑎2)0
𝐷0
𝑝′
𝐷𝑡
+ 𝐯′∇ • 𝑝0 + 𝜌0∇ • 𝐯′ + 𝜌0
(𝑎2
)′
(𝑎2)0
+ 𝜌′ ∇ • 𝑣0 =
=
𝜎0
𝑇0 𝜗.
+ 1 −
𝜎0 𝑝0
𝜌0 𝑇0 𝑚.′
= 𝜃.
′ . . . 15
𝑤ℎ𝑒𝑟𝑒
𝐷0
𝐷𝑡
=
𝜕
𝜕𝑡
+ 𝐯0 • ∇𝑖𝑠𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑡𝑖𝑎𝑙𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒𝑎𝑙𝑜𝑛𝑔𝑡ℎ𝑒
𝑠𝑡𝑟𝑒𝑎𝑚𝑙𝑖𝑛𝑒𝑠𝑜𝑓𝑡ℎ𝑒𝑚𝑒𝑎𝑛𝑓𝑙𝑜𝑤
82

83
Linearised Aeroacoustics Equations
𝑇ℎ𝑒 𝑎𝑏𝑜𝑣𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 𝑑𝑜 𝑛𝑜𝑡 𝑖𝑛𝑐𝑙𝑢𝑑𝑒 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑠
∇ • 𝜏′
𝑎𝑛𝑑 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 ℎ𝑒𝑎𝑡 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛
𝜎 𝜏: ∇𝑣 − ∇ • 𝑞
𝑇
′
𝑎𝑠 𝑡ℎ𝑒𝑠𝑒 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑖𝑒𝑠 𝑎𝑟𝑒 𝑣𝑒𝑟𝑦 𝑠𝑚𝑎𝑙𝑙
83
𝑊ℎ𝑒𝑛 𝑡ℎ𝑒 𝑔𝑎𝑠 𝑖𝑠 𝑝𝑒𝑟𝑓𝑒𝑐𝑡

84
Linearised Aeroacoustics Equations
 When the fluid incompressible, (a2)0 → ∞, in which case, the pressure equation is
degenerated
 The above equations are called Linearized gas dynamic equations or Linearized
Euler Equations (LEE)
 The equations do not describe global instabilities. They describe small
perturbations over and above a steady mean flow.
 They The equations are strictly valid for in-viscid flow. They neglect viscous stress,
dissipation & heat conduction
 These equations can be used for deriving acoustic pressure perturbation, which are
usually wave equations
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85
Acoustics in stagnant fluid
 In a stagnant fluid v0 = 0, p0 p0(x), it a constant, (say p∞). However permit density
ρ0 = ρ0 (x). This simplifies equans 13-15 to the following equations
 The above equations can be used to show that sound intensity and sound power are
related source terms (the right hand side terms)
𝜕𝜌′
𝜕𝑡
+ 𝐯′ • ∇𝜌 + 𝜌0∇ • 𝐯′ = 𝑚.′ . . . 16
𝜌0
𝜕𝐯′
𝜕𝑡
+ ∇𝑝′ = 𝑓′ . . . 17
1
(𝑎2)0
𝜕𝑝′
𝜕𝑡
+ 𝜌0∇ • 𝐯′ = 𝜃.′ . . . 18
85

86
Acoustics in stagnant fluid
 Multiplying equation (17) by v′ and equation (18) by p′/ρ0 and adding them gives
 In the above equation 1st term is the energy density. Integrating the above equation
over the (steady) control volume VV(t) gives
𝜕
𝜕𝑡
𝜌0
1
2
𝑣′2 +
1
2
𝑝′2
𝜌0(𝑎2)0
+ ∇ • 𝑝′𝑣′ = 𝑣′𝑓′ +
1
𝑝0
𝜃.′𝑝′
86

