Enhancing the focal-resolution of aeroacoustictime-reversal using a point sponge-layer damping

techniqueA. Mimani,a) C. J. Doolan, and P. R. Medwell

School of Mechanical Engineering, The University of Adelaide, Adelaide, South Australia5005, Australia

akhilesh.mimani@adelaide.edu.au, con.doolan@adelaide.edu.au,paul.medwell@adelaide.edu.au

Abstract: This letter presents the Point-Time-Reversal-Sponge-Layer(PTRSL) technique to enhance the focal-resolution of aeroacousticTime-Reversal (TR). A PTRSL is implemented on a square domaincentered at the predicted source location and is based on damping theradial components of the incoming and outgoing fluxes propagating to-ward and away from the source, respectively. A PTRSL is shown toovercome the conventional half-wavelength diffraction-limit; its imple-mentation significantly reduces the focal spot size to one-fifth of a wave-length for a monopole source. Furthermore, PTRSL reduces the focalspots of a dipole source to three-tenths of a wavelength, as compared tothree-fifths without its implementation.VC 2014 Acoustical Society of AmericaPACS numbers: 43.60.Tj, 43.60.Jn, 43.60.Fg, 43.20.Rz [CG]Date Received: April 17, 2014 Date Accepted: June 30, 2014

1. Introduction

Acoustic Time-Reversal (TR) is a robust method to accurately localize sound sources.1

However, during TR, the converging and diverging wave fronts interfere locally in thesource vicinity2,3 (due to energy-conservation4), and as a result, the size of the focalspot (yielding the source location) is at least a half-wavelength, even if the source ispoint-like, thereby limiting the TR resolution.2,3 Rosny and Fink2 implemented a time-reversed source (acoustic sink) during TR on a chaotic glass-plate cavity, to overcomethe half-wavelength diffraction limit. The sink was implemented at the predicted sourcelocation (obtained from the first-TR step) and emits outgoing waves that undergo a de-structive interference with the wave fronts converging at the source from the far-fieldTime-Reversal Mirror (TRM), thereby leading to an active cancellation. The imple-mentation of the sink reconstructed the near-field evanescent modes in the time-reversed acoustic field resulting in a sharp focal spot of considerably diminished size ofless than k/14, where k is the source wavelength. Conti et al.5 presented an algorithmbased on amplifying the near-field evanescent components termed as the Near-FieldTime-Reversal to obtain a sub-wavelength focusing of resolution k/20 in the audiblerange. Anderson et al.6 developed an energy current imaging method to enhance theresolution of sources in an elastic reconstruction space.

The implementation of a sink certainly enhances the focal-resolution,2,3 how-ever, an accurate a priori estimate of its strength, phase, and characteristics (monopoleor multipole nature) is important for effectively suppressing the diverging wave frontsfrom the TRM at the source vicinity. Since, TR is implemented using homogenous gov-erning equations,4,7,8 it is impossible to determine the strength and phase of the sink,rather only the location/characteristics of the aeroacoustic source can be predicted.8

a)Author to whom correspondence should be addressed.

J. Acoust. Soc. Am. 136 (3), September 2014 VC 2014 Acoustical Society of America EL199

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Furthermore, a TRM configuration completely enclosing the source or located in thenear-field5 is difficult in an aeroacoustics experiment conducted in a wind tunnel becausethe microphone Line Arrays (LAs) must be located outside the flow.4,8 In light of thesechallenges, this Letter presents a fundamentally new and an alternate two-step methodthat does not implement an active cancellation, rather it employs a passive radial damp-ing technique termed as the Point-Time-Reversal-Sponge-Layer (PTRSL) that mimics asink and requires a priori knowledge of the location and nature of the source8 (obtainedfrom first-TR step) to enhance the focal-resolution of aeroacoustic TR.

