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Enhanced focal-resolution of dipole sources using aeroacoustic time-reversal in a wind tunnel
Mechanical Systems and Signal Processing 72-73 (2016) 925–937
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Enhanced focal-resolution of dipole sources using aeroacoustic time-reversal in a wind tunnel A. Mimani a,n, D.J. Moreau b, Z. Prime b, C.J. Doolan b a b
School of Mechanical Engineering, The University of Adelaide, South Australia 5005, Australia School of Mechanical and Manufacturing Engineering, University of New South Wales, New South Wales 2052, Australia
a r t i c l e i n f o
abstract
Article history: Received 5 February 2015 Received in revised form 7 August 2015 Accepted 27 September 2015 Available online 28 November 2015
This paper presents the first application of the Point-Time-Reversal-Sponge-Layer (PTRSL) damping technique to enhance the focal-resolution of experimental flow-induced dipole sources obtained using the Time-Reversal (TR) source localization method. Experiments were conducted in an Anechoic Wind Tunnel for the case of a full-span cylinder located in a low Mach number cross-flow. The far-field acoustic pressure sampled using two line arrays of microphones located above and below the cylinder exhibited a dominant Aeolian tone. The aeroacoustic TR simulations were implemented using the time-reversed signals whereby the source map revealed the lift-dipole nature at the Aeolian tone frequency. A PTRSL (centred at the predicted dipole location) was shown to reduce the size of dipole focal spots to 7/20th of a wavelength as compared to one wavelength without its use, thereby dramatically enhancing the focal-resolution of the TR technique. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Aeroacoustic time-reversal Aeolian tone Flow-induced dipole sources Super-resolution Point-Time-Reversal-Sponge-Layer Acoustic sink
1. Introduction Acoustic Time-Reversal (TR) is a robust method used to accurately localize noise sources due to its ability to refocus waves exactly at the source location by back-propagating them along the same trajectory created during their emission [1]. Rosny and Fink [2] note that during TR, converging and diverging wave-fronts (formed due to energy-conservation [3,4]) interfere locally in the source vicinity. This interference leads to the breakdown of TR symmetry and, as a result, the size of the focal spot (the source location) is at least a half-wavelength (referred to as the diffraction limit), even if the source is point-like, thereby limiting the TR resolution [2]. Techniques to obtain a focal spot size significantly smaller than the halfwavelength limit [2] are subject matter of several investigations [2,5–14] and is often referred to as super-resolution of sources [6,12–14] or sub-wavelength TR focusing [5,9,11]. Rosny and Fink [2] reported the first experimental demonstration of super-resolution by implementing a time-reversed source (acoustic sink) during TR on a reverberant glass-plate cavity system using an ultrasonic pulse, i.e., a transient signal. The implementation of a sink at the predicted initial location of pulse (obtained from first-TR step) generated outgoing waves that are nearly equal in amplitude but opposite in sign to the diverging waves formed near the source; these wave fronts undergo destructive interference. Therefore, in effect, the sink absorbs the progressive wave-fronts converging at the source from all directions and produces a highly reduced focal spot due to the generation and reinforcement of near-field n
Corresponding author. E-mail addresses: [email protected] (A. Mimani), [email protected] (D.J. Moreau), [email protected] (Z. Prime), [email protected] (C.J. Doolan). http://dx.doi.org/10.1016/j.ymssp.2015.09.037 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
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evanescent waves. The new ‘super-resolved’ focal spot is of considerably larger amplitude and diminished size; less than λ=14; λ being the source-wavelength, thereby demonstrating a dramatic improvement in focal-resolution. The implementation of a sink has been shown to overcome the half-wavelength diffraction limit for an impulsive source [2,5,6]. However, its implementation for enhancing the focal-resolution of aeroacoustic sources of a time-harmonic nature [3,4,7,8], requires an accurate a-priori estimate of the source characteristics (monopole or multipole nature), its strength (volume-source strength Q 0 for an idealized monopole source or amplitude F D of the fluctuating point force for an idealized dipole source, see Refs. [4,7,8]) and its phase. The direct determination of strength/phase is impossible because the TR simulation numerically solves the homogenous governing equations [3,4,7,8,15,16], rather, only the location/characteristics of source can be predicted from the first-TR step. In light of these challenges, the authors have developed a fundamentally new and alternate approach that does not rely on cancellation using a time-reversed source; rather, the technique employs a passive radial damping approach termed as the Point-Time-Reversal-Sponge-Layer (PTRSL) that mimics a sink to enhance the focal-resolution of idealized monopole/dipole sources, either of a tonal or broadband nature located in different mean flow fields using aeroacoustic TR [7,8]. It is noted that the PTRSL was centred at the predicted source location and also accounted for the monopole/dipole nature of the source (obtained from first-TR step); therefore, in effect, the PTRSL enhanced the TR source map in vicinity of the predicted location [2,5] of idealized aeroacoustic sources. In particular, it was shown that PTRSL reduces the size of focal spots of a dipole to 0:3λ as compared to 0:6λ without its implementation and almost completely suppresses the side-lobes. Their TR simulations [7,8], however, used numerical or simulated data recorded during forward simulation at two Line Arrays (LAs) of nodes located on opposite sides of idealized sources. Despite the promising improvements in the TR source map possible by using PTRSL, it has not yet been applied to experimental aeroacoustic problems of practical importance [17–24]. In fact, application of TR in experimental aeroacoustics has received only limited attention [3] and certainly, this method has not yet been applied to characterize the source nature of flow-induced noise using experimental data. This paper therefore presents, for the first time, an application of aeroacoustic TR source localization method to investigate its suitability for characterizing the source nature of experimental flow-induced noise generated by a 2-D cylinder located in a wind tunnel providing a uniform cross-flow and subsequently, demonstrate the effectiveness of the numerical PTRSL damping technique to enhance the focal-resolution of experimental flow-induced noise sources.
2. Experimental set-up and test-model Experiments were conducted in the Anechoic Wind Tunnel (AWT) at the University of Adelaide. The AWT is nearly cubic having internal dimensions 1:4 m 1:4 m 1:6 m and its walls are acoustically treated with foam wedges providing a near reflection-free environment above 250 Hz. It contains a contraction-outlet of rectangular cross-section of height h¼75 mm and width w¼275 mm that produces a quiet, uniform test-flow. The maximum free-stream velocity of the jet and turbulence intensity at the contraction-outlet is U 1 ≈40 m s 1 and 0:33%; respectively [21–24]. The test-model, a full-span circular cylinder (of diameter D0 ¼ 4 mm) is secured between two side-plates attached to the contraction-outlet flange as indicated in Fig. 1(a) and (b), which shows the photograph and a schematic of the front-view, respectively, of the experimental set-up. It is noted that end-effects are reduced because the cylinder-span l ¼ 450 mm along the z (span-wise) direction is sufficiently beyond the width of contraction-outlet [22]. A schematic of the side-view of the experimental set-up and the co-ordinate system convention is shown in Fig. 1(c) where x is the stream-wise direction and y is the vertical direction. The origin ðx ¼ y ¼ 0Þ is taken on the axis of the contraction-outlet at its opening and the cylinder is located downstream at x ¼ 50mm; y ¼ 0: In this work, a set of experiments were carried out at U 1 ¼ f32; 24; 16g m s 1 with the flow issuing out from the contraction-outlet towards the positive x direction as indicated in Fig. 1(c).
3. Measuring and analyzing the far-field acoustic spectra Acoustic measurements were taken with two LAs of microphones aligned parallel to the flow and located 700 mm apart, on opposite sides and equidistant from the cylinder, i.e., at y ¼ 7 Ly ¼ 7 350 mm and are co-planar (located in the z ¼ 0 plane) as shown in Fig. 1(a) and (c). Each LA consists of 32 GRAS 40 PH 1/400 phase-matched microphones (mounted in a timber-frame) positioned such that the spacing between two consecutive microphones is 30 mm, therefore, the total array length equals 930 mm. The length of each LA measured upstream from the contraction-outlet (with four microphones located upstream) is given by Lx1 ¼ 96 mm whilst that measured downstream denoted by Lx2 ¼ 834 mm: Each of the 64 microphones were connected to a National Instruments PXI-8106 data acquisition system containing 4 PXI-4496 simultaneous sample and hold Analogue-to-Digital Converter cards. The data (acoustic pressure time-history) at the microphones are recorded at sampling frequency f s ¼ 65536 Hz for a sample time of 10 s: When a cylinder is immersed in uniform flow, vortices of alternate rotation are shed from either side of the cylinder into its wake [18]. This periodic shedding, known as a von Karman vortex street, occurs at a particular frequency, f a ; represented in a non-dimensional form by the Strouhal number St D0 ¼ f a D0 =U 1 ; based on D0 : The von Karman vortex street generates unsteady forces on the surface of the cylinder that support dipole sound sources known as the Aeolian tone [17,25]. Fig. 2
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Fig. 1. (a) Photograph of the experimental set-up: a full-span 4 mm circular cylinder held in a two-sided mounting frame attached to the contractionoutlet in the AWT and two LAs of microphones for recording far-field acoustic pressure. Schematic diagrams (not shown to scale) depicting the full-span cylinder secured in the mounting frame attached to the contraction-outlet depicting the (b) Side-view and (c) Front-View (showing the position of the two LAs of microphones).
