1

An experimental application of aeroacoustic time-reversal to the 1

aeolian tone 2

A. Mimani, a) 3

School of Mechanical Engineering, The University of Adelaide, Adelaide, South Australia 5005, 4

Australia 5

Z. Prime, D. J. Moreau and C. J. Doolan 6

School of Mechanical and Manufacturing Engineering, University of New South Wales, 7

Sydney, NSW 2052, Australia 8 9 Date received: 4th June, 2015 10 11 Running Title: Time-reversal and beamforming comparison 12 13 PACS numbers: 43.60.Tj, 43.60.Fg, 43.60.Jn, 43.28.Ra 14 15 16

ABSTRACT 17 18

This paper presents an experimental application of the aeroacoustic Time-Reversal (TR) 19

source localization technique for studying flow-induced noise problems and compares the TR 20

results with those obtained using Conventional Beamforming (CB). Experiments were conducted 21

in an Anechoic Wind Tunnel for the benchmark test-case of a full-span circular cylinder located 22

in subsonic cross-flow wherein the far-field acoustic pressure was sampled using two Line 23

Arrays (LAs) of microphones located above and below the cylinder. The source map obtained 24

using the signals recorded at the two LAs without modeling the reflective surfaces of the 25

contraction-outlet and cylinder during TR simulations revealed the lift-dipole nature of 26

aeroacoustic source generated at the Aeolian tone; however, it indicates an error of th3 20 of 27

Aeolian tone wavelength in the predicted location. Modeling the reflective contraction-outlet and 28

cylinder during TR was shown to improve the focal-resolution of the source and reduce side-lobe 29

a)Author to whom correspondence should be addressed. Electronic mail: akhilesh.mimani@adelaide.edu.au

Submitte

d Vers

ion

2

levels in the low-frequency range. The experimental TR results were shown to be comparable to 1

(a) the simulation results of an idealized dipole at the cylinder location in wind-tunnel flow and 2

(b) that obtained by Monopole and Dipole CB, thereby demonstrating the suitability of TR 3

method as a diagnostic tool to analyze flow-induced noise generation mechanism. 4

5 6 7

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Submitte

d Vers

ion

3

I. INTRODUCTION 1

Microphone array processing techniques can be used to provide important information on 2

the location and strength of the aeroacoustic sources. Such knowledge enables an understanding 3

of the underlying flow-induced noise generation mechanisms essential for reducing noise 4

emissions. A classical array processing technique is the Conventional Beamforming (CB) 5

method1 that has previously been successfully applied to several aeroacoustic problems such as 6

wind tunnel measurements of airfoil self noise2, jet-noise3, aircraft landing gear noise4, aircraft 7

fly-over tests5 and field measurements of wind turbine noise6. The CB method enables the 8

estimation of the direction of sound propagation from sources (typically, assumed to be of a 9

monopole nature) towards a microphone array by means of an appropriate Green’s function1. In 10

the conventional Delay-and-Sum Beamforming (DSB) method1, the acoustic pressure recorded 11

at the microphones are delayed and summed coherently to enhance the signal from the source 12

location and minimize it at a location different from the source. Therefore, the source location 13

can be theoretically determined by scanning over a grid of mesh points (or nodes) and 14

determining the location of the maximum. While in principle, the DSB method can be 15

implemented in the time-domain7, it is most often used in the frequency-domain because its 16

time-domain implementation is computationally expensive. A popular computationally efficient 17

variant of the DSB method is the cross-spectral CB method, implemented in the frequency-18

domain and differs from the DSB in that it can eliminate the microphone self-noise or the auto-19

correlation between microphones1,8. 20

The frequency-domain cross-spectral CB method1,8, however, does not allow the 21

visualization of the spatio-temporal evolution of the aeroacoustic fields, thus losing valuable 22

time-domain information. Furthermore, the CB method often makes a priori assumptions of the 23

Submitte

d Vers

ion

4

nature of the source (noted above), thereby immediately restricting the understanding of what the 1

true nature of the flow-induced noise source is. Also, it is difficult to directly account for the 2

effect of boundary conditions such as reflecting boundaries, diffracting surfaces and rigid-3

scatterers on the accuracy of the source localization through the use of the CB method. 4

An alternative array processing technique is acoustic Time-Reversal (TR) which is a 5

promising source localization method9. It finds application in diverse fields such as ultrasonics 6

for medical imaging, diagnostic and non-destructive testing9, long-range communication in 7

underwater acoustics10,11, in structural dynamics for health monitoring12 and localizing vibration 8

sources13, localizing sound sources in an outdoor urban area with many buildings14, in 9

propagation of water waves15, source identification in electromagnetics16 and recently, for 10

detecting gas-leakage in pipelines17. The acoustic TR method typically employs the following 11

two-step approach9. In the first-step, the acoustic pressure field radiated by source(s) is recorded 12

over an array of microphones either completely or partially enclosing it. During the second-step 13

(constituting TR), the recorded acoustic pressure data is reversed in time and is enforced at the 14

boundary nodes (corresponding to the microphone locations) as numerical sources in a high-15

resolution numerical algorithm that initiates back-propagation of acoustic waves that converge at 16

the source(s). The superior focusing quality of TR (in comparison to the CB method) is because 17

the back-propagated waves are simulated by numerically solving in reverse time, the same set of 18

equations that govern their emission from the source(s) without making a priori assumptions on 19

the source nature and also enables a visualization of the spatio-temporal evolution of the acoustic 20

fields (especially near the source). Indeed, the TR simulations are able to focus acoustic waves 21

(towards the source) by retracing them along the same trajectory created during their emission in 22

a homogenous stationary medium or a complex heterogeneous random medium with density or 23

Submitte

d Vers

ion

5

temperature gradients18, in the presence of rigid point-like cloud of scatterers19, thousands of 1

rigid vertical rods20 or a finite size rigid scatterer21. Furthermore, the TR method can take into 2

account, the effect of a reflecting surface22 and can be implemented (using limited channels or 3

transducers) in a reverberating environment such as a room23 or a chaotic mono-crystalline 4

silicon cavity24,25. 5

The application of TR in the field of Computational Aeroacoustics (CAA) has also 6

received considerable attention in recent works22,26-30. This includes the use of TR for localizing 7

a pulse in a uniform mean flow field22,26,27, characterizing the nature of idealized aeroacoustic 8

sources27-29, developments of techniques for improving its computational implementation29 as 9

well as the focal-resolution of sources30 and analysing the sound generated by a compressible 10

plane mixing-layer flow26. 11

A unique blend of numerical CAA simulation and data from experiments, therefore, gives 12

TR a potential advantage over CB to analyze flow-induced noise mechanisms; however, such an 13

experimental application of aeroacoustic TR has received only limited attention27. For instance, 14

Padois et al.27 presented an application of the TR technique to localize loudspeaker source(s) 15

(modeling a time-harmonic monopole or a dipole source) in a non-uniform shear mean flow 16

using the experimental acoustic pressure time-history measured over one Line Array (LA) of 17

microphones located above a wind tunnel test-section (and outside the flow). It is important to 18

note that Padois et al.27 essentially apply the TR technique to a simulated-experimental 19

aeroacoustics problem because noise is generated by loudspeakers even in the absence of flow. 20

Therefore, the application of aeroacoustic TR to an actual flow-induced noise problem has not 21

yet been demonstrated. 22

Submitte

d Vers

ion

6

In light of the background provided above, this paper presents, for the first time, an 1

application of the aeroacoustic TR method to an experimental flow-induced noise problem of the 2

benchmark test-case of a full-span circular cylinder placed in a subsonic cross-flow31-34. The 3

tonal sound generated by a cylinder in cross-flow is an important source of flow-induced noise 4

because cylindrical objects are found in a range of engineering applications such as rail 5

pantographs, automobile appendages, aircraft landing gear, periscopes, masts and antennas35-38. 6

Indeed, an analysis and review of the flow-induced noise mechanism due to cylindrical objects is 7

a subject matter of several investigations35-38. Therefore, the test-case of flow-induced cylinder 8

noise is considered here with a view to demonstrate the suitability of the TR method to analyse 9

such class of problems. The objective of this work is to present a detailed analysis of the spatio-10

temporal evolution of the time-reversed acoustic pressure fields and accuracy of the TR source 11

maps for characterizing the nature of flow-induced noise generated at the Aeolian tone using two 12

LAs of microphones located in the far-field at above and below the circular cylinder, the effect 13

of modeling the geometry of experimental set-up and ignoring mean flow during TR simulation 14

as well as the use of a single LA on source characterization. The experimental TR results are also 15

compared with those obtained using the cross-spectral CB method. 16

This paper is organized as follows. Section II briefly describes the experimental set-up. 17

Section III analyzes the far-field acoustic spectra generated by the circular cylinder in subsonic 18

cross-flow. Section IV describes the methodology for implementing the aeroacoustic TR 19

simulations, computation of the TR source maps and algorithm for the cross-spectral CB method 20

based on Monopole/Dipole steering vector formulation. Section V presents the results of the 21

forward and TR simulation of an idealized dipole source located in wind tunnel flow, the 22

analysis of which is intended to serve as a reference for interpretation of the experimental TR 23

Submitte

d Vers

ion

7

results. Section VI presents the experimental TR results obtained using the acoustic pressure data 1

of the Aeolian tone: an analysis of the spatio-temporal evolution of the time reversed acoustic 2

pressure fields and the TR source maps. Section VII presents the CB source maps obtained using 3

the experimental acoustic pressure data and compares them with the corresponding TR source 4

maps (obtained in Section VI without modeling the contraction-outlet facility and the cylinder). 5

The important contributions of this work are then summarized in the concluding Section VIII. 6

II. EXPERIMENTAL SET-UP AND TEST-MODEL 7

Experiments were conducted in the Anechoic Wind Tunnel (AWT) at the University of 8

Adelaide. The AWT is nearly cubic having internal dimensions 1.4 m 1.4 m 1.6 m and its 9

walls are acoustically treated with foam wedges providing a near reflection-free environment 10

above 250 Hz. It contains a contraction-outlet that produces a quiet, uniform test-flow and is of a 11

rectangular cross-section of height h = 75 mm and width w = 275 mm with flanges on the top 12

and bottom, each of height 40 mm.fh The maximum free-stream velocity of the jet and 13

turbulence intensity at the contraction-outlet is 140 m sU and 0.33%, respectively38,39. 14

The test-model, a full-span circular cylinder (of diameter 0 4 mmD ) is secured between 15

two side-plates attached to the contraction-outlet flange as indicated in Figs. 1(a) and (b), which 16

show a photograph and schematic of the front-view, respectively, of the experimental set-up. It is 17

noted that end-effects are minimized because the cylinder-span 450 mml along the z (span-18

wise) direction is sufficiently beyond the width of contraction-outlet39. A schematic of the side-19

view of the experimental set-up and the co-ordinate system convention is shown in Fig. 1(c) 20

where x is the stream-wise direction and y is the vertical direction. The origin 0x y is 21

taken on the axis of the contraction-outlet at its opening and the cylinder is mounted on a pivot-22

hole located downstream at 50 mm, 0. x y 23

Submitte

d Vers

ion

8

In this work, a set of experiments were carried out at 116, 24, 32 m sU with flow 1

issuing out from the contraction-outlet towards the positive x direction as indicated in Fig. 1(c). 2

III. MEASURING AND ANALYZING THE FAR-FIELD ACOUSTIC SPECTRA 3 4

Acoustic measurements were taken with two LAs of microphones aligned parallel to the 5

flow and located 700 mm apart, on opposite sides and equidistant from the cylinder, i.e., at 6

