AEROACOUSTICS - ICA

mustardarchaeologistMécanique

22 févr. 2014 (il y a 3 années et 4 mois)

106 vue(s)

A numerical approach to aerodynamic noise of aircraft wings
J.W.Delfs
Institut für Aerodynamik und Strömungstechnik,Technische Akustik
DLR - Deutsches Zentrum für Luft- und Raumfahrt e.V.
Lilienthalplatz 7,D-38108 Braunschweig,Germany
In viewof the need for quiet aircraft design,computational tools have to be developed,capable of describing the noise generation pro-
cess for airframe noise correctly.A respective simulation concept based on Computational Aeroacoustics (CAA) and linearized Euler
Equations is presented and discussed.In the context of airframe noise the question of linearity of sources is addressed.Employing the
proposed ”vortex test” simulation concept an aeroacoustic assessment of the thickness effect of airfoils in unsteady inflow conditions
is presented.
INTRODUCTION
The sound generated in unbounded unsteady subsonic
flowis marginal in intensity compared to the excess noise
produced when such unsteady flow interacts with finite
aerodynamic bodies.In airframe noise problems,such
unsteadiness is usually associated with turbulence.A
body may be understood as a disturbance to the dynam-
ics of the unsteady flow components,associated with a
corresponding change in the pressure field.This change
is usually large near inhomogeneities of the boundary
(edges,slots,humps,steps etc.) and it is accompanied
by the conversion of part of the hydrodynamic -usually
turbulent- pressure fluctuation into sound pressure.More-
over,sources of this kind represent particularly efficient
sound emitters for frequencies with respect to which the
geometric body components appear non-compact.All
these conditions are usually satisfied for the deployed
high lift devices on the wing of a typical modern civil
aircraft in approach.As verified in numerous flight- and
windtunnel tests the most intense sources of airframe
noise are indeed found near the leading-,trailing- and
side edges of the slat and flap when located by means
of microphone arrays [3] or elliptic mirrors [2].
The computation of airframe noise may be approached
in different ways.The most classical one is to solve an in-
homogeneous acoustic wave equation for a given source
term (e.g.Lighthill’s tensor as volume source,surface
pressure fluctuations as a surface source etc.).The issue
here is,that by definition all vortex dynamics (the orig-
inator to sound generation) lies buried inside the source
termand thus has to be known in advance.Typically,for
simple cases the source term is modelled or -if availabe-
extracted from DNS data.It is noted,that any domain,
where the wave equation is not satisfied (e.g.inside a
refracting shear layer) is to be understood as a source re-
gion,requiring some source term.
The second way to compute airframe noise involves
the direct numerical solution of the balance equations for
mass,momentum and energy and is usually refered to
as CAA-simulation.The characteristic difference to the
wave equation approach is,that apart from the acoustic
degrees of freedom,the vortical (and entropic) degrees
of freedom of the fluid dynamics is allowed for.In this
way the conversion process from vorticity to sound and
vice versa is incorporated into to theoretical description.
Although the CAA-approach is usually associated with
quite an increase in computation cost,it has important
advantages,namely a) that the sound generation is sim-
ulated,rather than modelled and b) that sound propaga-
tion through arbitrary flowfields is described properly.In
what follows a CAA-scheme is used to simulate unsteady
perturbations about a given steady mean flow field,along
with the sound generation near aerodynamic bodies in-
side this mean flow.
GOVERNINGEQUATIONS
For a given (quasi-) steady flowfield qqq
0
:



0

vvv
0

p
0

,
with  -density,vvv-velocity vector,p-pressure,the inviscid
dynamics of small perturbations q
qq

:





v
v
v


p


about
this basic floware governed by the linearised Euler equa-
tions for a thermally and calorically perfect gas:
d
0


dt

v
v
v
  



0


0



  
v
v
v







 
v
v
v
0

˙m

(1)

0 d
0
vvv

dt


0
v
v
v
   


 v
v
v
0



v
v
v
0
 


 v
v
v
0



 p


f
f
f

(2)
d
0
p

dt

vvv
 


 p
0

 p
0




vvv


 p





vvv
0

˙


(3)
Here,
d
0
dt
:


 t

v
v
v
0
  


 denotes the time derivative taken
along a streamline of the mean velocity field v
v
v
0
and 
is the isentropoc exponent (

1

4 for air under normal
conditions).The equations are dimensionless with the
following notation:time t

t

a


l,lengths x
i

x

i

l,
density 






,velocity vector v
v
v

v
v
v


a

,pressure
p

p




a
2

,a mass source density ˙m


˙m
 
l



a

,a
body force density fff

fff
 
l



a
2

and an energy (e.g.
heat) source density
˙

 
˙

 
l



a
3

,where a

 p

 is
the speed of sound.The asterisk denotes quantities with
dimensions and the quantities with an index  mean ref-
erences,which typically are freestreamvalues of the flow
quantities.
The boundary condition for acoustically hard walls is
equivalent with satisfying the kinematic flow condition
v
n
:

nnn
 
vvv

0

(4)
where n
n
n is the normal vector on the considered wall point.
The wall condition is not directly implemented into the
difference scheme but indirectly via the pressure gradi-
ent following[4].Taking the dot product of the momen-
tumequation (2) with n
n
n and respecting (4) at a wall point
yields:
 p

 n


0

vvv
0
vvv

vvv

vvv
0




vvv
0
vvv
0

::: nnn (5)
The normal derivative on the pressure is the product of
the momentum fluxes with the local curvature tensor of
the wall.The derivative vanishes when the wall is plane
or there is no flow present.
NUMERICAL SOLUTION SCHEME
The differential equation system (1-3) is solved nu-
merically subject to the given boundary and initial condi-
tions.The temporal discretization is done with the clas-
sical fourth order Runge-Kutta scheme (RK4).Spatial
gradients are approximated using the dispersion relation
preserving 7-point stencil finite difference scheme (DRP)
of Tam& Shen [5] and Tam& Dong[4],on curvilinear
(block-) structured grids,see e.g.[6].The physical grid
is given as node sequence in the indices i

j
xxx
i j

xxx



i



j

(6)
where  and  represent a uniform cartesian system and
assume integer values on the nodes.For fixed 

I the
grid variable  defines a grid line and vice versa.Since
the coefficients of the DRP scheme are defined for the
uniform computation grid 

i,

j the perturbation
equations (1-3) need to be transformed fromthe physical
domain to the computation domain 

.This is done by
replacing  by


MMM

(7)
where M
M
M





 is the metric of the transform.It is
obtained by inverting M
M
M

1





x
x
x





,which is avail-
able with high accuracy employing the DRP differencing
scheme along the grid lines.The metric is needed accu-
rately in order that the high resolution and accuracy prop-
erties of the DRP scheme would be transferred into the
physical space.Overlapped grid systems are well suited
for CAA [1] to facilitate gridding near complex geome-
try components,since these are the source locations of
airframe noise.
Very short wave length components of the signals
which cannot be represented physically correctly on the
given computation grid are suppressed with artificial se-
lective damping (ASD) due to Tam&Dong [7].For each
of the equations of the system (1-3) the same symmet-
ric linear,scalar damping operator
˜
D is introduced.The
source terms on the right hand side of (1-3) are identi-
fied with these damping terms acting as sinks rather than
sources:
˙m
 

˜
D


 
f
f
f
 

˜
D

v
v
v
 
(8)
˙

 

˜
D

p

with
˜
D
 



 xxx






2
D

   



 xxx






2
D

 
(9)
The coefficients of the 7-point stencil numerical opera-
tor D are given in [7];the subscripts on D indicate the
grid line direction along which the operator is to be ap-
plied.The damping coefficient  must be chosen such
that i) non-physical,i.e.purely numerically caused sig-
nal components are efficiently damped while affecting the
physical wave components as little as possible,and ii) no
numerical instability of the overall scheme is generated.
SIMULATION CONCEPT
Because of the wide range of turbulence scales,the
numerical prediction of airframe noise for technically
relevant flow Reynolds numbers is out of reach even on
today’s largest high performance computers.In order to
reduce the computational effort,it has become popular
to pre-compute (by CFD) the steady viscous mean
flow field qqq
0
and to simulate by CAA only the inviscid
perturbations qqq

.As in the wave equation approach
this again requires some modeling.The modeling is
however reduced to an appropriate excitation of vorticity
perturbations,rather than the whole aeroacoustic source.
Here,an approach is presented in which upstream of
the airframe component localized vorticity is introduced
into the flow field.In the course of the simulation the
mean flow convects the perturbation past the airframe
component upon which sound is generated.The acoustic
response is measured far from the body.This simulation
is repeated for varied geometry but the same initial test
vorticity.Comparison of the acoustic responses then
allows for a noise assessment of airframe components
in the following sense.The less efficient it converts
p'
FIGURE 1.Linear pressure field evolving from localized vor-
tex,seeded at t

0 and x

2.Above:no wedge,below:with
wedge at x

10.Wedge causing generation of sound.
vorticity into sound,the quieter the design.Note,that a
property of a body (in fact a response function) is sought,
rather than an absolute pressure level in dB.
Note on linearity
Before turning to examples a short note on the math-
ematical character of airframe noise sources is in place,
because the question arises,whether it is justified to use
linearized perturbation equations (1-3) for the descrip-
tion airframe noise generation or whether airframe noise
sources are fundamentally nonlinear.In order to address
this issue reference is made to Lilley’s equation,whose
wave operator is known to describe sound propagation
through parallel shear flows.It reads
d
dt

d
2

dt
2





 
a
2




2
t


 v
v
v:
:
:



a
2





2
t
 vvv:::

 vvv vvv

 (10)
where 



1
ln

p

p


is the acoustic variable,super-
script t means ”transpose” and  denotes terms of vis-
cous friction and entropy,which are negligible for high
Reynolds numbers.The remaining aeroacoustic source
term due to Lilley is therefore Q


2
t


 v
v
v:
:
:



 v
v
v

 v
v
v

.
Linearization of Lilley’s equation about a parallel mean
steady flow field vvv
0

u
0

y

0

0

shows that the linear
part of Q vanishes identically,of which follows immedi-
ately that sound generation in a parallel mean flow field
is a fundamentally nonlinear problem.
This in turn means that the evolution of a linear vortex
in a parallel shear layer is perfectly silent even while
initiating a Kelvin-Helmholtz instability.This situation
was simulated using the linearized Euler equations (1-3)
in a parallel shear layer with a realistic profile u
0

y

taken from a RANS-CFD simulation.The shearlayer is
located in

1

1

y

1

1;below there is no flow and
above there is a constant flow of Mach number M

0

4.
Localized vorticity is introduced at simulation time t

0
and position x

2

y


0

5.As a result a downstream
convecting unstable wave packet evolves representing
the classical Kelvin-Helmholtz instability.The upper
part of Fig.1 shows the respective linear pressure field
p

x

y

t

70

,which consists of purely hydrodynamic
(i.e.non-propagating) components indicating complete
silence and thus confirming the statement of Lilley’s
equation.Next,the effect of the presence of a wedge
on the pressure field is considered.For this purpose a
sharp vertical wedge with vertex at x

10

y


1

4 is
originally placed belowthe shearlayer and the simulation
is repeated for otherwise identical parameters.In this
case,shown in the lower part of Fig.1,sound is generated
by scattering of the wave packet’s linear hydrodynamic
pressure field at the wedge although the mean flow does
not even touch the wedge.The example shows that in
x
y
­1 ­0.5 0 0.5 1 1.5
­0.5
0
0.5
x
y
­3 ­2 ­1 0 1 2 3 4
­3
­2
­1
0
1
2
3
p':­0.1 ­0.05 ­0.025 ­0.01 0.01 0.025 0.05 0.1
v':­0.5 ­0.4 ­0.3 ­0.2 ­0.1 0.1 0.2 0.3 0.4 0.5
p'
v'
FIGURE 2.Vortex-airfoil interaction.above:v

t

3

5

,be-
low:p

t

3

5

.
the presence of edges the sound generation by vorticity
becomes a linear problem and it is reasonable to study
airframe noise generation using linear equations (1-3).
APPLICATION TOGUST-AIRFOIL
INTERACTION
The presented simulation concept for a computational
assessment of airframe noise using CAA is applied for
the example of noise generated by an airfoil due to in-
coming gusts.When a vorticity perturbation strikes the
leading edge of an airfoil,sound is generated.A 2D-
case is presented in Fig.2,showing the vertical veloc-
ity v

and the pressure field p

well after a test vortex
was seeded into the Mach-0.5 flow one chord length up-
stream the airfoil.The initial,circular perturbation ve-
locity field vvv

t

0

is derived from the stream function


x

y

t

0

0

21exp

ln

2

x

1

5

2

y
2

0

15
2

and is normalized such that the maximum perturbation
speed is equal to one.The upper part of Fig.2 shows the
divided vortex field,while the pressure distribution in the
0.0e+00 5.0e−04 1.0e−03
p
^
v’
max
cos
0.0e+00
4.0e−04
8.0e−04
1.2e−03
p
^
v’
max
sin
0 %
6 %
12 %
18 %
0.0e+00 5.0e−04 1.0e−03
p
^
v’
max
cos
0.0e+00
4.0e−04
8.0e−04
1.2e−03
p
^
v’
max
sin
0 %
6 %
12 %
18 %







x
y
FIGURE 3.Radiated pressure from airfoils of different thick-
ness in Mach-0.5 flow,referenced to free field impedance.
Above:Sr=1,Below:Sr=3;Bandwidth  Sr

0

2;[6].
lower part of the figure depicts the acoustic pressure prop-
agating away fromthe airfoil.
At DLR,a first CAA-design study on the influ-
ence of airfoil thickness on the sound generation in
”dirty” inflow conditions was carried out in three
dimensions[6].Joukowski-airfoils (span along z-
direction,flow along x) with different thicknesses
were subjected to the same localized test vortices with
v
v
v


t

0

y

x

0

exp

ln

2

x
2

y
2

z
2

0

1
2

.
Pressure time histories p

t

in the z

0-plane on
a circle with a radius of 1.5 chord lengths l around
the airfoil nose were Fourier-transformed.The Fig.3
shows radiation directivities for two Strouhal numbers
Sr

f l

U

with frequency f referenced w.r.t.the
freestream velocity U

.The transformed pressure is
related to the maximum initial perturbation velocity
v

max
and thus appears formally as an impedance.The
diagrams show that under the same inflow conditions
thick airfoils are much quieter than thin airfoils.
CONCLUSIONS
Inviscid perturbation equations may be used to
simulate the generation mechanism of airframe noise
employing high resolution CAA codes.Essential part
of airframe noise generation rests on linear dynam-
ics.The presented ”vortex-test” serves as a method
to support low-noise design of airframe components
theoretically/numerically.
ACKNOWLEDGMENTS
Part of the work,leading to the present paper was
sponsored by Deutsche Forschungsgemeinschaft DFG
as part of the SWING+ project,which is gratefully
acknowledged.
REFERENCES
1.Delfs,J.W.,AIAA Paper No.2001-2199 (2001).
2.Dobrzynski,W.;Nagakura,K.;Gehlhar,B.;Buschbaum,
A.,AIAA Paper No.98-2337 (1998).
3.Michel,U.;Helbig,J.;Barsikow,B.;Hellmig,M.,AIAA
Paper No.98-2336 (1998).
4.Tam,C.K.W.;Dong,Z.,Theoret.Comput.Fluid Dynamics,
Vol.6,pp.303–322,(1994).
5.Tam,C.K.W.;Shen,H.,AIAA Paper No.93-4325 (1993)
6.Grogger,H.A.;Lummer,M.;Lauke,Th.,AIAA Paper No.
2001-2137 (2001).
7.Tam,C.K.W.;Dong,Z.,Journal of Computational Acous-
tics,Vol.89,pp.439–461 (1993).

