(a primitive approach to) vortex sound theory and...
TRANSCRIPT
(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