350 mmyy L and are co-planar (located in the 0z plane which passes through the 7

mid-span of the cylinder) as shown in Figs. 1(a) and (c). Each LA consists of 32 GRAS 40PH 8

1/4” phase-matched microphones (mounted in a timber-frame) positioned such that the spacing 9

between two consecutive microphones is 30 mm, therefore, the total array length equals 930 mm. 10

The length of each LA measured upstream from the contraction-outlet is given by 1 96 mmxL 11

whilst that measured downstream denoted by 2 834 mm.xL It is noted that four microphones in 12

both top and bottom LAs are located upstream of the origin (i.e., the contraction-outlet opening) 13

with the microphone nearest to the contraction-outlet being 6 mm upstream as indicated in Fig. 14

1(c). The 64 microphones in the two LAs were connected to a National Instruments PXI-8106 15

data acquisition system containing 4 PXI-4496 simultaneous sample/hold Analog-to-Digital 16

Converter (ADC) cards. The data (acoustic pressure time-history) at each of the 64 microphones 17

are recorded at sampling frequency 162 Hzsf for a sample time of 10 s. 18

When a cylinder is immersed in uniform flow, vortices of alternate rotation are shed from 19

either side of the cylinder into its wake34. This periodic shedding, known as a von Karman vortex 20

street, occurs at a particular frequency, ,af represented in a non-dimensional form by the 21

Strouhal number 0 0 ,D aSt f D U based on 0.D The von Karman vortex street generates 22

Submitte

d Vers

ion

9

unsteady forces on the surface of the cylinder that support dipole sound sources known as the 1

Aeolian tone32,40. 2

Figure 2 shows the frequency spectrum (measured in terms of the Power-Spectral Density 3

(PSD)) of the acoustic pressure field due to flow-induced noise of the cylinder at 4

132, 24, 16 m sU measured at the microphone located in the bottom LA and positioned 5

24 mm downstream of the contraction-outlet opening. The occurrence of an Aeolian tone at 6

frequency 784, 1208, 1584 Hzaf in the spectrum for 116, 24, 32 m s ,U 7

respectively, (corresponding to 0

0.196, 0.201, 0.198 ,DSt respectively) suggests a dipole-like 8

nature of the aeroacoustic source expected from the vortex shredding process. This is in 9

agreement with the study of Norberg31 which shows that a circular cylinder is expected to 10

produce an Aeolian tone at 0

0.2DSt for Reynolds number 0

Re 4240, 6360, 8480D for 11

116, 24, 32 m s ,U respectively, based on 0 4 mm.D 12

For implementing the TR simulation, the acoustic pressure signals recorded at each 13

microphone were first band-pass filtered (using a high-order Finite Impulse Response (FIR) 14

filter) in rd1 3 octave bands with center frequencies Cf given by (a) 800 Hz, (b) 1250 Hz and 15

(c) 1600 Hz for the spectrum obtained at 1 116 m s , 24 m sU and 132 m s , respectively, 16

therefore, a Cf f implying that the one-third octave band considered, accounts for almost the 17

entire acoustic power contained in the spectrum. Furthermore, the background noise level 18

generated by the free-stream jet is insignificant in comparison to the spectrum generated due to 19

flow-induced noise38,39, especially around the Aeolian tone. Therefore, it is expected to have a 20

negligible contaminating effect on the TR (and CB) results; hence it was not removed from the 21

spectrum prior to band-pass filtering. 22

Submitte

d Vers

ion

10

IV. SOURCE LOCALIZATION TECHNIQUES: DESCRIPTION OF ALGORITHM 1

The algorithms for implementing the source localization techniques, namely, aeroacoustic 2

TR simulation and cross-spectral CB for characterizing the flow-induced noise source is 3

described in this section. 4

A. Aeroacoustic Time-Reversal (TR) simulation 5

The aeroacoustic TR simulation was implemented by numerically solving28-30 the 2-D 6

Linearized Euler Equations27 (LEE) using the Pseudo-Characteristic Formulation26,41 (PCF) on a 7

rectangular domain 1 3 2 , x x x y yL L x L L y L in reverse time t shown as follows. 8

0 0linear linear linear linear

0 0 0linear linear

0 0 0

linear linear 0

,2

1 ,21 ,2

p c X X Y Ytu U U p UX X v ut y c c xv vY Y Ut x

(1-3) 9

where linear 0 00 0

1 p uX c Uρ c x x

and linear 00 0

1 .p vY cρ c y y

(4-7) 10

The time-reversed acoustic pressure , ,p x y t signals (recorded during experiments) were 11

enforced at boundary nodes corresponding to the top and bottom LAs which initiates the back-12

propagation of acoustic wave into the 2-D domain. It is noted that 3 100 mm,xL therefore, the 13

upstream length of the domain for implementing TR simulations is greater than the upstream 14

length of the LAs. 15

In Eqs. (1-3), u and v denote acoustic velocities along the x and y directions, respectively, 16

30 1.19 kg m is the ambient density of the medium (air), the sound speed 1

0 345.75 m sc 17

(at ambient temperature 0 297.47 KT ) whilst 0U represents the spatially-developing shear 18

Submitte

d Vers

ion

11

mean flow profile of a 2-D free-jet issuing out from the contraction-outlet measured 1

experimentally using a pitot-tube (connected to a 10 torr baratron with a plastic tube of length 2

5 m and inner diameter equal to 5 mm). The experimental mean flow profile in the AWT was 3

modeled by 4

0 0 max1, 1 cosh sech sech ,2 2 2y y y

y yU U x y UL L L

(8) 5

6 where max max , , U U x U x and x are the maximum mean flow, steepness of 7

the shear-layer and half-thickness of the potential-core, respectively; their variation along x 8

direction (from the contraction-outlet opening) is modeled by an appropriate polynomial-fit 9

using constrained least-squares optimization shown as follows. 10

Downstream of the contraction-outlet, over the region 0,x 11

1 2 3 4 5

max 1 1 1 1 1

6 7 8 9 101 1 1 1 1

, 32 m s 31.1746 1.1549 2.9324 1.8821 2.4703 1.5011

1.4359 0.4891 0.4070 0.0504 0.0401 ,

U x x x x x x

x x x x x

(9) 12

2 3 4 5

1 1 1 1 1

6 7 8 9 101 1 1 1 1

17.8892 7.8857 8.7515 10.4626 39.7521 11.9655

14.3155 4.3221 2.2220 0.1335 0.2600 ,

x x x x x x

x x x x x

(10) 13

where 3

1 3

361.0635 10 .237.6565 10

xx

(11) 14

The variation in half-thickness of the potential-core x is given by 15

337.5 10 m, for 0 374 mm,x (12) 16

2 3 4 32 2 2 225.3448 8.3799 3.1940 2.3179 1.1574 10 ,x x x x x (13) 17

for 374 mm 524 mm,x where 18

3

2 3

434.4 1048.8699 10xx

(14) 19

Submitte

d Vers

ion

12

and 31 10 m, for 524 mm.x (15) 1

It is noted that in Eqs. (11) and (14), x is taken in m whilst the interpolated values of 2

1max , 32 m sU x and x given by Eqs. (9) and (13) are also in m. 3

For 0,x i.e., the region upstream of the contraction-outlet opening, 4

1 3max 32 m s , 150, 37.5 10 m.U (16) 5

Figure 3 depicts the variation of the spatially-developing shear mean flow of the free-jet at 6

132 m sU over the 2-D computational domain based on the model given by Eq. (8) and the 7

optimized interpolating polynomial functions obtained above. (The interpolating polynomial for 8

1max , 24 m sU x and 1

max , 16 m sU x were also similarly obtained using the constrained 9

least-squares optimization, although these are not shown here for brevity.) 10

It is important to mention that the mean flow direction was reversed towards the negative x 11

direction in Eqs. (1-5), i.e., 0 0U U for implementing the TR simulations which was 12

necessary to ensure the TR invariance26-30 of the governing 2-D LEE. 13

In Eqs. (1-3), the term linearX denotes the acoustic flux propagating towards the positive x 14

direction with a diminished speed of 0 0c U whilst linearX denotes the acoustic flux 15

propagating towards the negative x direction with an enhanced speed of 0 0c U during TR 16

simulations. Similarly, linearY denotes the fluxes propagating with sound speed 0c towards the 17

positive and negative y directions, respectively. Furthermore, the term 0U v x in Eq. (3) 18

denotes acoustic disturbances advected by the reversed mean flow towards the negative x 19

direction, the term 0v U y in Eq. (2) accounts for the refraction of acoustic waves through 20

Submitte

d Vers

ion

13

the shear-layer (along the y direction) whilst the term 0 0

0 0 0

U p Uuc c x

in Eq. (2) models 1

the effect of spatially-developing nature (along the x direction) of the free-jet flow on acoustic 2

wave propagation. 3

1. Computation of spatial derivatives, mesh-resolution and time-integration scheme 4

The spatial derivative of acoustic pressure and velocities in the opposing fluxes 5

linear linear, X Y of the PCF were computed using an overall upwind-biased FD scheme29 that is 6

formulated using a fourth-order, seven-point optimized upwind-biased FD scheme42 at interior 7

nodes and a seven-point optimised backward FD scheme at the boundary nodes43. In order to 8

increase the mesh-resolution, two equally spaced nodes were added between each pair of nodes 9

corresponding to the microphone locations. The acoustic pressure time-history at these two extra 10

nodes (required during TR) were obtained by interpolating the experimental data between each 11

pair of microphones using Lagrange polynomial interpolation44, resulting in a mesh-size 12

10 mmx along the x direction. Equal mesh-size 10 mmy was also considered along the y 13

direction. The efficiency of implementing the overall upwind-biased FD schemes is increased by 14

recasting them in the following matrix form29. 15

1 21 1, ,

Δ Δx x x x

R R (17a, b) 16

where nodes

1 2 3 , , ,..., T

N and represents both acoustic pressure and velocities. 17

Equations (17a) and (b) are used for computing the spatial derivatives in acoustic fluxes 18

propagating towards the positive and negative directions, respectively. (The rows and columns of 19

1R and 2R matrices are however, not shown for brevity.) It is noted that these overall 20

upwind-biased FD schemes make use of the opposite upwinding directions (or two different 21

Submitte

d Vers

ion

14

group of stencil points) at a given node i so that the inbuilt dissipation in the upwind-biased FD 1

schemes damps only the unresolved high frequency waves and does not induce spatially growing 2

oscillations with time, necessary to ensure the temporal stability and accuracy of TR 3

simulations29. The spatial derivative in the term 0U v x is computed using Eq. (17b) 4

because the direction of mean flow is reversed during TR simulations. 5

The maximum frequency that may be accurately propagated on this mesh is approximately 6

8255 Hz as determined from the Dispersion-Relation-Preserving42,43 range 1.5DRP of the 7

fourth-order optimized upwind-biased FD scheme and 10 345.75 m s .c The third-order Total-8

Variation-Diminishing Runge-Kutta scheme45 is used for time-integration with a time-step 9

51 1.5259 10 sSt f implying a 0.55, 0.56, 0.58CFL for 116, 24, 32 m s ,U 10

respectively, for the mesh-size considered. 11

2. Implementation of Anechoic Boundary Conditions (ABCs) 12

Enforcing the time-reversed acoustic pressure history at the two LAs (i.e., the Dirichlet 13

boundary conditions27) not only initiates the back-propagation of acoustic wave fronts into the 14

2-D domain which converge towards the aeroacoustic source location, but also generates 15

outgoing acoustic waves. In order to eliminate the spurious reflections due to these outgoing 16

waves and stabilize the 2-D TR simulations, it was crucial to implement the first-order Clayton–17