2. Methodology: Implementation of the point sponge-layer damping technique

The TR simulation was implemented by numerically solving8 the following set of 2-DLinearized Euler Equations9 (LEE) using the Pseudo-Characteristic Formulation7,10

(PCF) in reverse time ~t on a rectangular domain jyj � 0:5 m, jxj � 1:5 m with anechoicboundaries and enforcing time-reversed acoustic pressure ~pðx; y;~tÞ at the two LAs8

located at y¼6 0.5 m boundaries,

@~p=@~t ¼ �ðq0c0=2Þ ~Xþlinear þ ~X

�linear

� �þ ~Y

þlinear þ ~Y

�linear

� �n o; (1a)

@~u=@~t ¼ � 12

~Xþlinear � ~X

�linear

� �� ~v �@U0=@yð Þ; (1b)

@~v=@~t ¼ � 12

~Yþlinear � ~Y

�linear

� �� �U0ð Þ@~v=@x; (1c)

where

~X6

linear ¼ 6 c07U0 yð Þ� � 1

q0c0

@~p@x

6@~u@x

� �; ~Y

6

linear ¼ 6c01

q0c0

@~p@y

6@~v@y

� �: (2)

In Eqs. (1a)–(1c), ~u and ~v denote acoustic velocities along the x and y directions, respec-tively, �U0 represents the reversed mean flow toward the negative x direction, ~X

6

linear and~Y

6

linear denote a pair of opposing fluxes7,10 propagating toward the x and y directions, respec-tively, while the sound speed c0 ¼ 343:14 m s–1 and the ambient density q0 ¼ 1:21 kg m–3.The domain was divided into equally spaced nodes of mesh size Dx¼Dy¼ 0.005 m. Thesource location is predicted by determining the region(s) of maximum magnitude (termed asfocal spots) in the root-mean-square (RMS) time-reversed acoustic pressure field8 denotedby ~pTR

RMSðx; yÞ. The node at which the focal spot is maximum is termed the focal point. The~pTR

RMSðx; yÞ field is converted to dB scale (with respect to pref ¼ 2� 10�5 Pa), and the sourcemap denoted by ~pTR

dB ðx; yÞ is expressed relative to the focal point(s). The TR simulationswere carried out for a second-time wherein the Point-Time-Reversal-Sponge-Layer(PTRSL) damping was implemented (schematically shown in Fig. 1) over a small square do-main jx� x0j � nxDx, jy� y0j � nyDy centered at the predicted location (x0, y0) denoted byS obtained from the first-TR step. The underlying theory of a PTRSL is to damp the incom-ing and outgoing radial fluxes ~X

7

r , respectively, propagating toward and away from the pre-dicted location, by multiplying them with a smoothly varying 2-D Gaussian function givenby gPTRSLðx; yÞ ¼ 1� e–aPTRSLfðx�x0Þ2þðy�y0Þ2g that decays to 0 at S while gradually increas-ing to unity toward edges of the square domain. Here, aPTRSL is the damping-coefficientthat determines the steepness of the Gaussian function, thereby controlling the effectivedamping domain. It is noted that the angular fluxes ~X

6

h propagating along the counter-clockwise and clockwise directions, respectively, are however, undamped. The PTRSLdamping is implemented at all nodes of each of the four quadrants and an analytical deriva-tion of the transformations to be introduced in fluxes of the PCF is briefly presented. To thisend, a point P located on a node in the first quadrant (shown in Fig. 1) is considered. Theincluded angle h formed between the line joining P with S and the positive x axis varies from

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h¼ 0 to h¼ p/2 radian. The fluxes ~Xþr and ~X

þh at P are computed from the fluxes ~X

þlinear

and ~Yþlinear according to the following relations:

~Xþr ¼ ~X

þlinear cos hþ ~Y

þlinear sin h; (3a)

~Xþh ¼ � ~X

þlinear sin hþ ~Y

þlinear cos h: (3b)

The damped flux ~Xþr � gPTRSL and the undamped flux ~X

þh are re-projected

along the positive x and y directions to yield the following transformations in the~Xþlinear and ~Y

þlinear fluxes,

~Xþlinear!gPTRSLðxP;yPÞ ~X

þr cosh� ~X

þh sinh (4a)