Fig. 2. Acoustic spectrum of the cylinder at flow speed (a) U 1 ¼ 32 m s 1 ; (b) U 1 ¼ 24 m s 1 and (c) U 1 ¼ 16 m s 1 measured at the microphone located in the bottom LA and positioned 24 mm downstream of the contraction-outlet opening (taken as the origin).
shows the frequency spectrum (measured in terms of the Power-Spectral Density (PSD)) of the acoustic pressure field due to flow-induced noise of the cylinder at U 1 ¼ f32; 24; 16g m s 1 measured at the microphone located in the bottom LA and positioned 24 mm downstream of the contraction-outlet opening. The occurrence of an Aeolian tone at frequency f a ¼ f1584; 1208; 784g Hz in the spectrum for U 1 ¼ f32; 24; 16g m s 1 ; respectively, (corresponding to St D0 ¼ f0:198; 0:201; 0:196g; respectively) suggests a dipole-like nature of the aeroacoustic or flow-induced source because the vortex shredding process induces fluctuating lift force at the cylinder axis that results in dipole radiation characteristics. This physical phenomenon is in agreement with the study of Norberg [19] which shows that a circular cylinder is expected to produce an Aeolian tone at St D0 ¼ 0:2 in this Reynolds number range (ReD0 ¼ 4:2 103 8:5 103 ; based on D0 ). The acoustic pressure signal recorded at each microphone was band-pass filtered (using a high-order Finite Impulse Response (FIR) filter) in 1=3rd octave bands with centre frequencies f C given by (a) 1600 Hz, (b) 1250 Hz and (c) 800 Hz for the spectrum obtained at U 1 ¼ 32 m s 1 ; 24 m s 1 and 16 m s 1 ; respectively, therefore, f a f C implying that the onethird octave band considered, accounts for almost the entire acoustic power contained in the spectrum. Furthermore, the background noise level generated by the free-stream jet is insignificant in comparison to the spectrum generated due to
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flow-induced noise [21–23], especially around the Aeolian tone. Therefore, it is expected to have a negligible contaminating effect on the TR simulation and hence was not removed from the spectrum prior to band-pass filtering.
4. Implementation of TR and analyzing the source map The TR simulation was implemented by numerically solving [4,7,8,16] the 2-D Linearized Euler Equations (LEE) using the Pseudo-Characteristic Formulation [26] (PCF) on a rectangular domain ðLx1 þ Lx3 Þ rx r Lx2 ; Ly r yr Ly with anechoic boundaries [27] in reverse time t~ and enforcing the time-reversed acoustic pressure p~ x; y; t~ (recorded during experiments) only at boundary nodes corresponding to the two LAs, where Lx3 ¼ 100 mm: It is noted that the upstream length of the domain along the x direction given by ðLx1 þLx3 Þ ¼ 196 mm for implementing TR simulations is greater than the upstream length Lx1 ¼ 96 mm of the LAs. þ o ∂p~ ρ c0 n ~ þ ð1Þ X linear þ X~ linear þ Y~ linear þ Y~ linear ; ¼ 0 2 ∂t~ ∂u~ 1 þ ∂U 0 U 0 p~ ∂U 0 u~ þ ; ¼ X~ linear X~ linear v~ 2 ∂y c0 ρ0 c0 ∂x ∂t~
ð2Þ
∂v~ 1 þ ∂v~ ¼ Y~ linear Y~ linear ð U 0 Þ ; 2 ∂x ∂t~
ð3Þ
where 7 X~ linear ¼ 7 ðc0 8 U 0 Þ
1 ∂p~ ∂u~ 7 ρ0 c0 ∂x ∂x
and
7 Y~ linear ¼ 7 c0
1 ∂p~ ∂v~ 7 : ρ0 c0 ∂y ∂y
ð4–7Þ
7 7 In Eqs. (1–3), u~ and v~ denote acoustic velocities along the x and y directions, respectively, X~ linear and Y~ linear denote a pair of opposing fluxes [26] propagating towards the x and y directions, respectively, the sound speed c0 ¼ 345:75 m s 1 (at ambient temperature T 0 ¼ 297:47 K) whilst U 0 represents the spatially-developing shear mean flow of a free-jet modeled by 1 δ δy δ þy sech β sech β ; ð8Þ U 0 ¼ U 0 ðx; yÞ ¼ U max 1 þ cosh β 2 2Ly 2Ly Ly
where U max ¼ U max ðx; U 1 Þ; β ¼ βðxÞ and δ ¼ δðxÞ are the maximum mean flow, steepness of the shear-layer and halfthickness of the potential-core, respectively; their variation along x direction (from contraction-outlet) is modeled by an appropriate polynomial-fit using constrained least-squares optimization. The mean flow direction was reversed towards the negative x direction, i.e., U 0 - U 0 during TR, thereby ensuring the TR invariance [3,4,7,8,15,16] of the governing LEE. 7 7 The spatial derivative of acoustic pressure and velocities in the opposing fluxes X~ linear ; Y~ linear of the PCF were computed using an overall upwind-biased FD scheme [4,28] that is formulated using a fourth-order, seven-point optimized upwind-biased FD scheme [29] at interior nodes and a seven-point optimised backward FD scheme at the boundary nodes [30]. In order to increase the mesh-resolution, two equally spaced nodes were added between each pair of nodes corresponding to the microphone locations. The acoustic pressure time-history at these two extra nodes (required during TR) were obtained by interpolating the experimental data between each pair of microphones using Lagrange polynomial interpolation [31], resulting in a mesh-size Δx ¼ 10 mm along the x direction. Equal mesh-size Δy ¼ 10 mm was also considered along the y direction. The maximum frequency that may be accurately propagated on this mesh is approximately 8255 Hz as determined from the Dispersion-Relation-Preserving [29,30] range αDRP 1:5 of the fourth-order optimized upwind-biased FD scheme and c0 ¼ 345:75 m s 1 : The third order Total-Variation-Diminishing Runge-Kutta scheme [32] is used for time-integration with a time-step Δt ¼ 1=f S ¼ 1:5259 10 5 s implying a CFL ¼ f0:58; 0:56; 0:55g for U 1 ¼ f32; 24; 16g m s 1 ; respectively, for the mesh-size considered. In addition to enforcing the time-reversed p~ x; y; t~ data at boundary nodes of the two LAs, the first-order ClaytonEngquist-Majda (CEM) Anechoic Boundary Conditions (ABCs) were implemented at all four boundaries given by
∂p~ ∂p~
∂p~ ∂p~
¼ 0; ¼ 0; ð9a–dÞ 7 ðc0 8U 0 Þ 7 c 0 ∂x x ¼ 7 Lx ∂y y ¼ 7 Ly ∂t~ ∂t~ and at the four corner of the 2-D domain, the corner ABCs given by
∂p~ 1 ∂p~ c0 ∂p~
∂p~ 1 ∂p~ c0 ∂p~
¼ 0; ð10aÞ þ pffiffiffiðc0 U 0 Þ þ pffiffiffi
pffiffiffiðc0 þ U 0 Þ þ pffiffiffi
∂x ∂x ∂t~ ∂t~ 2 2 2 ∂y x ¼ Lx ;y ¼ Ly 2 ∂y x ¼
∂p~ 1 ∂p~ c0 ∂p~
pffiffiffiðc0 þ U 0 Þ pffiffiffi
∂x ∂t~ 2 2 ∂y x ¼
¼ 0; Lx ;y ¼ Ly
ð10cÞ
¼ 0;
ð10bÞ
¼ 0;
ð10dÞ
Lx ;y ¼ Ly
∂p~ 1 ∂p~ c0 ∂p~
þ pffiffiffiðc0 U 0 Þ pffiffiffi
∂x ∂t~ 2 2 ∂y x ¼ Lx ;y ¼
Ly
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were used, see Refs. [4,8,27,28]. These ABCs computationally model a 2-D free-space as the experiments were carried out in the AWT. Its implementation is necessary to eliminate the spurious numerical reflections generated at the free boundaries (due to the outgoing acoustic fluxes generated by enforcing the time-reversed experimental data at the top/bottom LAs), thereby ensuring temporal stability of 2-D TR simulations. Furthermore, the incoming normal acoustic fluxes (of the PCF 7 7 [26] of the 2-D LEE) were set to zero at the computational boundaries, i.e., X~ linear jx ¼ 8 Lx ¼ Y~ linear jy ¼ 8 Ly ¼ 0 to reinforce the ~ ABCs during TR simulation [4,8,28]. The boundary condition ∂v=∂x jx ¼ Lx ¼ 0 was also implemented to eliminate the incoming spurious numerical waves advected by the mean flow necessary for stabilizing the 2-D TR simulation [28]. Fig. 3(a–f) presents the time-snapshots of the spatio-temporal evolution of p~ x; y; t~ field at the Aeolian tone corresponding to U 1 ¼ 32 m s1 obtained using the band-pass time-reversed signals in 1=3rd octave band with f C ¼ 1600 Hz enforced at the boundary nodes corresponding to the top and bottom LAs. The two LAs, contraction-outlet and its flanges are shown by white lines, whilst the cylinder is represented by a circle O; the same symbolic convention is followed henceforth. During the initial time-instants, a simultaneous emission of acoustic fluxes from the top/bottom LAs is observed (shown in Fig. 3(a)) that propagate into the domain and are about to converge or undergo a constructive interference (shown in Fig. 3(b)). Fig. 3(c) shows that this is followed by formation of two instantaneous maxima regions (henceforth, referred to as the focal spots) near the cylinder-axis of nearly the same strength but opposite phase, indicating a dipolesource. The simulations reveal that at the source region, the width of the wave-fronts diminish whilst their amplitude significantly increase. Due to the conservation of energy [1–4] and absence of an acoustic-sink [2,5,6] during TR, the converging wave-fronts do not stop at the source but propagate beyond and interfere with flux emanating from the LAs [4] unlike a pulse [15,28]. In order to suppress the deteriorating flux-interference effect near the LAs, a Time-Reversal-SpongeLayer (TRSL) was used that damps the fluxes normally incident on the LAs implemented using the following transformations [4,16]. 7 7 ð11; 12Þ Y~ linear x; 7Ly 8nΔy -Y~ linear x; 7 Ly 8 nΔy GTRSL ðnÞ;
12 αTRSL
nTRSL n nTRSL 1
2
for n ¼ ½1; 2; …; nTRSL 1; αTRSL ¼ 3:5 is the damping coefficient whilst where GTRSL ð0Þ ¼ 0 and GTRSL ðnÞ ¼ e nTRSL ¼ 10 is the thickness of TRSL. Fig. 3(d–f) indicate a continuous formation of two instantaneous focal spots (of same strength but opposite phase) is observed throughout the TR simulations, although their instantaneous geometrical centre (taken as the dipole location) slightly varies over time. In order to enhance the foregoing discussion on the spatio-temporal evolution of p~ x; y; t~ field shown in Fig. 3, the reader is referred to Video 1 (see the Link below) which plays the corresponding TR simulation.