Engquist–Majda (CEM) ABC’s (see Ref. [46]) at all four computational boundaries ( yy L 18

and xx L ) and the corner ABC’s (see Ref. [47]) at the four corner nodes. The ABC’s at the 19

four boundaries were further reinforced by setting the incoming fluxes to zero at these 20

boundaries29, i.e., linear 0X at ,xx L respectively, and linear 0Y at .yy L The boundary 21

Submitte

d Vers

ion

15

condition 0~

xLxxv was also implemented to eliminate the incoming spurious numerical 1

waves advected by the fluid necessary for temporal stability of TR29. 2

3. Use of the Time-Reversal-Sponge-Layer (TRSL) damping or the Superposition technique 3

Due to the conservation of energy27 and absence of an acoustic-sink30,48 during TR 4

simulation of flow-induced noise sources (of a tonal or broadband nature), the converging wave-5

fronts do not stop at the source but propagate beyond and interfere with flux emanating from the 6

LAs located at the opposite boundary29 unlike the TR simulation of a transient signal such as a 7

pulse26. Therefore, in order to suppress the deteriorating flux-interference effect near the LAs, a 8

Time-Reversal-Sponge-Layer (TRSL) was implemented that damps the fluxes normally incident 9

on the LAs by multiplying them by a Gaussian damping function TRSLG n that smoothly decays 10

to zero across TRSLn nodes adjacent to and including the node on the LA boundary using the 11

following transformations29. 12

linear linear, , ,y y TRSLY x L n y Y x L n y G n (18a, b) 13

where 0 0TRSLG 14

and 2

1 2 1e

TRSLTRSL

TRSL

n nn

TRSLG n

for 1,2,..., 1 .TRSLn n (19a, b) 15

In Eq. (19b), TRSL is the damping coefficient taken equal to 3.5 whilst TRSLn (the thickness 16

of the TRSL) is taken equal to 10 in this work. The TRSL damping technique is however, used 17

during TR simulations only for Aeolian tone frequencies 1208, 1584 Hz.af The use of 18

TRSL is not preferred during TR simulations for 784 Hzaf because at such low-frequency 19

(for size of the computational domain considered along the y direction), the primary side-lobes 20

occur near the LAs and the TRSL generates spurious local maxima region near the LAs which 21

Submitte

d Vers

ion

16

tend to overwhelm the focal spots, thereby deteriorating the quality of source 1

localization/characterization. Therefore, in order to prevent the problems arising out of flux-2

interference at the LAs, the superposition technique29 is used to implement the TR simulations 3

for 784 Hz.af To this end, the time-reversed acoustic pressure fields given by Top , ,p x y t 4

and Bottom , ,p x y t obtained by implementing the TR simulations using a single LA located at 5

the top and bottom boundaries, respectively, are superposed, i.e., 6

Top Bottom, , , , , , ,p x y t p x y t p x y t (20) 7

to obtain the total time-reversed acoustic pressure field , ,p x y t which is equivalent to using 8

two LAs simultaneously. However, in comparison to the use of TRSL, the computational cost of 9

the superposition technique is high because it solves the same set of governing equations twice to 10

obtain the , ,p x y t field29. 11

4. Modeling the contraction-outlet walls, flanges and the cylinder during TR simulation 12

The flow-induced noise source generated at the cylinder location during the experiment 13

radiates acoustic waves which are refracted through the shear-layer. The wave fronts propagating 14

upstream of the mean flow are diffracted by the flanges of the contraction-outlet, undergo 15

reflection off its walls and subsequently propagate towards the non-reflective upstream, top and 16

bottom boundaries. (The diffraction phenomenon is pronounced in the low-frequency range49.) 17

As noted earlier, the TR simulation focuses acoustic waves by numerically solving the 18

same set of differential equations that govern their emission from the source and can also 19

incorporate appropriate boundary conditions. Indeed, the back-propagated acoustic waves during 20

TR simulation are focused at the source by retracing them along the same complex trajectory 21

created during their emission in the presence of multiple scatterers/diffraction objects and rigid 22

Submitte

d Vers

ion

17

reflecting surfaces. Therefore, with a view to account for the effect of wave diffraction by the 1

flanges and its reflections off the walls (of the contraction-outlet) on aeroacoustic source 2

localization, another set of TR simulations were carried out whereby the flanges and walls of the 3

contraction-outlet were modeled by appropriate rigid wall conditions implemented by setting 4

0, 0.04 m 0.08 m, 0u x y t and 0, 0.04 m 0.08 m, 0,u x y t (21a, b) 5

0.196 m 0, 0.04 m, 0,v x y t (22a, b) 6

respectively. The rigid cylinder was also modeled by setting the acoustic particle velocities to 7

zero at the node corresponding to the cylinder location, i.e., 8

0.05 m, 0, 0.05 m, 0, 0.u x y t v x y t (23a, b) 9

The TR source map (for the flow-induced dipole source at a given Aeolian tone) obtained 10

without modeling the contraction-outlet walls, flanges and the cylinder are compared with that 11

obtained with their modeling in an ensuing section. 12

5. Determining the location and nature of flow-induced noise sources 13

The TR simulation was implemented over a reverse time-interval 0, 10000t t 14

whereby the aeroacoustic source location/characteristics were obtained by determining the focal 15

spots in the Root-Mean-Square (RMS) time-reversed acoustic pressure field27-30 (computed over 16

the time-interval when a steady-state acoustic field is observed throughout the domain) denoted 17

by , .TRRMSp x y The focal spot maximum is termed the focal point. The ,TR

RMSp x y field is 18

converted to dB scale (with respect to 52 10 Parefp ) and the source map denoted by 19

dB ,TRp x y is expressed relative to the focal point(s) whose magnitude is taken as 0 dB, see 20

Mimani et al.29,30. 21

22

Submitte

d Vers

ion

18

B. Cross-spectral Conventional Beamforming (CB) 1

Cross-spectral CB using both Monopole and Dipole steering vectors was carried out in the 2

one-third octave bands containing the Aeolian tone frequencies (discussed earlier). To this end, 3

the acoustic pressure recorded at each microphone (during experiments) was first high-pass 4

filtered using a 6th order Butterworth filter having a cut-on frequency of 300 Hz. A cross-spectral 5

matrix, ,C f was then computed to estimate the cross-spectral density for each pair of 6

microphones by the Welch’s method50 using a Hanning window, Fast Fourier Transform (FFT) 7

size of 213 and 50% window overlap for a frequency resolution of 8 Hz and 150.7 effective 8

blocks (including the reduction due to window overlap). 9

For each frequency band within the upper and lower frequency limits of the one-third 10

octave band considered, the CB was carried out on a regular grid of 2.5 mm spacing over the 11

domain 96 mm 504 mmx and 300 mm 300 mm.y The steering vectors used from the 12

grid point m to the microphone n were constructed from the assumed Green’s function, , ,m na f 13

see Sarradj51. 14

, , 2 ,m n

m n

n

a fe f

a f (24) 15

where na f is the vector of Green’s functions from the grid point to all microphones. It is 16

noted that the monopole Green’s function is given by51 17

0

, e ,

4

m ni x xc

Mm n

m n

a fx x

(25) 18

Submitte

d Vers

ion

19

where 2 ,f f represents the analysis frequency, mx and nx are the co-ordinates of the grid 1

point and the microphone n, respectively. For a dipole source of orientation denoted by vector ,l

2

the Green’s function is given by52 3

0

, e .

4

m ni x xc

D m nm n

m n m n

x xa f lx x x x

(26) 4

The CB output was then calculated using the cross-spectral matrix and the steering vectors. 5

In order to reduce the effect of microphone self-noise, the diagonal removal procedure given by 6

Dougherty53 was used such that the beamforming output is 7

,1

Hn n n

NY f e f C f e fN

(27) 8

where N is the number of microphones, and the superscript H denotes the conjugate transpose. 9

The CB source maps were calculated by using a rectangular integration across these frequency 10

bins within the one-third octave frequency band of interest. 11

Finally, the effect of refraction of waves through the shear-layer is approximated by 12

shifting the grid by 0 ,M h where 0M is the Mach number and h is the perpendicular distance 13

within the flow (i.e., half-thickness of the potential-core 37.5 mm at the contraction-outlet 14

opening), see Padois et al.54. This approximation is considered sufficient due to the low Mach 15

number and the small height of the wind tunnel nozzle. 16

V. SIMULATION OF IDEALIZED DIPOLE IN WIND-TUNNEL FLOW: NUMERICAL 17

VALIDATION OF TR METHOD 18

This section considers the test-case of a simulation of an idealized (point-like) tonal dipole 19

source28-30 located at the cylinder location in a wind-tunnel flow. The objective of this simulation 20

based analysis is twofold: to investigate what effect does the modeling of the contraction-outlet 21

Submitte

d Vers

ion

20

walls and flanges have on (a) the directivity of the radiated RMS acoustic pressure field of an 1

idealized dipole source with its axis perpendicular to flow and (b) the corresponding TR source 2

maps. Since the flow-induced noise generated at the Aeolian tone of a cylinder is known to have 3

a lift-dipole type of source nature31-38, the aforementioned analysis will serve as a useful 4

reference for the interpretation of experimental TR results presented later, thus explaining its 5

significance. 6

A. Forward simulation 7

The forward simulation of the acoustic field radiated by an idealized dipole source of tonal 8

frequency 0f in a 2-D free-space was carried out by numerically solving the inhomogeneous 2-D 9

LEE (in the PCF) over a rectangular domain with ABCs (at all boundaries) given by29,55 10

20 0linear linear linear linear 0 1,2

p c X X Y Y c St

(28) 11

0 0 0 2linear linear

0 0 0 0

1 ,2

u U U p U SX X v ut y c c x

(29) 12

3linear linear 0

0

1 ,2

v v SY Y Ut x

(30) 13

where ± ±linear 0 0 linear 0

0 0 0 0

1 1, ,p u p vX c U Y cc x x c y y

(31a-d) 14

and setting28-30 15

1 2 0 0 0 3, 0, , δ δ sin 2 , , 0.DS x y S x y F x x y y f t S x y (32) 16

In Eq. (32), 2100 N mDF is the amplitude of the harmonic point-force simulating the 17

idealized dipole source, the known location is given by 0 054 mm, 0x y (taken at the node 18

nearest to the cylinder location 50 mm, 0x y ) whilst 0U is the spatially-evolving mean shear 19

flow (given by Eq. (8) for 132 m sU ). It is noted that the rectangular domain and the mesh 20

Submitte

d Vers

ion

21

size considered during forward simulations are identical to that considered for implementing 1

experimental TR simulations noted in Section IV.A. The forward simulations were implemented 2

for a sufficiently long time-interval given by 0, 10000t t wherein the zero initial conditions 3

were replaced by several periods of time-harmonic response and the acoustic pressure time-4

history was recorded at the top 0 350 mmy and bottom 0 350 mmy LAs. (Here, the 5

time-step 65.3059 10 st corresponding to a CFL number equal to 0.2.) 6

1. Without modeling the contraction-outlet walls and flanges 7

Figures 4(a), (c) and (e) show the RMS acoustic pressure field radiated by an idealized 8

dipole source (with its axis perpendicular to flow) of tonal frequency 0 750, 1500, 3000 Hz,f 9

respectively, located in a spatially-evolving mean shear flow obtained without modeling the 10

contraction-outlet walls and its flanges over the dynamic range 0, 80 dB . It is noted that the 11

known source location is represented by a circle O whilst the mean flow direction towards the 12

positive x direction (during forward simulation) is indicated by an arrow. These conventions are 13

followed throughout Section V. Furthermore, the two LAs, contraction-outlet and its flanges are 14

shown by thick white lines and the same symbolic convention is followed henceforth. 15