¼ ~Xþlinear�ðgPTRSLðxP;yPÞcos2hþsin2hÞþðgPTRSLðxP;yPÞ�1Þ� ~Y

þlinear coshsinh (4b)

� ~Xþlinear�ðgPTRSLðxP;yPÞcos2hþsin2hÞ (4c)

¼ ~Xþlinear�GX

PTRSL; (4d)

~Yþlinear!gPTRSLðxP;yPÞ ~X

þr sinh� ~X

þh cosh (5a)

¼ ~Yþlinear�ðgPTRSLðxP;yPÞsin2hþcos2hÞþðgPTRSLðxP;yPÞ�1Þ� ~X

þlinear coshsinh (5b)

� ~Yþlinear�ðgPTRSLðxP;yPÞsin2hþcos2hÞ (5c)

¼ ~Yþlinear�GY

PTRSL: (5d)

The crucial approximation made in obtaining Eqs. (4c) and (5c) is justified because~Yþlinear and ~X

þlinear are mutually perpendicular components. Furthermore, it may be verified

for a check of self-consistency of Eqs. (4c) and (5c), that the transformed fluxes ~Xþlinear and

~Yþlinear are almost undamped at nodes about h � p/2 line and h � 0 line, respectively.

Similarly, it can be shown that damping only the incoming radial flux ~X�r at P and re-

projecting the fluxes ~X�r � gPTRSL and ~X

þh along the negative x and y directions, yields

~X�linear ! GX

PTRSL � ~X�linear; (6)

~Y�linear ! GY

PTRSL � ~Y�linear: (7)

The radial component of acoustic disturbances �U0ð@~v=@xÞ advected by thereversed mean flow7 and ~vð�@U0=@yÞ due to non-uniform mean flow toward the

Fig. 1. (Color online) Schematic illustrating the implementation of PTRSL over a small square domain centeredat the predicted source location (S). Variation of the damping function gPTRSL (x, y) over the square domain isshown in the background.

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negative x direction at P is also damped and re-projected along with the undampedangular component on the x axis yielding

�U0@~v=@xð Þ ! GXPTRSL � �U0@~v=@xð Þ; (8)

~v �@U0=@yð Þ ! GXPTRSL � ~v �@U0=@yð Þ: (9)

The transformations shown in Eqs. (4c), (5c), (6)–(9) as well as an identical rangeof the included angle h are used to implement the radial damping at nodes located in thesecond, third, and fourth quadrants of the square domain, where h is measured withrespect to the negative x direction in the second and third quadrants, while in the fourthquadrant, h is measured with respect to the positive x direction. Furthermore, use of Eqs.(8) and (9) in the first and fourth quadrants signifies damping of radial components of theincoming acoustic disturbance whereas its implementation in the second and third quad-rants signifies damping of radial components of the outgoing acoustic disturbance. It isnoted that implementation of only Eq. (8) is necessary for a uniform mean flow whereasin case of a non-uniform mean flow field, Eq. (9) should also be implemented to take intoaccount, the refraction of waves through the shear-layer.9 Damping both the incomingand outgoing radial components in the square domain is also important to ensure stabil-ity of TR simulation for an arbitrarily large value of aPTRSL over large-durations.

The foregoing transformations induced in the fluxes ~X6

linear, ~Y6

linear,�U0ð@~v=@xÞ and ~vð�@U0=@yÞ over the square domain centered at S constitute thePTRSL damping and its implementation is used to enhance the focal-resolution char-acteristics of both, the idealized monopole as well as a dipole source. However, a smallmodification in the use of PTRSL is critical for enhancing the focal-resolution of amonopole source. Since, at the predicted monopole location S, an enhanced centralfocal spot is expected, ~pðx0; y0;~tÞ 6¼ 0 throughout TR simulation, hence, acoustic fluxes~X

6

linear and ~Y6

linear at S are computed without the use of PTRSL, therefore,

~X6

linearðx0; y0Þ 6¼ 0; ~Y6

linearðx0; y0Þ 6¼ 0; (10)

however, �U0ð@~v=@xÞjðx0;y0Þ ¼ ~vð�@U0=@yÞjðx0 ;y0Þ¼ 0. At the predicted dipole location