The TR simulation was implemented over t~ ¼ 0; 10000Δt whereby the aeroacoustic source location/characteristics is obtained by determining the focal spots in the Root-Mean-Square (RMS) time-reversed acoustic pressure field [3,4,7,8,16] ~ TR denoted by p~ TR RMS ðx; yÞ: The focal spot maximum is termed the focal point. The p RMS ðx; yÞ field is converted to dB scale (with TR 5 Pa) and the source map denoted by p~ dB ðx; yÞ is expressed relative to the focal point(s) whose respect to pref ¼ 2 10 magnitude is taken as 0 dB, see Refs. [4,7,8]. The TR source map shown in Fig. 4(a) exhibits two focal spots (of nearly the same magnitude, shape and size) located in proximity, thereby indicating the dipole-type source nature of the flow-induced noise generated at the Aeolian tone frequency [17–20] due to a cylinder in a cross-flow corresponding to U 1 ¼ 32 m s 1 : Indeed, due to the lift forces responsible for noise generation, the orientation of these focal spots is almost above and below the cylinder and thus, the source is termed a lift-dipole [19]. The geometrical centre of the two focal points is taken as the predicted dipole source location (denoted by a cross X) given by x0 ¼ 74 mm; y0 ¼ 10 mm in Fig. 4(a). This demonstrates that although the two LAs located on opposite sides of the full-span cylinder records partial boundary data/information [33], their use ensures a sufficient effective angular aperture [4,16] required during aeroacoustic TR for characterizing the lift-dipole source nature generated at the Aeolian tone and accurately localizing it within a distance 0:12λa of the true cylinder location, where λa ¼ c0 =f a : However, it is noted that accuracy of predicted location may possibly be improved by increasing the upstream/ downstream lengths of LAs (i.e., using more number of microphones) due to an increased effective angular aperture [4,34].
Furthermore, the source map in Fig. 4(a) is shown over the dynamic range 0; 10 dB and the reversed direction of mean flow is indicated by an arrow; the same symbolic convention and dynamic range is followed henceforth. The TR simulations were also implemented for U 1 ¼ f24; 16g m s 1 at the corresponding Aeolian tone using the two LAs. Fig. 4(c) and (e) presents the source map for U 1 ¼ f24; 16g m s 1 ; respectively, wherein the two focal spots located in proximity in each of the source maps again indicate a dipole nature. It is noted that predicted location of the lift-dipole source in Fig. 4(c) and (e) is given by x0 ¼ 94 mm; y0 ¼ 5 mm and x0 ¼ 134 mm; y0 ¼ 10 mm ; respectively, thereby indicating a prediction error of 0:16λa and 0:19λa ; respectively, of the true cylinder location. Furthermore, in Fig. 4(e), relatively large focal spots were obtained (due to the occurrence of Aeolian tone at low-frequency) which extend beyond the upstream length ðLx1 ¼ 96 mmÞ X2 of LAs whilst the primary side-lobes occur near the LAs. For these reasons, a larger upstream domain length ðLx1 þ Lx3 ¼ 196 mmÞ and the superposition technique [2,4], respectively, were used during TR for U 1 ¼ 16 m s 1 : The size of the focal spot(s) is quantified in terms of the following two metrics; the transverse spatial resolution (parallel to the LAs) and longitudinal spatial resolution (perpendicular to the LAs) and is taken as Full-Width at Half-Maximum
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Fig. 3. Spatio-temporal evolution of the time-reversed acoustic pressure field p~ x; y; t~ at the Aeolian tone corresponding to U 1 ¼ 32 m s 1 obtained using the experimental signals recorded at the top and bottom LAs (filtered in the 1=3rd octave band with f c ¼ 1600 Hz) at the reverse time-instants: (a) t~ ¼ 20Δt; (b) t~ ¼ 60Δt; (c) t~ ¼ 100Δt; (d) t~ ¼ 125Δt; (e) t ¼ 400Δt and (f) t~ ¼ 800Δt:
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Video S1. which plays the TR simulation of p~ x; y; t~ field at f a ¼ 1584 Hz corresponding to U 1 ¼ 32 m s 1 without implementing the PTRSL damping technique. A video clip is available online.
Supplementary material related to this article can be found online at doi:10.1016/j.ymssp.2015.09.037.
(FWHM) given by sum of the distances corresponding to 6 dB level considered on either side of the focal point [7,8]. The transverse and longitudinal resolution of dipole focal spots for f a ¼ f1584; 1208; 784g Hz expressed as a ratio of the corresponding source wavelength λa in the third and fifth columns (from left) of Table 1, respectively, indicate a nearly commensurate size of the focal spot(s); their average values are given by 1:03λa and 0:38λa ; respectively. It is noted that the average transverse resolution is significantly greater than the classical half-wavelength limit imposed by diffraction of converging/diverging wave-fronts near the source. This is because to obtain a classical half-wavelength diffraction limit, boundary data from a full angular aperture or equivalently, the use of a LA configuration that completely encloses the source is required during TR. The two LA configuration used here, however, provides only a partial angular aperture (θS 187:41 at the predicted dipole location), thereby explaining the observed average transverse resolution of the dipole focal spots.
