Indo-German Winter Academy 2007
Flow Produced Noise and its
Fluid Mechanical Treatment
Shyamprasad N R
IIT Madras
Guide: Dr. Stefan Becker
Special thanks to Prof. Biswas and Prof.Durst
Outline
•
What is Flow produced noise?
•
Motivation to study this subject.
•
Review of sound and waves.
•
Derivation of wave equation for fluids.
•
What is an acoustics analogy?
•
Lighthill’s Analogy.
•
Discussion of sources, dipoles and quadrupoles.
•
Computational Aeroacoustics.
•
Examples of aircraft noise reduction methods.
What is Flow Produced Noise?
• It refers to the sound generated as a by-product of fluid
motion and not to the vibration of a solid.
• Can be due to aerodynamic forces acting on surfaces or
due to turbulent fluid motion.
• Aeroacoustics is the branch of acoustics that studies this
type of sound or noise.
Why study this subject?
Why study this subject?
•The practical impetus behind studying this
subject was the need to understand aircraft noise.
Why study this subject?
• Includes propeller noise, boundary layer noise and
above all, noise of jets.
• Knowledge necessary in order to device methods of
reducing noise to tolerable levels.
Review of Waves and Sound
• A wave is a disturbance that travels through a medium
transporting energy from one location to another,
inducing oscillatory motion of the particles of the
medium.
• An acoustic or sound wave is a pressure fluctuation that
propagates through a fluid.
• What type of wave is sound? Transverse or
Longitudinal? Why?
Types of waves
Transverse wave
Longitudinal wave
Review of Waves and Sound
• Sound is a longitudinal wave.
• Transverse waves require a relatively rigid medium in
order to transmit their energy.
• If the medium is not rigid, as is the case with fluids, the
particles will slide past each other!
• An aside: Longitudinal earthquake waves go through and
hence believed that the earth has a molten core.
Derivation of Wave Equation for
Fluids
•
The wave equation is a second order linear PDE that describes the
propagation of a variety of waves including sound waves.
‘u’ is a scalar function and ‘c’ is the speed of the wave.
• We will now derive this equation, where ‘u’ will represent instantaneous
pressure or density of the fluid.
Derivation of Wave Equation for
Fluids
•
These equations are for a “uniform acoustic medium” at rest, that
is without any sources of matter or external forces.
Continuity equation:
..(1)
Momentum equation:
..(2)
•
Note that the second term in the momentum equation is different
from usual.
Derivation of Wave Equation for
Fluids
• Comes from the barotropic equation of state, after retaining
only the lowest order term is retained as the fluctuations are
of very small magnitude.
• This gives a linear variation between pressure fluctuation and
change in density , constant of proportionality being the speed
of sound.
• We know that the UAM at rest will experience stresses only in
the form of pure hydrostatic pressure, whose variation with
density is as mentioned above.
Derivation of Wave Equation for
Fluids
• The wave equation form comes by eliminating
the momentum density,
, from equations (1)
and (2).
• Wave equation:
•This is called the linear, lossless homogenous
wave equation.
Acoustic Analogy
•
Here the governing equations of motion of the fluid are coerced into a form
reminiscent of the inhomogeneous linear wave equation.
•
The term analogy refers to the idea of representing the aerodynamic
process acting as an acoustically source, by an acoustically equivalent
source term.
•
The source term will include all sound generating and propagating
mechanisms such wave scattering by turbulent eddies, local inhomogenities
etc.,
•
The most commonly used and quite famous analogy is Lighthill’s and it was
proposed in the 1950’s when jet noise was placed under scientific scrutiny.
•
Other analogies are due to Powell and Howe, who gave different
descriptions for this source term.
Lighthill’s Analogy
•
•
Consider a fluctuating fluid flow occupying a limited part of a very
large volume, the remainder of which is at rest.
The equations governing the fluctuations of density in the fluid will be
compared to those of an UAM at rest, which is what the fluid outside
the region of flow is.
Lighthill’s Analogy
• Exact equation of momentum:
• Where pij represents the hydrostatic and viscous
stresses or equivalently the surface forces and the
momentum change due to molecular input.
Lighthill’s Analogy
•
•
We know that a UAM at rest will experience stresses only in the
form of pure hydrostatic pressure, whose variation is proportional to
density variation, constant of proportionality being square of the
speed of sound.