Figures 4(a), (c) and (e) exhibit the expected directivity pattern due to an idealized dipole; 16

the RMS acoustic pressure is maximum along the dipole axis perpendicular to the flow, (i.e. 17

along the y direction) whilst a pressure nodal line is observed along the x axis. Furthermore, the 18

RMS acoustic pressure level observed at the two LAs increases with an increase in the tonal 19

frequency of the dipole source. 20

2. With the implementation of rigid conditions due to contraction-outlet walls and flanges 21

Figures 4(b), (d) and (f) show the RMS acoustic pressure field radiated by an idealized 22

dipole source of tonal frequency 0 750 Hz, 1500 Hz, 3000 Hz,f respectively, located in a 23

Submitte

d Vers

ion

22

spatially-evolving mean shear flow obtained with the modeling of the contraction-outlet walls 1

and its flanges over the dynamic range 0, 80 dB . 2

It is observed from Figs. 4(b), (d) and (f) that incorporation of the contraction-outlet 3

geometry during forward simulation significantly alters the directivity pattern of the radiated 4

acoustic pressure field which is explained in terms of the wave diffraction phenomenon49 at the 5

flanges. For tonal frequency 0 750 Hz,f the corresponding wavelength 0 461 mm is 6

significantly greater than the total height (150 mm) of the contraction-outlet facility taken as sum 7

of (a) the contraction-outlet height 75 mmh and (b) that of the flanges 2 80 mm.fh 8

Therefore, during forward simulation, the wave fronts are diffracted by the flanges and the 9

propagation of the diffracted wave fronts result in reflections from the rigid-walls of the 10

contraction-outlet and subsequently, a significant amount of acoustic power being received by 11

the non-reflective upstream boundary as observed from the RMS field shown in Fig. 4(b). 12

For tonal frequency 0 1500 Hz,f the corresponding wavelength 0 230.5 mm is 13

comparable to the total height of the contraction-outlet facility. Therefore, the wave diffraction49 14

phenomenon at the flanges is less pronounced. As a result, the wave reflection at the walls of the 15

contraction-outlet and the acoustic power intercepted by the upstream boundary is substantially 16

less. Rather, due to reflection of incident waves by the flanges, the downstream length of the 17

LAs intercept a larger amount of radiated acoustic flux in comparison to the upstream length as 18

may be observed from Fig. 4(d). 19

For tonal frequency 0 3000 Hz,f the corresponding wavelength 0 115.25 mm is 20

smaller than the total height of the contraction-outlet facility, therefore, the wave diffraction49 at 21

the flanges is negligible. Rather, the acoustic wave fronts propagating away from the source and 22

incident on the flanges are completely reflected from it. Indeed, the RMS directivity pattern 23

Submitte

d Vers

ion

23

shown in Fig. 4(f) resembles the directivity field due to a piston in a rigid baffle and as a result, 1

the downstream length of the LAs intercepts a further greater amount of radiated acoustic flux. 2

B. Aeroacoustic TR simulation 3

The aeroacoustic TR simulations were implemented by numerically solving Eqs. (1-3) and 4

using the simulated acoustic pressure data recorded at the top and bottom LAs during the 5

forward simulation of the idealized tonal dipole source when the contraction-outlet and its 6

flanges were modeled (corresponding to Figs. 4(b), (d) and (f)). The direction of the spatially-7

evolving mean shear flow at 132 m sU was reversed during TR. As discussed in 8

Section IV.A.5, the TR source map (of the idealized dipole source) was obtained by computing 9

the RMS acoustic pressure field when steady-state time-harmonic condition was developed 10

throughout the 2-D domain. The following set of TR simulation was carried out at each tonal 11

frequency: (a) without modeling the contraction-outlet walls and its flanges and (b) with their 12

modeling (by making use of Eqs. (21) and (22)) with a view to investigate the effect of rigid-wall 13

modeling of the contraction-outlet facility on characterization of the idealized dipole source. 14

1. Without modeling the contraction-outlet walls and flanges 15

Figures 5(a), (c) and (e) show the TR source maps of the idealized dipole source of tonal 16

frequency 0 750 Hz, 1500 Hz, 3000 Hz,f respectively, obtained using the top and bottom LAs 17

without modeling the contraction-outlet walls and its flanges over a dynamic range given by 18

0, 10 dB . It is observed that the TR source maps shown in Figs. 5(a), (c) and (e), exhibit two 19

focal spots (identical in terms of relative magnitude, shape, and size) that are located in 20

proximity, thereby confirming the dipole nature of the idealized source28-30. The geometrical 21

center of the two focal points (noted from their respective focal spots) is taken as the predicted 22

location of the dipole source and is denoted by a cross X in the source maps. Furthermore, the 23

Submitte

d Vers

ion

24

top and bottom LAs located at 350 mmy boundaries, respectively, are indicated by thick 1

white lines whilst the reversed direction of mean flow (during TR simulations) is indicated by an 2

arrow; the same dynamic range and symbolic conventions are followed henceforth. 3

The predicted location of the dipole source and the error (taken as the distance between the 4

predicted and known locations expressed as a fraction of a wavelength 0 corresponding to the 5

tonal frequency 0f ) is shown in the 2nd and 3rd columns, respectively, of Table I. These values 6

indicate a maximum error of 0 20 in the predicted source location, thereby demonstrating the 7

accuracy of the TR simulation (implemented without modeling the contraction-outlet walls and 8

its flanges). 9

2. With the implementation of rigid-wall conditions due to contraction-outlet walls and flanges 10

Figures 5(b), (d) and (f) show the TR source maps of the idealized dipole source of tonal 11

frequency 0 750 Hz, 1500 Hz, 3000 Hz,f respectively, obtained using the top and bottom LAs 12

with the incorporation of the contraction-outlet geometry modeled by implementing Eqs. (21) 13

and (22) during TR simulations. Figure 5(b) is comparable to Fig. 5(a); however, the location of 14

the idealized dipole source predicted in Fig. 5(b) is downstream of the known location, thereby 15

resulting in a large error 00.15 as shown in the 5th column of Table I. Furthermore, a significant 16

wave reflection at the rigid-walls of the contraction outlet is observed in Fig. 5(b) which is 17

explained by the following discussion. The wavelength 0 461 mm at tonal frequency 18

0 750 Hzf is significantly greater than the total height (150 mm) of the contraction-outlet 19

facility. Therefore, during TR, the back-propagated waves from the upstream LAs are 20

diffracted49 (at the flanges) which directs their propagation downstream of the contraction-outlet. 21

Furthermore, the back-propagated waves from the downstream LAs are also diffracted and 22

Submitte

d Vers

ion

25

propagate towards the rigid-walls of the contraction-outlet (whereby they are reflected) and the 1

non-reflective upstream boundary. Due to a continuous interaction of the waves back-2

propagating from the upstream and downstream LAs over the region downstream of the known 3

location, two instantaneous focal-spots are formed throughout TR yielding Fig. 5(b). 4

Figure 5(d) is however, significantly different from Fig. 5(c); it exhibits two localized 5

maxima regions formed near the flanges which makes it difficult to characterize the dipole 6

source nature. (It is for this reason that the predicted location and error are not shown in the 4th 7

and 5th columns, respectively, of Table I for 0 1500 Hz.f ) This observation is understood by 8

noting that wavelength 0 230.5 mm at tonal frequency 0 1500 Hzf is comparable to the 9

total height of the contraction-outlet facility. This signifies that the diffraction phenomenon49 of 10

waves (at the flanges) is less pronounced; as a result, the downstream lengths of the LAs receive 11

a larger portion of the radiated acoustic power during forward simulation as suggested by Fig. 12

4(d). During TR simulation, the waves back-propagating from downstream LAs and incident on 13

the flanges also experience a limited diffraction, rather, these are reflected from the flanges 14

towards downstream and continuously interferes with the waves converging at the source from 15

the downstream LAs resulting in formation of two instantaneous maxima regions near the 16

flanges throughout TR simulation yielding Fig. 5(d). Forward simulation were also carried out 17

for the idealized dipole of 0 1500 Hzf located further downstream at 0 080 mm, 0.x y 18

However, in this case, the TR source map obtained with the modeling of contraction-outlet 19

facility did not exhibit two maxima regions localized near the flanges; rather, the source map 20

showed two focal spots (located in proximity) similar to those shown in Fig. 5(b). This signifies 21

that when the dipole source wavelength is comparable to the total height of the contraction-outlet 22

facility, the effect of incorporating the contraction-outlet geometry during TR need not be 23

Submitte

d Vers

ion

26

counter-productive if the source is located sufficiently away from the flanges ensuring that span 1

(along the x direction) of the focal spots does not overlap with the flanges. 2

Figure 5(f) is similar to Fig. 5(e); it exhibits two focal spots, thereby confirming the dipole 3

nature. However, unlike Fig. 5(e), a small error of 00.09 is observed in the predicted source 4

location in Fig. 5(f) as indicated in Table I. This observation is explained by noting that the 5

wavelength 0 115.25 mm (at 0 3000 Hzf ) is smaller than the total height of the contraction-6

outlet facility. Therefore, during TR, the waves incident on the flanges from downstream LAs 7

undergo a negligible diffraction, rather, these are completely reflected from it. The interaction of 8

the reflected waves with those converging near the source (from the downstream LAs) 9

throughout the TR simulation and the relatively smaller size of the focal spots at 0 3000 Hzf 10

(so that it does not overlap with the flanges despite the proximity of the source to the flanges) 11

result in the formation of two instantaneous focal-spots yielding Fig. 5(f). Indeed, the focal spot 12

size (along the x direction) and the side-lobe levels in Fig. 5(f) are smaller in comparison to those 13

observed in Fig. 5(e), indicating an improved focal-resolution. 14

VI. EXPERIMENTAL TR RESULTS AND DISCUSSION 15

This section presents the experimental TR results obtained using the acoustic pressure data 16

of the Aeolian tone generated by the circular cylinder in cross-flow during aeroacoustics 17

experiments carried out in the AWT (described in Section II) and an analysis of these results. 18

A. TR simulation without modeling the contraction-outlet walls, flanges and cylinder 19

1. Spatio-temporal evolution of the time-reversed acoustic pressure field 20

Figure 6 shows the time-snapshots of the spatio-temporal evolution of the time-reversed 21

acoustic pressure field , ,p x y t obtained using the experimental acoustic pressure data 22

(recorded at the two LAs) of the flow-induced noise generated at the Aeolian tone 1584 Hzaf 23

Submitte

d Vers

ion

27

without modeling the contraction-outlet geometry and the cylinder during TR simulation. (The 1

colorbar in Fig. 6 represents the acoustic pressure in Pa.) It is noted that the cylinder (located at 2

0 050 mm, 0x y ) is represented by a circle O and this symbolic convention is followed 3

henceforth in the experimental TR results. 4

Figures 6(a) and (b) indicate a simultaneous emission of acoustic fluxes from the two LAs 5

during the initial time-instants that propagate into the domain and are about to converge or 6

undergo constructive interference near the cylinder location. Figure 6(c) shows that this is 7

followed by formation of two instantaneous maxima regions (i.e., the instantaneous focal spots) 8

about the cylinder-axis of nearly the same strength but opposite phase, indicating a dipole-9

source. The simulations reveal that at the source region, the width of the wave-fronts diminish 10

whilst their amplitude significantly increase. However, due to the conservation of energy27 and 11

absence of an acoustic-sink30,48 during TR, the converging wave-fronts do not stop at the source 12

but propagate beyond towards the LA located at the opposite boundary. By virtue of the TRSL 13

damping technique29 (implemented using Eqs. (18a) and (b)), the deteriorating flux-interference 14

effect near the LAs is suppressed. Figures 6(d-f) indicate a continuous formation of two 15

instantaneous focal spots throughout the TR simulations, although their instantaneous 16

geometrical center (taken as the instantaneous dipole location) slightly varies over time. To 17

enhance this discussion on the spatio-temporal evolution of the time-reversed acoustic pressure 18

field, the reader is also referred to Multimedia 1 which plays the corresponding TR simulation. 19