5. Enhancing the focal-resolution The spatial resolution of aeroacoustic TR focusing or the source maps obtained in Fig. 4(a), (c) and (e) is enhanced by implementing a Point-Time-Reversal-Sponge-Layer (PTRSL) damping technique [7,8]. To this end, the TR simulations were
carried out for a second-time wherein the PTRSL is implemented over a small square domain jx x0 j r nx Δx; y y0 r ny Δy (schematically shown in Fig. 1 of Ref. [7]) centred at the predicted location x0 ; y0 of the dipole source (denoted by S in Fig. 1 of Ref. [7]) obtained from the first-TR step. Its underlying theory is to gradually absorb the radial components of incoming progressive waves near the predicted source location and simultaneously suppresses the outgoing progressive waves, thereby tending to concentrate the acoustic power over a smaller region in the source vicinity and allowing only a limited fraction of the acoustic power to propagate away. A PTRSL is algorithmically implemented (at all nodes of each of the four quadrants of 8 the square domain) by damping the incoming and outgoing radial fluxes X~ ; respectively, propagating towards and away r
from the predicted location, by multiplying them with a smoothly varying 2-D Gaussian function given by 2 2 g PTRSL ðx; yÞ ¼ 1 e αPTRSL ðx x0 Þ þ ðy y0 Þ that decays to zero at S whilst gradually increasing to unity towards edges of the square domain. Here, αPTRSL is the damping-coefficient that controls the steepness of g PTRSL ðx; yÞ; thence, the effective damping domain; a study was conducted to identify its near-optimal value for a dipole such that the effective damping domain minimizes the focal spot size whilst suppressing the side-lobes [7,8]. It was found that αPTRSL ln 2=r 20dB yields the desired improvements where r 0dB is the distance between the predicted dipole location and a focal point. On the basis of this study, αPTRSL ¼ f180; 90; 45g m 2 whilst nx ¼ ny ¼ f18; 24; 28g for f a ¼ f1584; 1208; 784g Hz; respectively. The 7 angular fluxes X~ propagating along the counter-clockwise and clockwise directions, respectively, are however, undamped. θ
While using the top and bottom LAs simultaneously during TR simulation, a PTRSL is implemented at a point P in the first 7 7 ~ quadrant by introducing the following transformations in X~ linear ; Y~ linear ; U 0 ∂v=∂x ; v~ ∂U 0 =∂y ;
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Fig. 4. Comparison of TR source maps of the dipole source generated at the Aeolian tone obtained without the implementation of PTRSL (a, c and e) with those obtained with its implementation (b, d and f). Parts (a) and (b) are obtained at U ∞ ¼ 32 m s 1 ; f a ¼ 1584 Hz; parts (c) and (d) are obtained at U ∞ ¼ 24 m s 1 ; f a ¼ 1208 Hz whilst parts (e) and (f) are obtained at U 1 ¼ 16 m s 1 ; f a ¼ 784 Hz:
A. Mimani et al. / Mechanical Systems and Signal Processing 72-73 (2016) 925–937
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Table 1 Comparison of the transverse and longitudinal focal-resolution of the dipole source (generated at the Aeolian tone) obtained without and with the implementation of PTRSL. Flow speed U 1
32 24 16
m s1
Aeolian tone frequency f a ðHzÞ
1584 1208 784
Size of the focal spot(s) (relative to 6 dB) Transverse resolution (Parallel to the LAs)
Longitudinal resolution (Perpendicular to the LAs)
Without PTRSL
With PTRSL
Without PTRSL
With PTRSL
1:01λa 1:04λa 1:05λa
0:37λa 0:38λa 0:29λa
0:37λa 0:39λa 0:38λa
0:23λa 0:20λa 0:25λa
∂U 0 =∂x fluxes [7,8]. 7 7 7 X~ linear -X~ linear g PTRSL xP ; yP cos 2 θ þ sin 2 θ ¼ X~ linear GXPTRSL ;
ð13; 14Þ
7 7 7 Y~ linear -Y~ linear g PTRSL xP ; yP sin 2 θ þ cos 2 θ ¼ Y~ linear GYPTRSL ;
ð15; 16Þ
~ u~ þ cU0 0 ρ pc0 0
∂v~ ∂v~ ∂U 0 ∂U 0 -GXPTRSL U 0 ; v~ -GXPTRSL v~ ; ∂x ∂x ∂y ∂y U 0 p~ ∂U 0 U 0 p~ ∂U 0 -GXPTRSL u~ þ ; u~ þ c0 ρ0 c0 ∂x c 0 ρ0 c 0 ∂x
U0
ð17–19Þ
where the included angle θ formed between the line joining P with S and the positive x axis varies from θ ¼ 0 to θ ¼ π =2 radian (refer to Fig. 1 of Ref. [7]). The transformations shown in Eqs. (13)–(19) and an identical range of θ are used to implement the radial damping at nodes located in the second, third and fourth quadrants of the square domain, where θ is measured with respect to the negative x direction in the second and third quadrants, whilst in the fourth quadrant, θ is measured with respect to the positive x direction. At the predicted dipole location S, the following flux-condition
7 7 ∂v~
∂U 0
U 0 p~ ∂U 0
~ ~ X~ linear x0 ; y0 ¼ Y~ linear x0 ; y0 ¼ U 0 ¼ v ¼ u þ ¼ 0: ð20–26Þ ∂x ðx0 ;y0 Þ ∂y ðx0 ;y0 Þ c0 ρ0 c0 ∂x ðx0 ;y0 Þ is specified because ideally, p~ x0 ; y0 ; t~ ¼ 0 at the dipole centre during TR [7,8]. It is noted that when the superposition technique [2,4] (for U 1 ¼ 16 m s 1 ) is used during TR simulation, a small modification is introduced for implementing the PTRSL damping technique which is briefly described here. While computing the time-reversed acoustic pressure field p~ Top x; y; t~ obtained by TR simulation using only the top LA, the flux-transformation þ þ þ Y given by Eq. (15), i.e., Y~ linear -Y~ linear GPTRSL as well as Eq. (22), i.e., Y~ linear x0 ; y0 ¼ 0 are not implemented in each of the four quadrants. This is because the use of top LA generates a predominant back-propagating Y~ linear flux whilst a significantly þ þ ~ weaker Y linear flux is generated; therefore, the use of PTRSL to damp the radial components of Y~ linear flux is unnecessary. Y Based on similar reasoning, the flux-transformation given by Eq. (16), i.e., Y~ linear -Y~ linear GPTRSL as well as Eq. (23), i.e., Y~ linear x0 ; y0 ¼ 0 are not implemented in each of the four quadrants while computing the p~ Bottom x; y; t~ field obtained by TR simulation using only the bottom LA. In essence, a PTRSL mimics an acoustic sink [2] and is interpreted as a system of distributed time-reversed sources whose near-field components [2,9] reinforce at the predicted source location. Fig. 5 presents the spatio-temporal evolution of p~ x; y; t~ field at the Aeolian tone f a ¼ 1584 Hz corresponding to 1 at the corresponding time-instants (considered in Fig. 3) obtained with the implementation of PTRSL U 1 ¼ 32 m s damping centred at the predicted source location indicated in Fig. 4(a). Fig. 5(a) and (b) show a simultaneous emission of acoustic fluxes from the two LAs that propagate into the domain and are about to converge; these time-snapshots are qualitatively similar to Fig. 3(a) and (b), respectively. It is further noted that unlike Fig. 3(c), two instantaneous maxima regions are not formed in Fig. 5(c), rather the radial damping of the incoming fluxes induced by the PTRSL results in a somewhat localized maxima region in vicinity of the predicted dipole location due to a more efficient TR focusing. With a further progress in TR simulation, two instantaneous dipole focal spots are formed in Fig. 5(d) that are significantly smaller in size and their focal points are more compact (in comparison to Fig. 3(d)). Fig. 5(e) and (f) demonstrate a continuous formation of instantaneous dipole focal spots throughout TR simulation; however, their instantaneous geometrical centre does not oscillate (unlike Fig. 3(c–f) or Video 1). Furthermore, since the PTRSL significantly reduces the amount of acoustic flux diverging or propagating away from the source, its implementation results in a near-complete suppression of the sidelobes and enhanced focusing of the acoustic flux at the predicted location resulting in almost doubling of amplitudes at the two focal points as observed from Fig. 5(e). Video 2 (see the Link below) plays the corresponding TR simulation of the spatio-temporal evolution of p~ x; y; t~ field (shown in Fig. 5) with the implementation of PTRSL at the predicted location obtained from the TR first-step.
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Fig. 5. Spatio-temporal evolution of the time-reversed acoustic pressure field p~ x; y; t~ at the Aeolian tone corresponding to U 1 ¼ 32 m s 1 obtained with implementation of the PTRSL damping technique at the predicted location (shown in Fig. 4(a)) using the experimental signals recorded at the top and bottom LAs (filtered in the 1=3rd octave band with f c ¼ 1600 Hz) at the reverse time-instants: (a) t~ ¼ 20Δt; (b) t~ ¼ 60Δt; (c) t~ ¼ 100Δt; (d) t~ ¼ 125Δt; (e) t~ ¼ 400Δt and (f) t~ ¼ 800Δt:
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Video S2. which plays the TR simulation of p~ x; y; t~ field at f a ¼ 1584 Hz corresponding to U 1 ¼ 32 m s 1 with the implementation of PTRSL at the predicted dipole location. A video clip is available online.
Supplementary material related to this article can be found online at doi:10.1016/j.ymssp.2015.09.037.
