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VKI Lecture Series:Advances in Aeroacoustics
Fundamentals of Aeroacoustics I
Sheryl GraceBoston University
What is Aeroacoustics
Direct approach for analyzing aeroacoustic phenomenon
Integral methodsFree-space Green’s functionMonopole, dipole, quadrupoleLighthill’s analogy
Extensions of the analogyFfowcs-Williams and Hawkings Eq./Curle’s eq.Kirchhoff methodHowe’s analogy
Acoustically compact sources
Outline
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Sound produced by or in the presence of a fluid flow
What is aeroacoustics?
Free-space problems -- turbulence
Free-space problems with solid surfaces -- wings etc.
Bounded problems -- piping systems
Governing equations of fluid motion :
Direct approach
Continuity
Navier-Stokes
Definitions:
Sounds of interest : 10-130 dB 6.3X10-5 - 63 PaAtmospheric pressure at sea level 1X105 Pa
Energy
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Linearized Euler equations, mean flow denoted with 0 subscript
Direct approach: LEE
H20, airviscous effects : µ; 10-3, 10-5
thermal effects : κ; 10-6, 10-5
Neglected
For inviscid, nonheat-conducting, uniform mean flow: sound generated due to initial or boundary conditions.
Split unsteady velocity field into solenoidal (vortical)
&irrotational (acoustic)
parts
Acoustic/vortical splitting: Helmholtz Decomposition
Constant mean flow equations:
Vorticity is purely convectedCouples to the acoustic velocity only at solid boundaries
= 0
Unsteady pressure IS the acoustic pressure. No pressure associatedwith the vorticity.
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• Method has been used to compute interaction soundsee notes for references
• Popular applications: airfoil/gust, cascade/gust
Acoustic/vortical splitting: Comments
Vortical, acoustic, and entropic waves are decoupled in this approachNOT true when shocks occur or in swirling flow settings
Acoustic pressure associated with the irrotational portion of the flow which is driven by a coupling to the vortical and entropic portions of the field at solid surfaces.
Thus: VORTEX sound is a topic of great interest
Integral Methods
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Free-space Green’s function
observer source time of travelretarded time
How to use free-space Green’s function to find a solution to:
PIntegrate wrt time
Forced wave equation
Add a volume source (q) to the continuity equation and an external force (F) to the momentum equation, combine to form a wave equation… more details to come
quadrupoledipolemonopoleMonopole
For point monopole at the origin:
origin
concentric circles
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Source types cont.
Dipole
One extra step needed before integrating wrt to time: integration by parts using
Superposition of monopoles:
For point dipole at the origin:
x*+_
lθ As compared to
harmonic source:
Source types cont.
Quadrupole
Group the quadrupole terms into:
Superposition of pair of dipoles: tij = lihj dq/dt
+_
hAs compared to harmonic source:Superposition of monopoles
_+
l
+_
l
Longitudinal quadrupole (i=j) Lateral quadrupole
Four leaf clover… +_
l
_+
l h
h
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Far-field expansion: rules
Integration by parts to change independent variable in differential
1)
2)
3) Far-field expansion
4) Space differential to time differential (for far-field expansion)
Lighthill’s Equation
mean speed of soundLighthill stress tensor
mean density
creation of sound generation of vorticity
refraction, convection, attenuation, knowna priori
excess momentum transfer wave amplitude nonlinearity
mean density variations
attenuation of sound
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Forms of solution to Lighthill’s Eq.
Direct application of Green’s function
Far-field expansion, integration with respect to retarded time
Quadrupole like source!
What we can learn from far-field form
For low Mach number, M << 1 (Crow)
If the source is oscillates at a given frequency
The far-field approx to the source in Lighthill’s equation can be written as
Therefore the solution becomes
Scalings: velocity -- U, length -- L, f of disturbance -- U/L
Acoustic field pressure
Acoustic Power
fourth power of velocity
eighth power of velocity
Acoustic wavelength/source length >> 1
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Explicit dependence on vorticity
Low Mach number,high Reynolds number flow, Lighthill’s stress tensor dominated by
The double derivative of this term can be related to the vorticity
The solution to the wave equation with this source term becomes
dipole like termmust degenerate to quadrupole
in the far field
quadrupole term
Dipole like term can cause problems numerically for flows in free-space
Howe/Powell source term
Further comments…
Analogy is based on the fact that one never knows the fluctuating fluid flow very accurately
Get the equivalent sources that give the same effect
Insensitivity of the ear as a detector of sound obviates the need for highly accurate predictions
Just use good flow estimates…
Alternative wave operators that include some of the refraction etc. effects that can occur due to flow nonuniformity near the source have been derived: Phillips’ eq. , Lilley’s eq.