87
Acoustics in Stagnant Fluid
𝜕𝜔 𝜔
𝜕𝑡
− 𝜐0∇2
𝜔 𝜔 = 0
𝑝 𝜔 = 0, 𝑠 𝜔 = 0, ∇ × 𝑢 𝜔 = 𝜔 𝜔, ∇ • 𝑢 𝜔
= 0
 The fluctuations and the corresponding equations can be decomposed
into vorticity, entropy, and acoustic modes given below:
𝜕𝑠 𝑒
𝜕𝑡
− 𝜅0(𝛾 − 1)∇2
𝑝 𝑒 + 𝜅0∇2
𝑠 𝑒
𝑝 𝑒 = 0, 𝜔 𝑒 = 0, ∇ • 𝑢 𝑒 =
𝜕𝑠𝑒
𝜕𝑡
equation for
vorticity mode
equation for
entropy mode
87

88
𝜕2
𝑝 𝑎
𝜕𝑡2
− 𝑎0
2
∇2 𝑝 𝑎 = 𝜅0 𝛾
𝜕
𝜕𝑡
∇2 𝑝 𝑎
𝜕𝑠 𝑎
𝜕𝑡
− 𝜅0∇2 𝑠 𝑎 = 𝜅0(𝛾 − 1)∇2 𝑝 𝑎
𝜔 𝑎 = 0, ∇ • 𝑢 𝑎 =
𝜕𝑠 𝑎
𝜕𝑡
−
𝜕𝑝 𝑎
𝜕𝑡
equation for
acoustic mode
 Here a0 is the mean speed of sound. The subscripts ω, e and a denote the
vorticity, entropy and acoustic modes of u'
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89
 The vorticity and entropy advect at the speed of the fluid.
 The acoustic mode travels at the speed of sound.
 The decomposition of u into (uω, ue + ua) is Helmholtz decomposition into
solenoidal and dilatational modes
 The compressible affect the affects solenoidal (i.e., in-compressible) component.
Thus the components can not be separated into two modes and techniques of in-
compressible flow can not be used.
89

90
Hybrid methods - CAA for
practical applications
90

91
On Hybrid methods
 These methods use a known solution of the acoustic wave equation to find out
acoustic far field from known sound sources. These methods are computationally
efficient and fast.
 Integral methods neglect the variation in speed of sound or the flow velocity
between source and field point.
 Theoretically in integral methods there should be no source outside the surface.
However, for practical reasons, the domain needs be finite in size. This may lead to
large cut-off errors.
 The methods involve addition/subtraction of many large similar numbers leading to
numerical problems.
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92
Noise in Aerospace & its Position Stealth
 Hybrid methods use RHS which represent sources of noise. Most general or most
preferred aerodynamic sources do not exists.
 Field produced by sources is transported using acoustic analogy
 The most famous amongst these methods is Lighthill acoustic analogy
 Lilley’s and Mohring’s analogies are generalizations Lighthill’s analogy
 Curle analogy for in-compressible fluid
 Ffowcs-Williams and Hawkings are generalization of Lighthill’s analogy to explicitly
include boundaries and is the mostly widely method in engineering
21 Oct 2020
92

93
Lighthill’s Acoustic Analogy
 In this method NS equations are rearranged so that its left hand side becomes wave
operator applied to density perturbation or pressure perturbation. The RHS is
identified as the acoustic sources. The far-field is given in terms of a volume integral
of all sound sources.
 Lighthill's analogy does not require any assumption and hence all sources are taken
into account
 The sources can represent laminar or turbulent noise
 The wave operator in this method does not permit variation of density, speed of
sound and Mach number.
 Different mean flow conditions require many corrections to account for the sound-
flow interaction.
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94
𝜕2 𝜌
𝜕𝑡2
= ∇ • ∇ • (𝜌𝐯𝐯 + 𝑝𝐈 − 𝛕)
𝐴𝑓𝑡𝑒𝑟 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑛𝑔 𝑎∞
2 ∇𝜌 𝑓𝑟𝑜𝑚 𝑡ℎ𝑖𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑤𝑒 𝑔𝑒𝑡
𝜕2 𝜌
𝜕𝑡2
− 𝑎∞
2 ∇𝜌 = ∇ • ∇ • (𝜌𝐯𝐯 + (𝑝 − 𝑎∞
2 ∇𝜌)𝐈 − 𝛕)
Lighthill’s stress tensorwave operator
The LHS of this equation is the wave operator with a constant speed of sound of
a∞. The RHS is called the Lighthill’s stress tensor
94
𝑇𝑎𝑘𝑖𝑛𝑔 𝑡𝑖𝑚𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑐𝑜𝑛𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑒𝑞𝑛 𝑎𝑛𝑑 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑛𝑔 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 𝑜𝑓 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑔𝑖𝑣𝑒𝑠