Fig. 6. Comparison of the TR source maps along y-axis ð 0:3 m r y r 0:3 mÞ; i.e. the grid-line passing through the two focal points of the lift-dipole source obtained without the implementation of PTRSL with that obtained using this damping technique (at the predicted location) at flow speed (a) U 1 ¼ 32 m s 1 ; (b) U 1 ¼ 24 m s 1 and (c) U 1 ¼ 16 m s 1 :
Fig. 4(b) depict the TR source maps obtained with the implementation of PTRSL during the second-TR step at the predicted dipole location generated at f a ¼ 1584 Hz: A comparison of Fig. 4(b) with Fig. 4(a), indicates a significant reduction in the focal spot size, a more compact location of the two focal points and a near-complete suppression of the side-lobes, thereby enhancing the focal-resolution. This improvement is also demonstrated by comparing the source maps along the grid-line
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passing through the two focal points of the dipole (shown in Fig. 6(a)); the width of the focal-lobes obtained with the implementation of PTRSL is significantly smaller than that obtained without its use whilst the peaks of the former are much closer as compared to the latter. A similar improvement in TR focal-resolution obtained on implementing a PTRSL damping during the second-TR step is also demonstrated through a comparison of (1) the TR source maps shown in Fig. 4(d) and (f) with Fig. 4(c) and (e), respectively, and (2) the widths and proximity of dipole focal-lobes obtained with/without the implementation of PTRSL damping shown in Fig. 6(b) and (c). Table 1 quantifies the improvement in the focal-resolution for the flow-speeds considered. The corresponding transverse and longitudinal resolution values of the focal spots shown in the fourth and sixth column (from left) of Table 1, respectively, for U 1 ¼ f32; 24; 16g m s 1 demonstrates that use of PTRSL yields a commensurate reduction in the focal spot size. Indeed, the average values of transverse and longitudinal resolution given by 0:35λa and 0:23λa ; respectively, signify that PTRSL damping overcomes the conventional half-wavelength diffraction limit [2]. Furthermore, a 66% reduction in the transverse size and almost 40% reduction in longitudinal size indicate a much sharper focal-resolution. It is however, noted that since the effectiveness of the PTRSL damping depends on an accurate a-priori estimate of the predicted location of the source and its characteristics [7,8] (obtained from first-TR step or Fig. 4(a), (c) and (e)), in essence, it is a technique/an artefact for enhancing the TR source map in the proximity of the PTRSL centre, i.e. point S (refer to Fig. 1 of Ref. [7]) or the predicted location of the experimental flow-induced lift-dipole source.
6. Conclusions This paper has demonstrated, for the first time, robustness of the numerical PTRSL damping technique to enhance the TR focal-resolution of flow-induced dipole sources using experimental acoustic pressure data (recorded on top/bottom LAs of microphones) due to a classical test-case of the Aeolian tone [17–20] from a full-span cylinder in cross-flow. A PTRSL mimics an acoustic sink or a time-reversed source [2,5] and requires an accurate a-priori knowledge of the predicted source location as well as its characteristics (noted earlier); however, its implementation during aeroacoustic TR is relatively straightforward (by virtue of the PCF [26]) because it directly manipulates the acoustic fluxes, without requiring an estimate of the source strength and phase [7,8]; the technique is robust because it yields the desired improvements regardless of the tonal [7] or broadband [8] nature of the time-reversed signals. As such, the PTRSL damping is expected to have a strong impact in experimental aeroacoustic TR problems because its implementation can enhance the focal-resolution of dipole sources generated by cross-flow over a wall-mounted cylinder [21] or airfoil [24], full-span sharp-edged flat-plate [22,23] and possibly, the noise sources in a free-jet [35].
Acknowledgements The authors would like to thank Mr. Ric Porteous for providing experimental data of the mean flow and acknowledge the support of Australian Research Council (ARC) through grant DP 120102134 “Resolving the mechanics of turbulent noise production.”
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Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Enhanced focal-resolution of dipole sources using aeroacoustic time-reversal in a wind tunnel A. Mimani a,n, D.J. Moreau b, Z. Prime b, C.J. Doolan b a b
School of Mechanical Engineering, The University of Adelaide, South Australia 5005, Australia School of Mechanical and Manufacturing Engineering, University of New South Wales, New South Wales 2052, Australia
a r t i c l e i n f o
abstract
Article history: Received 5 February 2015 Received in revised form 7 August 2015 Accepted 27 September 2015 Available online 28 November 2015
This paper presents the first application of the Point-Time-Reversal-Sponge-Layer (PTRSL) damping technique to enhance the focal-resolution of experimental flow-induced dipole sources obtained using the Time-Reversal (TR) source localization method. Experiments were conducted in an Anechoic Wind Tunnel for the case of a full-span cylinder located in a low Mach number cross-flow. The far-field acoustic pressure sampled using two line arrays of microphones located above and below the cylinder exhibited a dominant Aeolian tone. The aeroacoustic TR simulations were implemented using the time-reversed signals whereby the source map revealed the lift-dipole nature at the Aeolian tone frequency. A PTRSL (centred at the predicted dipole location) was shown to reduce the size of dipole focal spots to 7/20th of a wavelength as compared to one wavelength without its use, thereby dramatically enhancing the focal-resolution of the TR technique. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Aeroacoustic time-reversal Aeolian tone Flow-induced dipole sources Super-resolution Point-Time-Reversal-Sponge-Layer Acoustic sink
1. Introduction Acoustic Time-Reversal (TR) is a robust method used to accurately localize noise sources due to its ability to refocus waves exactly at the source location by back-propagating them along the same trajectory created during their emission [1]. Rosny and Fink [2] note that during TR, converging and diverging wave-fronts (formed due to energy-conservation [3,4]) interfere locally in the source vicinity. This interference leads to the breakdown of TR symmetry and, as a result, the size of the focal spot (the source location) is at least a half-wavelength (referred to as the diffraction limit), even if the source is point-like, thereby limiting the TR resolution [2]. Techniques to obtain a focal spot size significantly smaller than the halfwavelength limit [2] are subject matter of several investigations [2,5–14] and is often referred to as super-resolution of sources [6,12–14] or sub-wavelength TR focusing [5,9,11]. Rosny and Fink [2] reported the first experimental demonstration of super-resolution by implementing a time-reversed source (acoustic sink) during TR on a reverberant glass-plate cavity system using an ultrasonic pulse, i.e., a transient signal. The implementation of a sink at the predicted initial location of pulse (obtained from first-TR step) generated outgoing waves that are nearly equal in amplitude but opposite in sign to the diverging waves formed near the source; these wave fronts undergo destructive interference. Therefore, in effect, the sink absorbs the progressive wave-fronts converging at the source from all directions and produces a highly reduced focal spot due to the generation and reinforcement of near-field n
Corresponding author. E-mail addresses: [email protected] (A. Mimani), [email protected] (D.J. Moreau), [email protected] (Z. Prime), [email protected] (C.J. Doolan). http://dx.doi.org/10.1016/j.ymssp.2015.09.037 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
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evanescent waves. The new ‘super-resolved’ focal spot is of considerably larger amplitude and diminished size; less than λ=14; λ being the source-wavelength, thereby demonstrating a dramatic improvement in focal-resolution. The implementation of a sink has been shown to overcome the half-wavelength diffraction limit for an impulsive source [2,5,6]. However, its implementation for enhancing the focal-resolution of aeroacoustic sources of a time-harmonic nature [3,4,7,8], requires an accurate a-priori estimate of the source characteristics (monopole or multipole nature), its strength (volume-source strength Q 0 for an idealized monopole source or amplitude F D of the fluctuating point force for an idealized dipole source, see Refs. [4,7,8]) and its phase. The direct determination of strength/phase is impossible because the TR simulation numerically solves the homogenous governing equations [3,4,7,8,15,16], rather, only the location/characteristics of source can be predicted from the first-TR step. In light of these challenges, the authors have developed a fundamentally new and alternate approach that does not rely on cancellation using a time-reversed source; rather, the technique employs a passive radial damping approach termed as the Point-Time-Reversal-Sponge-Layer (PTRSL) that mimics a sink to enhance the focal-resolution of idealized monopole/dipole sources, either of a tonal or broadband nature located in different mean flow fields using aeroacoustic TR [7,8]. It is noted that the PTRSL was centred at the predicted source location and also accounted for the monopole/dipole nature of the source (obtained from first-TR step); therefore, in effect, the PTRSL enhanced the TR source map in vicinity of the predicted location [2,5] of idealized aeroacoustic sources. In particular, it was shown that PTRSL reduces the size of focal spots of a dipole to 0:3λ as compared to 0:6λ without its implementation and almost completely suppresses the side-lobes. Their TR simulations [7,8], however, used numerical or simulated data recorded during forward simulation at two Line Arrays (LAs) of nodes located on opposite sides of idealized sources. Despite the promising improvements in the TR source map possible by using PTRSL, it has not yet been applied to experimental aeroacoustic problems of practical importance [17–24]. In fact, application of TR in experimental aeroacoustics has received only limited attention [3] and certainly, this method has not yet been applied to characterize the source nature of flow-induced noise using experimental data. This paper therefore presents, for the first time, an application of aeroacoustic TR source localization method to investigate its suitability for characterizing the source nature of experimental flow-induced noise generated by a 2-D cylinder located in a wind tunnel providing a uniform cross-flow and subsequently, demonstrate the effectiveness of the numerical PTRSL damping technique to enhance the focal-resolution of experimental flow-induced noise sources.