Hence the density fluctuations in a real flow must be exactly those
which would occur in a uniform acoustic medium subject to an
external stress system given by the difference between the effective
stress in the real flow and the UAM.
Lighthill’s Analogy
•
•
•
•
This analogy approach to the problem of aerodynamic sound production
is an exactly valid one.
The Tij term incorporates not only generation of sound but also its
convection with the flow, propagation with variable speed, gradual
dissipation by viscosity and conduction (process is not entirely adiabatic).
In practice the dissipation effects is a very slow process; in the
atmosphere only half the energy is lost in the first mile of propogation at
4kHz.
For flows in which temperature departs little from uniformity the
difference between
and
are unimportant.
Lighthill’s Analogy
•
Lets now rewrite the equations of fluid motion as equations of
propagation of sound in a UAM at rest due to externally applied
fluctuating stresses, Tij.
•
Note that the momentum equation is merely rewritten differently, it is
essentially the same.
Lighthill’s Analogy
• Physically, these state that a fluctuating flow in a limited
part of atmosphere otherwise at rest, generates same
fluctuations of density as would be produced in UAM at
rest by a system of externally applied stress, Tij.
• This view of sound as generated in a manner of forced
oscillation is suitable as the sound is a by-product of flow
and does not have a significant back reaction on it.
• Another advantage is that refraction or scattering of
sound as well as its generation are automatically
incorporated in Tij.
Lighthill’s Analogy
•
The inhomogenous wave equation form arises by eliminating the
momentum density as before.
..(3)
•
If we know the flow we can compute the source term (Tij ), which can
be plugged into the above equation to solve the wave equation.
Sources and Dipoles
• Mathematically, these are the elementary solutions of
the inhomogenous wave equation.
• These describe radiation generated at a mere point
singularity, but real sound, which is generated over a
region, can be described as continuous distribution of
such singularities.
• Physically, they are the source of the disturbance that
propagates in the medium as a pressure wave.
Sources and Dipoles
•
At a point source, it is the variation
of mass outflow that generates
sound.
•
In an acoustic dipole, sound is
generated by injection of
momentum rather than mass.
Equivalent to a force concentrated
at a point and varied in magnitude
and direction or both.
Sources and Dipoles
•
It is possible to derive the inhomogenous wave equation form for
distributed sources of mass or forces, from which the expressions
shown before are derived.
•
We will get
or
featuring on the RHS of the IHWE
for source and dipole respectively.
•
(3) on the other hand has a second derivative on the RHS, and its
solution represents distributed quadrupoles, of strength Tij per unit
volume.
Quadrupole
•
Without sources or dipoles arising from injection of mass or
momentum at internal boundaries, sound field is a quadrupole field
arising from stresses in the fluid in addition to those in UAM at rest.
Computational Aeroacoustics
• Why not use CFD methods to solve aeroacoustics
problems?
• There are issues that are relevant and unique to
aeroacoustics.
–
–
–
–
Large spectral bandwidth.
Acoustic-wave mean flow disparity.
Long propagation distance.
Radiation and outflow boundary conditions.
CAA
•
The spatial resolution requirement is determined by sound waves with
shortest wavelength.
•
Typically minimum of 6 to 8 mesh points per wavelength are required which
makes the number of mesh points in the computational domain enormous.
•
Developing finite difference schemes that give adequate resolution at 6 to 8
mesh points per wanvelength is also an important issue.
•
The rms velocity fluctuation associated with radiated sound is quire small(~
0.0001 times) compared to the typical flow velocity which puts it in the same
order of magnitude as the errors incurred in computation of the mean flow.
Noise reduction methods
•
•
•
The shape of the nose, windshield or canopy of an aircraft can greatly
affect the sound produced.
Much of the noise of a propeller aircraft is of aerodynamic origin due to
the flow of air around the blades. The helicopter main and tail rotors
also give rise to aerodynamic noise. This type of aerodynamic noise is
mostly low frequency determined by the rotor speed.
Fenestrons have between 8 and 18 blades. These are arranged in
varying distance, so that the noise is distributed over different
frequencies and thus appears quieter.
Noise reduction methods
•
In motorbikes mufflers are used in order to decrease noise
produced by exhaust gases.
•
Turbofan engines have a bypass that makes the operation quieter,
by decreasing the outlet jet velocity.