VKI Lecture Series: Advances in Aeroacoustics
Fundamentals of Aeroacoustics I
Sheryl Grace Boston University
Outline
What is Aeroacoustics
Direct approach for analyzing aeroacoustic phenomenon
Integral methods Free-space Green’s function Monopole, dipole, quadrupole Lighthill’s analogy
Extensions of the analogy Ffowcs-Williams and Hawkings Eq./Curle’s eq. Kirchhoff method Howe’s analogy
Acoustically compact sources
1 What is aeroacoustics?
Sound produced by or in the presence of a fluid flow
Free-space problems -- turbulence
Free-space problems with solid surfaces -- wings etc.
Bounded problems -- piping systems
Direct approach
Governing equations of fluid motion :
Continuity
Navier- Stokes
Energy
Definitions:
Sounds of interest : 10-130 dB 6.3X10-5 -63 Pa Atmospheric pressure at sea level 1X105 Pa
2 Direct approach: LEE
Linearized Euler equations, mean flow denoted with 0 subscript
Neglected
H20, air viscous effects : µ; 10-3, 10-5 thermal effects : κ; 10-6, 10-5
For inviscid, nonheat-conducting, uniform mean flow: sound generated due to initial or boundary conditions.
Acoustic/vortical splitting: Helmholtz Decomposition
Constant mean flow equations: Split unsteady velocity field into solenoidal (vortical) & irrotational (acoustic) parts = 0
Vorticity is purely convected Couples to the acoustic velocity only at solid boundaries
Unsteady pressure IS the acoustic pressure. No pressure associated with the vorticity.
3 Acoustic/vortical splitting: Comments
• Method has been used to compute interaction sound see notes for references • Popular applications: airfoil/gust, cascade/gust
Vortical, acoustic, and entropic waves are decoupled in this approach NOT 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
4 Free-space Green’s function
observer source time of travel retarded time How to use free-space Green’s function to find a solution to:
P Integrate 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
monopole dipole quadrupole Monopole
For point monopole at the origin:
concentric circles
origin
5 Source types cont.
Dipole
One extra step needed before integrating wrt to time: integration by parts using
For point dipole at the origin:
Superposition of monopoles:
_ + x θ * As compared to l harmonic source:
Source types cont.
Quadrupole
Group the quadrupole terms into:
Superposition of pair of dipoles: tij = lihj dq/dt _ +
h As compared to Superposition of monopoles harmonic source: _ Longitudinal_ quadrupole (i=j) Lateral quadrupole + + + _ _ l h l l Four leaf clover… + h l
6 Far-field expansion: rules
1)
2) Integration by parts to change independent variable in differential
3) Far-field expansion
4) Space differential to time differential (for far-field expansion)
Lighthill’s Equation
creation of sound generation of vorticity
refraction, convection, attenuation, known a priori
Lighthill stress tensor mean speed of sound
mean density
excess momentum attenuation of sound transfer wave amplitude nonlinearity mean density variations
7 Forms of solution to Lighthill’s Eq.
Quadrupole like source! Direct application of Green’s function
Far-field expansion, integration with respect to retarded time
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 wavelength/source length >> 1
Acoustic field pressure fourth power of velocity
Acoustic Power eighth power of velocity
8 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
Howe/Powell source term
The solution to the wave equation with this source term becomes
quadrupole term dipole like term must degenerate to quadrupole in the far field Dipole like term can cause problems numerically for flows in free-space
Further comments…
¾ Analogy is based on the fact that one never knows the fluctuating fluid flow very accurately
¾ Get the equivalent sources that give the same effect
¾ Insensitivity of the ear as a detector of sound obviates the need for highly accurate predictions
¾ Just use good flow estimates…
¾ 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.
¾ Using these is getting close to the direct calculation of sound.
9 Extensions of the analogy
FWH Equation
Need a more general solution when: Turbulence is moving Two distinct regions of fluid flow Solid boundaries in the flow
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:
10 FWH Equation cont. Multiply continuity and Navier Stokes equation by H Rearrange terms, add and subtract appropriate quantities Take the time derivative of the continuity and combine it with the divergence of the NS equation
Differential form of the FWH Eq.
Dipole type term
i i Quadrupole
Monopole type term
FWH Equation cont.
unsteady surface pressure Reynolds stress viscous stresses
i i
rate of mass transfer Only nonzero on the surface across the surface
11 FWH Equation cont.
Integral form of the FWH Eq.
Quadrupole
Dipole
Monopole
Square brackets indicate evaluation at the retarded time
If S shrinks to the body dipole = fluctuating surface forces monopole = aspiration through the surface
Curle’s Equation
When the surface is stationary the equation reduces to
Curle’s Equation
12 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
BEM computes unsteady pressure on the wing surface
Curle’s Eq.