2. Experimental set-up and test-model Experiments were conducted in the Anechoic Wind Tunnel (AWT) at the University of Adelaide. The AWT is nearly cubic having internal dimensions 1:4 m 1:4 m 1:6 m and its walls are acoustically treated with foam wedges providing a near reflection-free environment above 250 Hz. It contains a contraction-outlet of rectangular cross-section of height h¼75 mm and width w¼275 mm that produces a quiet, uniform test-flow. The maximum free-stream velocity of the jet and turbulence intensity at the contraction-outlet is U 1 ≈40 m s 1 and 0:33%; respectively [21–24]. The test-model, a full-span circular cylinder (of diameter D0 ¼ 4 mm) is secured between two side-plates attached to the contraction-outlet flange as indicated in Fig. 1(a) and (b), which shows the photograph and a schematic of the front-view, respectively, of the experimental set-up. It is noted that end-effects are reduced because the cylinder-span l ¼ 450 mm along the z (span-wise) direction is sufficiently beyond the width of contraction-outlet [22]. A schematic of the side-view of the experimental set-up and the co-ordinate system convention is shown in Fig. 1(c) where x is the stream-wise direction and y is the vertical direction. The origin ðx ¼ y ¼ 0Þ is taken on the axis of the contraction-outlet at its opening and the cylinder is located downstream at x ¼ 50mm; y ¼ 0: In this work, a set of experiments were carried out at U 1 ¼ f32; 24; 16g m s 1 with the flow issuing out from the contraction-outlet towards the positive x direction as indicated in Fig. 1(c).
3. Measuring and analyzing the far-field acoustic spectra Acoustic measurements were taken with two LAs of microphones aligned parallel to the flow and located 700 mm apart, on opposite sides and equidistant from the cylinder, i.e., at y ¼ 7 Ly ¼ 7 350 mm and are co-planar (located in the z ¼ 0 plane) as shown in Fig. 1(a) and (c). Each LA consists of 32 GRAS 40 PH 1/400 phase-matched microphones (mounted in a timber-frame) positioned such that the spacing between two consecutive microphones is 30 mm, therefore, the total array length equals 930 mm. The length of each LA measured upstream from the contraction-outlet (with four microphones located upstream) is given by Lx1 ¼ 96 mm whilst that measured downstream denoted by Lx2 ¼ 834 mm: Each of the 64 microphones were connected to a National Instruments PXI-8106 data acquisition system containing 4 PXI-4496 simultaneous sample and hold Analogue-to-Digital Converter cards. The data (acoustic pressure time-history) at the microphones are recorded at sampling frequency f s ¼ 65536 Hz for a sample time of 10 s: When a cylinder is immersed in uniform flow, vortices of alternate rotation are shed from either side of the cylinder into its wake [18]. This periodic shedding, known as a von Karman vortex street, occurs at a particular frequency, f a ; represented in a non-dimensional form by the Strouhal number St D0 ¼ f a D0 =U 1 ; based on D0 : The von Karman vortex street generates unsteady forces on the surface of the cylinder that support dipole sound sources known as the Aeolian tone [17,25]. Fig. 2
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Fig. 1. (a) Photograph of the experimental set-up: a full-span 4 mm circular cylinder held in a two-sided mounting frame attached to the contractionoutlet in the AWT and two LAs of microphones for recording far-field acoustic pressure. Schematic diagrams (not shown to scale) depicting the full-span cylinder secured in the mounting frame attached to the contraction-outlet depicting the (b) Side-view and (c) Front-View (showing the position of the two LAs of microphones).
Fig. 2. Acoustic spectrum of the cylinder at flow speed (a) U 1 ¼ 32 m s 1 ; (b) U 1 ¼ 24 m s 1 and (c) U 1 ¼ 16 m s 1 measured at the microphone located in the bottom LA and positioned 24 mm downstream of the contraction-outlet opening (taken as the origin).
shows the frequency spectrum (measured in terms of the Power-Spectral Density (PSD)) of the acoustic pressure field due to flow-induced noise of the cylinder at U 1 ¼ f32; 24; 16g m s 1 measured at the microphone located in the bottom LA and positioned 24 mm downstream of the contraction-outlet opening. The occurrence of an Aeolian tone at frequency f a ¼ f1584; 1208; 784g Hz in the spectrum for U 1 ¼ f32; 24; 16g m s 1 ; respectively, (corresponding to St D0 ¼ f0:198; 0:201; 0:196g; respectively) suggests a dipole-like nature of the aeroacoustic or flow-induced source because the vortex shredding process induces fluctuating lift force at the cylinder axis that results in dipole radiation characteristics. This physical phenomenon is in agreement with the study of Norberg [19] which shows that a circular cylinder is expected to produce an Aeolian tone at St D0 ¼ 0:2 in this Reynolds number range (ReD0 ¼ 4:2 103 8:5 103 ; based on D0 ). The acoustic pressure signal recorded at each microphone was band-pass filtered (using a high-order Finite Impulse Response (FIR) filter) in 1=3rd octave bands with centre frequencies f C given by (a) 1600 Hz, (b) 1250 Hz and (c) 800 Hz for the spectrum obtained at U 1 ¼ 32 m s 1 ; 24 m s 1 and 16 m s 1 ; respectively, therefore, f a f C implying that the onethird octave band considered, accounts for almost the entire acoustic power contained in the spectrum. Furthermore, the background noise level generated by the free-stream jet is insignificant in comparison to the spectrum generated due to
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flow-induced noise [21–23], especially around the Aeolian tone. Therefore, it is expected to have a negligible contaminating effect on the TR simulation and hence was not removed from the spectrum prior to band-pass filtering.
4. Implementation of TR and analyzing the source map The TR simulation was implemented by numerically solving [4,7,8,16] the 2-D Linearized Euler Equations (LEE) using the Pseudo-Characteristic Formulation [26] (PCF) on a rectangular domain ðLx1 þ Lx3 Þ rx r Lx2 ; Ly r yr Ly with anechoic boundaries [27] in reverse time t~ and enforcing the time-reversed acoustic pressure p~ x; y; t~ (recorded during experiments) only at boundary nodes corresponding to the two LAs, where Lx3 ¼ 100 mm: It is noted that the upstream length of the domain along the x direction given by ðLx1 þLx3 Þ ¼ 196 mm for implementing TR simulations is greater than the upstream length Lx1 ¼ 96 mm of the LAs. þ o ∂p~ ρ c0 n ~ þ ð1Þ X linear þ X~ linear þ Y~ linear þ Y~ linear ; ¼ 0 2 ∂t~ ∂u~ 1 þ ∂U 0 U 0 p~ ∂U 0 u~ þ ; ¼ X~ linear X~ linear v~ 2 ∂y c0 ρ0 c0 ∂x ∂t~
ð2Þ
∂v~ 1 þ ∂v~ ¼ Y~ linear Y~ linear ð U 0 Þ ; 2 ∂x ∂t~
ð3Þ
where 7 X~ linear ¼ 7 ðc0 8 U 0 Þ
1 ∂p~ ∂u~ 7 ρ0 c0 ∂x ∂x
and
7 Y~ linear ¼ 7 c0
1 ∂p~ ∂v~ 7 : ρ0 c0 ∂y ∂y
ð4–7Þ
7 7 In Eqs. (1–3), u~ and v~ denote acoustic velocities along the x and y directions, respectively, X~ linear and Y~ linear denote a pair of opposing fluxes [26] propagating towards the x and y directions, respectively, the sound speed c0 ¼ 345:75 m s 1 (at ambient temperature T 0 ¼ 297:47 K) whilst U 0 represents the spatially-developing shear mean flow of a free-jet modeled by 1 δ δy δ þy sech β sech β ; ð8Þ U 0 ¼ U 0 ðx; yÞ ¼ U max 1 þ cosh β 2 2Ly 2Ly Ly
where U max ¼ U max ðx; U 1 Þ; β ¼ βðxÞ and δ ¼ δðxÞ are the maximum mean flow, steepness of the shear-layer and halfthickness of the potential-core, respectively; their variation along x direction (from contraction-outlet) is modeled by an appropriate polynomial-fit using constrained least-squares optimization. The mean flow direction was reversed towards the negative x direction, i.e., U 0 - U 0 during TR, thereby ensuring the TR invariance [3,4,7,8,15,16] of the governing LEE. 7 7 The spatial derivative of acoustic pressure and velocities in the opposing fluxes X~ linear ; Y~ linear of the PCF were computed using an overall upwind-biased FD scheme [4,28] that is formulated using a fourth-order, seven-point optimized upwind-biased FD scheme [29] at interior nodes and a seven-point optimised backward FD scheme at the boundary nodes [30]. In order to increase the mesh-resolution, two equally spaced nodes were added between each pair of nodes corresponding to the microphone locations. The acoustic pressure time-history at these two extra nodes (required during TR) were obtained by interpolating the experimental data between each pair of microphones using Lagrange polynomial interpolation [31], resulting in a mesh-size Δx ¼ 10 mm along the x direction. Equal mesh-size Δy ¼ 10 mm was also considered along the y direction. The maximum frequency that may be accurately propagated on this mesh is approximately 8255 Hz as determined from the Dispersion-Relation-Preserving [29,30] range αDRP 1:5 of the fourth-order optimized upwind-biased FD scheme and c0 ¼ 345:75 m s 1 : The third order Total-Variation-Diminishing Runge-Kutta scheme [32] is used for time-integration with a time-step Δt ¼ 1=f S ¼ 1:5259 10 5 s implying a CFL ¼ f0:58; 0:56; 0:55g for U 1 ¼ f32; 24; 16g m s 1 ; respectively, for the mesh-size considered. In addition to enforcing the time-reversed p~ x; y; t~ data at boundary nodes of the two LAs, the first-order ClaytonEngquist-Majda (CEM) Anechoic Boundary Conditions (ABCs) were implemented at all four boundaries given by