 
   
          
            
              
       
            
        
       
 


   
    
     
 
   
      
         
  
     

   
 
    
     
  
       
        
 


 


    
 






  
  
       
   
      


    
  
 
      
      
    
 

   


     
          
 
 

 
  
  
 

  
  
   


     
 
      
         
 
0
30
60
90
120
102
106
110
114
118
 (deg)
SPL(dB)
 
   
     
       




  

   
 
   
      
   
    
     
   
      
   
    
  
   
       
  
    
    
   
  

    


 
   
      
   
  
  

   

       
  
  
     
     
    
    
       
  
    
       
    

 
      
   


  
    
   

   


   


  
     
    
       
  
  
 
     
    






  

   
     
  
  
     
       



  
 
     
  


 
The Role of Computational Aeroacoustics in Thermoacoustics
P.J.Morris
a
and S.Boluriaan
a
a
Department of Aerospace Engineering,Penn State University,University Park,PA 16802,USA
Thermoacoustic devices use the phase relationship between the pressure and particle velocity in the Stokes boundary layer near
surfaces to transport heat from cold to hot heat exchangers.For thermoacoustic devices to be optimized,an improved understanding
of unsteady minor losses is required.In this paper a parallel numerical simulation of the minor losses in a sudden expansion in a
resonator is described.The Navier-Stokes equations are discretized in space and time with high-order accurate numerical schemes.
These schemes,that are also used in computational aeroacoustics,minimize numerical dispersion and dissipation errors.A high
amplitude standing wave is generated in a resonator with a sudden change in cross-sectional area.The details of the unsteady ßow in
the vicinity of the sudden expansion are provided.It is shown that mean recirculating ßowregions are established in the two sections of
the resonator.The average pressure losses across the expansion are determined and the relative contributions of the Bernoulli pressure
and the total pressure losses due to the generation of vorticity are estimated.
INTRODUCTION
Thermoacoustic devices can be either prime movers or
heat pumps.Swift [1] provides a description of the basic
physical processes involved as well as examples of dif-
ferent thermoacoustic engines.Recently built thermoa-
coustic engines,such as the Stirling heat engine designed
by Backhaus and Swift [2],have efÞciencies that rival the
common internal combustion engine.This is achieved
through the use of a traveling acoustic wave,that has the
correct pressure/volume phasing of the Stirling cycle,in
one part of the engine.This wave is maintained at a high
amplitude by a standing wave in another section of the
engine.To prevent a net mean ßow in the traveling wave
loop of the device,that reduces the systemÕs efÞciency,a
Òjet-pumpÓ is used.The average minor losses across the
jet-pump eliminate the mean ßow.Minor losses are well-
documented for steady ßows (see Idelchik [3]):however,
this is not the case for the unsteady ßowin a thermoacous-
tic engine.The purpose of the research described here is
to address this lack of understanding.
TECHNICAL APPROACH
In order to describe the interaction between the acous-
tic wave in the resonator and the resonator walls the
Navier-Stokes equations are used.They are written in a
generalized coordinate system.The total energy equation
is used as well as the equation of state for a perfect gas.
The coefÞcient of viscosity is related to the thermody-
namic properties using SutherlandÕs formula.A Prandtl
number of 0.72 is used to relate the coefÞcients of viscos-
ity and thermal conductivity.No turbulence model is used
in the present simulations.The equations are discretized
using the Dispersion Relation Preserving algorithm of
Tam and Webb [4] in space and a fourth-order Runge-
Kutta scheme in time.The computer code is written in
Fortran 90 with the Message Passing Interface (MPI) as
the parallel implementation.A domain decomposition
method is used,in which the physical domain is decom-
posed into sub-domains and message passing is only em-
ployed at the sub-domain boundaries.In addition,a par-
allel multiblock grid structure is used.This is appropriate
for the present problem of a resonator with two sections
of different cross sections.The geometry and computa-
tional domain used in the present two-dimensional sim-
ulations are shown in Fig.1.The lengths are nondimen-
sionalized by the length of the larger resonator.Differ-
ent blocks are used for the grids in the two parts of the
resonator in order to insure the grid orthogonality.The
1.0
0.5
0.03
0.05
Line Source
0.025
0.025
A
0.003
B
y
x
FIGURE 1.Sketch of the computational domain.Not to scale.
calculations are performed on a PC cluster.The compu-
tational time on 32 processors is 2.8µsec/grid point/time
step.A companion experimental resonator (see Doller et
al.[5]) is driven by either a shaker or a loudspeaker.In
order to model the effect of the driver,a source term is
introduced into the continuity equation.It has a Gaussian
distribution in the x-direction and is located a distance of
0.05 fromthe closed end of the larger channel,as shown
in Fig.1.Asource termis also introduced into the energy
equation to insure that only acoustic disturbances are gen-
erated.No slip and no penetration conditions are applied
at all walls.Either adiabatic or isothermal wall boundary
conditions are enforced.
RESULTS AND DISCUSSION
After an initial transient period,a standing wave is es-
tablished in the smaller resonator channel.In the com-
panion experiment,the resonant frequencies are estab-
lished with a broadband excitation.In the present calcu-
lations,a single frequency that generates a quarter wave-
length standing wave in the smaller channel is used.Mor-
ris et al.[6] show how the system settles into a periodic
state after approximatelytwenty periods.Near the change
in the cross section,there is a periodic shedding of vor-
tices associated with the jet-like part of the cycle.In-
stantaneous streamtraces are shown in Fig.2.The ßow is
symmetric about the channel centerline.This is due to the
plane wave excitation.Two pairs of vortices of equal rota-
tion sense are seen.In addition,a pair of vortices with the
opposite sense formin between them.The vortices move
away fromthe contraction by a process of mutual induc-
tion.In addition,a net mean ßow is generated.Immedi-
ately at the contraction there is a very small mean ßow
on the channel centerline towards the smaller channel.
A stronger centerline mean ßow away from the contrac-
tion is observed in the larger channel.These mean ßows
-
0
.
0
3
-
0
.
0
2
-
0
.
0
1
0
0
.
0
1
0
.
0
2
0
.
0
3
0
.
5
0
.
5
1
0
.
5
2
0
.
5
3
0
.
5
4
0
.
5
5
0
.
5
6
0
.
5
7
0
.
5
8
0
.
5
9
FIGURE 2.Streamtraces of the instantaneous ßow in the res-
onator.
are associated with two recirculating regions that Þll both
parts of the resonator.The instantaneous pressure differ-
ence between point A in the larger channel and point B
in the smaller channel (shown in Fig.1) reaches a steady
value after the transient period.This pressure difference
has contributions from the Bernoulli pressure (see Wang
and Lee [7]) and losses due to the generation of vorticity.
The former is a second order effect associated with the
Þnite amplitude of the acoustic pressure and particle ve-
locity.ItÕs contribution is estimated to be 25%of the total
pressure difference.Thus,the larger contribution may be
associated with the generation of vorticity at the sudden
expansion/contraction.
The simulations described here represent a prelimi-
nary examination of the ability of numerical simulations,
based on methods from computational aeroacoustics,to
aid in the understanding and optimization of ßuid dy-
namic phenomena in thermoacoustic devices.Much work
remains to be done.In particular,more realistic three-
dimensional geometries that match the actual jet pumps
should be examined.The acoustic driver needs to be
modeled more accurately.Also,at high amplitudes,the
boundary layers may be alternately laminar or turbulent.
This is caused by the periodic variation of the pressure
gradient from favorable to adverse.This is a very chal-
lenging turbulence modeling problem.Some preliminary
efforts by the authors suggest that an unsteady Reynolds-
averaged Navier-Stokes method could be useful.
ACKNOWLEDGEMENTS
This work was supported by the OfÞce of Naval Re-
search.
REFERENCES
1.G.W.Swift,Journal of the Acoustical Society of America
84,1145 (1988).
2.S.Backhaus and G.W.Swift,Nature 399,335 (1999).
3.I.E.Idelchik,Handbook of Hydraulic Resistance,3rd ed.
(Begell House,New York,1994).
4.C.K.W.Tamand J.C.Webb,J.Computational Physics 107,
262 (1993).
5.A.Doller,A.A.Atchley,and R.Waxler,Journal of the
Acoustical Society of America 108,2569 (2000).
6.P.J.Morris,S.Boluriaan,and C.M.Shieh,Computa-
tional Thermoacoustic Simulation of Minor Losses Through
a Sudden Contraction and Expansion,AIAA/CEAS Paper
2001/2272,2001.
7.T.G.Wang and C.P.Lee,in Nonlinear Acoustics,edited by
M.F.Hamilton and D.T.Blackstock (Academic Press,New
York,1998),Chap.6,pp.177Ð204.
Aeroacoustic studies and tests performed to optimize
the acoustic environment of the Ariane 5 launch vehicle
D. Gly, G. Elias and C. Bresson
Office National d’Etudes et de Recherches Arospatiales
29, avenue de la Division Leclerc
BP 72, 92320 Chtillon Cedex - France
To decrease the acoustic levels inside the Ariane 5 fairing and to reduce the excitation applied to the payload, it was necessary to
investigate a method for reducing substantially the acoustic environment during the lift-off. The highest-noise radiating regions were
identified by analyzing the signals from a microphone array implemented on the launch vehicle. It is to our knowledge the first time such
a technique is used on a launch vehicle. The MARTEL facility was then used to characterize the efficiency of a horizontally extension of
the lateral flues. Based on test results and applying similarity criteria, it was possible to determine the length of this extension necessary
to achieve the required noise attenuation inside the fairing. The acoustic measurements made on the launch vehicle after the flue
extension construction in Kourou confirmed the reduced scale predictions obtained at MARTEL facility.
INTRODUCTION
The more and more powerful launch vehicles, such as Ariane 5,
involve an increase of the external Sound Pressure Level (SPL).
As the confort of launch vehicle at lift-off may become a quite
important commercial argument for the customers in the future
therefore, a continuous effort must be kept in order to obtain
“low” acoustic levels in the payload bay. The acoustic analyses
made during the V503 qualification flight of Ariane 5 confirmed
the possibility to reduce the acoustic levels inside the fairing
when the launch vehicle goes through the altitude from 10 to 20
meters. This study has been supported by the Ariane Program
and the Research & Technology program of CNES.
MEASUREMENTS DURING V503 FLIGHT
For the V503 flight, a microphone array has been implemented
around the fairing to perform noise source localization (figure 1).
For the launcher altitude of 20 m, the acoustic map shows only
three main source locations appearing, one in the middle of the
uncovered central engine flue, and the two others at the Solid
Rocket Booster (SRB) flue outlets.
Figure 1: Acoustic sources localization V503 flight. Alt. 20 m
Based on the relative levels of the SRB and central engine
sources and on the NR index, it appeared necessary to reduce the
acoustic sources at the SRB flue outlets by at least 6 dB to
achieve the required noise reduction inside the fairing. Thus,
CNES decided to perform tests at the MARTEL facility,
conducted by ONERA, in order to find a solution allowing to
decrease the acoustic levels in the payload bay and then to
improve the specification applied to the payload.
TESTS AT THE MARTEL FACILITY
MARTEL facility, installed in CEAT at Poitiers
University, has been developed as part of the Research and
Technology program lead by CNES 1. The air-hydrogen
combustor generates subsonic or supersonic jets, cold or
hot, up to 1800 m/s and 2100 K. A test campaign has been
carried out with a 1/47 mock-up representative of the ELA3
launch pad. Only a single jet being available in MARTEL,
the mock-up simulates only the half part of the Ariane 5
pad with its SRB flue (figure 2).
Figure 2: Ariane 5 launch pad mock-up in MARTEL facility
The absolute acoustic levels measured in MARTEL facility
are not representative of the full scale but the relative levels
between several test configurations can be extrapolated.
The first approach to obtain noise reduction was to improve
the efficiency of the water injection devices 2.
Unfortunately, the gains were too low to implement this
solution. The second method investigated was an extension
of the SRB flues. In practice, three extensions were tested
namely 10, 15 and 30 meters at full scale.
The tests were conducted for several simulated altitudes,
between 0 and 20 m. The noise spectra measured for the
three extensions are shown in the figure 3. These results
were obtained for a critical altitude of 10 m. In each case, a
noise reduction was observed over a wide frequency band.
The noise reduction increases with the flue extension
length. However, based on results not presented herein, the
reduction decreases as the launcher climbs. Indeed, the
extension of the flue gradually masks the jet near the final
section.
SRB flue outlet
SRB flue outlet
Central flue
No flue extension (V503)
10-meter flue extension
15-meter flue extension
30-meter flue extension
dB
Third octave band (Hz)
5 dB
Figure 3: Flue extensions effect
The noise attenuations are due to the masking effect of the flue
and to the change in the direction of the jet subsequent to the
horizontal extension of the flue. The main jet emission direction
is thus farther from the top part of the launcher. The
extrapolation of the attenuation predictable at full is based on
similarity criteria. Applying a 2 dB safety margin to take into
account the measurement error and the reproducibility of the tests
shows that a 30-meter horizontal extension of the SRB flues
satisfies the requirement.
MEASUREMENTS DURING V504 FLIGHT
EADS, CNES and ARIANESPACE decided to perform the flue
extension on the Kourou site just before the V504 flight, as seen
in figure 4.
Figure 4: SRB flue extension in Kourou site
The comparison between V503 and V504 flights appears in
figure 5, where the mean SPL of flush-mounted microphones
located on the fairing has been plotted.
Figure 5: V503-V504 Ariane 5 flights. Acoustic data comparison
Before the lift-off, the central engine is running alone and no
difference is observed. In contrast, a 5 dB reduction appears from
the lift-off up to an altitude of 60 meters. It is interesting
to quote that, with the new flues, the contribution of the
SRB to the noise during this period seems to have
disappeared, the levels being the same before and after their
ignition. This result is confirmed by noise source
localization, presented for V504 flight in figure 6, where no
acoustic source remains at the SRB flue outlets at an
altitude of 20 meters.
Figure 6: Acoustic sources localization V504 flight. Alt. 20 m
The acoustic measurement performed in Kourou fully
agrees with the predictions made using the MARTEL
facility and the final noise reduction objective has been
reached.
CONCLUSIONS
The test campaigns conducted by ONERA on the
MARTEL facility allowed efficient solutions to be found
for optimizing the noise level during lift-off of the Ariane 5
launch vehicle. The SRB flues were extended by 30 meters
just before V504 flight. The acoustic measurements made
during this flight confirmed the predictions. An attenuation
of about 5 dB was measured on the fairing and the
localization of the noise sources using a microphone array
on board has shown that the noise source at the SRB flue
outlet has been dramatically eliminated. The MARTEL
facility and expertise in advanced signal processing
techniques prove to be excellent performing tools for
successfully conducting experimental studies which can be
extrapotated at full scale.
ACKNOWLEDGMENTS
This study was supported by CNES (Ariane Program). The
authors thank the companies EADS and Arianespace for their
help. We would like to extend a special note of thanks to the
MARTEL team.
REFERENCES
1
H. Foulon, D. Gly, J. Varnier, E. Zoppellari, Y. Marchesse
MARTEL: Simulation of Space Launchers Aeroacoustic
Ambience
12th European Aerospace Conference (AAAF/CEAS)
Paris (France), November 29-30 - December 1, 1999
2
D. Gély, G. Elias, C. Bresson, H. Foulon, S. Radulovic
Reduction of supersonic jet noise. Application Ariane 5
6
th
AIAA/CEAS Aeroacoustic conference.
Lahaina (Hawaii – USA), 12-14 june 2000
V504 flight:
30-meter extension flue
SRB flue outlet
SRB flue outlet
Central flue
Altitude (m)
V503 Flight
5 dB
V504 Flight
0 10
40 100
SRB IGNITION
Influence of compartment size on radiated sound power
level of a centrifugal fan
Leping Feng
MWL, Department of Vehicle Engineering, KTH, SE-100 44 Stockholm, Sweden
e-mail: fengl@fkt.kth.se
The influence of the compartment size on the radiated sound pressure level of a centrifugal fan is investigated experimentally.
The measurement set-up consists of a commercial centrifugal fan and a cavity with adjustable walls and ceiling. The inflow
condition is adjusted indirectly by adjusting the geometry of the box, in order to avoid the difficulties to describe and measure
the inflow conditions quantitatively. The tests are performed for a range of typical situations of ventilation systems. The sound
pressure levels in a few typical positions are measured in a semi-aechoic room. Some useful results, and an empirical relation
between the sound power level and the geometry of the cavity in a certain range, are obtained from the measurements.
INTRODUCTION
The radiated sound power level of a centrifugal fan
is strongly influenced by the inflow condition. An
example of this is that a fan usually radiates 3 dB or
more sound power when located inside a ventilation
system than in a free condition. There are several
different parameters that may influence the inflow
conditions. In this paper, we only deal with one
situation: the change of the cross section of the
compartment where the fan is located.
DESCRIPTION OF TEST SET-UP
The tests were performed in the semi-anechoic
room of the Department, with the test set-up shown in
Figure 1. The positions of the walls and roof are
adjustable to make variable cross sections of the fan
compartment. The centrifugal fan tested is SAMI GS
with 11 blades manufactured by ABB.
Figure 1 Illustration of the test set-up
Figure 2. Illustration of microphone positions
The opening of the outlet, which is a 0.3 X 0.3 m
square duct, is covered with a perforated panel
(perforated ratio ∼ 30%) in order to make the fan
working in practical working point. Two microphones
are employed to register the sound pressure levels.
One is located at the same plane of the outlet, 0.65 m
from the centre of the duct. Another is located at the
centre line of the inlet side, 2 meters away from the
compartment (see Figure 2). The tests are performed
in four different (motor) speeds: 700, 1000, 1500 and
1900 rpm. Eleven cross sections of the compartment,
varying from 0.22 to 0.596 m
2
, or from 2.443 to 6.622
times of the area of the outlet duct, are tested.
SOME RESULTS
In order to check the general trend of the sound
pressure/power level in function of the size of the
compartment, the registered sound pressure levels at
the four different rotating speeds are normalised.
That is, the total sound pressure levels at different
rotating speeds are set to be equal to that of the sound
pressure level when the rotating speed is 1000 rpm.
The sound pressure level at each frequency band is
then calculated as
Mic. 1
(outlet side)
Mic. 2
(inlet side)
Duct
Fan
compartment
2
m
0.5 m
60
65
70
75
80
85
90
95
100 200 400 800 1600 3150 A-
weghted
Frequency, Hz
outlet
inlet
Figure 3 Typical spectra at the two positions
)(
1 rki
normalised
i
LLLL −+=
(1)
where subscript i denotes 1/3 octave band number,
1k and r are rotating speed.
Typical spectra of sound pressure levels measured
at the two microphone positions are shown in Figure 3.
They have different shapes. The differences at low
frequencies could be due to flow, since the flow speed
at outlet side is much higher than that at inlet side. The
high frequency components, on the other hand, might
be due to the interaction between the flow and the
perforated panel. Since the microphone at the inlet
side is directly pointed to the fan and the cross section
at this side is much larger, the signal registered by
microphone 2 might more correctly reflect the changes
of the fan due to the size change of the compartment.
The test situations of the cross section are
normalised by the area of the outlet duct in order to get
a non-dimensional measure. Figure 4 & 5 shows the
third octave band sound pressure level at microphone 2
as a function of the normalised area. As a general
tendency, sound pressure levels decrease when the area
is increased, with the value dependent on frequencies.
Results from microphone 1 show the same tendency.
65
70
75
80
85
90
2 3 4 5 6 7
Normalised area
100
125
160
200
250
315
400
500
630
Figure 4 Sound pressure levels at microphone 2
as a function of area: low frequencies
65
70
75
80
85
90
95
2 3 4 5 6 7
Normalised area
800
1000
1250
1600
2000
2500
3150
4000
5000
A
Figure 5 Sound pressure levels at microphone 2
as a function of area: high frequencies
In order to get a general picture of the influence of
the cross section on the radiated sound pressure level,
linear regression is made for all measured data, in
decibel, according to equation
BsAL += 72 ≤≤ s
(2)
where s is the normalised area and A and B regression
coefficients. Figure 6 shows the coefficient B, which is
the increase of the sound pressure level when the size
of the compartment increases the area of one cross
section of the outlet duct. For both microphones, this
value is almost always negative, indicating that
reducing the compartment size will increase sound
pressure levels at all frequency bands. Although there
is a big difference for 1/3 octave band values, the
coefficient B is almost same for A-weighted sound
pressure levels measured at both microphone positions.
CONCLUSIONS
Reducing cross section of the fan compartment will
increase the radiated sound power level. This increase
seems not as big as we expected.
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
100 200 400 800 1600 3150 A-weghted
Frequency, Hz
Figure 6 Regression coefficients B
Solid: inlet side; Dotted: outlet side
Effects of Blade Material on Sound Radiation by Attached
Cavity in Unsteady Flow
S. Kovinskaya
a

and

E. Amromin
b
a
Seagate Technology, 10323 West Reno, Oklahoma City, OK73127, USA
b
Mechmath LLC, 2109 Windsong,Edmond,OK73034, USA
Sound radiation by a cavity attached to a blade under unsteady flow excitation is analyzed. It is shown that c
avity
volume oscillations and radiated sound power are sensitive to variations in ratio of blade material Young’s modulus to product of
fluid density on square of flow speed. These variations change both frequencies and levels of peaks in spectra of radiated sound.
MATHEMATICAL FORMULATION OF
PROBLEM