13 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
Curle acoustic calc
using our CL
analytic result (in appendix) our acoustic calc
using our CL
Curle acoustic calc
using our CL
Purely analytic acoustic calc (based on
analytic CL
14 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
=
angle between flow because direction and R
= unit vector in the direction of R
FWH Eq. Moving Coordinate Frame (cont)
Volume element may change as moves through space
τ τ density at = 0 Volume element affected by Jacobian of the transformation
When control surface moves with the coordinate system … becomes ratio of the area elements of the surface S in the two spaces Retarded time is calculated from
15 FWH Eq. Moving Coordinate Frame (cont)
When f is rigid: uj = Vj When body moves at speed of fluid: Vj = vj
τ Square brackets indicate evaluation at the retarded time e
> 1 for approaching subsonic source Doppler shift < 1 for receding subsonic source accounts for frequency shift heard when vehicles pass
Comparison of turbulent noise sources
Stationary turbulence (low M) Moving turbulence (high M)
16 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
17 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
18 Example of usefulness of explicit ω dependence
Spinning vortex pair
rate of travel
position *a *a
vorticity associated with each one
velocity associated with each one
source term
expanded about s
Sound from spinning vortex pair The governing equation The source term
The general solution using the free-space Green’s function
Perform the integration:
1)
2) Note that *a
19 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
Spinning vortex pair discussion Acoustic pressure from spinning vortex pair
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 correction needed for computations
20 Comparison of calculation methods
Familiar spiral pattern Calculated vs. analytical
Calculated: • Analytical source with Oseen correction • Second order finite difference in space and time • First order characteristic type radiation boundary conditions
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
21 Acoustically compact source
Alternative method of defining integral form of wave eq.
Meaning of compact
Given: Characteristic length scale of the source region : L Wavelength of the sound : λ If : L/λ << 1 we say that the source region is acoustically compact
22 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 that involved the normal derivative of the Green’s function
We can construct a Green’s function such that on the surface,
A straight forward method exists to construct such functions when the source region is acoustically compact
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
Receiver x Source x
Source y Receiver y
23 Construct the Compact Green’s function free space correction Assume the form
Far-field expansion + Compact assumption
Note this looks like the potential function for freestream flow
The correction term could then be analogous to the perturbation potential required 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
It is argued that the main equation is still satisfied to 1st order because
Construction cont.
Add the perturbation potential to get first order approx to compact Green’s function
Surface normal condition gives:
φ∗ Therefore, j signifies the perturbation potential necessary to produced the appropriate geometry when there is a unit freestream flow in the jth direction impinging on the body
Final form of the compact Green’s function
Kirchhoff vector
24 Example: Circular Cylinder
¾Source of sound near a circular cylinder
¾Source is located near the y1-y2 plane
¾No “perturbation potential” needed in the
y3 direction:
¾Need to calculate the required perturbations in the other two directions
Consider potential flow past a cylinder from elementary fluid mechanics Cylinder formed by superposition of freestream and doublet y
x freestream in doublet at necessary strength x-direction the origin relation
2D Potential flow for flow past a cylinder freestream in doublet at x-direction the origin +
flow past = cylinder
25 2D Potential flow for flow past a cylinder
From this 2D solution
We infer the perturbation parts of the Kirchhoff vectors
Recall:
Compact Green’s function for the cylinder
Example: Dipole near rectangular wing
¾Dipole near rigid strip
¾Dipole is located near trailing edge centerline
¾No “perturbation potentials” needed in the
y1 & y3 directions:
¾Need to calculate the required perturbations in the other direction
Governing wave equation (single frequency disturbance amplitude f2 at x1 = L)
General solution:
Simplifies to
26 Example: Dipole near rectangular wing (cont.)
Details for defining the potential flow solution are given in the notes. The transformation (in the complex plane) from the rectangle to a cylinder is used.
Kirchhoff vector component:
Solution for the dipole near the rectangular wing has the form:
where the compact Green’s function is
Example: Dipole near rectangular wing (cont.)
Final solution:
Dipole in free space radiates similarly, but no factor:
27 Other forms of the compact Green’s function
Symmetric form
Time domain form
Exercise: Compact Green’s function for sphere
Coordinates x2 ϑ θ x1= r sin cos ϑ θ x2= r sin sin θ ϑ r x3= r cos ϑ x1
x3
Potential function for a freestream flow in x3 direction
Potential function for a sphere with flow in the x3 direction:
For sphere with radius a
28 Exercise: Compact Green’s function for sphere
Coordinates x2 ϑ θ x1= r sin cos ϑ θ x2= r sin sin θ ϑ r x3= r cos ϑ x1
x3 Third component of Kirchhoff vector is :
Exercise: Dipole near sphere
x2
Dipole in x1 direction
Oscillating hamonically with radial f1 frequency ω
x1 On x1-axis at x1=L x3 Find the far-field sound
29 Exercise: Dipole near sphere
x2
f1
x1
x3
Recall representation of dipole from previous example (in x2 direction)
Exercise: Dipole near sphere
x2
f1
x1 x3
30 31