95
𝜕2
𝜌′
𝜕𝑡2
− 𝑎∞
2
∇𝜌′ = ∇ • ∇ • (𝜌𝐯𝐯 + (𝑝′ − 𝑎∞
2
∇𝜌′)𝐈 − 𝛕)
 In terms of p′ = p−p∞ and ρ′ = ρ−ρ∞, the equation becomes
 RHS is Lighthill’s stress tensor
 p′−a2
∞ρ′ represents entropy perturbation s′
 The equation forms the basis of Lighthills aero acoustic analogy. ρ′ is acoustic
signal (note the wave operator) only in domains where the time averages v = 0, p
= p∞, ρ = ρ∞
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96
 The equation is an exact equation as it stands.
 ρ′ may be interpreted as acoustic signal only in domains where the time averages v =
0, p = p∞, ρ = ρ∞ which is true in cases such as turbulent jet radiating sound in
quiecent air having mean density and pressure ρ∞ and p∞ respectively.
 The wave operator on LHS is ”self adjoint” operator. This means that temporally
amplified solutions without external forcing are excluded, i. e. hydrodynamic
instabilities are excluded
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97
 This equation is analogous to an acoustic problem in a ficticious nonmoving medium
and hence it can be called ”aero acoustic analogy”
 Since LHS is wave operator, RHS can be called as the aero acoustic sources
 Three sources in Lighthill’s stress tensor T are:
 Flactuation in velocities: ρv'v' (e.g. turbulence)
 Change in entropy p′−a2
∞ρ′ (e.g. temperature fluctuations due to combustion or heat
addition)
 Viscous friction stresses τ (usually considered to be unimportant in most cases
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97

98
 The source term (∇·∇·T) represents the double divergence of a tensor. Thus it
represents the leading order term of the multipole expansion of the tensor or order
4 (a quadrupole)
 The above equation can be written in terms of pressure instead of the density to
obtain the pressure form of Lighthill’s analogy:
 Thus for cold flow both the forms of Lighthill’s wave equation are almost the same
1
𝑎∞
2
𝜕2
𝑝′
𝜕𝑡2 − Δ𝑝′ = ∇ • ∇ • (𝜌𝐯𝐯 − 𝛕) +
1
𝑎∞
2
𝜕2
𝜕𝑡2 (𝑝′2
− 𝑎∞
2
𝜌′)
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99
 The major problem with this method is that the sound source is not compact in
supersonic flow
 This requires large extension to the computational domain till the sound source is
fully decayed, which further requires keeping track of retarded time-effect of long
record of the time-history of the converged solutions of the sound source
 For realistic problems, this method is likely to require terabytes of data as storage
99

10
Lilley’s equation: Analogy for Shear Flows
 One of drawback of Lighthill’s equation that the refraction at shear layers is
considered as a source
 This happens because these effects appear implicitly in the Lighthill’s stress tensor,
even though they are not generating any sound
 Lilley analogy takes care of this problem
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10
 Lilley's equation modifies Lighthill's equation without additional assumptions
 Lilley’s wave operator is not self-adjoint, which menas the eigenmodes get amplified
in time and they represent hydrodynamic instabilities
 Lilley's equation produces refraction in parallel shear flows. Therefore Lilley’s source
term is not a true source.
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10
 Lilley’s equation has some demerits
 the equation not only describe acoustic, but also hydrodynamic instability
 the methodology requires numerical solution, Analytical solution being not possible.
 in quiecent medium Lilley’s equation does not reduce to the simple wave equation, which it
should.
102