Prediction of sound radiation by cavitating
blades/hydrofoils is currently based on model tests.
Because of differences between flow-induced sound in
model and full-scale flows, extrapolations of
experimental data to full-scale conditions are not
completely satisfactory, especially in low-frequency
band [1].
Selection of appropriate similitude criteria is an
important problem that can be clarified by the
numerical analysis. A realistic analysis must take into
account simultaneous oscillations of cavity thickness
and length under periodical excitations of incoming
flow, and represents a nonlinear problem with a
varying boundary. Theory [2] allows such analysis for
elastic blades with the use of 2-D numerical modeling.
For a blade (hydrofoil) at a given time-averaged angle
of attack, periodical perturbation of incoming flow can
be caused by turbulence. The perturbation magnitudes
are much smaller than free-stream speed. The vibration
of blade with attached cavity in unsteady flow (Fig.1)
is described in 2D approach by equation for beam in
bending motion:
F
U
t
V
h
x
V
JiE
x
2
*)1(
2
2
2
2
2
2
2
￿
￿￿ ￿
￿
￿
￿
￿
￿
￿
￿
￿
(1)
Here U is free-stream speed; V is transverse
displacement of the blade; ￿ is its loss factor; E is its
Young’s modulus; h and J are thickness and inertia
moment of its sections; ￿* and ￿ are densities of blade
material and water; F is the hydrodynamic load
coefficient. The coefficient F depends on velocity
potential ￿ that is a solution of Laplace equation
0￿￿￿
with the following boundary conditions:
yt
V
S
￿
￿
￿
￿
￿ ￿
1
;
yt
BV
S
￿
￿
￿
￿￿
￿
￿
2
)(
(2)-(3)
3
2
2
2
)(1
2
S
S
t
FU
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿￿￿￿￿
￿
￿
(4)
Here B is the cavity thickness, S
1
is a cavitation-free
blade surface, S
2
is the cavity surface, S
3
is projection
of S
2
on down surface of blade; ￿ is potential of time-
averaged flow around the blade. Cavitation number
￿=2￿P/(￿U
2
), where ￿P is a difference between
pressure in incoming flow and within the cavity. The
system (1)-(4) must be completed by Joukovski-Kutta
condition that defines the blade lift. For periodic
excitations that correspond to boundary condition
￿￿￿￿x￿
x￿￿
￿0 and ￿￿￿￿y￿
x￿￿
￿Ae
i￿t
(where A=const;
￿ is excitation frequency), Eq.(1) can be rewritten as
2
)1(
2
2
2
2
2
2
F
R
C
hV
S
x
V
iJK
x
t
￿￿
￿
￿
￿
￿
￿
￿
Here C is the blade’s chord (see Fig. 1); K=E/(￿U
2
),
R=￿*/￿,

and
St
=￿C/U are dimensionless parameters
that affect vibration.
C
L
U
a
Fig. 1. Flow around a blade section.
Mechanical boundary conditions of rigid clamping
in a middle part of the blade (where V=dV/dx=0) and
of free edges (where d
2
V/d
2
x=d
3
V/dx
3
=0) are sufficient
for integration of Eq. (1) and determination of blade
vibration. Estimating cavitation-induced sound
radiation, it is important to keep in mind that an
oscillation of the cavity volume D is its principal cause,
and sound power level Sp~d
2
D/dt
2
[1]. This volume,
however, depends on blade elastic properties, because
boundary conditions for ￿ include V. Therefore, sound
power can be found by computing d
2
D/dt
2
after solving
Eqs.(1)-(4). For any periodical excitation of incoming
flow, the cavity thickness oscillates simultaneously
with the cavity length L. As a result, the cavity volume
oscillation accumulates both length and thickness
oscillation. The volume oscillation is nonlinear and has
high harmonics. Therefore, the blade response is multi-
frequency, and significant nonlinear effects appear [3].
Numerical Analysis
Although the current analysis is assigned to
simplified flow geometry, this analysis is able to clarify
some physical aspects. There are five dimensionless
parameters in Eqs. (1)-(4): St, ￿, R, L/C and K. The
similitude by using R=￿*/￿, St and ￿ is evidently
attainable for model tests, but there are at least three
effects that are different for full-scale and model flows.
First, the ratio of cavity length L to the blade chord C
depends on blade’s scale; this is an implicit viscosity
effect [4]. Second, real incoming flow spectra are
broad band and usually unknown [5]. Third, the blade
admittance in the actual flow affects its sound
radiation. This effect can be modeled by keeping K and
R, but it is usually impossible for model tests to fix
both K=E/(￿U
2
) and R=￿*/￿.
-3
0
3
6
0.2 0.6 1 1.4
Log S
t
Lo
g
S
p
Fig..2. Effect of cavity length on sound radiation S
P
by steel
hydrofoil NACA-0015. Solid curve –computation for
L/C=0.75, dashed – for L/C=0.15. X -measurements for
L/C=0.75, ￿ - for L/C=0.15
The incoming flow spectrum is accepted as white
noise in the presented computations. A computed
cavity response at every frequency ￿ includes the
response on excitation at the same frequency (first
harmonic), second harmonic of its response on
excitation at ￿/2, third harmonics of the response at
￿/3, etc. One can see in Fig.2 that the prominent
frequencies of sound radiation are found satisfactory in
numerical analysis based on Eqs.(1)-(4).
Fig. 3. Effect of blade material on sound radiation (Sp) of
cavitating hydrofoil NACA-0015. The curve 15-S
corresponds to incoming flow speed 15m/s for steel hydrofoil
(K=0.9￿10
8
), 9-S (marked by *) –to 9m/s for steel hydrofoil
(K=2.5￿10
9
), 9-A (marked by o) -to 9m/s for aluminum
hydrofoil (K=0.9￿10
8
). Cavity length L=0.6C.
Selection of the similitude criterion for modeling of
a material effect must depend on frequency band (St
values). For large St, the ratio ￿*/￿ is more influent,
but K=E/(￿U
2
) is more important for moderate and low
frequencies. The dependencies of sound power from
frequency for different values of K are plotted in Fig.3.
The performed analysis allows conclusion that cavity
volume oscillations and the radiated sound power are
sensitive to variations of K. These variations change
both frequencies and levels of spectrum peaks.
REFERENCES
1.
Blake WK. Mechanics of Flow-Induced Sound and
Vibration. Academic Press, 1986
2.
Amromin E & Kovinskaya S. Journal of Fluids and
Structures, 2000, v14, p735-751.
3.
Koinskaya S, Amromin E & Arndt R.E.A. Seventh
International Congress on Sound and Vibration,
Garmisch-Partenkirchen, 2000, vIII,p1417-1424
4.
Amromin E. Applied Mechanics Reviews. 2000, v53,
p307-322
5.
Arndt R.E.A. ONR 23
rd
Symposium on Naval
Hydrodynamics, Val-de-Reul, 2000
Line source radiation over inhomogeneous ground
using an extended Rayleigh integral method
F.-X.Bécot
a,b
,P.J.Thorsson
b
and W.Kropp
b
a
Transport and Environment Laboratory - INRETS,F-69675 Bron,France – becot@inrets.fr
b
Department of Applied Acoustics - Chalmers,S-41296 Gothenburg,Sweden
The method presented in this paper is proposed as an alternative to standard boundary integral equations for the sound radiation
of a line source over grounds of arbitrary impedance and profile.Valid for any kind of source,it takes advantage of the Rayleigh
integral formulation to yield a minor computational effort for flat surfaces.The calculation time,optimized according to the Fresnel
zone principle,is expected,however,to be similar to boundary element methods for the case of non-flat grounds.The extended
Rayleigh integral method is validated here for multipole sources radiating over homogeneous grounds.This proves its reliability for
the prediction of strongly directional sound fields.
INTRODUCTION
The general case of a source radiating above a ground
of arbitrary impedance and profile is usually handled
by integral equation methods,often to the expense of
the computational effort.Therefore,an original method
for such cases has been developed on the basis of the
Rayleigh integral method for flat surfaces (see also [1]).
Like BE methods,to which it is an alternative,this
method is valid for sound propagation above grounds of
arbitrary impedance and profile,and it handles any kind
of primary source.
Firstly,the boundary value problem is briefly derived.
The specificity of the present work is explained in a sec-
ond part.Finally,numerical examples are presented to
prove the reliability of the method.
THEORETICAL BASIS
The main idea is to estimate the sound field above an
arbitrary impedance ground fromthe pressure field of the
same source radiating above either rigid or totally soft
ground (this approach is also that of the study in [2]).To
account for the ground effects,a number of sources are
placed at the ground level.Thus,at a point x
r
in the half
space above the surface ,the total radiated pressure can
be written
p

x
r

Q
0
G
0

x
s

x
r



Q



G



x
r

d (1)
where G
0

x
s

x
r

is the free-field Green’s function at a
point x
r
due to a source located at x
s
.G



x
r

are the
analogue Green’s functions for the sources located at a
point  of the ground.According to the Rayleigh inte-
gral for a flat surface,if a rigid,respectively soft,primary
boundary condition is chosen,they represent monopoles,
respectively dipoles,on the ground surface.
The desired boundary condition on the ground surface is
expressed using the definition of the normal acoustical
impedance of the ground,p

Zv
n
.Including the corre-
sponding Green’s functions for the velocity,the boundary
value equation of the problemcan be expressed as


Q



G



x

Z

x

G
(v,y)



x

d

Q
0

G
0

x
s

x

Z

x

G
0
(v,y)

x
s

x

(2)
where the superscript

v

y

indicates the velocity Green’s
functions in the y

direction,normal to the surface at the
point x.The amplitude of the sources on the ground,
Q



,are the unknwowns of this integral equation.
Eq.(2) holds for any shape of the surface .However in
the following,only sound propagation over flat surfaces
will be examined because it results in a major simplifica-
tion of the problem.For uneven terrains though,the com-
putational effort using the present method is expected to
be equivalent to that resulting fromBE approaches.
THE EXTENDED RAYLEIGH
INTEGRAL METHOD
Eq.(2) can be simplified if,for instance,a rigid pri-
mary boundary condition is assumed to be fulfilled.In
this case,according to the Rayleigh integral,G
0
(v,y)
is
zero on the ground surface.G
(v,y)
is also zero at all points
of the surface,except for x

,which represents a sin-
gularity.
Thus,in Eq.(2),the evaluation of the integral at the
singular points is performed by determining the Cauchy
principal value:it is zero for G



x

and a finite value for
G
(v,y)



x

.At other points of the surface,a numerical
integration,for instance,using a Gauss-Legendre quadra-
ture,can be performed with arbitrary accuracy as long as
the singularity itself is not chosen.As a result,the bound-
ary value problemcan be formulated as

quad

Q



G



x

d

jZ

x

Q

x

2

Q
0
G
0

x
s

x

(3)
Once the source strengths are determined,the pressure
field including the ground effects,can be calculated at any
point in the above half space.
NUMERICAL EXAMPLES
The method has also been presented in [1] and was
shown to yield good predictions of relative pressure fields
for monopoles above homogeneous and inhomogeneous
grounds.Special attention is paid here to the case of a
high order source radiating above homogeneous surfaces.
As in [1],receiving points are placed on a quarter circle
of radius 1.2 m,from the perpendicular vertical to the
surface (0 degree) to directly on the ground (90 degrees).
The source is placed on a 1.2 m radius quarter circle op-
posite to the receivers,with an angle of 5 degrees (low
source position) or 45 degrees (high source position) with
the direction of the surface.This geometry allows the in-
vestigation of the near field and the far field of the source.
For the discretization of the ground surface,a number of
10 elements per wavelength was chosen,on a portion of
ground corresponding to the first Fresnel zone,to opti-
mize the calculation time.Furthermore,a 10:th order
Gauss-Legendre quadrature was used to insure good con-
vergence of the solution.First,normalised pressure fields
0
30
60
90
-30
-20
-10
0
10
Receiving angles (°)
Lp relative free field (dB)
Extended Rayleigh
Exact soft solution
0
30
60
90
−10
0
10
20
30
40
Receiving angles (°)
Lp relative free field (dB)
Extended Rayleigh
Exact soft solution
FIGURE 1.Radiation above totally soft,flat ground (Z=0):
60:th order multipole,high position,f =5kHz (right) – dipole in-
cluding positive and negative order,lowposition,f =1kHz (left).
froma high order line source are compared with exact an-
alytical solutions available for the radiation above totally
soft surfaces (see Fig.1).The good correspondance ob-
tained for both low and high source positions proves this
method to be reliable for the radiation fromsuch sources.
Secondly,to test the method for sound propagation above
partially soft ground (Z finite and different from 0),a
dipole pressure field is reproduced by the superposition
of two monopoles pressure fields,which were obtained
according to [2].(Simulation of a source of higher order
than 10 would fail due to numerical limitations).Mathe-
matically,the obtained solution is equivalent to a dipole
including both negative and positive orders (cf Fig.1,
left).The pressure fields fromthese two sources are com-
puted using the Extended Rayleigh integral method.As
solutions in [2] are accurate for rather rigid grounds,a
normalised acoustical admittance of  =0.2 is chosen.As
a guideline,the exact analytical solution for rigid ground
is also shown in Fig.2 (left).Despite discrepencies
0
30
60
90
-5
0
5
10
15
20
25
30
35
Receiving angles (°)
Lp relative free field (dB)
Extended Rayleigh
Exact rigid solution
Modified Chandler-Wilde
0
30
60
90
-10
0
10
20
30
40
Receiving angles (°)
Lp relative free field (dB)
Extended Rayleigh
Modified Chandler-Wilde
Exact rigid solution
FIGURE 2.Dipole radiation above a flat surface of acoustical
admittance  =0.2,f =1kHz,lowsource position – left:arbitrary
length of ground,right:Fresnel zone principle.
for steep incidence angles,which were expected due to
the limitations of solutions from [2],the agreement is
fairly good.In Fig.2 (right),limitations of the Fresnel
zone principle are exemplified due a considered portion
of ground,which is too small.Thus,the method seems
applicable to sound propagation due to the superposition
of sources above finite impedance grounds,at least for
grazing angles of incidence.
CONCLUSIONS
Due to a substantially lower computational effort for
flat surfaces,the Extended Rayleigh integral method was
proved to be advantageous regarding standard BE ap-
proaches.This applies for any source type (or super-
position of sources),radiating above arbitrary impedance
grounds.
ACKNOWLEDGMENTS
The authors wish to thank Région Rhône-Alpes and
the Swedish Transportation and Communication Re-
search Board (KFB) for their financial support.
REFERENCES
1.Bécot,F.-X.,Thorsson,P.J.,and Kropp,W.,“Noise prop-
agation over inhomogeneous ground using an extended
rayleigh integral method”,in Proceedings of inter.noise
2001,The Hague,The Netherlands,2001.
2.Chandler-Wilde,S.N.,Hothersall,D.C.,“Efficient calcu-
lation of the green function for acoustic propagation above
a homogeneous impedance plane”,Journal of Sound and
Vibration,180,705-724 (1995).
Experimental Investigations on Rijke Tube
Y. Zhu, K. Liu, M. Chen, J. Tian
Institute of Acoustics, Chinese Academy of Sciences
17 Zhongguancun St. P. O. Box 2712Beijing 100080, P. R. China
To investigate the principle of thermoacoustic interaction, a series of experiments on a heat duct (Rijke tube) are studied. The
influence of heat source location and temperature on the sound pressure and frequency in Rijke tube is provided. Besides the
linear characteristics, the nonlinear phenomena in Rijke tube are presented, such as instantaneous character, heat source
temperature saturation, and how the variation of inlet velocity, outlet acoustical condition, and outlet temperature affects the
acoustic field in tube. The results show that the acoustic change in Rijke tube can be highly nonlinear.
INTRODUCTION
Compared with normal steady combustors, the
pulsing combustors are highly efficient, energy saving,
and cause little pollution [1,2]. Rijke tube pulsating
combustor is a main kind of pulsating combustor. And
Rijke oscillations have been observed in industrial gas
furnaces, burner and rocket engine. The investigation
on Rijke combustor has not only the theoretical
significance, but also prospects for engineering
application [3]. However, the detailed mechanism
causing Rijke oscillation remains to be explained, and
it relates to aerodynamics, combustion and acoustics
This paper presents the experimental results of the
influence of heat source location, temperature on sound
pressure and frequency. The nonlinear phenomena are
also illustrated.
Experimental Apparatus
The Rijke tube is composed of a copper tube (55cm
long) and an earthenware pipe, (40cm long) with an
inner diameter of being both 5cm. They are connected
by a nut. The pipe is held vertically. The heater is made
of heating wire winding on quartz cross. The diameter
of the plane heated gauze is the same as the internal
diameters. The temperature of the heat source is
adjusted by a voltage regulator. The heat source is
placed in the tube, attaching a pair of thermo-couples
which measures the temperature near the heat source.
This temperature is obtained from a thermometer. The
voltage, current, sound pressure level and sound
frequency are also measured.
EXPERIMENTAL RESULTS
The characteristics of heater position
At the same voltage (the voltage V=99.6V, the electric
power W=371W), by changing the heat source
location, the oscillation region can be obtained. At this
electric power, the thermo-acoustic oscillation will
occur only when the heater is located in a given region
where 3cm <L<34cm, nearly from 1/30 to 1/3 of the
total tube length in the lower half of the tube. The
sound pressure level and the frequency will vary with
the location of the heat source. The maximum acoustic
oscillation occurs at L=14cm.
The character of heater temperature