∂p~ ∂p~
∂p~ ∂p~
¼ 0; ¼ 0; ð9a–dÞ 7 ðc0 8U 0 Þ 7 c 0 ∂x x ¼ 7 Lx ∂y y ¼ 7 Ly ∂t~ ∂t~ and at the four corner of the 2-D domain, the corner ABCs given by
∂p~ 1 ∂p~ c0 ∂p~
∂p~ 1 ∂p~ c0 ∂p~
¼ 0; ð10aÞ þ pffiffiffiðc0 U 0 Þ þ pffiffiffi
pffiffiffiðc0 þ U 0 Þ þ pffiffiffi
∂x ∂x ∂t~ ∂t~ 2 2 2 ∂y x ¼ Lx ;y ¼ Ly 2 ∂y x ¼
∂p~ 1 ∂p~ c0 ∂p~
pffiffiffiðc0 þ U 0 Þ pffiffiffi
∂x ∂t~ 2 2 ∂y x ¼
¼ 0; Lx ;y ¼ Ly
ð10cÞ
¼ 0;
ð10bÞ
¼ 0;
ð10dÞ
Lx ;y ¼ Ly
∂p~ 1 ∂p~ c0 ∂p~
þ pffiffiffiðc0 U 0 Þ pffiffiffi
∂x ∂t~ 2 2 ∂y x ¼ Lx ;y ¼
Ly
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were used, see Refs. [4,8,27,28]. These ABCs computationally model a 2-D free-space as the experiments were carried out in the AWT. Its implementation is necessary to eliminate the spurious numerical reflections generated at the free boundaries (due to the outgoing acoustic fluxes generated by enforcing the time-reversed experimental data at the top/bottom LAs), thereby ensuring temporal stability of 2-D TR simulations. Furthermore, the incoming normal acoustic fluxes (of the PCF 7 7 [26] of the 2-D LEE) were set to zero at the computational boundaries, i.e., X~ linear jx ¼ 8 Lx ¼ Y~ linear jy ¼ 8 Ly ¼ 0 to reinforce the ~ ABCs during TR simulation [4,8,28]. The boundary condition ∂v=∂x jx ¼ Lx ¼ 0 was also implemented to eliminate the incoming spurious numerical waves advected by the mean flow necessary for stabilizing the 2-D TR simulation [28]. Fig. 3(a–f) presents the time-snapshots of the spatio-temporal evolution of p~ x; y; t~ field at the Aeolian tone corresponding to U 1 ¼ 32 m s1 obtained using the band-pass time-reversed signals in 1=3rd octave band with f C ¼ 1600 Hz enforced at the boundary nodes corresponding to the top and bottom LAs. The two LAs, contraction-outlet and its flanges are shown by white lines, whilst the cylinder is represented by a circle O; the same symbolic convention is followed henceforth. During the initial time-instants, a simultaneous emission of acoustic fluxes from the top/bottom LAs is observed (shown in Fig. 3(a)) that propagate into the domain and are about to converge or undergo a constructive interference (shown in Fig. 3(b)). Fig. 3(c) shows that this is followed by formation of two instantaneous maxima regions (henceforth, referred to as the focal spots) near the cylinder-axis of nearly the same strength but opposite phase, indicating a dipolesource. The simulations reveal that at the source region, the width of the wave-fronts diminish whilst their amplitude significantly increase. Due to the conservation of energy [1–4] and absence of an acoustic-sink [2,5,6] during TR, the converging wave-fronts do not stop at the source but propagate beyond and interfere with flux emanating from the LAs [4] unlike a pulse [15,28]. In order to suppress the deteriorating flux-interference effect near the LAs, a Time-Reversal-SpongeLayer (TRSL) was used that damps the fluxes normally incident on the LAs implemented using the following transformations [4,16]. 7 7 ð11; 12Þ Y~ linear x; 7Ly 8nΔy -Y~ linear x; 7 Ly 8 nΔy GTRSL ðnÞ;
12 αTRSL
nTRSL n nTRSL 1
2
for n ¼ ½1; 2; …; nTRSL 1; αTRSL ¼ 3:5 is the damping coefficient whilst where GTRSL ð0Þ ¼ 0 and GTRSL ðnÞ ¼ e nTRSL ¼ 10 is the thickness of TRSL. Fig. 3(d–f) indicate a continuous formation of two instantaneous focal spots (of same strength but opposite phase) is observed throughout the TR simulations, although their instantaneous geometrical centre (taken as the dipole location) slightly varies over time. In order to enhance the foregoing discussion on the spatio-temporal evolution of p~ x; y; t~ field shown in Fig. 3, the reader is referred to Video 1 (see the Link below) which plays the corresponding TR simulation.
The TR simulation was implemented over t~ ¼ 0; 10000Δt whereby the aeroacoustic source location/characteristics is obtained by determining the focal spots in the Root-Mean-Square (RMS) time-reversed acoustic pressure field [3,4,7,8,16] ~ TR denoted by p~ TR RMS ðx; yÞ: The focal spot maximum is termed the focal point. The p RMS ðx; yÞ field is converted to dB scale (with TR 5 Pa) and the source map denoted by p~ dB ðx; yÞ is expressed relative to the focal point(s) whose respect to pref ¼ 2 10 magnitude is taken as 0 dB, see Refs. [4,7,8]. The TR source map shown in Fig. 4(a) exhibits two focal spots (of nearly the same magnitude, shape and size) located in proximity, thereby indicating the dipole-type source nature of the flow-induced noise generated at the Aeolian tone frequency [17–20] due to a cylinder in a cross-flow corresponding to U 1 ¼ 32 m s 1 : Indeed, due to the lift forces responsible for noise generation, the orientation of these focal spots is almost above and below the cylinder and thus, the source is termed a lift-dipole [19]. The geometrical centre of the two focal points is taken as the predicted dipole source location (denoted by a cross X) given by x0 ¼ 74 mm; y0 ¼ 10 mm in Fig. 4(a). This demonstrates that although the two LAs located on opposite sides of the full-span cylinder records partial boundary data/information [33], their use ensures a sufficient effective angular aperture [4,16] required during aeroacoustic TR for characterizing the lift-dipole source nature generated at the Aeolian tone and accurately localizing it within a distance 0:12λa of the true cylinder location, where λa ¼ c0 =f a : However, it is noted that accuracy of predicted location may possibly be improved by increasing the upstream/ downstream lengths of LAs (i.e., using more number of microphones) due to an increased effective angular aperture [4,34].
Furthermore, the source map in Fig. 4(a) is shown over the dynamic range 0; 10 dB and the reversed direction of mean flow is indicated by an arrow; the same symbolic convention and dynamic range is followed henceforth. The TR simulations were also implemented for U 1 ¼ f24; 16g m s 1 at the corresponding Aeolian tone using the two LAs. Fig. 4(c) and (e) presents the source map for U 1 ¼ f24; 16g m s 1 ; respectively, wherein the two focal spots located in proximity in each of the source maps again indicate a dipole nature. It is noted that predicted location of the lift-dipole source in Fig. 4(c) and (e) is given by x0 ¼ 94 mm; y0 ¼ 5 mm and x0 ¼ 134 mm; y0 ¼ 10 mm ; respectively, thereby indicating a prediction error of 0:16λa and 0:19λa ; respectively, of the true cylinder location. Furthermore, in Fig. 4(e), relatively large focal spots were obtained (due to the occurrence of Aeolian tone at low-frequency) which extend beyond the upstream length ðLx1 ¼ 96 mmÞ X2 of LAs whilst the primary side-lobes occur near the LAs. For these reasons, a larger upstream domain length ðLx1 þ Lx3 ¼ 196 mmÞ and the superposition technique [2,4], respectively, were used during TR for U 1 ¼ 16 m s 1 : The size of the focal spot(s) is quantified in terms of the following two metrics; the transverse spatial resolution (parallel to the LAs) and longitudinal spatial resolution (perpendicular to the LAs) and is taken as Full-Width at Half-Maximum
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Fig. 3. Spatio-temporal evolution of the time-reversed acoustic pressure field p~ x; y; t~ at the Aeolian tone corresponding to U 1 ¼ 32 m s 1 obtained using the experimental signals recorded at the top and bottom LAs (filtered in the 1=3rd octave band with f c ¼ 1600 Hz) at the reverse time-instants: (a) t~ ¼ 20Δt; (b) t~ ¼ 60Δt; (c) t~ ¼ 100Δt; (d) t~ ¼ 125Δt; (e) t ¼ 400Δt and (f) t~ ¼ 800Δt:
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Video S1. which plays the TR simulation of p~ x; y; t~ field at f a ¼ 1584 Hz corresponding to U 1 ¼ 32 m s 1 without implementing the PTRSL damping technique. A video clip is available online.