10
Mohring’s wave equation: Analogy for
Potential Flows
 Mohring’s equation can be obtained without simplification by using equations of
flow motion
 LHS Mohring’s equation is the wave equation for sound hence RHS can be
considered as acoustic sources
 RHS comprises of the motion related aero acoustic source and entropy related
source
 The equation explicitly shows that the vorticity is necessary for the generation of
aerodynamic sound
103

10
Mohring’s wave equation: Analogy for
Potential Flows
 The equation correctly describes propagation in the potential domain of the flow,
which is normally the largest part in external flow.
 Its LHS is a self-adjoint operator
 Mohring’s wave operator is reduced to the classical wave equation for quiecient
medium without flow
 Green’s function for this equation unknown
 There problems in implementing boundary condition on surfaces
104

10
Curle’s Equation: Noise from Steady
Objects in Low Speed Flows
 Curle’s equation is suitable for simulating sound field generated by a body
immersed in a flow field.
 Three sources in this equation are :
 The sound field produced by the free fluid (same as in Lighthill’s equation)
 The force acting on the surface of the object immersed in the fluid. If the body surface is
rigid and non-moving,this is zero
 The source due to dispalcment of fluid. This source is also zero, if the body is not moving
105

10
Ffowcs Williams and Hawkings Analogy
(FWH Analogy)
 This is a modification Lighthill analogy, which avoids volume integral
 It assumes that the source region bounded by (FW-H surface) is limited
 Surface integrals involves monopole and dipole sources and also quadruploes
 Unlike Kirchhoff method, the sources are derived from Navier-Stokes equations
through Lighthill's analogy and hence it offers insight into various types of sources
and their contributions
106

10
Ffowcs Williams and Hawkings Analogy
(FWH Analogy)
 FWH analogy is widely used and preferred method in prediction of acoustics in
aerospace and turbomachinery
107

10
Methods of simulations in Aeroacoustics
CAA
Direct
methods
Hybrid
methods
CFD
analysis
Sound
generation
Transport
methods
Resolved
sources
(LES,DES
Reconstruct
sources
(RANS+
SNGR/SATIN
Acoustic
perturbation
equan
Computatio
nal
transport
Analytical
transport
Linearise
Euler
equation
Wave
equation
Acoustic
analogy
(FW-H)
Kirchhoff
integral
108

10
Two practical applications of
CAA
109

11
Direct Method for Predicting Noise of a
Gun
 Aim: Find out noise produced by a gun using CFD and improve the design to reduce
the noise
 Motivation : silensers in the guns are importatnt. They protect jawans being deteted
 Use ALE based Euler simulations for simulating the unsteady flow
 Find out unsteady pressure (called noise here) at given locations
110

11
Governing Equations and Assumptions
 User Euler simulations for modeling physics
 Fluid is Ideal gas with R = 287.1 kJ/kgk, with specific hear ratio = 1.4
 Simulated motion of the bullet starts from all burn point.
 One-degree-of-freedom (1 DOF) simulation is used for bullet movement
 For simplification, though diameter of bullet is smaller than barrel diameter, leak
between the bullet and the barrel is ignored
111

11
Strategy of Solution
 As acoustic signals are not small, hybrid methods can not be used.
 ALE based CFD which permit movement of control volume (meshes) is used
 Octree meshes which permit random motion of walls is used.
 The domain is divided into isotropic rectangular cells Cartesian cells (root-cells). The
root cells and their children are divided recursively till the geometry or the solution
accuracy is acceptable
112

11
Simplified Computational Domain and
Meshes
Solver produces octree meshes to capture the geometry of
computational domain and change in the pressure
113

11
Magnified View of Meshes
Sectional view in the plane of symmetry. Some root cells get
subdivided up to 4 level, redacting their volume by a factor of
(1/8)4=1/256
114

11
Unsteady Flow Field(mach Nos)
t = 0.222 msecs t = 0.477 msecs
t = 0.986 msecs t = 5.261 msecs
Bullet (shown in white) travels the barrel in less that 0.47 msec.
115