With the increase of voltage and electric power, the
temperature of heat source should increase from 140C°
to 340C°. When the heat source temperature is 140C°,
only even harmonic can be activated. By increasing the
electric power, all the harmonics will be stimulated. In
the mean time, the oscillation frequency will rise. It
shows that not only the intensity but also the frequency
of sound is strongly dependent on heat source
temperature.
The instantaneous character of Rijke
tube

Since the heater is in the tube and the top of the tube
is closed, the convective air doesn’t exist. When the
temperature of heater rises up to a constant, the cover
is put off. Then the acoustic pressure oscillation will
appear immediately. With the air flowing and the
temperature decreaseing, the sound will attenuate, and
come to be silent finally. The frequency varies greatly
according to different heater temperature. With V=50V,
L=12cm, and different heater temperature, the
frequency spectrum which is measured at t=0 (first
second to sound) is as below.
The frequency spectrum becomes abundant, when the
temperature increases. When Tc=200C°, only even
harmonic can be stimulated. Along with the
temperature rising, odd harmonic appears. When
Tc=240C°, the odd harmonics in frequency spectrum
are 30dB lower than the even harmonics in adjacence.
The odd harmonics will not appear until temperature
reaches a standard value. Therefrom, the odd
harmonics increase quickly, and the pressure level of
every harmonic decreases in turn. But in this condition,
the sound pressure level of fundamental harmonic is
always a little lower than that of second harmonic. In
this experiment, the second harmonic is the easiest to
be stimulated, and its pressure level is the highest. The
frequency of fundamental and second harmonic is still
in proportion to heat temperature. The second order
harmonic frequency rises from 385Hz to 400Hz, with
the heater temperature increasing from 200C° to
400C°. This trend is the same as it in the steady state.
The influence of flow velocity on the
sound
When the electric power is settled, steady natural
convection is established in the tube. Then the inlet
area is changed. Due to restriction on flow, the sound
pressure level decrease, and the excitation of high
harmonics are suppressed. Although the heater
temperature is nearly the same, the sound pressure
level and frequency spectrum change greatly. The
fundamental harmonic frequency also decreases by
5Hz. Greater intensity sounds are provided when the
velocity is increased. But if the flow velocity is too
large, the acoustic oscillation will not be maintained.
The heat source temperature saturation
The heat source is at 1/4 of the tube length. The
oscillation will be maintained with the heat source
temperature increasing. But when the electric power
rises from 306W to 528W, the sound pressure level
will not rise accordingly, instead it will decrease a
little. The saturation phenomena are that the sound
pressure level does not increase with electric power in
proportion. There is a maximum when heater is at a
given location.
The influence of outlet condition
A piece of sound absorbing material is placed near the
top of the tube, and the heat temperature is relatively
low. If the acoustic oscillation has been excited, the
absorption function can’t destroy the energy balance in
the tube and suppress the instability. But if the initial
condition is silent, the sound absorbing material will
play an important role, and the thermo-acoustic
oscillation will not be established, even when the
heater temperature is the same.
The influence of the outlet temperature
At lower heater temperature, if some cold air is
blown to the tube outlet vertically, the sound will be
suppressed at once. When cold air is blown again, the
oscillation will be stimulated. At high heater
temperature, when cold air is blown too, the oscillation
will stop for a while. Then the oscillation is excited
again, but the temperature at the outlet will increase.
Keep this for several times, and the temperature will
rise until it reaches a constant, then it is maintained.
These phenomena are very interesting, but the reason
isn’t found.

0
50
100
150
200 400 600 800 1005 1205 1405 1605
f(Hz)
Lp(dB)
Figure 1. frequency of maximum oscillation
0
50
100
150
189 199 225 306 343 426 528
electric power
sound pressure level
(d
B
)
Figure 2. heater temperature saturation
Conclusions
In order to investigate the mechanism of thermo-
acoustic oscillation and to provide a theoretical base
for pulsating combustion, the influence of parameters
on the sound pressure and frequency in Rijke tube is
studied experimentally. Some interesting nonlinear
phenomena are observed and reported.
REFERENCES
1.C.-C. Hantschk, D. Vortmeyer, “Numerical simulation of self-
excited thermoacoustic instabilities in a Rijke tube”, J. Sound
and Vibration, 277(3),511-522 (1999)
2.S.Karpov, A.Prosperetti, “Linear thermoacoustic instability in
the time domain”, J. Acoust. Soc. Am, 103(6), 3309-3317 ,1998
Experimental study of the thermal sources contribution

to the acoustic emission of supersonic jets


Y. Gervais
a
, Y. Marchesse
a
and H. Foulon
b


a
Laboratoire d Etudes Aérodynamiques, Université de Poitiers, 40, av. du Recteur Pineau, 86022 Poitiers, France

b
Centre d Etudes Aérodynamiques et Thermiques, Université de Poitiers, 43 rue de l aérodrome, 86036 Poitiers
Cedex, France

An experimental investigation was conducted in order to determine the effect of jet temperature in supersonic jet noise. Jet
velocitie
s from 900 m/s to 1700 m/s and static temperatures from 330 K to 1110 K were used. Acoustic results (Acoustic
power, directivity analysis) showed that heating the jet leads to a decrease of jet noise. In a second part, mean and fluctuating
temperatures in
jets are investigated. Therefore, a two beam Schlieren system based on the measurement of angular beams
deflection across the flow is developed. The mean temperature is obtained by the Abel transform using the Gladstone
approximation. Fluctuating temperatu
res are estimated by statistical processes on beam deflections. Finally the Schlieren
method is successfully applied on jets approaching space launcher conditions.




Studies related to supersonic jet noise have received
considerable attention to reduce a
coustic environment
in the vicinity of space launcher. In the work described
here, we take an interest in the temperature
dependence of jet noise. Therefore, in a first part,
acoustic measurements (Acoustic power level,
directivity) carried out on two jets
with the same jet
velocity for different temperatures are introduced
(Table
1
). Afterwards, mean and fluctuating
temperature in the flow are measured in order to
provide a better understanding of their influence.

ACOUSTICS MEA
SUREMENTS

The experiments were performed on
MARTEL
facility
[
1
] fit out with a 50 mm diameter nozzle designed to
provide a perfectly expanded jet for stagnations
conditions P
i
=30 Bar and T
i
=1900 K (Jet 1 on table
1
).


Table
1
. Jet test conditions.


Jet

V
j
(m/s)

T
s
(K)

1

1700

860

2

1700

1110


Far field sound pressure, directivity and acoustic
power (L
w
) have been measured with 12 microphones
(1/4) located on
semi circle (R=84D) centered on the
nozzle exit.

Table
2
. Acoustic Power Level.

Jet

L
w
(dB)

1

124.9

2

119.4


20
50
80
110
140
125
130
135
140
145
150
 (°)
OASPL (dB - Ref. 2e-5 Pa)

FIGURE
1
.
Effects of temperature on directivity, (
O
),
jet 1 (T
s
=860 K) ; (

), jet
2 (T
s
=1110 K)

It appears that the noise radiated by jet 1 is more
important than the noise of jet 2 (Tab.
2
) which
presents a higher temperature. Moreover, one notices
that this can be observed whatever the direction of
observation
(Fig.
1
) except in the upstream direction
where the broadband shock associated noise is
dominant for the non perfect expanded jet.

The effects of temperature are many
-
sided as it
influences the fluctuating stress Reynolds
(
u
i
u
j
) and
also the entropy fluctuating source (p
-

c
0
2
) in
Lighthill s stress tensor [
2
]. Nevertheless, our results
confirm that in the case of high exhaust speed, it
mainly appears that the increased contribution of noise
due to
entropy fluctuations source is compensated by
the decreased contribu
tion from the Reynolds s stress.
0
0.5
1
1.5
300
400
500
600
700
800
900
T(r) (K)
r/D
0
0.5
1
1.5
2
300
400
500
600
700
800
900
1000
1100
r/D
0
0.5
1
1.5
2
0
20
40
60
80
100
120
TRMS (r) (K)
r/D
0
1
2
0
20
40
60
80
100
120
r/D
X=3D
X=4D
X=6D
X=8D
X=10D
X=12D
a
b
c
d

FIGURE
2
.
Mean temperature, jet 1 (a) and jet 2 (b) ; Quadratic temperature, jet 1 (c) and jet 2 (d)
MEASUREMENTS OF MEA
N AND
FLUCTUATING TEMPERAT
URE

The temperature has been estimated with Schlieren
optical method based on the measurements of angular
deflections of two perpendicular crossed
LASER
beams
through the flow (details of this method can be found
in [
3
]). Mean transverse angular deflections may be
written as an Abel s integral of the form :






  












=






(
1
)

where r and y denote radial and axial distances of
LASER
beam from the centerline and n the refracti
ve
index of the medium. An inversion of this relation
allows an estimation of the refractive index and
temperature profile with Gladstone relation :














=




(
2
)

where J is constant and depends on the flow
charac
teristics. Quadratic temperatures may also be
estimated from a statistical process with the two
fluctuating longitudinal beam deflections



,



and





,


the turbulent length scales in the x, y
and z directions
:



































(
3
)

After a satisfactory validation was carried out in a
subsonic flow with classical thermocouple data,
temperature are estimated on the two former jets (Fig.
2
). One notices axial temperatures similar to
theoretical values and profiles spreading out due to
the extent of the mixing layer (Fig.
2
.a and
2
.b).
Fluctuating profiles show that a peak appears i
n the
middle of the mixing layer corresponding to a
maximum of turbulence (Fig.
2
.c and
2
.d). Quadratic
temperatures estimated in jet 2 are more important
than those measured in jet 1 because of the use of the
same r
atio of length scale in the two jets which is not
adapted in jet 2. Unfortunately, none previous work
propose value of these length scale.

It appears that fluctuating temperature doesn t greatly
affect the noise level. Moreover, the role of mean
temperatu
re is not certain. Indeed, the identification of
the physical phenomenon at the origin of the
experimental results is delicate. As one can t separate
the two contributions in Lighthill s tensor, it is then
difficult to conclude that the temperature influen
ces
the first term while the second one remains unaffected.

ACKNOWLEDGMENTS

This work was supported by CNES.

REFERENCES

1
.

H. Foulon, D. Gely, J. Varnier, E. Zoppellari and Y.
Marchesse.,
MARTEL facility : Simulation of sp
ace
launcher aeroacoustic ambiance
, 29 Nov.

1 Dec. 1999,
12
th
European Aerospace Conference (AAAF/CEAS).

2
.

J. Lighthill.,
On sound generated aerodynamically,
Proceedings of the Royal Society of London,
A211
, pp.
654
-
68
7 (1952).

3
.

M.R. Davis,
Measurements in a subsonic turbulent jet
using a quantitative Schlieren technique,
J. Fluid. Mech.
46
(3), pp. 631
-
656, 1971.

On the Use of the Divergence Theoremin the Derivation
of Curle's Formula for the Amplitude of Aerodynamic
Sound
A.Zinoviev
a
a
Department of Mechanical Engineering,Adelaide University,North Terrace,Adelaide,5005,Australia
Curles formula establishes that the amplitude of aerodynamic noise radiated by turbulent flow near solid boundaries depends
upon the surface distribution of the total pressure in the fluid.In this work,the mathematical algorithm,used by Curle in the
derivation of his formula,is analyzed.Anewunderstanding of the use of the divergence theoremin this algorithm,different from
the traditional one,is proposed.This new understanding leads to the conclusion that the amplitude of density fluctuations in the
acoustic wave radiated by turbulent flow in the presence of a solid body depends on the surface distribution of density and its
normal derivative rather than pressure.The newexpression for the amplitude of density fluctuations is shown to take the formof
the well-known formula,which claims that a potential field is a sum of three fields generated by three kinds of sources:volume
distribution of sources,surface distribution of monopoles,and surface distribution of dipoles.
FORMULATION OF CURLES
THEORY
Lighthill [1] and Curle [2] showed that the amplitude
of density fluctuations
0
ρρ−
in an acoustic wave
radiated by turbulent flow in the presence of solid
boundaries is determined by a sumof two integrals:
( )











∂∂

=−
∫∫
S
i
i
V
ij
ji
dS
r
P
x
d
r
T
xx
c
yy
2
2
0
0
4
1
π
ρρ
(1)
where
ρ
is the density of the fluid,
0
ρ
is the density
of the fluid at equilibrium,
0
c
is the speed of sound in
the fluid at rest,
yxr −=
,
( )
321
,,xxx=x is the
coordinate of the observation point,
( )
321
,,yyy=y is
the coordinate of the source point.
ij
T
is Lighthills stress tensor determining
turbulence.
,
2
0 ijijjiij
cpvvT
ρδρ −+= (2)
where
i
v
is the
i-
th component of the velocity of fluid
particles,
ij
p
is the compressive stress tensor in the
fluid,and
ij
δ
is the Kroneckers symbol.
i
P
is determined by
,
ijji
plP −= (3)
where
j
l are the direction cosines of the outward
normal
n
from the fluid,i.e.
( )
n
=
321
,,lll,and
ij
p is
the compressive stress tensor in the fluid.
The first term in Equation (1) has been obtained by
Lighthill in his famous work [1].It describes the
generation of sound by turbulence in volume V without
boundaries.
The second term in Equation (1) describes the
generation of sound that occurs in turbulent flow on
the solid boundaries S.It states that the sound is
generated by a layer of dipoles on the solid boundaries
and its amplitude is determined by the surface
distribution of the total pressure.Equation (1) can be
considered the fundamental result of Curles work.
According to Curle,the second term in Equation (1)
can be simplified,if the following conditions are
satisfied:
,2,πλλ <<>> Lx (4)
where
λ
is a typical wavelength of the sound
generated,x is the coordinate of the observation point,
and L is the largest dimension of the solid object.If the
conditions (4) are true,the surface integral in Equation
(1) can be written as
( )
,
1
2
0
tF
t
x
x
c
i
i


(5)
where the i-th component
( )
tF
i
of the total force
acting upon the fluid is determined by
( ) ( ) ( )

=
S
ii
dStPtF yy,.(6)
Equations (5) and (6) represent the well-known
result of Curle.They state that sound generated by
turbulent flow on the surface of a small solid object
has dipole characteristics and its amplitude is
proportional to the total force exerted upon the fluid by
the object.
Curles Use of the Divergence Theorem
While deriving Equation (1) Curle [2] used the
divergence theorem to make possible the following
transformations of volume integrals into surface
integrals:
( )
,
1
∫∫


=












S
j
ij
i
V
j
ij
i
r
dS
y
T
ld
ry
T
y
y
y (7)
( )
,
1
∫∫
=








S
ijj
V
ij
j
r
dS
Tld
r
T
y
y
y
(8)
where V is the total volume external to the solid
boundaries and S is the surface area of the solid
boundaries.
FORMULATION OF THE
DIVERGENCE THEOREM
Formulation of the divergence theorem with two
extensions and necessary proofs and definitions can be
found,for instance,in the book by Kellogg [3].
The divergence theorem can be written as:
( )
∫∫
++=