S, however, it is necessary to implement PTRSL on all the fluxes [as ~pðx0; y0;~tÞ ¼ 0during TR], therefore,

~X6

linear x0; y0ð Þ ¼ ~Y6

linear x0; y0ð Þ ¼ �U0@~v=@xð Þj x0;y0ð Þ ¼ ~v �@U0=@yð Þj x0 ;y0ð Þ ¼ 0: (11)

The PTRSL therefore, gradually absorbs the radial components of incomingprogressive waves near the predicted source location and simultaneously suppresses theoutgoing progressive waves, thereby tending to concentrate the acoustic power over asmaller region in the source vicinity and allowing only a limited fraction of the acous-tic power to propagate away. In this manner, it mimics an acoustic sink.2,3 Indeed, thePTRSL damping may be interpreted as a system of distributed sources whose near-field components2,3 reinforce at the predicted source location.

3. Simulation results, discussion and conclusions

The source maps of idealized monopole and dipole sources, each of a tonal frequencyf¼ 3 kHz obtained without implementation of the PTRSL are compared with thoseobtained with its implementation over the sub-domain jxj � 0:5 m, jyj � 0:5 m. (Thecenter of the domain set as the origin was taken as the known source location.) Threedifferent mean flow profiles were considered for each of the monopole/dipole test-cases; (a) stationary medium, (b) uniform mean flow of Mach number M0¼ 0.3, and(c) fully developed non-uniform (sheared) mean flow modeled as a 2-D free-jet9 givenby U0ðyÞ ¼ 0:3c0sech2½bðy=2LyÞ�, where b¼ 20. Figures 2(a), 2(c), and 2(e) depict themonopole source maps obtained without the implementation of PTRSL while Figs.2(b), 2(d), and 2(f) depict the monopole source maps obtained with its implementation

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centered at the predicted location. It is noted that a stationary medium is considered inFigs. 2(a) and 2(b), uniform mean flow is considered in Figs. 2(c) and 2(d), while non-uniform mean flow is considered in Figs. 2(e) and 2(f). The reversed direction of themean flow is indicated by an arrow in Figs. 2(c)–2(f), while the thick white lines signifythe presence of LAs at the y¼60.5 m boundaries.8 The same conventions are also fol-lowed for the dipole source maps. A systematic study was carried out at different tonalfrequencies for monopole/dipole sources to identify a near-optimal value of aPTRSLsuch that the effective damping domain minimizes the size of focal spot(s) and alsosuppresses the side-lobes. It was found that for a monopole aPTRSL � ln 2=r2

�3 dB andfor a dipole aPTRSL � ln 2=r2

0 dB yields the desired improvements in the source maps.Here, r–3 dB is the half-width of the monopole focal spot at �3 dB along the transversedirection while r0 dB is the distance between the predicted dipole location and a focalpoint. At f¼ 3 kHz, (regardless of the mean flow), aPTRSL¼ 1000 for a monopole andaPTRSL¼ 500 for a dipole while nx¼ ny¼ 20 for both sources. Figures 2(b), 2(d), and2(f) demonstrate a significant reduction in size of the near circular focal spot (monop-ole location) and a simultaneous reduction in the relative magnitudes and size of theside-lobes obtained on implementing the PTRSL. The TR simulations of a monopolesource located in a stationary medium (a) without the use of PTRSL shown in Mm. 1and (b) with its use shown in Mm. 2 may be referred to enhance the understanding ofPTRSL. The colorbar in Mm. 1 and Mm. 2 indicates ~pðx; y;~tÞ in Pa.

Mm. 1. TR simulation without the implementation of PTRSL. This is a “avi” file (11.1 Mb)[URL: http://dx.doi.org/10.1121/1.4890204.1].

Mm. 2. TR simulation with the implementation of PTRSL. This is a “avi” file (9.4 Mb)[URL: http://dx.doi.org/10.1121/1.4890204.2].