The spatio-temporal evolution pattern of the , ,p x y t field obtained (without modeling 20

the contraction-outlet geometry and the cylinder during TR) using the experimental acoustic 21

pressure data recorded at 1208, 784 Hzaf were found to be similar to that shown in Fig. 6. 22

However, the size of the instantaneous dipole focal spots during TR simulation at 23

Submitte

d Vers

ion

28

1208, 784 Hzaf were successively larger (due to low-frequency range) than those observed 1

at 1584 Hzaf in Figs. 6(c-f). 2

2. Analyzing the TR source maps: Effect of Aeolian tone frequency 3

Figures 7(a), (c) and (e) show the TR source maps due to the full-span circular cylinder 4

obtained using the experimental acoustic pressure data (recorded at the top and bottom LAs) at 5

the Aeolian tone 784, 1208, 1584 Hz,af respectively, without modeling the contraction-6

outlet walls, its flanges and the cylinder during TR simulation. It is noted that Figs. 7(a), (c) and 7

(e) present the results of experiments carried out at 116, 24, 32 m s ,U respectively, in the 8

one-third octave bands with 800, 1250, 1600 Hz,Cf respectively. Figures 7(a), (c) and (e) 9

exhibit a pair of focal spots (of nearly the same magnitude, shape and size) located in proximity. 10

The experimental TR source maps shown in Figs. 7(a) and (e) are observed to be similar to the 11

TR source maps of the idealized dipole source of comparable tonal frequency shown in Figs. 5(a) 12

and (c), respectively. This indicates that the flow-induced noise generated at the Aeolian tone 13

due to a cylinder in a cross-flow has a dipole-type nature and is qualitatively similar to an 14

idealized dipole source (whose axis is perpendicular to the mean flow). Indeed, due to the lift 15

forces responsible for noise generation, the orientation of the focal spots in Figs. 7(a), (c) and (e) 16

is almost above and below the cylinder and thus, the source is termed a lift-dipole31. 17

Table II shows the predicted location (indicated by X) of the flow-induced dipole source 18

obtained in Figs. 7(a), (c) and (e) corresponding to 784, 1208, 1584 Hz,af respectively, and 19

the error which is taken as the distance of the predicted location from the cylinder expressed as a 20

fraction of the corresponding wavelength a of an Aeolian tone. The average error in prediction 21

is 0.16 ,a while the error is largest at the lowest Aeolian tone 784 Hzaf and smallest at the 22

Submitte

d Vers

ion

29

highest Aeolian tone 1584 Hz.af This observation is explained as follows. The span-wise 1

coherence length scale of vortex shedding31 is larger at lower free-stream velocity 2

116 m s ;U hence, it is possible that the two co-planar LAs of microphones located in the 3

x-y plane passing through the cylinder mid-span and oriented along a direction perpendicular to it 4

will receive phase information from sections along the cylinder span that are similar to a dipole 5

source placed further downstream (of the cylinder) and thus gives a positional error in the TR 6

source map. In other words, a greater length scale of span-wise coherence at low-frequency 7

indicates a 3-D nature of the flow-induced noise problem, therefore, the use of 2-D TR 8

simulation somewhat under-represents the problem of low-frequency Aeolian tone generation 9

which may be more accurately modeled by the computationally intensive 3-D TR simulation. 10

On the other hand, at higher free-stream velocity 132 m s ,U this positional error in 11

the TR source map will be reduced as the span-wise coherence length scale will be smaller, 12

forming a larger number of compact acoustic dipole sources that radiate individually along the 13

cylinder span which reduces the phase error on the two co-planar LAs of microphones at a given 14

time, thereby indicating that the use of 2-D TR simulation is able to model the high-frequency 15

Aeolian tone generation problem with a relatively greater accuracy. Furthermore, it is noted that 16

accuracy of predicted location may possibly be improved by increasing the upstream and 17

downstream lengths of LAs, (i.e., by using more number of microphones) due to an increased 18

effective angular aperture at the source location29. 19

Table II also indicates that the average Sound Pressure Level (SPL) at the predicted focal 20

points of the dipole source in Figs. 7(a), (c) and (e) increases with an increase in the frequency of 21

Aeolian tone. This is in agreement with the PSD spectrum of the full-span (see Fig. 2) which 22

shows that the highest SPL for 132 m s .U 23

Submitte

d Vers

ion

30

Figures 7(a), (c) and (e) indicate that with an increase in the Aeolian tone, the longitudinal 1

and transverse size of the dipole focal spots decreases and a more compact orientation of the 2

focal points are obtained. In particular, Fig. 7(a) exhibits focal spots of a relatively larger size 3

(along the flow-wise direction) due to the occurrence of low-frequency Aeolian tone which 4

extend beyond the upstream length of LAs whilst in Figs. 7(c) and (e), the transverse span of the 5

focal spots extends up to a distance close to the upstream end of the LAs. For this reason, a 6

larger upstream domain length was used during TR simulations. The size of the dipole focal 7

spots is quantified in terms of the following two metrics; the transverse spatial resolution 8

(parallel to the LAs) and longitudinal spatial resolution (perpendicular to the LAs) and is taken 9

as Full-Width at Half-Maximum (FWHM) given by sum of the distances corresponding to 10

6 dB level considered on either side of a focal point29,30. The average transverse and 11

longitudinal focal-resolution of the dipole focal spots is quantified in the 2nd and 4th columns, 12

respectively, of Table III for different Aeolian tone frequency in terms of the wavelength a 13

corresponding to .af It is observed from Table III that the average transverse and longitudinal 14

size of the focal spots are nearly commensurate with ;a their average values are given by 15

1.03 a and 0.38 ,c respectively, for the given LA configuration. These average values of the 16

focal-resolution signify the conventional half-wavelength diffraction limit30,48 (at the source 17

vicinity) of the TR method. Furthermore, it is noted that dipole focal spots at 18

1208, 1584 Hzaf span a length along the flow-wise direction (i.e., the transverse size) 19

which includes the contraction-outlet flanges because the cylinder is mounted at pivot-hole 20

which is in proximity to the contraction-outlet. At 784 Hz,af the transverse size of the focal 21

spots is largest, however, its transverse span does not overlap the flanges because at such low-22

Submitte

d Vers

ion

31

frequency, the distance (along y direction) between the focal points is greater than the total 1

height (155 mm) of the contraction-outlet facility. 2

B. TR simulation with the modeling of contraction-outlet walls, flanges and cylinder 3

1. Spatio-temporal evolution of the time-reversed acoustic pressure field 4

Figure 8 shows the spatio-temporal evolution of the time-reversed acoustic pressure field 5

, ,p x y t obtained using the experimental acoustic pressure data (recorded at the two LAs) of 6

the flow-induced noise generated at the Aeolian tone 1584 Hzaf with the modeling of the 7

contraction-outlet geometry and the cylinder during TR simulation. (The reader is also referred 8

to Multimedia 2 which plays the corresponding TR simulation.) It is noted that the reverse time-9

instants shown in Figs. 8(a-f) are the same as those considered in Figs. 6(a-f), respectively. 10

Figures 8(a) and (b) are similar to Figs. 6(a) and (b), respectively. These time-snapshots 11

indicate a simultaneous back-propagation of acoustic fluxes from the two LAs that are about to 12

converge near the cylinder. Figure 8(c) shows that the waves back-propagating from the LAs 13

downstream of the contraction-outlet impinge and are about to be reflected from the flanges 14

unlike Fig. 6(c). It is noted that because the Aeolian tone wavelength 218 mma (at 15

1584 Hzaf ) is comparable to the total height of the contraction-outlet facility given by 155 16

mm (indicated earlier), the incident acoustic waves undergo a limited diffraction49 about the rigid 17

flanges. Due to limited acoustic fluxes diffracted towards the rigid-walls of the contraction-18

outlet, almost negligible reflection off its walls is observed in Fig. 8(d). Rather, instantaneous 19

local maxima regions are observed near the flanges in Fig. 8(d) due to interaction of the waves 20

reflected from the flanges and that incident towards it from the LAs downstream of the 21

contraction-outlet. Figure 8(e) shows the formation of two instantaneous focal spots 22

(downstream of the cylinder) and is similar to Fig. 6(e). Figure 8(f) is similar to Fig. 8(d) and 23

Submitte

d Vers

ion

32

indicates the formation of two instantaneous local maxima regions near the flanges at different 1

time-instants during TR simulation. 2

The spatio-temporal evolution pattern of the , ,p x y t field obtained (with modeling of 3

the contraction-outlet geometry and the cylinder during TR) at 1208, 784 Hzaf were 4

however, found to be significantly different from Fig. 8. This is explained as follows. At 5

784 Hz,af the acoustic waves radiated from the flow-induced dipole (at the cylinder location) 6

undergo a significant diffraction about the flanges because the corresponding Aeolian tone 7

wavelength 441 mma is significantly greater than the total height (155 mm) of the 8

contraction-outlet facility49. As a result, the upstream LAs intercept a significant amount of the 9

outgoing acoustic fluxes during experiments (as was explained through numerical simulation in 10

Fig. 4(b)). Therefore, during TR simulation, the back-propagated waves from the upstream LAs 11

also undergo a significant diffraction (about the flanges) which directs their propagation 12

downstream of the contraction-outlet. Furthermore, the low-frequency back-propagated waves 13

from the downstream LAs are also diffracted about the flanges and propagate towards the walls 14

of the contraction-outlet and the non-reflective upstream boundary. Therefore, due to interaction 15

of the waves continuously back-propagating from the upstream and downstream LAs over the 16

region downstream of the cylinder, two instantaneous focal-spots are formed throughout the TR 17

simulation. Indeed, the spatio-temporal evolution of the , ,p x y t field at 784 Hzaf obtained 18

without and with the modeling of rigid conditions is observed to be somewhat similar. 19

2. Analyzing the TR source maps: Effect of Aeolian tone frequency and implementation of the 20

rigid-wall conditions 21

Figures 7(b), (d) and (f) show the TR source maps due to the full-span circular cylinder 22

obtained by enforcing the time-reversed experimental acoustic pressure data (at the two LAs) 23

Submitte

d Vers

ion

33

corresponding to the Aeolian tone 784, 1208, 1584 Hz,af respectively, with the modeling of 1

the contraction-outlet geometry and the cylinder during TR simulations by making use of Eqs. 2

(21-23). Figure 7(b) is comparable to Fig. 7(a); however, location of the flow-induced dipole 3

source predicted in Fig. 7(b) is further downstream of the cylinder, thereby indicating a greater 4

error. Furthermore, Fig. 7(b) shows a significant wave reflection at the rigid-walls of the 5

contraction outlet due to the low-frequency waves diffracted about the flanges and incident on it 6

and is observed to be similar to Fig. 5(b) which shows the TR source map of the idealized dipole 7

of 0 750 Hzf obtained with the modeling of contraction-outlet walls and flanges. Figure 7(d) is 8

also observed to be comparable to Fig. 7(b); however, at this Aeolian tone wavelength 9