+


+


SV
dSlZlYlXdV
z
Z
y
Y
x
X
321
,(9)
or,alternatively,
( ) ( )
,div
∫ ∫
⋅⋅=
V S
dSdV nrFrF (10)
where V is the volume of a regular region of space
bounded by a surface S,
( )
ZYX,,=F,functions X,Y
and Z are continuous in V and have partial derivatives
of the first order which are continuous in the interiors
of a finite number of regular regions of which V is the
sum,and the volume integral in the left-hand part of
Equation (9) is convergent.
A set of points is said to be bounded if all its points
lie in some sphere.
Equations (7) and (8) can be shown to take a form
equivalent to (9).Differentiation is carried out with
respect to the source point y.
A VIEWOF THE DIVERGENCE
THEOREMIN CURLES WORK
A common case where a solid object is surrounded
by a fluid with turbulent flow is considered below.It is
assumed that turbulence occupies a finite region of
space.
The volume integrals in Equations (7) and (8) can be
evaluated in the following way.As stated above,for
the divergence theorem to hold the volume V must be
bounded by the surface S.Consequently,it is not
sufficient to consider the surface S of the solid body in
the surface integrals in Equations (7) and (8).Instead,
the integration must be carried out over a surface
enclosing all turbulence.
The integral over such a surface can be evaluated as
follows.If the size of the surface is large,there is no
turbulence on the surface and the integral over the
surface disappear and the volume integrals in
Equations (7) and (8) are equal to zero.
As a result of the above consideration the second
termin Equation (1) will take the following form:
( )
( )










+


∫∫
yy dS
r
l
x
dS
yr
lc
S
i
i
S
i
i
ρ
ρ
11
2
0
(11)
Equation (11) shows that the amplitude of density
fluctuations in a sound wave generated on the surface
of a rigid body depends upon the surface distribution
of density and its normal derivative rather than
pressure.Thus,Equation (5) will not hold and the
amplitude of aerodynamic noise cannot be determined
only by the force exerted upon the fluid by the object.
Equation (11) together with Equation (1) is a direct
consequence of the well-known formula from the
theory of potential [4],which establishes that the
solution of a linear differential equation can be
represented as a sum of three potentials:a) potential of
a volume distribution (Lighthills solution);b)
potential of a simple layer,or a layer of monopoles,
(first term in (11));c) potential of a double layer,or a
layer of dipoles,(second termin (11)).
According to Curles theory,the term describing the
layer of monopoles vanishes due to the boundary
conditions on the surface of a solid immoveable
object,and the sound radiated has dipole
characteristics.However,it needs to be noted that
investigation of the properties of the sound determined
by Equation (11) is outside the scope of this work.
ACKNOWLEDGMENT
The author is grateful to Professor Colin H.Hansen
for his support and encouragement.
REFERENCES
1.Lighthill,M.J.,Proc.Roy.Soc.A,211,564  586
(1952).
2.N.Curle,Proc.Roy.Soc.A,231,505  514 (1955).
3.O.D.Kellogg,Foundations of the Potential Theory,
Berlin,Verlag von Julius Springer,1929 pp.84  121.
4.G.A.Korn and E.M.Korn,Mathematical Handbook for
Scientists and Engineers,McGraw-Hill,1968,p.488.


On the Use of Linear and Non-linear Source Terms in
Aeroacoustics - A Comparison of Different Approaches

Ricardo E. Musafir

School of Engineering & PEM/COPPE – Universidade Federal do Rio de Janeiro
C.P. 68503, Rio de Janeiro, 21945-970, Brazil
rem@serv.com.ufrj.br

The equations of Lighthill, Lilley and Howe are compared with respect to the use of terms linear and non-linear in the
fluctuations. It is shown that apparent inconsistencies are due to the fact that the forms of the equations normally used correspond
to different levels of approximation, having thus different ranges of application. It is important that this be properly considered
when choosing the equation to model a particular problem.



INTRODUCTION

A marked difference between the successful
aeroacoustic approaches of Lighthill [1], Lilley [2] and
Howe [3] is the use made of terms linear and non-
linear in the fluctuations. Lighthill’s analogy, in its
original form, is given by a linear equation with a
source function containing terms linear and non linear
in the fluctuations. Being based on the equation for an
homogeneous medium at rest, the equation is generally
valid although, unless direct simulation is intended, it
cannot be solved without the introduction of
simplifications in the source function, which amounts
to abandoning propagation effects which are hidden
therein. Lilley’s equation is based on the identification
of source terms as those involving non-linear
interaction of fluctuating quantities, while linear terms
are associated with sound propagation. Although the
source identification scheme is independent on mean
flow assumptions, the assembling of a wave equation
with these properties is possible only if the mean flow
is no more complex than a parallel shear flow. Howe’s
equation, on the other hand, chooses to define as
sources the terms involving vorticity and entropy
inhomogeneities. As a consequence, both sides of the
equation, which, as Lighthill’s, is generally valid,
present terms linear as well as non linear in the
fluctuations. This, of course, generates a difficulty to
its solution, which is usually faced by linearizing in the
fluctuations the wave operator and, frequently, also the
source function. Thus, in practical terms, one is left
with the picture of three linear wave equations with
forcing functions sometimes described by both linear
and non-linear terms, or, instead, exclusively by one
type or the other. In what follows, the differences in
these approaches will be examined. For simplification,
an inviscid isentropic flow will be assumed.

EQUATIONS AND SOURCE TERMS

Lighthill’s equation can be written


2
ρ/∂t
2
− ∇
2
p = ∇.∇.(ρvv) (1)

where ρ is density, p is pressure and v is velocity. The
source term above involves linear and non-linear parts,
usually called – density fluctuations neglected – shear
and self noise terms, respectively. Some of the source
density fluctuations, however, refer to convection of
sound waves by the mean flow and are relevant if the
equation is to be transformed into a convected wave
equation. The linear term 2ρ∂U
i
/∂x
j
∂u
j
/∂x
i
, where U is
the mean and u the fluctuating part of v, includes, even
for constant density, interaction of the sound field with
the mean flow.
Lilley’s equation is obtained when all variables are
decomposed into mean and fluctuating parts (which
will be noted, except for v, by the suffix
0
and primes,
respectively) and all linear terms are transferred to the
left side. A relevant step in its derivation is

[D
0
{(ρc
2
)
–1
D
0
p/Dt}/Dt – ∇.(∇p/ρ)]’ =

∇.∇.(uu)’ + ∇.(u.∇U) + ... (2)

where D
0
/Dt = ∂/∂t + U.∇, c is sound speed and


p
0
= 0 was assumed. The omitted terms, –∇.(u∇.u)’
and [–D
0
{u.∇p/(ρc
2
)}/Dt]’, refer to the interaction of
sound with turbulence and can be neglected. The linear
term on the right side can be eliminated with the use of
the momentum equation, yielding, in turn, linear and
non-linear components. A linear wave equation with an
exclusively non-linear source function can be obtained
only if the mean flow is, at most, a parallel shear flow.
In this case, the resulting equation for the logarithmic
pressure π, defined by dπ = d

p/(ρc
2
), is third order.
Only if there is no mean shear a second order equation


with this property can be obtained.
Howe’s equation is based on the equation for the
time derivative of the velocity potential φ, in an
irrotational homoentropic mean flow (a situation for
which all field variables can be written in terms of φ).
An intermediate step in its derivation is

∂ [(ρc
2
)
–1
Dp/Dt]/∂t – ∇.(∇p/ρ) = ∇.(v.∇v)

= ∇.(w x v) + ∇
2
|v|
2
/2 (3)

where w = ∇ x v.
In order to have the chosen dependent variable, the
stagnation enthalpy B = h + |v|
2
/2, where h is the
enthalpy, in the first term on the left, one has to add to
(3) derivatives of a multiple of the momentum
equation, what answers for refraction-like terms,
containing the factor c
-2
Dv/Dt, in both sides of the
resulting equation. Alternative approaches, where this
factor is replaced by –∇h/c
2
or –∇ρ/ρ, have been
derived by Doak, Mohring and Musafir, having been
discussed in [4]. Although Howe’s equation is
generally valid (as are the alternative ones), it is
usually not employed in a form more complex than

c
0
-2
D
0
2
B
/
Dt
2
– ∇
2
B = ∇.(w x v – T

∇S) (4)

where T is temperature, S is entropy and all non-linear
terms in the operator, as well as the terms containing
c
-2
Dv/Dt, have been neglected. In most applications the
source function is also linearized in the fluctuations,
being frequently reduced to ∇.(w’ x U). The relevance
of the entropy terms is discussed in [4, 5]. Indeed, it is
frequent to neglect altogether convection effects in the
operator, the equation being then effectively reduced to
Lighthill’s equation with the density removed form the
source function (or, actually, taken as constant therein).

DISCUSSION

Equations (2) and (3) can be seen as different stages in
the process of obtaining a convected wave equation by
shifting parts of the source term in (1) to the left side.
The approximation leading to equation (4), for the
case of a uniform mean flow, makes Howe’s operator
basically identical to the corresponding form of
Lilley’s, although the right hand side, in the former
equation, contains linear terms, while in the latter it is
given in terms of quadratic quantities only. A partial
explanation is that, while Lilley’s equation holds to
second order in the fluctuations (it can be shown, by
changing the dependent variable to π + π
2
/2, that the
combined effect of the neglected second order terms is
actually of third order [6]), the approximate Howe’s
equation (4), holds only to first order. Being less
‘exact’, however, it is subject to less restrictions on the
mean flow than Lilley’s.
The fact that, in a free shear flow, the homogeneous
form of Lilley’s equation is satisfied to first order in
the fluctuations supports the idea of second order terms
as responsible for sound generation in this case [2, 6,
7]. In the presence of solid boundaries, however, the
first order coupling of vorticity and sound modes gives
rise to linear source terms [7]. The fact that Howe’s
formulation is employed preferably for problems
involving surface generated vorticity justifies then the
use of the linear source term only. For free flows,
however, the non-linear terms would have to be
included.
That approximations in Howe’s equation can be
tricky can be exemplified with Mohring’s exact form
of the equation, which, for homoentropic flows, is [8]

ρD(c
-2
DB
/
Dt)/Dt – ∇.(ρ∇B) = ∇.(ρw x v) (5)

Since equation (5) is, in the appropriate limit,
equation (1) with ∇.[ρ∇|v|
2
/2 + v∇.(ρv)] shifted to the
left side and, for homoentropic flows with no external
sources, ∇ρ = ∇p/c
2
= – ρc
-2
Dv/Dt, one can conclude
that density fluctuations were not completely removed
from the source function in Howe’s approach,
persisting still in the terms containing c
-2
Dv/Dt.
The analysis shows that the forms of the equation of
Lilley and Howe normally used correspond to different
levels of approximation and so, cannot be strictly
compared. Howe’s equation seems to be closer to
Lighthill’s than is usually thought. The choice of the
equation to be used in a particular problem must rely
on the satisfactory degree of approximation. This
suggests that different equations will always coexist in
Aeroacoustics.

REFERENCES


1. Lighthill, M. J., Proc. Royal Soc. London, A211, pp.
564-587 (1952).
2. Lilley, G.M., Lockheed Georgia 4th Month. Progr.
Report, Contract F-33615-71-C-1663, Appendix, 1971.
3. Howe, M. S., J. Fluid Mech. 71(4), pp. 625-673 (1975).
4. Musafir, R.E., “On the Use of the Stagnation Enthalpy
as an Acoustic Variable”, in Proc. 7th Int. Congr. Sound
and Vibration
, edited by G. Guidati et all., Garmish-
Partenkirchen , 2000, pp. 1275-1282.
5. Aurégan, Y., Starobinski, R.,
J. S. V.
216(3), pp. 521-
527 (1998).
6. Goldstein, M.E., Ann. Rev. Fluid Mech. 16, pp. 263-285
(1984).
7. Goldstein, M.E., Aeroacoustics, MacGraw-Hill, N.Y.,
1976.
8.
Mohring, W., Obermeier, F. “Vorticity, the Voice of
Flows”, in Proc. 6th Int. Congr. Sound and Vibration
edited by Finn Jacobsen, Coppenhagen, 1999, pp. 3617-
3626.

Analysis of Sound Filed of Ultrasonic Transducer in Air
with Temperature Variation by FDTD
N.Endoh, Y.Tanaka, and T.Tsuchiya
Department of Electrical, Electronics and Information Engineering,
Kanagawa University, 221-8686 Yokohama, Japan
endoh@cc.kanagawa-u.ac.
The Finite Difference Time Domain (FDTD) method is proposed for calculation of acoustical characteristics of an ultrasonic
transducer in air. An aerial-sonar projected a 40kHz pulse sound whose pulse-width was about 0.15ms. The sound pressure
field of sonar was calculated in constant temperature of 30 degrees. We also calculated sound pressure field in air with
temperature variation of 20 degrees. There was little difference between these two contour patterns. FDTD method enabled
the visualization of propagation pulse projected from the transducer as a function of propagation time. The reflected echo
signals from the target were also calculated as a function of its height. The amplitude and propagation time of the reflected
pulse changed a little with temperature variation. These results show the validity of the FDTD method.
INTRODUCTION
An aerial-sonar for an automobile is very useful to
detect the object in the rear of a car. In this paper, the
Finite Difference Time Domain (FDTD) method [1] is
proposed to calculate the propagation of sound in air.
The recent development of computer system enables
the FDTD to be applied in the acoustics.[2] To
confirm the validity of the FDTD, the sound pressure
field of sonar was calculated in constant temperature
of 30 degrees when the 40kHz pulse was projected. A
series of propagated pulse waveforms were obtained
because the FDTD was capable of calculating the
instantaneous sound pressure along the propagation of
pulse. The reflected echo signals from the target were
also detected in air with temperature variation.
FDTD CALCULATION METHOD
Wave Equation for FDTD
The basic equations of the FDTD method, which is
taking account of attenuation, are given as follows:
2
1
y
x
v
p v
c t x y￿
￿
￿ ￿
￿ ￿ ￿
￿ ￿ ￿
(1)
x
x
v p
v
t x
￿ ￿
￿ ￿
￿ ￿ ￿
￿ ￿
(2)
y
y
v
p
v
t y
￿ ￿
￿
￿
￿ ￿ ￿
￿ ￿
(3)
where
p
is sound pressure,
v
is the particle velocity,
￿
is the density and
t
is time. The second part of the
right hand side in Eqs. (2) and (3) show an attenuation
of the medium. If the only plane sinusoidal wave
propagates in x-direction, substitutions of two
equations into Eq. (1) yields the next equation
2 2
2 2 2
0
P
j P
x c c
￿ ￿￿
￿
￿
￿ ￿ ￿
￿
￿ ￿
￿ ￿
￿ ￿
(4)
￿
￿
0 1 2
exp ( )P P j j x￿ ￿￿ ￿ ￿ (5)
where P
0
is the constant and
￿
1
and
￿
2
are the wave
number and attenuation constant, respectively. The
velocity of sound c and resistance coefficient are
obtained where
￿￿
is angular frequency:
2 2
1 2
/c ￿ ￿ ￿￿ ￿
(6)
1 2
2 2
1 2
2
c
￿ ￿
￿ ￿
￿ ￿
￿
￿
(7)
The finite differential equations are obtained as a
function of discrete positions x, y in space and a
discrete time t as shown below. [3]
1
1/2 1/2
1/2 1/2
(,) (,)
[ ( 1/2,) ( 1/2,)
(,1/2) (,1/2)],
n n
n n
p x x
n n
y y
C
p i j p i j
v i j v i j
v i j v i j
￿
￿ ￿
￿ ￿
￿
￿
￿ ￿ ￿
￿ ￿ ￿ ￿
1/2 1/2
1 2
(,1/2) (,1/2) (,1) (,),
n n n n
y v y v
v i j C v i j C p i j p i j
￿ ￿
￿ ￿
￿ ￿ ￿ ￿ ￿ ￿ ￿
￿ ￿
1/2 1/2
1 2
( 1/2,) ( 1/2,) ( 1,) (,),
n n n n
x v x v
v i j C v i j C p i j p i j
￿ ￿
￿ ￿
￿ ￿ ￿ ￿ ￿ ￿ ￿
￿ ￿
where