Using these is getting close to the direct calculation of sound.
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Extensions of the analogy
FWH Equation
Turbulence is moving Two distinct regions of fluid flowSolid boundaries in the flow
Need a more general solution when:
Define a surface S by f = 0 that encloses sources and boundaries (or separates regions of interest)
Surface moves with velocity V
Heavy side function of f : H(f)
Rule:
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FWH Equation cont.
Multiply continuity and Navier Stokes equation by HRearrange terms, add and subtract appropriate quantitiesTake the time derivative of the continuity and combine
it with the divergence of the NS equation
Dipole type term
Monopole type term
Quadrupole
Differential form of the FWH Eq.
i i
Only nonzero on the surface
i i
FWH Equation cont.
Reynolds stress
unsteady surface pressure
viscous stresses
rate of mass transfer across the surface
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FWH Equation cont.
Dipole
Monopole
Quadrupole
Integral form of the FWH Eq.
Square brackets indicate evaluation at the retarded time
If S shrinks to the body dipole = fluctuating surface forcesmonopole = aspiration through the surface
Curle’s Equation
When the surface is stationary the equation reduces to
Curle’s Equation
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Example: 3D BVI acoustics validation
We have a BEM for calculating the near field (surface forces)
?? Is the acoustic calculation correct??(contributed by Trevor Wood -- not in the notes)
Example: 3D BVI acoustics validation
Curle’s Eq.
BEM computes unsteadypressure on the wing surface
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Example: 3D BVI acoustics validation (cont.)
far-field expansion
integration of pressure is lift
interchange space and time derivatives
Acoustic pressure in non-dimensional form
Example: 3D BVI acoustics validation results
our CL vs. t
analytic result(in appendix)
our acoustic calcusing our CL
Curle acoustic calcusing our CL
Curle acoustic calcusing our CL
Purely analytic acoustic calc (based on analytic CL
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FWH Eq. Moving Coordinate Frame
Introduce new Lagrangian coordinate
Inside integral, the δ function depends on τ now and
Where the additional factor that appears in the denominator is
=
= unit vector in the direction of R
becauseangle between flow direction and R
FWH Eq. Moving Coordinate Frame (cont)
Volume element may change as moves through space
density at τ = τ0
Volume element affected by Jacobian of the transformation
Retarded time is calculated from
When control surface moves with the coordinate system … becomes ratio of the area elements of the surface S in the two spaces
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FWH Eq. Moving Coordinate Frame (cont)
When f is rigid: uj = VjWhen body moves at speed of fluid: Vj = vj
Square brackets indicate evaluation at the retarded time τe
Doppler shift
accounts for frequency shift heard when vehicles pass
> 1 for approaching subsonic source< 1 for receding subsonic source
Comparison of turbulent noise sources
Stationary turbulence (low M) Moving turbulence (high M)
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Stationary turbulence (low M)
Far-field form
From before…. Pressure goes as fourth power of velocity and power as eighth power of velocity
Moving turbulence (high M)
Pressure goes as scaled fourth power of velocity
Power goes as eighth power scaled by
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Comments on Kirchhoff method
• Solve the homogeneous wave equation using the free-space Green’s function approach
• All sources of sound and nonuniform flow regions must be inside the surface of integration
• FWH is same if the surface is chosen as it is for the Kirchhoff method
• FWH superior• Based on the governing equation of motion (not wave equation)• Valid in the nonlinear region
Howe’s acoustic analogy
Howe formulated an analogy based on the total enthalpy
The wave equation that is formulated :
In the far field, away from sources of sound:
Good for thermal sources such as temperature fluctuations on a surface
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Example of usefulness of explicit ω dependence
Spinning vortex pair
rate of travel
position
vorticity associated with each one
velocity associated with each one
source term
expanded about s
*a*a
Sound from spinning vortex pairThe governing equation The source term
The general solution using the free-space Green’s function
Perform the integration:
1)
2) Note that *a
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Solution for spinning vortex pair (cont.)