In order to quantify the enhancement in source maps, two metrics are defined;the transverse and longitudinal spatial resolution given by the focal spot size paralleland perpendicular to the LAs, respectively. The focal spot size is taken as Full-Widthat Half-Maximum (FWHM) given by sum of the distances corresponding to the level�6 dB considered on either side of the focal point (0 dB). The transverse and longitudi-nal resolution values expressed as a ratio of source wavelength (k¼ 0.11 m atf¼ 3 kHz) for the monopole shown in Table 1 indicate a dramatic enhancement of thefocal-resolution obtained on implementation of PTRSL, regardless of the mean flow.The average values of transverse and longitudinal resolution given by 0.21k and 0.20k,respectively, obtained by using PTRSL signify that its implementation overcomes theconventional half-wavelength diffraction limit,2,3 thereby generating a sharper mono-pole focal spot. It is however, noted that the side-lobes are influenced by the nature of

Fig. 2. (Color online) Comparison of the monopole source maps obtained without the implementation ofPTRSL (a), (c), and (e) with those obtained with its implementation (b), (d), and (f).

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flow, in particular, the non-uniform (sheared) mean flow produces side-lobes of rela-tively larger magnitude and size located in upstream of the reversed mean flow.

Figures 3(a), 3(c), and 3(e) depict the dipole source maps obtained without theimplementation of PTRSL while Figs. 3(b), 3(d), and 3(f) depict the dipole sourcemaps obtained with its implementation at the predicted source location. The geometri-cal center of the two focal points is taken as the predicted dipole source location8 inFigs. 3(a), 3(c), and 3(e) and is indicated by a cross X. Its location demonstrates accu-racy of the predicted dipole location. Figures 3(b), 3(d), and 3(f) indicate a significantreduction in size of the two focal spots, a more compact location of the two focalpoints and suppression of the side-lobes. The transverse and longitudinal resolutionvalues for the dipole source are shown in Table 1. The average values of transverseand longitudinal resolution given by 0.33k and 0.18k, respectively, obtained by usingPTRSL signify that its use results in nearly 50% reduction in size of the focal spots,thereby generating a much sharper resolution. Unlike the monopole, the side-lobes indipole source maps are, however, marginally influenced by the nature of flow.

A frequency-parametric study was carried out to investigate the effectivenessof PTRSL at tonal frequencies f¼ {2, 1, 0.5} kHz in enhancing the focal-resolution ofmonopole and dipole sources located in sheared mean flow. (The source maps arehowever, not shown here.) For a monopole source, aPTRSL¼ {500, 100, 60},nx¼ ny¼ {20, 45, 60} while for a dipole source, aPTRSL¼ {300, 50, 15}, nx¼ ny¼ {25,60, 80} at f¼ {2, 1, 0.5} kHz, respectively. The transverse and longitudinal focal-resolution values at these tonal frequencies (quantified in terms of a fraction of the cor-responding source wavelength) shown in Table 1 demonstrates that use of PTRSLresults in a commensurate reduction in the size of the focal spot(s).

The implementation of the PTRSL damping is therefore, shown to signifi-cantly reduce the size of focal spot(s) and suppress side-lobes, thereby enhancing thefocal-resolution characteristics of aeroacoustic TR. Its implementation overcomes theconventional half-wavelength diffraction-limit2,3 for a monopole source because ityields an enhanced focal spot of size comparable to that obtained using an acousticsink.2 The effectiveness of PTRSL is demonstrated using test-cases of tonal monopole

Table 1. Comparison of the focal-resolution (transverse and longitudinal) obtained without and with the imple-mentation of PTRSL for a tonal monopole and dipole source.