2
/
p
C c t x
￿
￿ ￿ ￿
.
In these equations, superscripts show the time and i
and j are the grid-numbers in the x and y directions in
space, respectively. For simplification,
￿
x =
￿
y. In
this paper, the resistance coefficient
￿
that is
proportional to the particle velocity is ignored because
of low attenuation in air.
FIGURE 1. Snapshots of sound propagation.￿(3.2ms,
4.8ms and 5.6ms after radiation of the pulse sound)
FIGURE 2. Received waveform. (Height of object is 0.2m,
0.4m and 0.7m from top to bottom)
Calculation Results
The FDTD method calculated the sound field in air
changing the sound velocity from 349 to 361m/s. A
sound source placed at 50cm above the earth projected
a pulse sound of 40kHz. Target was at x=1.5m. We
decided that the increments in space
￿
x=
￿
y=0.8mm
and in time
￿
t=0.8
￿
s for obtaining accurate results.
To eliminate the reflection wave from the outer
boundary of the calculation space, Mur’s first order
absorbing boundary conditions were provided. The
contour patterns were almost the same in various
temperatures from 30 to 50 degrees. Figure 1 shows
the propagating sound to and from the target in
constant temperature of 30 degrees. It is clearly shown
that there are not only direct and reflected echo pulses
but also diffracted wave behind the target. Figure 2
show the receiving echo signals of the target as a
function of its height. There is always the same echo
pulse at t=10.2ms from the corner of the target and the
earth. The first echo from the upper-left corner of the
target increases with its height. The amplitude and
propagation time of the echo pulse changed a little
with temperature variation. The maximum amplitude
of the first echo decreased with temperature. When
the temperature was 50 degrees, traveling time to and
from the target became 0.1ms shorter than at 30
degrees.
CONCLUSION
The FDTD method calculated the acoustical
characteristics of an aerial-sonar. It projected a 40kHz
pulse whose pulse-width was about 0.15ms. The
sound pressure field of sonar was almost the same in
various temperatures from 30 to 50 degrees. We also
calculated the reflected echo signal from the target as a
function of its height. The amplitude and propagation
time of the reflected echo pulse changed a little with
temperature variation. These results show the validity
of the FDTD method.
ACKNOWLEDGMENTS
The authors wish to thank Professor T.Anada at
Kanagawa university for his useful and constructive
comments.
REFERENCES
1. K.S.Yee, K.Shlager and A.H.Chang, IEEE Trans. Ant. Prop., 14,
302 (1966)
2. R.A.Stephen, J. Acoust. Soc. Am. 87, 1527 (1990)
3. N.Endoh, F.Iijima and T.Tsuchiya, Jpn. J. Appl. Phys. 39, 3200-
3204 (2000)
0.0 0.5 1.0 1.5 2.0
0.7
0.5
0.3
0.1
Height[m]
0.0 0.5 1.0 1.5 2.0
0.7
0.5
0.3
0.1
Height[m]
0.0 0.5 1.0 1.5 2.0
0.7
0.5
0.3
0.1
Height[m]
Range [m]
-0.2
0.0
0.2
Amplitude[arb.]
10.6
10.4
10.2
10.0
9.8
Time[ms]
-0.2
0.0
0.2
-0.4
-0.2
0.0
0.2
0.4
0.2m
0.4m
0.7m
Numerical calculations of sound propagation over ground
surfaces
O. Kr. Ø. Pettersen
a
, V. Henriksen
a
, M. Bjørhus
b,d
, U. Kristiansen
c
, G. Taraldsen
a
a
SINTEF Telecom and Informatics - Acoustics, 7465 Trondheim, Norway
b
SINTEF Applied Mathematics - Numerical Simulation, 0314 Oslo, Norway
c
Institute of Telecommunications - Acoustics, NTNU, 7034 Trondheim, Norway
d
TTYL, 0212 Oslo
The purpose of this study has been to develop a numerical model for sound propagation in both air and ground over long
distances. A set of partial differential equations (PDEs) for wave propagation in a porous medium with a rigid frame has been
used. By introducing Chebyshev spectral collocation in the spatial variable the equations have been transformed to a set of
ordinary differential equations (ODEs) in time. Domain decomposition is used to divide the computational domain into sub
domains of manageable sizes. The time integration is performed by a second order explicit Runge-Kutta method. Spectral
collocation can only be used if the computational domain is a square of a certain dimension. To be able to calculate sound
propagation in complex shaped domains a mapping is performed from the physical coordinates to the coordinates of a square
domain. An approximation to the exact open boundary condition is introduced at the outer boundaries of the computational
domain. Simulation results are compared to analytical results, calculations done with a Nordic calculation model and results from
outdoor measurements.
THE DIFFERENTIAL EQUATIONS
Sound propagation in a porous medium with a rigid
frame is modeled by Eqs. (1) and (2).
0ρ ￿￿
￿
￿
￿
￿￿ v
v
R
t
p
)1(

2
￿￿￿
￿￿
￿
￿
￿
vc
t
p
)2(
Here
p
is the air pressure,
v
a vector containing the
particle velocity components and
R
the flow resistivity.
Furthermore is
￿
´ the equivalent density and
c
´ the
equivalent sound speed for the porous medium and
they are given by Eq. (3) and (4) respectively.
￿
s
k
0
ρ
ρ'￿ )3(
s
k
c
c ￿' )4(
Here is
￿
0
the air density, k
s
the structure factor,
￿
the
porosity and c the sound speed.
SPECTRAL COLLOCATION
For a two dimensional Cartesian coordinate system
Eqs. (1) and (2) can be written as Eq. (5).
0
21
￿￿
￿
￿
￿
￿
￿
￿
￿
￿
CU
U
A
U
A
U
yxt
)5(
The vector
U
is defined by
U
= [u v p] with u and v
being the horizontal and vertical particle velocity
components and p the air pressure. Through the
introduction of spectral collocation in the spatial
variable a discretization in space is achieved. This
means that the PDEs are transformed to ODEs in time.
0
ρρ
1
￿
￿
￿
￿
￿
￿
￿
upDu
R
t
x
)6(
0
ρρ
1
￿
￿
￿
￿
￿
￿
￿
vpDv
R
t
y
)7(
￿ ￿

2
￿￿
￿￿
￿
￿
￿
vDuDp
yx
c
t
)8(
The matrixes D
x
and D
y
are the derivative matrixes in
the two spatial directions (see [1] for more details
about derivative matrixes). The velocity components
and pressure values at the discretization points, which
are called collocation points in spectral collocation, are
contained in the matrixes u, v and p. The ODEs are
integrated using a second order Runge-Kutta method.
DOMAIN DECOMPOSITION
Domain decomposition is introduced to be able to
calculate sound propagation over long distances. The
computational domains are divided into a set of smaller
non-overlapping subdomains. All subdomains contain
only air or ground and are in a sense homogenous. To
ensure satisfaction of the differential equations on the
boundaries between subdomains, a correctional method
is introduced. In this method the boundary values are
corrected between each timestep in the Runge-Kutta
time integration. These corrections are calculated on
the basis of the physical boundary conditions,
continuous normal particle velocity and air pressure,
and implemented via characteristic boundary
conditions [2, 3].
OPEN BOUNDARIES
In order to reduce reflections from the outer boundaries
of the computational domain, an approximation to the
exact open boundary conditions is implemented at the
outer boundaries. The implemented approximation is a
low order approximation, but it gives very good results
for normal incidence.
MAPPING OF DOMAINS
A mapping of non-square domains is performed in
order to be able to calculate sound propagation over
different terrain profiles. In this mapping the physical
coordinates of the domain is mapped onto a square.
This leads to a transformation of the spatial variables.
NUMERICAL EXPERIMENTS

Some numerical experiments were performed in order
to test the proposed model. A two dimensional
computational domain with 66 subdomains (3x22) and
the dimensions 15x110 meters (height x length) was
used. In each subdomain 51 collocation points were
used in each spatial direction. This means a resolution
of 3.44 points per wavelength for a 1000 Hz signal. A
point source was implemented by trigging one
collocation point with an air pressure that varied with
time. A bandlimited signal between approximately 100
and 850 Hz was radiated from this point source. The
power spectral density of this signal is shown in Fig. 1.
The sound speed and damping of the signal in free
field was calculated from a simulation with only air
domains. The sound speed between 5 and 80 meters
from the source is shown in Fig. 1. The damping of the
signal over the same distance is shown in Fig. 2.
The results are in good accordance with the expected
sound speed and damping of 344 m/s and 12dB within
the frequency band of the signal (100 to 850 Hz).
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
340
340.5
341
341.5
342
342.5
343
343.5
344
344.5
345
Frequency in Hz
Sound speed in m/s
Figure 1. Sound speed between 5 and 80 meters from the
source.
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
9
10
11
12
13
14
15
Frequency in Hz
Damping in dB
Figure 2. Damping between 5 and 80 meters from the
source.
Results from simulation where ground surfaces are
introduced show equally promising results. Some of
these results will be given in the oral presentation. See
also [4] for more results.
REFERENCES
1.
Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A.,
Spectral methods in fluid dynamics. Springer-Verlag, New
York, 1988, pp. 68-70.
2.
Bjørhus, M., SIAM J. Sci. Comput., No. 16, May 1995, pp.
542-549.
3.
Bjørhus, M., A computational model for outdoor sound
propagation. Part II: A spectral collocation method, SINTEF
Report STF42 A00XYZ, SINTEF Applied Mathematics,
Norway, 2000, pp. 11-13.
4.
Henriksen, V., Numerical calculations of sound propagation
over ground surfaces, Diploma Thesis (in Norwegian), NTNU,
2000.
Power
spectral
density
of source
signal
0
0.013
Analytical Prediction of Aeroacoustic Cavity Oscillations
D. B. Bliss, L. P. Franzoni, and M. A. Cornwell
Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, USA
High-speed flow over cavities in vehicle surfaces can produce intense tonal pressure fluctuations. This problem has been of
concern for decades, especially for aircraft, but the underlying physical mechanisms have not been well understood. A fairly
simple analytical model has been developed that improves understanding of the oscillation mechanisms in shallow rectangular
cavities. The model describes the phenomenon, and gives reasonable agreement with experimental data. The analysis is
constructed from waves that exist in a system having a finite thickness shear layer dividing two acoustic media, one at rest and
one in motion. The wave types can be interpreted in physical terms. The shear layer thickness has an important effect on the
speed of convective waves. Only two convective waves and three acoustic waves suffice to represent the basic phenomenon.
Mass addition and removal at the rear bulkhead due to shear layer oscillation plays an important role in the process. Conditions
on energetics and shear layer motion at the rear of the cavity must be satisfied simultaneously for the oscillation to occur.
INTRODUCTION
As shown in Fig. 1, high-speed flow over cavities
or cutouts in vehicle structural surfaces frequently
produces intense tonal pressure fluctuations. The
high pressure levels can jeopardize the integrity of
nearby structural components, internal stores, and
sensitive instrumentation. Refs [1,2] describe the
phenomenon and propose a physical explanation of
the oscillation mechanism. A semi-empirical formula
for resonance frequencies has been developed [3].
Model studies and flight tests have provided
information about the relationship of geometric and
aerodynamic variables to the aeroacoustic
phenomenon. Recently, CFD methods have provided
simulations of the oscillation phenomenon [4,5].
Nevertheless, the underlying physical mechanisms
remain not well understood. The development of
simple but effective means to reduce unsteady
pressure levels continues to be a major challenge,
especially given constraints on cavity geometry and
size restrictions on suppression devices.
FIGURE 1. High-speed flow over a shallow cavity.
The oscillation occurs because disturbances on the
shear layer impinge on the trailing edge, causing
unsteady mass addition and removal at the rear
bulkhead. This effect appears much like an
oscillating pseudo-piston at the rear bulkhead,
producing waves that propagate forward in the cavity
and also radiate to the exterior. The waves reflect
from the front bulkhead and also excite convective
disturbances on the shear layer. The oscillation is
sustained because these aeroacoustic waves and the
shear layer convective waves interact with each other
and the cavity boundaries, in particular at the trailing
edge, in such a way as to draw energy from the mean
flow.
The present study was inspired by the analytical
approach suggested in Refs [1,2], but only now
carried to completion. This relatively simple analysis
method has been refined, and an important model for
shear layer structure has been added. The results
exhibit the physical behavior of cavity oscillations,
and there is good agreement with data, especially for
supersonic flow. The most important aspect of the
results is that they provide physical insight into the
primary wave mechanics of the oscillation
mechanism, and emphasize the important role of
mass addition and removal at the cavity trailing edge.
FORMULATION
The analysis is based on the various harmonic
wave solutions that can exist for a 2-D acoustic
domain of infinite extent above a rigid boundary and
below an idealized shear layer. Above the shear
layer is a high-speed compressible flow, see Fig. 2.
FIGURE 2. Infinite shear layer dividing moving and
stationary acoustic media for wave analysis.
D
L
M
￿
U
￿
LE TE
waves
trailing edge
mass addition
and removal
shear layer
shear layer
D
M
￿
U
￿
waves
￿
H￿￿￿ ￿
+￿
-￿
damping ￿￿￿￿ 0
￿(z)e
i(￿t-￿x)
z
x
Governing equations are solved in the acoustic
region (wave equation), the compressible flow region
(convective wave equation), and the shear layer
region. Extending Ref.[6], the shear layer is analyzed
in an idealized form, as a linear Mach number profile,
with disturbances governed by the compressible
Euler equations (inviscid, rotational flow). All waves
are isentropic, with uniform sound speed.
The compressible flow region requires special
treatment to identify the physically realistic solutions
that have only outward radiation. A rigid boundary is
placed above the compressible flow at H, far from the
shear layer. The convective wave equation is
modified to include a small amount of damping, ￿.
For very small damping, as H ￿￿￿￿, the reflected
waves from the upper boundary become negligible.
In each region, harmonic pressure wave solutions
p = P(z)e
i(￿t-￿￿x)
are found. At each interface, the
pressure and normal velocity are matched, giving sets
of homogeneous equations for the pair of arbitrary
constants associated with P(z) for each region. These
equations are expressed in matrix form. The
determinant gives a transcendental dispersion relation
D(￿,￿) =0 relating real frequency ￿ to complex
wavenumber￿￿ = ￿
R
+ i￿
I
. Contour plots of the
magnitude of D(￿,￿) are used to locate the zeros in
the complex ￿-plane. The roots are strongly
dependent on frequency and Mach number. Roots
associated with vorticity convection also depend
strongly on shear layer thickness ￿.
Typically five roots play the most important role
in cavity oscillations: a conjugate pair of amplifying
and decaying acoustic waves propagating upstream
under the shear layer; a conjugate pair of amplifying
and decaying vorticity convection waves propagating
downstream on the shear layer; and a real-valued root
having fast acoustic propagation downstream. The
remaining infinite set of roots are higher acoustic
modes in the z-direction under the shear layer. At
lower cavity oscillation frequencies, these roots are
evanescent, having small real parts and imaginary
parts that are integer multiples of ￿. As frequency
increases, the smaller of these roots become
propagating waves with higher mode structure.
The solution is assembled by summing waves to
satisfy boundary conditions at points in the cavity, as
shown in Fig. 3. The Kutta condition is applied at the
leading edge, normal velocities must vanish on the
front bulkhead, and normal velocities equal the
pseudo-piston velocity at the rear bulkhead. A linear
system is then solved for the wave amplitudes. The
pressure on the cavity floor was calculated, along
with the mechanical impedance of the pseudo-piston,
and the trailing edge shear layer deflection. Using
only the five basic roots typically gives good results.
FIGURE 3. Assembling the cavity from waves
satisfying the appropriate boundary conditions.
RESULTS
For two cases, each given an assumed frequency,
Figure 4 shows that realistic pressure mode shapes
are predicted for a cavity having L/D = 4. At the
assumed frequencies, the real part of pseudo-piston
impedance is plotted versus L/D in Figure 5. The
large negative real part around L/D = 4 shows that a
maximum amount of energy is extracted from the
free stream flow to drive the trailing edge mass
addition and removal process at this L/D. Figure 6
shows the corresponding trailing-edge shear layer
deflections versus L/D. Maximum deflections are
obtained around L/D = 4. Overall, these results show
that the frequencies are correctly chosen for L/D = 4.
There is also reasonable agreement with experiment.
FIGURE 4. Typical pressure modes for L/D = 4.
FIGURE 5. Pseudo-piston impedance versus L/D.
FIGURE 6. TE shear layer deflection versus L/D.
REFERENCES
1.H. Heller and D. Bliss, Progress in Aeronautics
and Astronautics, MIT Press, 45, 281-296 (1976).
2.H. Heller and D. Bliss, USA AFFDL, TR-74-133.
3.H. Heller, et al, USA AFFDL, TR-70-104 (1970).
4.N. Sinha, et al, AIAA Paper 2000-1968.(2000).
5.C. Rowley, et al, AIAA Paper 2000-1969 (2000).
6. Williams, et al, AIAA J.,15, No.8, 1159-66 (1977)
shear layer
D
L
M
￿
U
￿
LE
TE
waves
trailing edge
mass addition
and removal
modeled by
pseudo-piston
Ue
i￿t
￿
Kutta
Condition
Satisfy
BC's
0
1
2
3
4
5
6
0.6
0.8
1
1.2
1.4
TE Shear Layer
Displ. vs L/D
Mach 1.2
Mode 1
0
1
2
3
4
5
6
-1.5
-1
-0.5
0
0.5
1
Real Impedance
versus L/D
Mach 1.2
Mode 1
0
1
2
3
4
5
6
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Mach 2.0
Mode 2
Real Impedance
versus L/D
0
1
2
3
4
5
6
0
0.2
0.4
0.6
0.8
Mach 2.0
Mode 2
TE Shear Layer
Displ. vs L/D
Mach 1.2
Mode 1
x/D
P/P
LE
Mach 2.0
Mode 2
x/D
P/P
LE
Statistics of the meteorological conditions favourable to
propagation according to the three definitions
K. Rudno-Rudziński
Institute of Telecommunication and Acoustics, Wrocław University of Technology,
Wybrzeże Wyspiańskiego 27, 50370 Wrocław, Poland, krr@zakus.ita.pwr.wroc.pl
Frequencies of the meteorological condition favourable to propagation according to the definitions of CONCAWE, ISO and
NMPB were compared. Territories in Poland in different climate zones were chosen, in day and night time. Polar graphs of the
conditions favourable to propagation show a lot of variability caused by the differences in the definitions. Variability caused
by these differences is stronger than the one coming from the climate and the direction of propagation influences.
INTRODUCTION
The three classifications of meteorological condi-
tions are likely the most popular in environmental
acoustics: CONCAWE [1], ISO [2] and IMPB [3]. The
aim of this work was to compare the statistics of the
conditions favourable to propagation (CFTP) accord-
ing to above mentioned definitions.
DEFINITIONS
In the ISO definition the wind velocity and direction
decides that the conditions are favourable. In the re-
maining two the atmospheric stability is the second
main factor besides of the wind. In CONCAWE the
Pasquill classification and the wind velocity compo-
nent in the direction of propagation are used, in the
NMPB some practical stability classes are employed
together with the wind velocity and direction. In spite
of some similarity in wind and stability classes, there
are differences between CONCAWE and NMPB in
joining them in the outcoming classes of propagation
conditions.
ANALYSED DATA
Meteorological observation from the three stations
lying in different climatic region were chosen for this
research (Table 1)[4]. The data set contains the results
of observation made in three-hour cycles during the
one whole year. Only the observations with the wind
below 5 m/s were included (67 % in Łeba, 76 % in
Warszawa and 85 % in Wrocław).
Table 1.
Three meteorological stations
Localisation Main climate influences Wind rose type
Łeba seaside breeze
Warszawa continental bipolar
Wrocław oceanic uniform
STATISTICS
The calculated frequencies (percent) of the CFTP
versus the direction of propagation are shown on the
radar graphs with the angular resolution of 30
o
.
Frequencies for the day time and the night time are
shown on the Figure 1 and Figure 2 for Wrocław sta-
tion as an example. Graphs of CFTP for the ISO defi-
nition correspond to the smoothed wind rose (180
°
re-
versed). Frequencies according to CONCAWE are
higher than to ISO, for the day and the night. For all
definitions, frequencies according to NMPB are lowest
for the day and highest for the night. Such a regularity
occurs in the all stations.
0
10
20
30
40
0
30
60
90
120
150
180
210
240
270
300
330
CONCAWE
ISO
NMPB
Wrocław
day
FIGURE 1.
Percent

