VKI Lecture Series:Advances in Aeroacoustics

Vortex Sound in Bounded Flows

Sheryl GraceBoston University

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 termdominates in the far field

quadrupole term

Dipole like term can cause problems numerically

Howe/Powell source term

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:

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

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

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

from Gradshtyen and Ryzhik

therefore

Solution for spinning vortex pair (cont.)Finally:

the sine integral is similar…..

4) Putting it back together… recall

The part:

which in polar notation

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

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:

Integral form of solution to Howe’s analogy

Use the same methodology as was usedto generate the FWH eq.

Multiply Crocco’s equation and continuity equationby H and recombine to get:

Apply the free-space Green’s function to arrive at the general form of the solution

Integral form of solution to Howe’s analogy (cont.)

When the normal derivative of G vanishes on S and S coincides with a rigid body

Sound produced by body vibration

Sound produced by vorticity(G takes care of body modificationSound produced by friction forces on body

For high Reynolds number and a stationary surfaces

Integral solution for compact rigid bodiesCompact Green’s function, satisfies normal derivative condition

For high Reynolds number and a stationary surfaces

velocity of centroid of body

Example: Blade-vortex interaction (2D)

vorticity velocity

source term

compact Green’s function in for strip in 2D

recognize from rectangularwing example

In 2Donly j component

Need dG/dy2

1) Identify important quantities

Example: Blade-vortex interaction (cont.)

4) Evaluate the integral in an approximate sense for very close approachAlso, focus on the noise contribution from the leading edge

the Kirchhoff vector becomes:

and it has derivatives:

2) Plug everything into

to get

3) Evaluate the integrals in y

the vortex passes the leading edge, at time

Example: Blade-vortex interaction (cont.)

4) Substitude Y and apply the Heavyside function to get

5) Perform the integration (steps outlined in notes)

Pressure signature:

The term implies that

Comments

The Howe-Powell source which explicitly shows the dependence of sound on vorticity (and the Howe analogy) offers method for acoustic evaluation of many classical type problems.

More complex problems must be done numerically, Lighthill source may be easier to calculate.

It may also be easier to simply numerically compute the wave equation rather than using the integral forms.

High speed flows with high mean flow gradients require an alternative wave equation

Example of complex flow situation:Low speed flow past a cavity (underwater applications)

Cavity flow

Calculated with method similar to spinning vortex problemincompressible, viscous flow solution : FLUENTfinite difference calculation of acoustic pressure

Cavity flow

Streamwise velocity component from FLUENT

M = 0.26, L/D = 4.0, Laminar

Cavity flow

Source term

Cavity flow

Calculated acoustic pressure