11
Unsteady Pressure wrt Time
t = 0.222 msecs t = 0.477 msecs
t = 0.986 msecs t = 5.261 msecs
t = 0.986 msecs t = 5.261 msecs
Within 5 msec the barrel in the barrel is close to atmospheric pressure.
116

11
Unsteady Velocity wrt Time
117

11
Hybrid Method for Predicting Noise of an
Aerospace Vehicle
 Aim: Find out jet-noise of an aerospace vehicle at sea level using CFD-CAA
methodology
 Jet-noise is an ideal condidate for Hybrid CAA simulations.
 Unfortunately, LES is complex and expensive for a large aerospace object. Thus
Unsteady Reynolds Averaged Navier Stokes (URANS) can be used for simulating
unsteady flow for find out noise sources (monpoles, dipoles and quadrupoles)
 Use FWH analogy for transmission of noise
118

11
Computational Domain
Outer boundary is 6 vehicle lengths away in the upstream
and side-ward directions and 12 in downstream direction.
Aerospace
vehicle
Downstream
boundary
119

12
Surface Discretisation
No of surface cells (quads on the launch
vehicle surface) ~ 36880 respectively.
120

12
Volume Meshes (cut section)
No of volume cells are ~ 12 million. Volume cells are
clustered near the surface
121

12
Magnified View of Meshes Near the
Surface
Complexity in the
geometry forced
hybrid meshes.
Structured meshes
reduced the cell
count. unstructured
mesh captured the
complex geometric
features
122

12
Meshes Near the Nozzle Exit
Large no of cells capture near the virtual surface
capture acoustic sources
123

12
Boundary Conditions for CFD Simulations
124

12
Calculation of Noise Sources
 Compressible Navier Stokes implicit and second order accurate solver which uses
AUSM scheme for convective fluxes is used
 Spalart Allmaras Edward turbulence model is used
 Dual time stepping is used for calculating unsteady flow at a physical time of 0.2
seconds. The flow field was sampled at deltat = 0.1 millisecond
125

12
Calculation of Noise Sources and Noise
 This enables capturing of the noise sources of the maximum frequency of 5000 Hz
 FWH source terms are calculated and used in FWH acoustic analogy
126

12
Steady State Pressure Variation
Steady state pressure variation in the plane of
symmetry
127

12
Contours of Noise Levels
128

12
Contours of Noise Levels
129

13
Closure
 CAA is a multiscale problem. LES is recommended in the direct method of solution if
high fidelity solutions are required. The diecretisation schemes must be less
dissiptative and dispersive
 For real life problems, hybrid methods for simulating aeroacoustics are
recommended
 An example of predicting noise around an aerospace vehicle has been illustrated. It
used RANS for generating sources of noise and popular FWH aero acoustic analogy
for transmission of noise.
130

13
Closure
 We did not study formulae / methods useful for hand calculations
 We also did not elaborate
 The numerical methods called Dispersion-Relation Preserving Schemes (DRP) schemes
recommended for direct solution of aero acoustic problems
 Hybrid methods and mathematics supporting them
 Characteristics based, Asymptotic, Buffer zone, and Perfectly Matched Layer boundary
conditions
 Transdusers and receivers for acoustic signals
131

13
References
1 Marvin E. Goldstein Aeroacoustics, NASA SP 346, 1974
2 Hardin, J.C. and Lamkin, S. L. Aero acoustic Computation of Cylinder Wake Flow, AIAA Journal, 22(1), 51-
57, 1984
3 Jan Delfs Basics of Aeroacoustics-Lecture notes
4 Edited by CLAUS WAGNER,
THOMAS HUTTL, and PIERRE
SAGAUT
Large-Eddy Simulation for Acoustics -
Cambridge University Press 2007
5 S.W. Rienstra & A. Hirschberg An Introduction to Acoustics - Lecture notes. Eindhoven University of
Technology, 12 Apr 2017
132

13
www.zeusnumerix.com
+91 72760 31511
Abhishek Jain
abhishek@zeusnumerix.com
Thank You !

Aero Acoustic Field & its Modeling @ Zeus Numerix

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