of

CFTP for Wrocław in the day time
according to the three definitions.
0
10
20
30
40
50
60
70
0
30
60
90
120
150
180
210
240
270
300
330
CONCAWE
ISO
NMPB
Wrocław
night
FIGURE 2.
Percent

of CFTP for Warsaw in the night time
according to the three definitions.
Figure 3 shows frequencies of the CFTP conditions
according to NMPB for three stations, for the day (in-
side) and for the night (outside).
Table 2 shows the d
ifference between the maximum and
the minimum frequency and between the mean frequencies
for night and day.
0
10
20
30
40
50
60
70
0
30
60
90
120
150
180
210
240
270
300
330
Łeba
Warszawa
Wrocław
day
night
NMPB
FIGURE 3.
Percent

of

CFTP according to NMPB for the
three stations in the night time and the day time.
Table 2.
Difference between maximum and minimum fre-
quency and between the mean frequencies for night and day
station method
max-min
(day)
max-min
(night)
mean (night)-
mean (day)
CONC 20,1 24,8 -2,0
ISO 12,7 25,2 0,0
Łeba
NMPB 10,7 15,8 41,2
CONC 10,1 7,5 7,5
ISO 13,8 9,7 0,0
Warszawa
NMPB 11,3 10,0 39,8
CONC 3,2 21,2 7,8
ISO 14,9 22,5 0,0
Wrocław
NMPB 12,8 13,0 34,4
The difference between mean frequency of the CFTP
for the night time and the day time equals to zero for
ISO definition, ranges from –2 to 7,8 % for CON-
CAWE and from 34,4 to 41,2 % for NMPB.
The difference between the maximum frequency and
the minimum frequency of the CFTP (for all the
propagation directions in a given location) ranges from
3,2 to 24,8 %.
CONCLUSIONS
The difference between mean frequency of the CFTP
for the night time and the day time equals to zero for
ISO definition, ranges from –2 to 7,8 % for CON-
CAWE and from 34,4 to 41,2 % for NMPB.
In the case of ISO definition the obvious reason of
the lack of the difference between day and night is the
neglect of the atmospheric stability.
The lack of the difference in frequency of the CFTP
between night and day indicates the same outdoors
noise attenuation what is inconsistent with the experi-
ence.
From this point of view, NMPB gives likely the best
classification method for the meteorological conditions
of sound propagation outdoors.
Additional calculations indicates that CONCAWE
could be corrected to better differentiate day and night
conditions, maintaining precision of wind velocity
quantification resulting from wind vector.
REFERENCES
1.

K. J. Marsch, Appl. Acoustics,
15
, 411-428 (1982)
2.

Bruit des infrastructure routiere - methode de calcul
incluant… CERTU, CSTB, LCPC, SETRA (1997)
3.

ISO 1996, 9613
4.

K. Rudno-Rudziński, Meteorological conditions of sound
propagation outdoors in Poland, in
Proceedings of the
Sixth ICSV
, Techn. Univ. of Denmark 1999, pp.749-756
Split Mufflers for Improved Aerodinamic for
Ventilation Systems
O.V.Plitsina, V.T.Plitsin, M.N.Kucherenko
Togliatti Polytechnical Institute, Belorusskaya 14, 445667 Togliatty, Russia
Split mufflers suggested by the authors provide equal wide-band noise reduction in air conduits of large cross-section,
have stable characteristic in two-phase flows. Aeroacoustic test of split designs is presented. Their acceptability for ventilation
systems with aerodynamic limitations is shown. Effective noise reduction and aerodynamic resistance decrease may be achieved
by the splits orientation in parallel to the larger side of design’s cross-section.
Split mufflers were suggested for noise control of
fans mounted in systems with large cross–section ducts
(having characteristic dimensions which are larger than
the wavelength of propagating sound). In order to
obtain muffler characteristics the aeroacoustic stand for
model (1 : 10) experiments was used.
The stand is mounted in three separate rooms
with 400 mm thickness brick walls (Fig. 1).
FIGURE 1. Scheme of aeroacoustic stand
High pressure fan 1 entering into the stand has direct–
current motor. The motor is supplied from voltage
control source 23. It allows to change flow of air in the
air conduit 4 from 0 to 1000 m³/h. Fan’s aerodynamic
noise is reduced in the tubular muffler 3 joined with
the fan by the flexible insertion 2. In order to measure
flow of air the diaphragm 6 having manometer tubes 7
and micromanometer 19 are used. In order to measure
pressure of air passing through the diaphragm 6 the U–
tube manometer 18 is joined to manometer tube 5.
The air conduit 4 is joined with inlet of the reverberant
chamber 8 in which the sound columns 11 and the
condenser microphone 10 on the support are mounted.
The air conduit with the replacement sections 13 is
joined to outlet (in direction of stream) of the
reverberant chamber 8. There are manometer tubes 12
for the U–tube manometer 18 joining in the
replacement sections. The air conduit with the
replacement sections is ended with trumpet in inlet of
the reverberant chamber 14. Air goes out the chamber
14 to room through the nozzle 17. The condenser
microphone 16 is placed into the chamber 14 on the
support. Set of the replacement sections 13 allows to
mount various length split mufflers between the
reverberant chambers 8 and 14. The reverberant
chambers 8 and 14 are welded 4 mm thickness plate
steel structures having ribs. All chambers’ surfaces are
not parallel. The chambers have the doors 9 and 15.
The doors are fulfilled from steel corners sheathed by 4
mm thickness plate steel with packing over the
perimeter by sponge rubber. Each chamber is mounted
on rubber shock dampers. Leads input is fulfilled in
the chambers (to the microphones 10, 16 and sound
columns 11) through packing glands. The chambers
have nozzles for insertion of starting pistol used in
reverberant time defining. The instrumentation is
placed in the room situated between the rooms with the
reverberant chambers. In acoustic testing the signal is
transferred from the rose noise oscillator 21 to the
amplifier 20 joint with the sound columns 11. Sound
oscillations arising in the reverberant chamber 8 are
sensed by microphone 10, sound oscillations
transmitted through the testing split muffler are sensed
by microphone 16. The signal is transferred from the
microphones to the B&K complex of the measuring
apparatus 22 for precise laboratory and nature testing.
The complex allows to obtain noise characteristic in
frequency bands of set width in the such form as
digital presentation, curve on the recorder or on the
cathode – ray oscillograph.
Stand’s chambers was attested before mufflers
testing with taking into consideration requirements of
ISO documents.
It was tested on the stand: the model of one
section muffler; the model of two sections muffler with
the same sections; the model of two sections muffler
having sections with opposite orientation splits.
The model of one section muffler has built–up
steel body of rectangular shape (Fig. 2).
The model of two sections muffler with the
same sections is successive arrangement of the
mufflers with horizontal splits. The replacement air
conduit 273 mm long is placed between the mufflers.
The model of the muffler having sections with opposite
orientation splits includes the muffler with horizontal
splits, replacement air conduit 273 mm long and the
muffler with ten vertical splits. Section’s vertical and
horizontal splits are identical.
According to acoustic characteristics split
mufflers provide even broad –band noise reduction.
Minimum effectiveness of the one section muffler
(design is not optimized) is 4 dB; minimum
effectiveness of the muffler having two sections with
opposite orientation splits is 8 dB. The acoustic
characteristic of muffler including the same sections
coincides with the characteristic of one section muffler.
Experiments show that influence of air stream moving
with speed to 18 m/s through the models of muffler on
acoustic characteristics is insignificant and it may be
disregarded.
In obtaining aerodynamic resistance in the models
the air speed was being changed from 3 to 18,5 m/s
and the flow was being changed from 160 to 1000 m³/h
correspondingly. Experimental data was twelve
measurements averaged. Confidence interval was
calculated for probability 0,95 and instrumental error
2,5 %.
According to experiment (Fig. 3) friction and
local resistance pressure loss in the split muffler ∆P, Pa
is
∆P = k Q²,
where k – characteristic of resistance, Pa/(m³/h)²;
Q – flow of air, m³/h.
The number of splits influences k value significantly.
Its decrease from 13 (muffler with horizontal splits) to
10 (muffler with vertical splits) results in k reduction
from 1,87∙10
–3
to 1,2∙10
–3
Pa/(m³/h)² in one section
muffler. Correspondingly the aerodynamic resistance
decreases on 36%. In adding section the pressure loss
increases in 2 time in case of the same splits
orientation and in 1,6 times in case of opposite
orientation. Hence, in order to increase aerodynamic
resistance the split muffler should be designed with
rectangular shape cross–section breaked up in parallel
its larger side. In estimating acceptability of pressure
loss it was took into consideration design difference of
split muffler (bars) and baffle–type silencer (baffles
with thickness about 200 mm). In increasing split
muffler rectangular cross – section in 1,5 times and
fulfilling suggested splits orientation pressure loss in
split mufflers and in baffle–type silencer are the same.
Thus split mufflers may be used in systems with
aerodynamic limitations.
FIGURE 2. Model of one section muffler with horizontal s
p
lits
FIGURE 3. Pressure loss-air flow relation: 1- muffle
r
with horizontal splits; 2- muffler with vertical splits.
FIG. 2

FIG. 1
1
e
f
f
f
T
LL
T
L
4
kR
r
f











FIG. 4
FIG. 3
Acoustic Analysis Aimed At Characterizing
Combustion Instability In Premixed Burners

M. Annunziato
a
, G. Coppola, M. Presaghi
a
, R. Presaghi, G. Puglisi, F. Romanello
a


a
ENEA Ingegneria ed Impianti di Generazione di Energia, 00060 Roma, Italia

New method to determine combustion characteristics by means of acoustic noise generated by dimensional variations of the
flame due to unstable operation of the burner. The method allows to test efficiency of a burner and to deeply understand the
causes that originate some problems related to the combustion.


METHOD OF ANALYSIS

A method to characterize burners used in energy
generation plants is presented. The method has
demonstrated to be particularly useful to put into
evidence the anomalies presented by premixed burners
in some operation regimes. In premixed burners, fuel
and air are premixed before their entering combustion
chamber. Differently from other burners, the premixed
ones operate at lower temperatures, thus emitting less
quantity of pollutants, but, although having these
positive characteristics, they are affected by
unpredicatble operation criticities, that cause a
lowering in efficiency, environmental pollution and
dangerous structural stresses [1]. The method is based
on the principle that the burner represents a true
acoustic cavity, containing both “flame fluid” and
“exhaust fluid”. The different mechanical impedances
and the dimensions of such fluids, determine the
resonance frequency of the cavity. Acoustic effects of
the said cavity move the noise band and center it on the
resonance value. Consequently, by monitoring the
latter parameter, it is possible to detect its cause: the
variation in flame’s activity.
FIGS. 1 and 2 represent a schematic view of two
dimensional aspects of the flame due to the up
mentioned phenomenon. When dimensional variation
in the flame occurs with no variation in the quantities
of air and fuel entering the burner, a flame instability
[2] is experienced. Such instabilities are due to a series
of causes that are originated by the reciprocal
interaction of many parameters; for instance, pressure
fluctuations generated by acoustic noise, act on
combustion chemistry and determine chaotic
evolutions in the phenomenon. However, the principle
on which the method is based remains a valid one.
The photographic series of FIG.3 (a-f) shows a
typical evolution of the flame in instability conditions.
The most important information that are possible to
gain from noise are the thermal power variation and the
dimensional variation of the flame, being the former
correlated to noise amplitude and the latter to its
dominant frequency. With the aim of evidencing the
effects induced by the flame on the cavity’s resonance
frequency, a simplified formula is reported, based on
the principle [3] that the resonant acoustic wave
propagates itself into a cavity in a time that equals ¼ of
its period. The simplification concerns the disposition
of the two fluids (flame and exhaust gas) that occupy
volumes that are adjacent, not in communication, and
having each constant temperature, T
f
e T
e
(FIG. 4).

(1)

f
r
= resonance frequency [Hz]; L
f
=flame length[m];
L=cavity length [m]; k=adiabatic index;
R= specific gas constant [J/kg K= m
2
/s
2
K].
140

150

160

170

180

190

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

flame lenght [m]

T
f
= 1400°C

T
e
= 550°C
L = 1m
FIG. 5

Frequency (Hz)

Frequency

Time (ms)
FIG. 6
min Amplitude Max
FIG. 8

The trend of f
r
versus L
f
is shown in FIG.5,
assuming k and R to be the same for both fluids, that is
not relevant for our aim. The instrumental analysis of
the phenomenon has been thus conducted on the basis
of the up mentioned considerations.
Processed signals are originated by two different
types of sensors: acoustic-electric transducer
(microphone) and pressure gauge connected to
combustion chamber. Both the signals allowed to catch
the combustion instability phenomenon, also
considering that the microphone signal contained
some noise coming from external environment.










FIG. 6 represents a three-dimension diagram (time,
frequency, amplitude) obtained by the wavelet
transform technique. In this figure many frequency
excursions are noticed, while only once it appears an
event associated to high instability. This shows how
not always an high variation in the thermal power of
the flame is associated to a change in its dimensions.
Similar results have been obtained when
processing the signal by means of a frequency
discriminator. One example is reported in FIG.7:
diagram a) reproduces combustion noise, while b)
represents the frequency deviation of the noise itself. It
can be evidenced how b) has a trend similar to the
envelopment of a) and presents noticeable variations
also at times in which a) assumes reduced values. In
this way, signal b) has the peculiarity to detect the
flame’s instability ranges, this in advance to the
statistical occurrence of strong instabilities. FIG. 8
shows the front panel of the final software device
(realized by LabView), based on the up mentioned
discriminator. The instrument in the upper right corner
of the figure shows a bar graph generated by five
different operation conditions of the burner,
respectively having the following values of : 1.1; 1.2;
1.3; 1.4; 1.5 (  meaning the ratio between introduced
air quantity and steichiometric one). Values of  of 1.1
and 1.5 are the operation limits of the investigated
burner. The highest bar is referred to the most unstable
condition of the flame; the third condition appears to
be the most stable one, then the bars start to increase
again their height. The bars represent the energy of the
signals shown in FIG. 7, b), that have been previously
filtered as to eliminate the components that are not
correlated to the phenomenon. Asymmetry in
hystogram is due to the different instability types
(mixtures close to the steichiometric value and
mixtures with air eccess). Every other device of the
front panel is used to set the parameters of the analysis
or to control their different phases.


CONCLUSIONS


This study has been carried out to detect a
methodology able to give more information when
compared to the already known ones; this has been
obtained considering the burner to act as a resonant
cavity and thus generating the idea to detect the
induced effects on the combustion generated acoustic
noise: that is the variation in dominant frequency of the
noise itself. On the basis of the obtained results, the
method appears to be suitable for carrying out analysis
aimed at the knowledge of burner efficiency and the
one of its control system.


