(A Primitive Approach to)

Vortex Sound Theory

and Application to Vortex Leapfrogging

Christophe Schram

von Karman Institute for Fluid Dynamics

Aeronautics and Aerospace

& Environmental and Applied Fluid Dynamics depts

ERCOFTAC course – Computational Aeroacoustics

Plan

� Lighthill’s analogy

� The quadrupolar character of low-Mach turbulence noise (M8 law) is

not an accident! There has been premeditation!

� Vortex Sound Theory, choice of the source term

� Powell’s analogy, Mohring’s analogy, conservative formulation

� Application to vortex pairing described by wrong flow models

If your flow model is wrong… just tell your analogy that it’s correct!

Lighthill’s aeroacoustical analogy :

concept

� Wave propagation region: linear

wave operator applies

sourceregion

observerin uniformstagnant

fluid

propagation regionuniform fluid at rest

V

S

x

yNo source

mass momentum

� Turbulent region: fluid mechanics

equations apply

� The problem of sound produced by a

turbulent flow is, from the listener’s point of

view, analogous to a problem of propagation

in a uniform medium at rest in which

equivalent sources are placed.

No external forces� no external dipole!

Lighthill’s analogy: formal

derivation

Lighthill’s aeroacoustical analogy:

how to make it useful ?� Reformulation of fluid mechanics equations, and use of arbitrary speed c0 :

� Definition of a reference state:

sourceregion

observerposition

propagation regionuniform fluid at rest

V

S

x

y

Lighthill’s tensor

Exact… and perfectly useless!

with

� Aeroacoustical analogy :

Sound produced by free isothermal turbulent flows

at low Mach number

� Solution using Green’s fct

� Purpose: simplify the RHS

� High Reynolds number

� Isentropic

� Low Mach number

integral solution

sound scattering at boundaries

� Using free field Green’s fct

Quadrupolar source

No monopole!

No boundary � no external forces� no external dipole!

Acoustic scale:

Lighthill’s M 8 law

� Integral solution:

� Scaling law:

D U0

λ

Flow time scale:

Spatial derivative:

Acoustical power:

� Conservation principles

� Powell’s analogy

� Mohring’s analogy

� Conservative formulation

� Application to vortex pairing

Vortex Sound Theory

Lighthill’s analogy: some issues for

free subsonic flows

� Spatial extent of source term fora localized distribution of vorticity(Oseen vortex):

∝ 1 / r

� Alternative formulation of the analogy : Vortex Sound Theory

� Yields a more localised source term

� Allows reinforcing the quadrupolar character of free turbulence

Invariants of incompressible, inviscid vortex flows

in absence of external forces (Saffman, 1992)

� Circulation:

� Impulse (momentum):

� Kinetic energy:

where C is a closed material line.

vanishes if the force f derives of a single-valuedpotential, and for inviscid flows.

vanishes in absence of non-conservative body forces.

conserved quantity using the same assumptions.

Vortex Sound Theory: Powell’s analogy for free flows

� Vectorial identity:

� Momentum equation becomes:

� Similar manipulation as for Lighthill’s analogy:

∝M 2

� Retaining leading order terms in M2:

� Integral solution using free field Green’s function and first order Taylor

expansion of the retarded time:

conservation of kinetic energy

conservation of impulse

� Powell’s integral formulation:

low Mach isentropic

Vortex Sound Theory: Möhring’s analogy for free flows

� Starting from Powell’s integral formulation:

� Using vectorial’s identity:

� By substitution:

� Using Helmholtz’s vorticity transport equation:

� Möhring’s integral formulation:

conservation of kinetic energy

� We have derived two (formally) equivalent formulations of the Vortex

Sound Theory:

� Powell’s analogy:

� Mohring’s analogy:

Vortex Sound Theory: 2 solutions for the same problem

� Although formally equivalent, these two formulations do not yield the same

numerical robustness!

The choice of a source term affects the numerical performance of the prediction!