Size of the Focal Spot(s) (relative to �6 dB)

Transverse Resolution(Parallel to the LAs)

Longitudinal Resolution(Perpendicular to the LAs)

Source nature and Tonalfrequency

Mean flowprofile

WithoutPTRSL

WithPTRSL

WithoutPTRSL

WithPTRSL

Monopole

f¼ 3 kHz Stationary medium 0.54k 0.19k 0.45k 0.19kf¼ 3 kHz Uniform flow 0.52k 0.20k 0.42k 0.18k

f¼ 3 kHz Non-uniform flow 0.59k 0.25k 0.41k 0.22kf¼ 2 kHz Non-uniform flow 0.58k 0.22k 0.40k 0.22kf¼ 1 kHz Non-uniform flow 0.56k 0.20k 0.44k 0.20k

f¼ 0.5 kHz Non-uniform flow 0.51k 0.22k 0.47k 0.21k

Dipole

f¼ 3 kHz Stationary medium 0.63k 0.33k 0.40k 0.19kf¼ 3 kHz Uniform flow 0.58k 0.32k 0.38k 0.18kf¼ 3 kHz Non-uniform flow 0.67k 0.35k 0.38k 0.18kf¼ 2 kHz Non-uniform flow 0.65k 0.33k 0.37k 0.19kf¼ 1 kHz Non-uniform flow 0.63k 0.31k 0.40k 0.18k

f¼ 0.5 kHz Non-uniform flow 0.58k 0.32k 0.41k 0.18k

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and dipole sources9 located in different types of flow fields. The PTRSL damping tech-nique mimics an acoustic sink;2 however, its implementation during aeroacousticTR6–8 is robust and straightforward (by virtue of the PCF10) because it is based ondamping the radial components of the incoming and outgoing acoustic fluxes on asponge-layer domain centered at the predicted source location, without requiring anestimate of the source strength and phase. The implementation of PTRSL will have astrong impact in experimental aeroacoustic TR for enhancing the focal-resolution offlow-induced noise sources such as a dipole source generated by flow over uniform cyl-inder, tandem cylinders, airfoil, rod-airfoil, or a sharp-edged flat-plate.11

Acknowledgments

The authors would like to thank Dr. Zebb Prime for helpful suggestions and acknowledgethe Australian Research Council for their support through grant DP 120102134.

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2J. de Rosny and M. Fink, “Overcoming the diffraction limit in wave physics using a time-reversal mirrorand a novel acoustic sink,” Phys. Rev. Lett. 89, 124301 (2002).

3E. Bavu, C. Besnainou, V. Gibiat, J. D. Rosny, and M. Fink, “Subwavelength sound focusing using atime-reversal acoustic sink,” Acta. Acust. Acust. 93, 706–715 (2007).

4T. Padois, C. Prax, V. Valeau, and D. Marx, “Experimental localization of an acoustic source in a wind-tunnel flow by using a numerical time-reversal technique,” J. Acoust. Soc. Am. 132, 2397–2407 (2012).

5S. G. Conti, P. Roux, and W. A. Kuperman, “Near-field time-reversal amplification,” J. Acoust. Soc.Am. 121, 3602–3606 (2007).

6B. E. Anderson, R. A. Guyer, T. J. Ulrich, P. Y. Le Bas, C. Larmat, M. Griffa, and P. A. Johnson,“Energy current imaging method for time reversal in elastic media,” Appl. Phys. Lett. 95, 021907 (2009).

7A. Deneuve, P. Druault, R. Marchiano, and P. Sagaut, “A coupled time-reversal/complex differentiationmethod for aeroacoustic sensitivity analysis: Towards a source detection procedure,” J. Fluid Mech. 642,181–212 (2010).

8A. Mimani, C. J. Doolan, and P. R. Medwell, “Multiple line arrays for the characterization of aeroacous-tic sources using a time-reversal method,” J. Acoust. Soc. Am. 134, EL327–EL333 (2013).

9C. Bailly and D. Juve, “Numerical solution of acoustic wave propagation problems using linearizedEuler equations,” AIAA J. 38, 22–29 (2000).

10J. Sesterhenn, “A characteristic-type formulation of the Navier-Stokes equations for high order upwindschemes,” Comput. Fluids. 30, 37–67 (2000).

11W. K. Blake, Mechanics of Flow-Induced Sound and Vibration (Academic Press, New York, 1986), Vol.1, pp. 64, 65, 219–283.

Fig. 3. (Color online) Comparison of the dipole source maps obtained without the implementation of PTRSL(a), (c), and (e) with those obtained with its implementation (b), (d), and (f).

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