REFERENCES

1. Lefebrve A. H., Gas Turbine Combustion, Mc Grow-
Hill, New York, 1983.
2. McManus K. R., Poinsot T. and Candel S. M., A Review
of Active Control of Combustion Instabilities, Progr.
Energy Combustion Science, 1993, vol 19, pp 1-29.
3. Javorskij B.M., Detlaf A. A., Spravoènik po fisike, Mir
pubblisher, 1977

Time
FIG. 7
Max
min
Frequency
Amplitude
Higher mode cylindrical radiator
for an aerial intense ultrasound source
H.Yamane
a
and T.Otsuka
b
a
Department of Electrical Engineering,College of Science and Technology,Nihon University
1-8 Surugadai,Chiyoda-ku Tokyo 101-8308,JAPAN
b
Department of Electrical and Electronic Engineering,College of Industrial Technology,
Nihon University 1-2-1 Izumicyo,Narashino,Chiba 275-8575,JAPAN
This work deals with a development of a cylindrical radiator for industrial applications such as collecting fine oil mist
that floats in air.The thickness of the cylindrical radiator was selected so as to be within the range of the flexural
vibration.The radiator was driven by a Bolt-Clamped Langivin transducer(BLT).The stripe mode vibration was
obtained on the radiator while the flexural vibration appeared along the radiator axis.The resonance frequency of
the cylinder itself and the ultrasound of a radiator in the cylinder were adjusted at the same frequency to produce a
high intensity ultrasound field.As a result,the sound pressure level in the cylinder up to 170 dB was obtained.This
makes it possible to use the system in the industrial application of collecting and/or coagulating fine oil mist floating
in air.
CYLINDRICAL RADIATOR
The frequency of the (m,s) mode in the cylindri-
cal radiator
1)
is given by
f
ms
= α
ms
c
2πr
(1)
where m,s −1 are the number of nodal diam-
eters and the number of nodal circles,α
ms
is the
eigenvalue given by
m
α
ms
J
m

ms
)−J
m+1

ms
) = 0,
where J
m
,J
m+1
are cylindrical Bessel functions of
the m-th and (m+1)-th order and c is the velocity
of sound,r is the radius of cylinder.
The m-th order of the resonance frequency of
the flexural vibration on the cylinder surface
2)
is
determined for the number of m mode lines as
f
m
=
N
m
C
M
h
2π(r +h/2)
2
(2)
where N
m
=
m(m
2
−1)

m
2
+1
,C
M
is the material con-
stant and determined from
￿
=
￿
E
12ρ(1−ν
2
)
￿
,where
E is the Young’s module of elasticity,ρ is the den-
sity,ν is the Poisson’s ratio and h is the cylinder
thickness.
The cylinder radius at the (m,s) mode is ob-
tained using Eq.(1)
r = α
ms
c
2 π f
ms
(3)
The thickness h is determined by the following
methodology:
Substitute f
ms
for f
m
fromEq.(2) into Eq.(3) and
the resulting thickness h is obtained by the following
expression
h = −2
￿
r −
N
m
C
M
2πf
ms
￿
+
￿
￿
r −
N
m
C
M
2πf
ms
￿
2
−r
2
(4)
SOUND PRESSURE DISTRIBUTION IN-
SIDE THE CYLINDER
The Root-Mean-Square value of the sound pres-
sure level distribution inside the cylinder is given by
P = J
m

ms
r) ∙ cos mθ (5)
Table 1 gives the dimension of the cylindrical
radiator calculated from Eqs.(3) and (4).
Table 1.Details of cylindrical radiator
(m,s) Inside radius Thickness Length Frequency
r [cm] h[cm] l [cm] f[kHz]
(11,5) 7.715 0.48 9.30 20.90
Figure 1 shows the sound pressure level distri-
bution inside the cylinder as calculated by equation
(5).There are 11 nodal diameters and four nodal
circles.
Figure 1.Sound pressure level distribution inside
the radiator at the frequency of 20.9 kHz by
Eq.(5).
DRIVING UNIT
The schematic of the driving unit is shown in
Figure 2.The BLT at the frequency of 20 kHz is
connected to a half wavelength exponential horn,
the cylindrical radiator is driven by the horn.
nodal lines
Figure 2.Cylindrical radiator for the frequency of
20.9 kHz.The stripe mode can be observed on the
inner surface of the radiator.
ACOUSTIC CHARACTERISTICS
The sound pressure level distribution was mea-
sured along the radius with the maximum sound
pressure levels occurring on the central axis inside
the cylinder.
0 2 4 6 8
0
0.5
1
Distance from center axis of cylindrical
radiator [cm]
Normalized sound pressure
A
Figure 3.Normalized sound pressure distribution.
The sound pressure level was measured with a
1/4 ” condenser microphone with the probe tube
when the electric power of 20 Wwas applied to the
BLT.The probe tube is 20 cm long,has 0.15 cm
inner diameter and is 0.05 cm thick.The measured
sound pressure level distribution is in good agree-
ment with the calculations.
Fig.4 shows the linearity of the acoustic power
as measured with a 1/8 ” 4138 microphone.The
microphone was located at the point A in Figure
3.The sound pressure is proportional to one half
the input electric power and 5.64 kPa (169 dB) was
obtained at 200 Wof input power.
10 100
1000
2000
3000
4000
5000
6000
7000
160
170
Electric input power [W]
Sound pressure [Pa]
Sound pressure level [dB]
Figure 4.Linear characteristic of the sound
pressure versus the input electric power.
CONCLUSIONS
The cylindrical radiator with radial flexural vi-
bration was designed to operate at the frequency of
20.9 kHz.The radiator is made of titanium and the
sound pressure levels of 169 dB were obtained at
200 Wof electric power input.
This system can be used in industrial applica-
tions as a tool that would enable attraction and
coagulation of the fine free-floating oil mist.
REFERENCES
1) McLanchlan,N.W.,Bessel function for engineer-
ing.Oxford University Press,London 1955.
2) Timoshenko,S.P.,Young,D.H.,& Weaver,W.
Jr.,Vibration Problems in Engineering.John Wiley
& Sons,New York 1974.
Figure 1. Geometry of the system
A propagation of acousto-gravity waves in
atmosphere has been studied rather intensively
[1-3]. These waves are considered often as one
of basic channels of energy exchange between
lithosphere and ionosphere caused by different
disturbances of both seismic and space nature
(see, for example, [4-6] and references therein).
The main consequence is the exponential growth
of amplitude of the oscillatory velocity of
particles with an altitude. Certainly, such a
growth is limited by the account of such factors
as viscosity of air, and also of convective
nonlinearity in equations of motion. As it is
shown below, the very important factor is also
mutual coupling of acousto-gravity waves with
another excitations in the ground, one of them is
the Rayleigh wave in the Earth crust.
Generally, an influence of the sound in air on a
propagation of Rayleigh waves in a substrate is
very little, because of a great difference of the
densities of environments. However, such an
influence, as it was marked in [6], becomes
essential in the point of intersection of
dispersion branches of the Rayleigh wave in a
substrate with the volume branch of the sound
wave in the air when the phase velocities of
waves become equal. Such a point refers to as a
coupling point, or a point of synchronism. In a
vicinity of such a point, the mutual influence of
waves becomes strong. Particularly, this
concerns to a case when the gravity force is
essential and the acoustic waves in atmosphere
become of acousto-gravity type. It is impossible
to investigate the wave subsystems in air and
ground separately in a vicinity of a synchronism
point. In this case, the aero - hydrodynamic
system in atmosphere and the system of theory
of elasticity in ground should be solved jointly.
The propagation in an atmosphere of the
seismically excited sound waves causes
oscillations of the ionosphere - atmosphere
boundary. As a result, Doppler reflections of
electromagnetic waves from ionosphere layers at
frequencies 10 and 25 MHz were observed [7].
Those observations demonstrated that the
Rayleigh waves can propagate to considerable
distances from a source (about 5000km). As this
effect was detected rather far from a seismic
source, it is apparent that in such conditions the
Rayleigh wave propagated as an own wave of a
layered system. During such propagation, the
conditions of a synchronism with waves in
atmosphere can be reached. Therefore,
understanding the features of interaction of a
Rayleigh wave with acousto-gravity waves in
vicinity of a point of a synchronism is important
to see over the total physical picture. However,
in this case the problem becomes more complex,
as it is necessary to take into account for both
waves under equal conditions. We should solve
the coupled acousto-gravity - sound problem by
a self-consistent way.
Variations of transverse structure of coupled
acousto-gravity - Rayleigh waves in multilayered
system Earth - atmosphere
G. Burlak, V.Grimalsky, S. Koshevaya
Center for Research on Engineering and Applied Sciences, Autonomous State University of
Morelos,Z.P. 62210, Cuernavaca, Mor., Mexico.E-mail: gburlak@uaem.mx
The transverse distribution of coupled acousto-gravity waves in atmosphere and a Rayleigh wave in a layered
system Earth - atmosphere is investigated. The special attention is given to a mode of strong coupling o
f
waves in a solid substrate with the oscillations in upper medium (atmosphere) when the phase velocities o
f
waves are close (coupling point, or a point of synchronism). Is shown that in this point the frequency shift
induced by the mutual influence of waves becomes maximal. In vicinity a coupling point, a strong change o
f
the cross structure of wave in air takes place.
0
1
2
3
4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(a)
kh
Re(x)
0
1
2
3
0
0.02
0.04
0.06
0.08
0.1
(b)
kh
Im(x)
0
1
2
3
4
0.1
0.15
0.2
0.25
0.3
0.35
(c)
kh
v, km/sec
v
gr

v
ph

0
1
2
3
4
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
(c)
kh
Re(κ+)
Figure 2. Dependencies of real and imagine
parts of a wave frequency (a,b), phase and group
velocities (c), and the transverse wave number
κ
+
of the acoustic-gravity wave in the air (d)
In this report, the transverse distribution of
coupled acousto-gravity wave with a Rayleigh
wave in a layered system Earth - atmosphere is
investigated. The motion equations has form[8]:
,)( zgpvv
t
v
∇+−∇=∇+


ρ
ϖϖ
ϖ
k
iki
x
p
t
u


=


2
2
γ
(1)
The special attention is given to a mode of
strong coupling of waves in a solid substrate
with a sound in upper medium (atmosphere)
when the phase velocities of waves are close
(coupling point, or a point of synchronism). We
use the dimensionless parameters x=ωh/c
t1
and
y=kh. In these variables, the dispersion equation
has a general form
),()2(),()/(
2
2
1
0
2
22
yxFxyxDcxy
n
ηκ
γ
ρ
+=−
+
(2)
Thus, the dispersion equation of sound waves in
examined system can be presented as
)2)(,(),(
2
2
0
ηκδκ +=
+
yxFxyxD
(3)
where δ=ρ
0
/γ, κ
0
=y
2
-x
2
/c
0n
2
, c
0n
=c
0
/c
1t
. For a
contact soil - air, it occurs δ=10
-3
<<1. At δ=0,
equation (3) splits into two independent
branches: a volume wave in the air and a normal
mode in a layered ground. We are interested in
the modes, which can intersect with a volume
wave in the air. Note that second the lowest
branch of acousto-gravity wave [4] does not
have intersection with the Rayleigh wave. This
branch is not studied here. At δ≠0, the waves
become coupled. However, as δ<<1, this
coupling is weak and it results in the small
frequency shift. We found that in this point the
frequency shift induced by the mutual influence
waves becomes maximal. In vicinity a coupling
point, a strong change of a cross structure of
wave in air arises. The strong dependence of a
factor of the exponential growth of an oscillatory
velocity of acousto-gravity waves with an
altitude due to parameters of a solid substrate is
demonstrated. It is shown that the frequency
band of the exponential growth is limited on
high frequencies. As a result, a wave becomes
localized near the surface of the Earth at higher
frequencies. The point of the change from
increase to decreasing is determined by a point
of a synchronism with a Rayleigh wave in the
ground substrate. Fig.2 presents the
dependencies of real and imagine parts of a
wave frequency (a,b), phase and group velocities
(c), and the real part of coefficient κ
+
of the
wave in the air(d) of a coupled wave in the
system Earth - air versus the value of y=kh for
pointed above parameters. These dependencies
are obtained by a direct numerical solution of the
dispersion equation. It is seen from the Fig.2 (b)
that in the left side from coupling point the
attenuation of waves is unessential but it
increases sharply in the right hand side from this
point yh=2.6. This increase is due to a more
deep penetration of the wave into the soil where
a viscosity is not equal to zero. Also one can see
from the Fig.2(d) that in the left side from the
point of coupling the most part of the wave is
delocalized (Re κ
+
<0 on z<0; in the right side
from this point the wave is concentrated mainly
in the solid substrate z>0. The account of
viscosity in soil causes weak wave attenuation
but does not change a general physical picture of
the wave interaction.
ACKNOWLEDGEMENT
This work is supported by CONACYT project
#35455-A.
REFERENCES
1. Hocke, K. and Schlegel, K., Ann. Geophys.
12, 917 (1996).
2. Hunsucker, R.D., Rev. of Geophys. and Space
Phys. 20, 293 (1982).
4. Gossard, E. E. and Hooke, W. H., ''Waves in
the Atmosphere'' (Elsevier, Oxford, 1975).
5. lngersoli, A.P., Kanamori, H. and Dowling,
T.E., Geophys. Res. Lett. 21, 1083 (1994).
6. Ewing, W.M., W.S.Jardetsky, W.S. and Press,
E., ''Elastic Waves in Layered Media'' (McGraw-
Hill, New York, 1957).
7. Weaver, P.F., P.C.Yuen, P.C., G.W.Prolls,
G.W. et al., Nature, 226, 1239 (1970).
8. Brekhovskikh, L.M., ''Waves in Layered
Media'' (Pergamon Press, Oxford, 1985).
Measurements of Aerodynamic Velocity Fields With An
Acoustical Probe
V. Dewailly, F. Cohen Tenoudji, J.P. Frangi and M. de Billy
Laboratoire Environnement et Développement, Université D. Diderot, Tour 33-43, Case postale 7087,
2 place Jussieu, 75251 Paris Cedex 05, France.
e-mail : dewailly@ccr.jussieu.fr
A system using the displacement of a three-axis ultrasonic anemometer probe in the air flow within a wind tunnel is used to probe
the local variations of the flow velocity. The three velocity components are calculated from the sound travel times between
transducer pairs placed in opposition (three pairs ; 44 kHz frequency). For each probe location, 400 velocity vector data values are
measured at 32Hz. The average velocities and their fluctuations along the main axis of the flow and in the transverse plane are
pictured as images coded as false colors or vector arrows. The spatial resolution of the 22 cm x 22 cm images is in the order of 1
cm after correction of the bias brought by a particular probe orientation. The technique shows good reproducibility in the air flow
characterization. Acoustical results are compared with those obtained by an optical technique for different air flow situations.
The sound scattering by a moving fluid is an important
problem which has received a considerable experimental and
theoretical interest since many years [1-3]. The interaction of
acoustic waves with a turbulent flow is used to visualize the
velocity field near a square outlet at moderate Reynolds
number. This experimental method is not intrusive and does
not necessitate additional matter. The average field velocity
fluctuations - which are generated by the turbulence - are
plotted in a plane perpendicular to the flow axis and in the
direction of the air flow. The patterns obtained with and
without a grid (located at the outlet) are compared and
analyzed.
EXPERIMENTAL SET-UP
A sonic thermometer – anemometer (type CSAT 3 from
Campbell-Society) is used for the measurements. It includes
three non – coplanar pairs of ultrasonic transducers operating
about 40kHz and is attached to an X-Y linear displacement
equipment which lets the transmitters to move accurately by
step of 10mm which determines the spatial resolution. All the
displacements are driven by a computer. The jet flow is
produced by a rotating fan placed in the central section
situated between the square (22x22cm
2
) inlet and outlet.
X
Y
Z
Gr i d
F a n
Ai r f l o w
The components of the velocity vector are recorded
simultaneously in three orthogonal directions. The z-
component (or axial velocity) coincides with the axis of the
propeller (see Fig.1). Because of the fluctuation of the
velocity of propagation of the sound in the turbulent medium,
the arrival time varies and for each probe location 400
velocity vector data values are measured at 32Hz. The
turbulence is supposed to be stationary. The measurements
are achieved in a (X,Y) plane located at 4cm from the outlet.
RESULTS AND ANALYSIS
In figure 2-a are plotted the diagrams of the velocity fields
(represented by vector arrows) obtained in the X-Y plane and
without any grid (vertical and horizontal scales are in cm).
0
5
1 0
1 5
2 0
2 5
0
2
4
6
8
1 0
1 2
1 4
1 6
1 8
2 0
On the picture we observe only one vortex the origin of
which is the rotation of the fan. No such behavior is noticed
in the figure 3-a which represents the velocity field diagram
when a grid is introduced between the outlet and the
anemometer. The dimensions of the grid are 1x1x1cm
3
.

No
periodicity characteristic of the grid may be extracted from
FIGURE 1 : Schematic diagram of the experimental
set-up.
FIGURE 2-a : Velocity field in the X-Y plane
(without grid).
this plot. In figures 2-b and 3-b are showna gray-scale
distribution of the z-component velocity measured with and
without the grid respectively. In the center of the first picture,
0
2
4
6
8
1 0
1 2
1 4
0
2
4
6
8
1 0
1 2
1 4
1 6
1 8
we may the presence of negative values of the axial velocity
which result from the conjugated effects: the existence of
high velocity components near the periphery and the presence
of a virtual “tube” in which the z-component velocity value is
zero.
The second figure confirms that there is no more negative
component of the velocity.
CONCLUSION
The measurements previously described and limited to the
comparison of results obtained with and without grid may be
extended to the velocity fields measurements behind
axisymmetric targets. Temporal inter-correlation and auto-
correlation should be calculated to get additional
characteristics of the fields velocity and to gather more
informations on the energy exchange between the different
components of the velocity.
REFERENCES
1. Petrossian, A., and Pinton, J.F., J. Phys. II France 7, 801-
812 (1997).
2. Baudet, C., Ciliberto, S. and Pinton, J.F., Phys. Rew. Lett.,
67, 193-195 (1991).
3. Wen-Shyang Chiu, Lauchle, G.C. and Thompson, D.E., J.
Acoust. Soc. Am., 85, 641-647 (1989).
FIGURE 2-b : Gray representation of the
axial velocity (without grid).
FIGURE 3-a : : Velocity field in the X-Y
plane (with grid).
FIGURE 3-b : Gray representation of the
axial velocity (with grid).