Vortex Sound Theory for axisymmetrical flows

� Coordinate of a vortex element:

� General form of velocity and vorticity:

� Powell’s analogy becomes:

� Möhring’s analogy becomes:

� Vortex pairing = inviscid interaction (Biot-Savart)

� Vortex leapfrogging: periodic motion

� Vortex merging : requires core deformation

Application: vortex ring pairing

� One of the mechanisms of sound production in subsonic jets

� Can be easily stabilized and studied at laboratory scale

� Generic interaction showing how the reciprocal

exchange of impulse |b| two vortex elements

produces a quadrupole in far field

for each vortex ring

Vortex pairing: U0 = 5.0 m/s

2D and 3D models of vortex ring leapfrogging

� 2D model (σ << d << R0): locally planar interaction, neglects vortex stretching:

� 3D model (σ << d = O(R0) ): accounts for vortex stretching:

2D model: vortex trajectories and flow invariants

� Two cases considered: d / R0 = 0.1 and 0.3 .

� Locus of the vortex cores:

secular term

� d / R0 = 0.3d / R0 = 0.1�

2D 3D

� Flow invariants:

d / R0 = 0.1� d / R0 = 0.3

2D

2D

2D model: sound prediction

� Powell’s analogy:

� Möhring’s analogy:

secular term

secular term

� Conclusion: failure of both Powell’s and Möhring’s analogies when applied to a flow model that does not respect the conservation of momentum and kinetic energy.

Möhring’s solution: reinforcement of

conservation assumptions

� Using Lamb (1932) identities:

� Imposing further conservation of impulse:

� We obtain:

� Imposing further conservation of kinetic energy:

2D� 3D

2D

3D�

d / R0 = 0.1

d / R0 = 0.3

Generalization of Möhring’s solution

� Using Lamb (1932) identities:

will disappear insubsequent derivations

� Second correction: subtraction of the vortex centroid axial coord.

from the axial coord. of each vortex element.

� Doesn’t harm if impulse is conserved, since

� Improves numerical stability.

� Imposing further conservation of impulse and kinetic energy:

conservative formulation

Robustness of different formulations

of Vortex Sound Theory

� Investigation of the robustness of Powell’s form, Möhring’s form and of the conservative form when the flow data is perturbed.

� Perturbation = addition of random noise to

� coordinates,

� circulation.

� Effects on conservation of impulse and kinetic energy ?

� Effects on sound prediction ?

� Relation |b| both ?

Reference casevortex trajectory

d / R0 = 1impulse

kinetic energysound production

Effect of perturbation on flow invariants

Perturbation of coordinates Perturbation of circulation

impulse

kinetic energy kinetic energy

impulse

Effect on sound prediction

perturb. of coordinates Powell

� Möhring

� perturb. of circulation Powell

� Möhring

�

perturbation of coordinates� perturbation of circulation×

conservative form

Application to PIV data

PIV results: flow invariants� Low frequency fluctuations of about 10%.

� Increasing scatter due to growing instabilities.

5.0 m/s 5.0 m/s 5.0 m/s

34.2 m/s 34.2 m/s 34.2 m/s

34.2 m/s

5.0 m/s

PIV results: acoustical source terms

5.0 m/s

34.2 m/s

5.0 m/s

34.2 m/s

Acoustic predictions

� 2nd time derivative : 4th order polynomial fit over moving interval� acoustical source term.

5.0 m/s

PIV

—— 3D model

�

34.2 m/s

PIV

—— 3D model

�

� Case U0 = 34.2 m/s: order of magnitude OK, but quite different frequency content.

� Case U0 = 5.0 m/s: good agreement between predictions obtained from PIV data and 3D leapfrogging model.

