VKI Lecture Series: Advances in

Fundamentals of Aeroacoustics I

Sheryl Grace Boston University

Outline

ƒ What is Aeroacoustics

ƒ Direct approach for analyzing aeroacoustic phenomenon

ƒ Integral methods Free-space Green’s function Monopole, dipole, quadrupole Lighthill’s analogy

ƒ Extensions of the analogy Ffowcs-Williams and Hawkings Eq./Curle’s eq. Kirchhoff method Howe’s analogy

ƒ Acoustically compact sources

1 What is aeroacoustics?

Sound produced by or in the presence of a flow

Free-space problems --

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 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: 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 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 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: ƒ Turbulence is moving ƒ Two distinct regions of fluid flow ƒ 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 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

ƒ Turbulence in low Mach number uniform flow --- eighth power

ƒ Turbulence in low Mach number variable density flow --- sixth power

ƒ Turbulence in high Mach number flow --- third power

ƒ Turbulent fluctuations in a 2D low Mach number uniform flow --- seventh power

ƒ Simple source --- fourth power

ƒ Simple source in 2D--- third power

ƒ Dipole in low Mach number uniform flow --- sixth power

ƒ 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 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