286 mm,a the diffraction effect is relatively less pronounced. As a result, due to reflection at 10

the flanges, localized region of maximum acoustic pressure (whose relative magnitude is slightly 11

smaller than the focal points) is formed near the flanges whilst the predicted location of the 12

dipole is relatively closer to the cylinder. Figure 7(f) is however, significantly different from its 13

counterpart Fig. 7(e). It exhibits two local maxima regions near the flanges (due to reasons noted 14

in the discussion of Fig. 8) which makes it difficult or nearly impossible to characterize the 15

dipole source nature and is indeed qualitatively similar to Fig. 5(d). 16

The 4th column of Table II indicates that incorporation of the rigid-wall model of the 17

contraction-outlet geometry and the cylinder (during TR) results in a greater error in the 18

predicted location of the flow-induced dipole source at the low-frequency Aeolian tone whilst 19

the error in prediction reduces at a higher frequency. Furthermore, the 5th column of Table II 20

indicates that the average SPL at the focal points also marginally increases on incorporation of 21

the rigid-wall model of the contraction-outlet geometry and the cylinder. The 3rd column of 22

Table III indicates a significant improvement in the average transverse focal-resolution of the 23

Submitte

d Vers

ion

34

dipole focal spots whilst the average longitudinal focal-resolution shown in 5th column of Table 1

III indicates only a marginal change when the rigid-wall modeling is implemented during TR. 2

In view of the formation of two local maxima regions (near the flanges) in Fig. 7(f) which 3

makes the characterization of the flow-induced dipole source impossible, the predicted source 4

location, average SPL and the average transverse and longitudinal focal-resolution corresponding 5

to 1584 Hzaf are not shown in Tables II and III, respectively. This demonstrates that when the 6

Aeolian tone wavelength 218 mma is comparable to the total height (155 mm) of the 7

contraction-outlet geometry and the cylinder in cross-flow (inducing dipole sources) is mounted 8

in proximity to the contraction-outlet opening, the rigid-wall modeling of the contraction-outlet 9

geometry during TR is counter-productive. 10

C. Effect of neglecting the convective effect of mean flow during TR simulation 11

The effect of neglecting the convective effect of mean flow during TR simulations on the 12

accuracy of the source localization is investigated in this section. To this end, the TR simulation 13

was implemented by considering a stationary medium, i.e., by setting the mean flow field 14

0 , 0U x y in Eqs. (1-3) and enforcing , , yp x y L t at nodes of the two LAs 15

corresponding to the Aeolian tone 1584 Hzaf whereby the source map shown in Fig. 9(a) is 16

obtained. It is observed that Fig. 9(a) is nearly identical to the corresponding TR source map 17

shown in Fig. 7(e) obtained by considering the mean flow (modeled by Eq. (8)) during 18

simulation. Indeed, the predicted location of the dipole source, the average SPL at the focal 19

points as well as the focal-resolution obtained in Fig. 9(a) is the same as that obtained in Fig. 7(e) 20

indicated in the 3rd row of Tables II and III. The only (minor) difference apparent in Figs. 7(e) 21

and 9(a) is that primary side-lobe levels are marginally lower in the former due to the 22

incorporation of the mean flow. This signifies that for such low Mach number flows when 23

Submitte

d Vers

ion

35

0 0.1M (approximately), incorporation of the mean flow in the governing LEE may be 1

altogether ignored during TR simulations without affecting the accuracy of source localization. 2

D. Comparison of source map obtained using one LA of microphones with that obtained 3

using the two LA configuration 4

The effect of using a limited angular aperture of the Time-Reversal Mirror (TRM) during 5

aeroacoustic TR simulations on the characterization of flow-induced noise source generated at 6

the Aeolian tone is examined. To this end, the TR simulations were implemented by enforcing 7

the time-reversed acoustic pressure data (obtained at the Aeolian tone 1584 Hzaf ) at only the 8

top LA28 and the corresponding TR source map is presented in Fig. 9(b). (It is noted that the 9

TRSL damping technique was not used at nodes near the bottom LA in this case.) Figure 9(b) 10

exhibits a large elongated focal spot28 which extends from the top LA boundary to the bottom 11

LA boundary resulting in a poor resolution of the flow-induced dipole source. The predicted 12

location of the source (obtained by determining the focal point) is given by 13

0 094 mm, 60 mm,x y thereby indicating a localization error equal to 0.34 .a The source 14

strength (at the focal point) is estimated to be 87.3 dB. Therefore, due to its limited angular 15

aperture, the use of only one LA (for intercepting/recording the radiated acoustic far-field) 16

during aeroacoustic TR simulations is insufficient for characterizing the dipole nature of the 17

flow-induced source and also results in a significant error in the predicted location28. Rather, as 18

shown earlier28,29, the use of two LAs located at the top and bottom boundaries on the opposite 19

sides of the cylinder records minimum boundary data (due to its relatively larger angular 20

aperture) required for accurately characterizing the lift-dipole nature of the flow-induced source 21

and localizing it within an average accuracy of 0.16 a (see Table II). Therefore, the present 22

results (Figs. 7(e) and 9(b)) experimentally corroborates the simulation based results of the 23

Submitte

d Vers

ion

36

characterization of an idealized dipole source located in a uniform flow field reported in a 1

previous work28. 2

VII. CB RESULTS AND COMPARISON WITH TR SOURCE MAPS 3

The source maps obtained by the CB method are analyzed and compared with the 4

corresponding TR source maps (obtained without modeling the contraction-outlet facility and the 5

cylinder) in this section with respect to the source characterization, accuracy of the predicted 6

location, strength and focal-resolution of the flow-induced dipole source. 7

A. Monopole steering vector 8

Figures 10(a), (c) and (e) present the CB source maps obtained using the Monopole 9

steering vector formulation (henceforth, termed as Monopole CB source maps) for Aeolian tone 10

frequency 784, 1208, 1584 Hz,af respectively. A pair of nearly identical focal spots located 11

in proximity in each of these source maps indicates the dipole source nature of the flow-induced 12

noise generated at the Aeolian tone due to a circular cylinder in cross-flow. It is observed that the 13

dipole focal spots in Figs. 10(a), (c) and (e) are similar to those exhibited by the corresponding 14

TR source maps shown in Figs. 7(a), (c) and (e), respectively. Indeed, the only noticeable 15

difference between the Monopole CB and the TR source maps is that side-lobe levels are 16

significantly lower in the former. 17

The predicted location of the dipole source, error (expressed as a fraction of the 18

corresponding Aeolian tone wavelength) and the average SPL observed at the focal points in the 19

Monopole CB source maps are shown in the 2nd and 3rd columns, respectively, of Table IV. An 20

average error of 0.15 a is observed in the predicted location of the dipole whilst the average 21

SPL (at the focal point) increases with an increase in the free-stream velocity .U The 2nd and 3rd 22

columns of Tables II and IV indicate that the error in the predicted location as well as the 23

Submitte

d Vers

ion

37

average SPL obtained (at different Aeolian tone frequencies) by the TR and the Monopole CB 1

methods, respectively, are comparable. This similarity is also highlighted in Table V which 2

compares the transverse and longitudinal focal-resolution of the dipole focal spots obtained by 3

the TR and Monopole CB. It is observed from Table V that the average transverse and 4

longitudinal focal-resolution of the focal spots obtained by the TR method is given by 1.03 a 5

and 0.38 ,a respectively, whilst those obtained by the Monopole CB method are given by 6

0.93 a and 0.38 ,a respectively, thereby indicating a similar focal spot size at a given Aeolian 7

tone frequency. 8

In light of the foregoing quantitative comparisons of the source characteristics, accuracy of 9

predicted location, source strength and focal-resolution, it is therefore, concluded that both TR 10

and Monopole CB method yield similar source maps for a given LA configuration. 11

B. Dipole steering vector 12

Figures 10(b), (d) and (f) present the CB source maps obtained using the Dipole steering 13

vector formulation (henceforth, termed as Dipole CB source maps) for Aeolian tone frequency 14

784, 1208, 1584 Hz,af respectively. Unlike the TR and Monopole CB source maps, the 15

Dipole CB source maps exhibit a central focal spot (representing the flow-induced dipole source) 16

flanked by side-lobes on either sides. The central focal spot is of an elongated shape oriented 17

along the x direction and its size decreases with an increase in the Aeolian tone frequency. The 18

predicted location of the dipole source obtained from the Dipole CB method (taken as the focal 19

point of the central focal spot) and the error in predicted location is shown in the 4th column of 20

Table IV whilst the average source strength (at the focal point) is tabulated in the 5th column. An 21

average error of 0.13 a is observed in the predicted location which is comparable to that 22

obtained by the Monopole CB and the TR method. Furthermore, the average SPL at source 23

Submitte

d Vers

ion

38

location at different Aeolian tone frequencies obtained by the Dipole CB are slightly higher than 1

those predicted by the Monopole CB and about 6 dB higher than those predicted by TR method. 2

The transverse and longitudinal focal-resolution of the central focal spot exhibited in the 3

Dipole CB source maps is shown in the 4th and 7th columns of Table V wherein it is observed that 4

the size of focal spots is nearly commensurate with respect to the Aeolian tone frequencies 5

784, 1208, 1584 Hz.af The average transverse and longitudinal focal-resolution of the 6

central focal spot is given by 0.94 a and 0.36 ,a respectively, thereby indicating that the Dipole 7

CB method yields a focal spot whose size is comparable to the focal spot size obtained by the 8

Monopole CB and the TR method. 9

VIII. CONCLUSIONS 10

This paper has demonstrated, for the first time, an experimental application of aeroacoustic 11

Time-Reversal (TR) for a benchmark test-case of a full-span circular cylinder31-34 located in 12

subsonic cross-flow. The spatio-temporal evolution of , ,p x y t field and the corresponding TR 13

source map obtained using the experimental acoustic pressure data recorded at top and bottom 14

microphone Line Arrays (LAs) (without modeling the contraction-outlet and the cylinder) 15

indicate the lift-dipole nature of the aeroacoustic source generated at the Aeolian tone31-38 16

frequency. This observation is consistent with the classically known result that the interaction of 17

a rigid body (typically, smaller than or comparable to a wavelength) with a flowing medium 18

induces local surface stresses which are of equal magnitude but act in opposite direction to the 19

surrounding fluid resulting in a dipole source nature33,40. The TR source maps indicate a small 20

error, approximately 0.15 a in the predicted location of the dipole which is attributed to the 21

partial angular aperture of the two LAs or possibly, the use of the computationally simpler 2-D 22

Linearized Euler Equation (LEE) solver that under-represents the experimental noise generation 23

Submitte

d Vers

ion

39

phenomenon, especially at low Aeolian tone frequency. The average transverse and longitudinal 1

TR focal-resolution of the flow-induced dipole focal spots given by c and 0.4 ,c 2

respectively, signifies the half-wavelength diffraction limit of the TR method30,48. 3

It is shown that modeling the contraction-outlet and the cylinder during TR simulations 4

marginally improves the transverse focal-resolution of the lift-dipole (given by 0.83 a ) and also 5

reduces the side-lobe levels in the source map corresponding to the Aeolian tone 6