3) Compute the integral
using
make a change of variables
assume the observer distance r is much larger than the acoustic wavelength
Acoustic pressure from spinning vortex pair
Acoustic pressure from spinning vortex pair
Spinning vortex pair discussion
Dependence on distance
Power dependence on velocity 2D : 7th power
When one uses the Lighthill form : not explicit with ω
Source term for incompressible flow becomes Oseen correctionneeded for computations
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Comparison of calculation methods
Familiar spiral pattern Calculated vs. analytical
• Analytical source with Oseen correction• Second order finite difference in space and time• First order characteristic type radiation boundary conditions
Calculated:
Sound power scalings
Far-field behavior …. dependence on velocity
Turbulence in low Mach number uniform flow --- eighth power
Turbulence in low Mach number variable density flow --- sixth power
Turbulence in high Mach number flow --- third power
Turbulent fluctuations in a 2D low Mach number uniform flow --- seventh power
Simple source --- fourth power
Simple source in 2D--- third power
Dipole in low Mach number uniform flow --- sixth power
Dipole in 2D --- fifth power
Directivity in notes
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Acoustically compact source
Alternative method of defining integral form of wave eq.
Meaning of compact
Given:Characteristic length scale of the source region : LWavelength of the sound : λ
If :L/λ << 1 we say that the source region is acoustically compact
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Consider alternative Green’s function
Integral form of solution to analogy was formed using the free-space Green’s function
For the case of solid boundaries in the fluid, surface integrals thatinvolved the normal derivative of the Green’s function
A straight forward method exists to construct such functions when the sourceregion is acoustically compact
We can construct a Green’s function such that on the surface,
This method is closely related to the method of matched asymptotic expansions:Solve the Laplace equation not the Helmholtz equation.
Construction done in frequency domain
Transform of the Green’s function wave equation gives
Added constraint. G must still be causal.
Reciprocal relation
Source y
Receiver x Source x
Receiver y
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It is argued that the main equation is still satisfied to 1st orderbecause
The correction term could then be analogous to the perturbation potentialrequired to create potential flow past the object (reciprocal idea)
However, this means that the correction function satisfies Laplace Eq. and by its definition it satisfies the normal condition
Construct the Compact Green’s function
Assume the form free space correction
Note this looks like the potential function for freestream flow
Far-field expansion +Compact assumption
Construction cont.
Add the perturbation potential to get first order approx to compact Green’s function
Surface normal condition gives:
Therefore, φ∗j signifies the perturbation potential necessary to produced the
appropriate geometry when there is a unit freestream flow in the jthdirection impinging on the body
Final form of the compact Green’s function
Kirchhoff vector
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Example: Circular Cylinder
Source of sound near a circular cylinder
Source is located near the y1-y2 plane
No “perturbation potential” needed in the y3 direction:
Need to calculate the required perturbationsin the other two directions
Consider potential flow past a cylinder from elementary fluid mechanics
Cylinder formed by superposition of freestream and doublet
freestream in x-direction
x
y
doublet at the origin
necessary strengthrelation
2D Potential flow for flow past a cylinder
freestream in x-direction
doublet at the origin
+
=flow pastcylinder
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2D Potential flow for flow past a cylinder
From this 2D solution
We infer the perturbation parts of the Kirchhoff vectors
Compact Green’s function for the cylinder
Recall:
Example: Dipole near rectangular wing
Dipole near rigid strip
Dipole is located near trailing edge centerline
No “perturbation potentials” needed in the y1 & y3 directions:
Need to calculate the required perturbationsin the other direction
Governing wave equation (single frequency disturbance amplitude f2 at x1 = L)
General solution:
Simplifies to
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Solution for the dipole near the rectangular wing has the form:
Kirchhoff vector component:
where the compact Green’s function is
Details for defining the potential flow solution are given in thenotes. The transformation (in the complex plane) from the rectangle to a cylinder is used.
Example: Dipole near rectangular wing (cont.)
Example: Dipole near rectangular wing (cont.)
Final solution:
Dipole in free space radiates similarly, but no factor:
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Other forms of the compact Green’s function
Symmetric form
Time domain form
Exercise: Compact Green’s function for sphere
x1
x2
x3
ϑ
θ
x1= r sinϑ cosθ
r
x2= r sinϑ sinθ
x3= r cosϑ
Potential function for a sphere with flow in the x3 direction:
For sphere with radius a
Potential function for a freestream flow in x3 direction
Coordinates
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x1
x2
x3
ϑ
θ
x1= r sinϑ cosθ
r
x2= r sinϑ sinθ
x3= r cosϑ
Exercise: Compact Green’s function for sphere
Third component of Kirchhoff vector is :
Coordinates
Exercise: Dipole near sphere
x1
x2
x3
f1
Dipole in x1 direction
Oscillating hamonically with radialfrequency ω
On x1-axis at x1=L
Find the far-field sound
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Exercise: Dipole near sphere
x1
x2
x3
f1
Recall representation of dipole from previous example (in x2 direction)
x1
Exercise: Dipole near sphere
x1
x2
x3
f1