Acoustic measurements

o90

m9.0

=

=

θ

x

f (kHz)

SP

L(d

BR

e20

µPa)

0 2.5 5 7.5 10 12.5-20

-10

0

10

20

30

40

50

60

70

80

background noise

f (kHz)

SP

L(d

BR

e20

µPa)

0 2.5 5 7.5 10 12.50

10

20

30

40

50

60

70

80unexcitedexcited

EXC+1P+2P

2P

2P1P+2P

2P

1P+2P

1P+2P

EXC+1P+2P

EXC+1P+2P2P

2P

f (kHz)

SP

L(d

BR

e20

µPa)

0 2.5 5 7.5 10 12.50

10

20

30

40

50

60

70

80unexcitedexcited

Surprise:

double pairing

EXC+1P

1P

1P

1P

EXC+1P

EXC+1P

f (kHz)S

PL

(dB

Re

20µP

a)

0 2.5 5 7.5 10 12.50

10

20

30

40

50

60

70

80single

spectrum

EXC+1P

1P

1P

1P

EXC+1P

EXC+1P

f (kHz)S

PL

(dB

Re

20µP

a)

0 2.5 5 7.5 10 12.50

10

20

30

40

50

60

70

80single

spectrum

� For some acquisitions: absence of double pairing.� Quite good agreement between prediction and measurement, for first

and third pairing harmonics.

EXC+1P+2P

2P

2P1P+2P

2P

1P+2P

1P+2P

EXC+1P+2P

EXC+1P+2P2P

2P

f (kHz)

SP

L(d

BR

e20

µPa)

0 2.5 5 7.5 10 12.50

10

20

30

40

50

60

70

80average

spectrum

Intermittence of double pairing

and comparison prediction - measurement

Summary

� Aeroacoustical analogies allow extracting a maximum of acoustical

information from a given description of the flow field

� Assuming a decoupling between the sound production and propagation, the

analogies provide an explicit integral solution for the acoustical field at the

listener position

� Improves numerical robustness

� Permits drawing scaling laws

� Some formulations make the dominant character of the source appear more

explicitly, and allow making useful approximations.

� Without approximations, the analogy is useless!

A few references

� A. Pierce, Acoustics: an Introduction to its Physical Principles and

Applications,

McGraw-Hill Book Company Inc., New York, 1981.

� S.W. Rienstra and A. Hirschberg, An Introduction To Acoustics (corrections),

Report IWDE 01-03 May 2001, revision every year or so…

� M.E. Goldstein, Aeroacoustics, McGraw-Hill International Book Company,

1976.

� A.P. Dowling and J.E. Ffowcs Williams, Sound and Sources of Sound, Ellis

Horwood-Publishers, 1983.

� D.G. Crighton, A.P. Dowling, J.E. Ffowcs Williams, M. Heckl and F.G.

Leppington, Modern Methods in Analytical Acoustics, Springer-Verlag

London, 1992.

� And of course: the VKI Lecture Series…

VKI Lecture Series - 2-4 Dec 2014

Fundamentals of Aeroengine Noise

Tue 2 Wed 3 Thu 4

9:00

↓

10:15

Innovative architectures

N. Tantot

(SNECMA)

Combustion noise

M. Heckl

(Univ. Keele)

Subsonic jet noise prediction

C. Bailly

(ECL)

10:45

↓

12:00

Aeroengine – airframe integration

and aeroacoustic installation

effets

T. Node-Langlois (Airbus)

Aeroengine nacelle liner design

and optimization

G. Gabard

(ISVR)

An introduction to supersonic jet

noise

C. Bailly

14:00

↓

15:15

Fundamentals of aeroacoustic

analogies

C. Schram

(VKI)

Analytical methods for

turbomachinery noise prediction

(con’td)

M. Roger

Experimental methods applied to

jet noise

M. Felli

(INSEAN)

15:45

↓

17:00

Analytical methods for

turbomachinery noise prediction

M. Roger

(ECL)

Visit VKI Laboratories

Advanced analysis techniques

(wavelets, LSE, POD) for noise

sources identification

R. Camussi

(Univ. Roma3)

www.vki.ac.be