784, 1208 Hzaf occurring in the low-frequency range; however, it yields a larger error in 7

the predicted location. For the Aeolian tone 1584 Hzaf occurring at a higher frequency (when 8

the wavelength of back-propagated acoustic waves becomes comparable to the total height of the 9

contraction-outlet), incorporation of the contraction-outlet geometry and the cylinder is shown to 10

be counter-productive because it leads to the formation of localized maxima regions near the 11

flanges of the contraction-outlet, thereby making it difficult to characterize the dipole source 12

nature. It is therefore concluded that modeling of the facility geometry and full-span cylinder 13

test-model during TR simulations is not necessary to improve the characterization of dipole 14

source generated at the Aeolian tone in this case. However, the use of 3-D TR simulation is 15

likely to improve the source resolution further. 16

The experimental TR source maps for the Aeolian tone obtained with/without modeling the 17

contraction-outlet and the cylinder were found to be similar to the TR source map of an idealized 18

dipole source placed at the cylinder location (during forward simulations) in the wind-tunnel 19

flow for the counterpart cases, thereby validating the aeroacoustic TR simulation for 20

localizing/characterizing flow-induced dipole sources. 21

The source maps obtained by the TR simulation implemented without modeling the 22

contraction-outlet and the cylinder were shown to be comparable to those obtained using 23

Submitte

d Vers

ion

40

Monopole and Dipole Conventional Beamforming (CB) in terms of accuracy of the predicted 1

location, source strength as well as the transverse and longitudinal focal-resolution of dipole 2

focal spots (see Tables II, IV and V). Furthermore, the shape, size and orientation of the dipole 3

focal spots in the TR source maps were observed to be nearly identical to those of the 4

corresponding Monopole CB source maps for the given LA configuration. These results, 5

therefore, demonstrate the suitability of the TR technique as a diagnostic tool to study/analyze 6

experimental flow-induced noise generation mechanism36-39 and as an alternate approach to the 7

more popularly used CB method1. 8

ACKNOWLEDGMENTS 9

The authors would like to thank Mr. Ric Porteous for providing experimental data of the 10

mean flow velocity and acknowledge the support of Australian Research Council (ARC) for this 11

work through grant DP 120102134 “Resolving the mechanics of turbulent noise production.” 12

REFERENCES 13 14 1D. H. Johnson and D. E. Dudgeon, Array Signal Processing (Prentice Hall, Englewood Cliffs, New 15 Jersey, 1993), pp. 1-512. 16 2T. F. Brooks and W. M. Humphreys, “A deconvolution approach for the mapping of acoustic sources 17 (DAMAS) determined from phased microphone arrays,” J. Sound Vib. 294, 856-879 (2006). 18 3T. Suzuki, “Identification of multipole noise sources in low Mach number jets near the peak frequency,” 19 J. Acoust. Soc. Am. 119, 3649-3659 (2006). 20 4W. Dobrzynski, R. Ewert, M. Pott-Pollenske, M. Herr, and J. Delfs, “Research at DLR towards airframe 21 noise prediction and reduction,” Aerosp. Sci. Technol. 12, 80-90 (2008). 22 5J. Hald, Y. Ishii, T. Ishii, H. Oinuma, K. Nagai, Y. Yokokawa, and K. Yamamoto, “High-resolution fly- 23 over beamforming using a small practical array,” in 18th AIAA/CEAS Aeroacoustics Conference, 24 AIAA Paper 2229, Colorado, USA (2012), pp. 1-16. 25 6S. Oerlemanns, M. Fisher, T. Maeder, and K. Kgler, “Reduction of wind turbine noise using optimized 26 airfoils and trailing-edge serrations,” AIAA J. 47, 1470-1481 (2009). 27 7I. Rakotoarisoa, J. Fischer, V. Valeau, D. Marx, C. Prax, and L.-E. Brizzi, “Time-domain delay-and- 28 sum beamforming for time-reversal detection of intermittent acoustic sources in flows,” J. Acoust. Soc. 29 Am. 136, 2675–2686 (2014). 30 8E. J. G. Arcondoulis, C. J. Doolan, A. C. Zander, and L. A. Brooks, “Design and calibration of a small 31 aeroacoustic beamformer,” in 20th International Congress on Acoustics, ICA 2010, Paper 453, Sydney, 32 Australia (2010), pp. 1-8. 33 9M. Fink, D. Cassereau, A. Derode, , C. Prada, P. Roux, M. Tanter, J. L. Thomas, and F. Wu, “Time- 34 reversed acoustics,” Rep. Prog. Phys. 63, 1933-1995 (2000). 35 10T. Shimura, Y. Watanabe, H. Ochi and H. C. Song, “Long-range time reversal communication in deep 36

Submitte

d Vers

ion

41

water: Experimental results,” J. Acoust. Soc. Am. 132, EL49-EL53 (2012). 1 11P. Roux and M. Fink, “Time reversal in a waveguide: Study of the temporal and spatial focusing,” 2 J. Acoust. Soc. Am. 107, 2418-2429 (2000). 3 12H. W. Park, H. Sohn, K. H. Law and C. R. Farrar, “Time reversal active sensing for health monitoring 4 of a composite plate,” J. Sound Vib. 302, 50-66 (2007). 5 13D. Vigoureux and J.-L. Guyader, “A simplified time reversal method used to localize vibrations sources 6 in a complex structure,” Appl. Acoust. 73, 491-496 (2012). 7 14D. G. Albert, L. Liu, and M. L. Moran, “Time reversal processing for source location in an urban 8 environment,” J. Acoust. Soc. Am. 118, 616-619 (2005). 9 15A. Przadka, S. Feat, P. Petitjeans, V. Pagneux, A. Maurel, and M. Fink, “Time reversal of water waves,” 10 Phys. Rev. Lett. 109, 064501 (2012). 11 16G. Lerosey, J. D. Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of 12 electromagnetic waves,” Phys. Rev. Lett. 92, 193904 (2004). 13 17A. O. Maksimov and Yu. A. Polovinka, “Time reversal technique for gas leakage detection,” J. Acoust. 14 Soc. Am. 137, 2168–2179 (2015). 15 18C. Hudin, J. Lozada, and V. Hayward, “Spatial, temporal, and thermal contributions to focusing 16 contrast by time reversal in a cavity,” J. Sound Vib. 333, 1818-1832 (2014). 17 19A. Derode, M. Tanter, A. Tourin, L. Sandrin, and M. Fink, “Numerical and experimental time-reversal 18 of acoustic waves in random media,” J. Comput. Acoust. 9, 993-1003 (2001). 19 20A. Derode, A. Tourin, and M. Fink, “Limits of time-reversal focusing through multiple scattering: 20 Long-range correlation,” J. Acoust. Soc. Am. 107, 2987-2998 (2000). 21 21J.-M. Parot, “Localizing impulse sources in an open space by time reversal with very few transducers,” 22 Appl. Acoust. 69, 311–324 (2008). 23 22A. Mimani, C. J. Doolan, and P. R. Medwell, “Aeroacoustic time-reversal in the presence of a reflecting 24 surface,” in 43rd International Congress on Noise Control Engineering, Internoise 2014, Paper 96, 25 Melbourne, Australia (2014), pp. 1-10. 26 23G. Ribay, J. de Rosny, and M. Fink, “Time reversal of noise sources in a reverberation room,” J. 27 Acoust. Soc. Am. 117, 2866–2872 (2005). 28 24C. Draeger and M. Fink, “One-channel time reversal in chaotic cavities: Theoretical limits,” J. Acoust. 29 Soc. Am. 105, 611–617 (1999). 30 25C. Draeger, J-C Aime, and M. Fink, “One-channel time reversal in chaotic cavities: Experimental 31 results,” J. Acoust. Soc. Am. 105, 618–625 (1999). 32 26A. Deneuve, P. Druault, R. Marchiano, and P. Sagaut, “A coupled time-reversal/complex differentiation 33 method for aeroacoustic sensitivity analysis: towards a source detection procedure,” J. Fluid Mech. 642, 34 181-212 (2010) 35 27T. Padois, C. Prax, V. Valeau, and D. Marx, “Experimental localization of an acoustic source in a wind- 36 tunnel flow by using a numerical time-reversal technique,” J. Acoust. Soc. Am. 132, 2397-2407 (2012). 37 28A. Mimani. C. J. Doolan, and P. R. Medwell, “Multiple line arrays for the characterization of 38 aeroacoustic sources using a time-reversal method,” J. Acoust. Soc. Am. 134, EL327-EL333 (2013). 39 29A. Mimani, Z. Prime, C. J. Doolan, and P. R. Medwell, “A sponge-layer damping technique for 40 aeroacoustic time-reversal,” J. Sound Vib. 342, 124-151 (2015). 41 30A. Mimani, C. J. Doolan, and P. R. Medwell, “Enhancing the focal-resolution of aeroacoustic time- 42 reversal using a point-sponge-layer damping technique,” J. Acoust. Soc. Am. 136, EL199-EL205 43 (2014). 44 31C. Norberg, “Fluctuating lift on a circular cylinder: Review and new measurements,” J. Fluid Struct. 45 17, 57-96 (2003). 46 32O. Phillips, “The intensity of Aeolian tones,” J. Fluid Mech. 1, 607-624 (1956). 47 33W. K. Blake, Mechanics of Flow-Induced Sound and Vibration, Vol. I: General Concepts and 48 Elementary Sources (Academic Press, New York, 1986), Chaps. 2 and 4. 49 34C. H. K. Williamson, “Vortex dynamics in the cylinder wake,” Annu. Rev. Fluid Mech. 28, 477-539 50 (1996). 51

Submitte

d Vers

ion

42

35C. Cheong, P. Joseph, Y. Park, and S. Lee, “Computation of aeolian tone from a circular cylinder using 1 source models,” Appl. Acoust. 69, 110-126 (2008). 2 36S. Becker, C. Hahn, M. Kaltenbacher, and R. Lerch, “Flow-induced sound of wall-mounted cylinders 3 with different geometries,” AIAA J. 46, 2265-2281 (2008). 4 37R. Porteous, D. J. Moreau, and C. J. Doolan, “A review of flow-induced noise from finite wall- 5 mounted cylinders,” J. Fluid Struct. 51, 240-254 (2014). 6 38D. J. Moreau and C. J. Doolan, “Flow-induced sound of wall-mounted finite length cylinders,” AIAA J. 7 51, 2493-2502 (2013). 8 39D. J. Moreau, L. A. Brooks, and C. J. Doolan, “Broadband trailing edge noise from a sharp-edged 9 strut,” J. Acoust. Soc. Am. 129, 2820-2829 (2011). 10 40N. Curle, “The influence of solid boundaries upon aerodynamic sound,” Proc. R. Soc. London A231, 11 505-514 (1955). 12 41J. Sesterhenn, “A characteristic-type formulation of the Navier-Stokes equations for high order upwind 13 schemes,” Comput. Fluids 30, 37-67 (2001). 14 42M. Zhuang and R. F. Chen, “Applications of high-order optimized upwind schemes for computational 15 aeroacoustics,” AIAA J. 40, 443-449 (2002). 16 43C. K. W. Tam, “Computational aeroacoustics: Issues and methods,” AIAA J. 33, 1788-1796 (1995). 17 44K.E. Atkinson, An Introduction to Numerical Analysis (Wiley, Singapore, 2004), Chap. 3, pp. 131-150. 18 45S. Gottlieb and C-W. Shu, “Total variation diminishing Runge-Kutta schemes,” Math. Comput. 67, 19 73-85 (1998). 20 46R. Clayton, B. Engquist, “Absorbing boundary conditions for acoustic and elastic wave equations,” 21 Bull. Seism. Soc. Am. 67, 1529-1540 (1977). 22 47B. Engquist and A. Majda, “Radiation boundary conditions for acoustic and elastic wave calculations,” 23 Comm. Pure Appl. Math. 32, 313–357 (1979). 24 48J. de Rosny and M. Fink, “Overcoming the diffraction limit in wave physics using a time-reversal 25 mirror and a novel acoustic sink,” Phys. Rev. Lett. 89, 124301 (2002). 26 49P. M. Morse and K. U. Ingard, Theoretical Acoustics (Princeton University Press, New Jersey, 1986), 27 Chap. 8, pp. 449-463. 28 50P. D. Welch, “The use of fast fourier transform for the estimation of power spectra: A method based 29 on time averaging over short, modified periodograms,” IEEE T. Acoust. Speech. 15, 70-73 (1967). 30 51E. Sarradj, “Three-dimensional acoustic source mapping with different beamforming steering vector 31 formulations,” Adv. Acoust. Vib. 2012, 292695 (2012). 32 52Y. Liu, A. R. Quayle, A. P. Dowling, and P. Sijtsma, “Beamforming correction for dipole 33 measurement using two-dimensional microphone arrays,” J. Acoust. Soc. Am. 124, 182-191 (2008). 34 53R. P. Dougherty, “Beamforming in acoustic testing,” in Aeroacoustic Measurements, edited by T. J. 35 Muller (Springer, Berlin, 2002), Chap. 2, pp. 62-97. 36 54T. Padois, C. Prax, and V. Valeau, “Numerical validation of shear flow corrections for beamforming 37 acoustic source localisation in open wind-tunnels,” Appl. Acoust. 74, 591-601 (2013). 38 55A. Mimani, C. J. Doolan, and P. R. Medwell, “Enhancing the resolution characteristics of aeroacoustic 39 time-reversal using a point-time-reversal-sponge-layer,” in 20th AIAA/CEAS Aeroacoustics Conference, 40 AIAA Paper 2316, Atlanta, USA (2014), pp. 1-37. 41

42 43 44 45 46 47 48 49 50

Submitte

d Vers

ion

43

TABLES 1 2 3

TABLE I Comparison of the predicted location of the idealized dipole source of tonal 4 frequencies 0 750, 1500, 3000 Hzf and error in prediction obtained by the TR method 5 (implemented using simulated data recorded at the top and bottom boundaries during forward 6 simulations) without modeling the contraction-outlet with those obtained with their modeling 7 during TR. 8 9

10

TR without modeling the contraction-outlet

TR with the modeling of contraction-outlet Tonal

frequency 0f Predicted location

0 0,x y (in mm)

Error

Predicted location 0 0,x y (in mm)

Error

750 Hz 64, 0 00.02 124, 0 00.15

1500 Hz 44, 0 00.04

3000 Hz 54, 0 0 64, 0 00.09

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Submitte

d Vers

ion

44

TABLE II Comparison of the predicted location of the dipole source generated at different 1 Aeolian tone corresponding to 116, 24, 32 m s ,U

error in prediction and the average 2 source strength (at the dipole focal points) obtained using the TR method (considering a full-3 length of the two LAs) without modeling the contraction-outlet and the cylinder with those 4 obtained with their modeling during TR . 5 6

7

TR without modeling the contraction-outlet and the cylinder

TR with the modeling of contraction-outlet and the cylinder

Aeolian tone

frequency af

Predicted location 0 0,x y (in mm)

and Error

Average

SPL at focal

points

(in dB)

Predicted location 0 0,x y (in mm)

and Error

Average SPL

at focal

points

(in dB)

784 Hz 134, 10 and 0.19 a 61.2 154, 10 and 0.24 a 62.0

1208 Hz 94, 5 and 0.16 a 80.1 84, 0 and 0.12 a 80.4

1584 Hz 74, 10 and 0.12 a 91.3

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

31

Submitte

d Vers

ion

45

TABLE III Comparison of the transverse and longitudinal focal-resolution of the dipole focal 1 spots generated at the Aeolian tone corresponding to 116, 24, 32 m sU

obtained using TR 2 simulations without modeling the contraction-outlet and the cylinder with those obtained their 3 modeling by considering the full length of the two LAs. 4 5

6

Size of the focal spots (relative to 6 dB) Transverse resolution

(Parallel to the LAs) Longitudinal resolution

(Perpendicular to the LAs) Aeolian tone

frequency af

Without modeling the contraction-

outlet and the cylinder

With the modeling of contraction-

outlet and the cylinder

Without modeling the contraction-

outlet and the cylinder

With the modeling of contraction-

outlet and the cylinder

784 1.05 a 0.79 a 0.38 a 0.41 a

1208 1.04 a 0.87 a 0.39 a 0.39 a

1584 1.01 a 0.37 a

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Submitte

d Vers

ion

46

TABLE IV Predicted location of the dipole source generated at different Aeolian tone 1 corresponding to 116, 24, 32 m s ,U

error in prediction and the average source strength (at 2 the dipole focal points) obtained using Monopole and Dipole CB. 3 4 5 6

Monopole steering vector CB

Dipole steering vector CB Aeolian

tone frequency

af Predicted location 0 0,x y (in mm)

and Error

Average

SPL at

focal points

(in dB)

Predicted location 0 0,x y (in mm)

and Error

Average

SPL at

focal points

(in dB)

784 Hz 109.0, 11.3 and 0.14 a 64.0 101.5, 7.5 and 0.12 a 66.9

1208 Hz 104, 3.75 and 0.19 a 83.6 99, 2.5 and 0.17 a 85.6

1584 Hz 75.3, 7.5 and 0.12 a 96.9 71.5, 7.5 and 0.1 a 98.9

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Submitte

d Vers

ion

47

TABLE V Comparison of the transverse and longitudinal focal-resolution of the dipole focal 1 spots generated at the Aeolian tone corresponding to 116, 24, 32 m sU

obtained using the 2 TR method (without modeling the contraction-outlet and the cylinder) with those obtained using 3 Monopole and Dipole CB. 4 5

6

Size of the focal spots (relative to 6 dB) Transverse resolution

(Parallel to the LAs) Longitudinal resolution

(Perpendicular to the LAs)

Aeolian tone

frequency af TR Monopole CB Dipole CB TR Monopole CB Dipole CB

784 1.05 a 0.94 a 0.93 a 0.38 a 0.36 a 0.35 a

1208 1.04 a 0.89 a 0.90 a 0.39 a 0.41 a 0.37 a

1584 1.01 a 0.95 a 0.98 a 0.37 a 0.36 a 0.36 a

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

Submitte

d Vers

ion

48

FIGURE CAPTIONS 1 2

FIG. 1. (Color Online) (a) Photograph of the experimental set-up: A full-span 4 mm circular 3 cylinder held in a two-sided mounting frame attached to the contraction-outlet in the AWT and 4 two LAs of microphones for recording far-field acoustic pressure. Schematic diagrams (not 5 shown to scale) depicting the full-span cylinder secured in the mounting frame attached to the 6 contraction-outlet depicting the (b) Side-view and (c) Front-view (showing the position of the 7 two LAs of microphones). 8 9 FIG. 2. (Color Online) Acoustic spectrum of the cylinder at flow speed (a) 132 m s ,U

10 (b) 124 m sU

and (c) 116 m sU measured at the microphone located in the bottom LA 11

and positioned 24 mm downstream of the contraction-outlet opening (taken as the origin). 12 13 FIG. 3. (Color Online) Spatially-developing (evolving) non-uniform shear mean flow profile 14 modeling a subsonic free-jet issuing out of the contraction-outlet in the AWT with the free-15 stream velocity 132 m sU

at the contraction-outlet opening. The top-hat profile occurring at 16 and near the contraction-outlet opening 0x gradually evolves within the mixing region 17 wherein the potential-core collapses and the jet rapidly spreads into the ambient stationary fluid 18 producing a developing free-shear layer. 19 20 FIG. 4. (Color Online) RMS plot depicting the directivity of the radiated acoustic pressure field 21 due to idealized dipole of tonal frequencies: 0 750 Hzf in (a) and (b), 0 1500 Hzf in (c) and 22 (d), 0 3000 Hzf in (e) and (f) simulated through excitation by a point force along the y 23 direction at 0 054 mm, 0x y (almost co-incident with the cylinder location) in the spatially-24 developing shear mean flow with 132 m sU

(shown in Fig. 3). Parts (a, c and e) are obtained 25 without modeling the contraction-outlet walls and flanges whilst the parts (b, d and f) are 26 obtained with their modeling. 27 28 FIG. 5. (Color Online) TR source maps of the idealized dipole source of tonal frequencies: 29

0 750 Hzf in (a) and (b), 0 1500 Hzf in (c) and (d), 0 3000 Hzf in (e) and (f) obtained 30 using time-reversed acoustic pressure history (recorded during forward simulation) at the top and 31 bottom LAs. Parts (a, c and e) are obtained without modeling the contraction-outlet walls and 32 flanges whilst the parts (b, d and f) are obtained with their modeling. The idealized dipole was 33 simulated through excitation by a point force along the y direction at 0 054 mm, 0x y (almost 34 co-incident with the cylinder location) and implementation of rigid-wall conditions due to 35 contraction-outlet walls and flanges. 36 37 FIG. 6. (Color Online) Spatio-temporal evolution of the time-reversed acoustic pressure field 38 , ,p x y t at the Aeolian tone corresponding to 132 m sU

obtained using the experimental 39

signals recorded at the top and bottom LAs (filtered in the rd1 3 octave band with 1600 Hzcf ) 40 without modeling the contraction-outlet walls, flanges and the cylinder at reverse time-instants: 41 (a) 20 ,t t (b) 60 ,t t (c) 100 ,t t (d) 125 ,t t (e) 400t t and (f) 800 .t t 42

Submitte

d Vers

ion

49

FIG. 7. (Color Online) Experimental TR source maps of the lift-dipole source generated at the 1 Aeolian tone: 784 Hzaf corresponding to 116 m sU

in (a) and (b), 1208 Hzaf 2 corresponding to 124 m sU

in (c) and (d), 1584 Hzaf corresponding to 132 m sU 3

in (e) and (f) obtained using the time-reversed signals at the top and bottom LAs. Parts (a, c and 4 e) are obtained without modeling the contraction-outlet walls, flanges and the cylinder whilst the 5 parts (b, d and f) are obtained with their modeling. 6 7 FIG. 8. (Color Online) Spatio-temporal evolution of the time-reversed acoustic pressure field 8 , ,p x y t at the Aeolian tone corresponding to 132 m sU

obtained using the experimental 9

signals recorded at the top and bottom LAs (filtered in the rd1 3 octave band with 1600 Hzcf ) 10 with the implementation of rigid-wall conditions modeling the contraction-outlet walls, flanges 11 and the cylinder at reverse time-instants: (a) 20 ,t t (b) 60 ,t t (c) 100 ,t t (d) 12

125 ,t t (e) 400t t and (f) 800 .t t 13 14 FIG. 9. (Color Online) Experimental TR source map of the lift-dipole source generated at the 15 Aeolian tone 1584 Hzaf obtained using (a) time-reversed signals at the top and bottom LAs 16 by considering a stationary medium (zero mean flow), i.e., ignoring the convective effect of 17 spatially developing shear mean flow during TR simulation and (b) time-reversed signals at only 18 the top LA but considering the spatially developing shear mean flow during TR simulation. 19 20 FIG. 10. (Color Online) Experimental source maps of the lift-dipole source generated at the 21 Aeolian tone corresponding to (a) 116 m s ,U

(c) 124 m sU and (e) 132 m sU

22 obtained using monopole steering vector beamforming. Experimental source maps of the lift-23 dipole source generated at the Aeolian tone corresponding to (b) 116 m s ,U

24 (d) 124 m sU

and (f) 132 m sU obtained using dipole steering vector beamforming. 25

26

Submitte

d Vers

ion