Supplementary material related to this article can be found online at doi:10.1016/j.ymssp.2015.09.037.
(FWHM) given by sum of the distances corresponding to 6 dB level considered on either side of the focal point [7,8]. The transverse and longitudinal resolution of dipole focal spots for f a ¼ f1584; 1208; 784g Hz expressed as a ratio of the corresponding source wavelength λa in the third and fifth columns (from left) of Table 1, respectively, indicate a nearly commensurate size of the focal spot(s); their average values are given by 1:03λa and 0:38λa ; respectively. It is noted that the average transverse resolution is significantly greater than the classical half-wavelength limit imposed by diffraction of converging/diverging wave-fronts near the source. This is because to obtain a classical half-wavelength diffraction limit, boundary data from a full angular aperture or equivalently, the use of a LA configuration that completely encloses the source is required during TR. The two LA configuration used here, however, provides only a partial angular aperture (θS 187:41 at the predicted dipole location), thereby explaining the observed average transverse resolution of the dipole focal spots.
5. Enhancing the focal-resolution The spatial resolution of aeroacoustic TR focusing or the source maps obtained in Fig. 4(a), (c) and (e) is enhanced by implementing a Point-Time-Reversal-Sponge-Layer (PTRSL) damping technique [7,8]. To this end, the TR simulations were
carried out for a second-time wherein the PTRSL is implemented over a small square domain jx x0 j r nx Δx; y y0 r ny Δy (schematically shown in Fig. 1 of Ref. [7]) centred at the predicted location x0 ; y0 of the dipole source (denoted by S in Fig. 1 of Ref. [7]) obtained from the first-TR step. Its underlying theory is to gradually absorb the radial components of incoming progressive waves near the predicted source location and simultaneously suppresses the outgoing progressive waves, thereby tending to concentrate the acoustic power over a smaller region in the source vicinity and allowing only a limited fraction of the acoustic power to propagate away. A PTRSL is algorithmically implemented (at all nodes of each of the four quadrants of 8 the square domain) by damping the incoming and outgoing radial fluxes X~ ; respectively, propagating towards and away r
from the predicted location, by multiplying them with a smoothly varying 2-D Gaussian function given by 2 2 g PTRSL ðx; yÞ ¼ 1 e αPTRSL ðx x0 Þ þ ðy y0 Þ that decays to zero at S whilst gradually increasing to unity towards edges of the square domain. Here, αPTRSL is the damping-coefficient that controls the steepness of g PTRSL ðx; yÞ; thence, the effective damping domain; a study was conducted to identify its near-optimal value for a dipole such that the effective damping domain minimizes the focal spot size whilst suppressing the side-lobes [7,8]. It was found that αPTRSL ln 2=r 20dB yields the desired improvements where r 0dB is the distance between the predicted dipole location and a focal point. On the basis of this study, αPTRSL ¼ f180; 90; 45g m 2 whilst nx ¼ ny ¼ f18; 24; 28g for f a ¼ f1584; 1208; 784g Hz; respectively. The 7 angular fluxes X~ propagating along the counter-clockwise and clockwise directions, respectively, are however, undamped. θ
While using the top and bottom LAs simultaneously during TR simulation, a PTRSL is implemented at a point P in the first 7 7 ~ quadrant by introducing the following transformations in X~ linear ; Y~ linear ; U 0 ∂v=∂x ; v~ ∂U 0 =∂y ;
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Fig. 4. Comparison of TR source maps of the dipole source generated at the Aeolian tone obtained without the implementation of PTRSL (a, c and e) with those obtained with its implementation (b, d and f). Parts (a) and (b) are obtained at U ∞ ¼ 32 m s 1 ; f a ¼ 1584 Hz; parts (c) and (d) are obtained at U ∞ ¼ 24 m s 1 ; f a ¼ 1208 Hz whilst parts (e) and (f) are obtained at U 1 ¼ 16 m s 1 ; f a ¼ 784 Hz:
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Table 1 Comparison of the transverse and longitudinal focal-resolution of the dipole source (generated at the Aeolian tone) obtained without and with the implementation of PTRSL. Flow speed U 1
32 24 16
m s1
Aeolian tone frequency f a ðHzÞ
1584 1208 784
Size of the focal spot(s) (relative to 6 dB) Transverse resolution (Parallel to the LAs)
Longitudinal resolution (Perpendicular to the LAs)
Without PTRSL
With PTRSL
Without PTRSL
With PTRSL
1:01λa 1:04λa 1:05λa
0:37λa 0:38λa 0:29λa
0:37λa 0:39λa 0:38λa
0:23λa 0:20λa 0:25λa
∂U 0 =∂x fluxes [7,8]. 7 7 7 X~ linear -X~ linear g PTRSL xP ; yP cos 2 θ þ sin 2 θ ¼ X~ linear GXPTRSL ;
ð13; 14Þ
7 7 7 Y~ linear -Y~ linear g PTRSL xP ; yP sin 2 θ þ cos 2 θ ¼ Y~ linear GYPTRSL ;
ð15; 16Þ
~ u~ þ cU0 0 ρ pc0 0
∂v~ ∂v~ ∂U 0 ∂U 0 -GXPTRSL U 0 ; v~ -GXPTRSL v~ ; ∂x ∂x ∂y ∂y U 0 p~ ∂U 0 U 0 p~ ∂U 0 -GXPTRSL u~ þ ; u~ þ c0 ρ0 c0 ∂x c 0 ρ0 c 0 ∂x
U0
ð17–19Þ
where the included angle θ formed between the line joining P with S and the positive x axis varies from θ ¼ 0 to θ ¼ π =2 radian (refer to Fig. 1 of Ref. [7]). The transformations shown in Eqs. (13)–(19) and an identical range of θ are used to implement the radial damping at nodes located in the second, third and fourth quadrants of the square domain, where θ is measured with respect to the negative x direction in the second and third quadrants, whilst in the fourth quadrant, θ is measured with respect to the positive x direction. At the predicted dipole location S, the following flux-condition
7 7 ∂v~
∂U 0
U 0 p~ ∂U 0
~ ~ X~ linear x0 ; y0 ¼ Y~ linear x0 ; y0 ¼ U 0 ¼ v ¼ u þ ¼ 0: ð20–26Þ ∂x ðx0 ;y0 Þ ∂y ðx0 ;y0 Þ c0 ρ0 c0 ∂x ðx0 ;y0 Þ is specified because ideally, p~ x0 ; y0 ; t~ ¼ 0 at the dipole centre during TR [7,8]. It is noted that when the superposition technique [2,4] (for U 1 ¼ 16 m s 1 ) is used during TR simulation, a small modification is introduced for implementing the PTRSL damping technique which is briefly described here. While computing the time-reversed acoustic pressure field p~ Top x; y; t~ obtained by TR simulation using only the top LA, the flux-transformation þ þ þ Y given by Eq. (15), i.e., Y~ linear -Y~ linear GPTRSL as well as Eq. (22), i.e., Y~ linear x0 ; y0 ¼ 0 are not implemented in each of the four quadrants. This is because the use of top LA generates a predominant back-propagating Y~ linear flux whilst a significantly þ þ ~ weaker Y linear flux is generated; therefore, the use of PTRSL to damp the radial components of Y~ linear flux is unnecessary. Y Based on similar reasoning, the flux-transformation given by Eq. (16), i.e., Y~ linear -Y~ linear GPTRSL as well as Eq. (23), i.e., Y~ linear x0 ; y0 ¼ 0 are not implemented in each of the four quadrants while computing the p~ Bottom x; y; t~ field obtained by TR simulation using only the bottom LA. In essence, a PTRSL mimics an acoustic sink [2] and is interpreted as a system of distributed time-reversed sources whose near-field components [2,9] reinforce at the predicted source location. Fig. 5 presents the spatio-temporal evolution of p~ x; y; t~ field at the Aeolian tone f a ¼ 1584 Hz corresponding to 1 at the corresponding time-instants (considered in Fig. 3) obtained with the implementation of PTRSL U 1 ¼ 32 m s damping centred at the predicted source location indicated in Fig. 4(a). Fig. 5(a) and (b) show a simultaneous emission of acoustic fluxes from the two LAs that propagate into the domain and are about to converge; these time-snapshots are qualitatively similar to Fig. 3(a) and (b), respectively. It is further noted that unlike Fig. 3(c), two instantaneous maxima regions are not formed in Fig. 5(c), rather the radial damping of the incoming fluxes induced by the PTRSL results in a somewhat localized maxima region in vicinity of the predicted dipole location due to a more efficient TR focusing. With a further progress in TR simulation, two instantaneous dipole focal spots are formed in Fig. 5(d) that are significantly smaller in size and their focal points are more compact (in comparison to Fig. 3(d)). Fig. 5(e) and (f) demonstrate a continuous formation of instantaneous dipole focal spots throughout TR simulation; however, their instantaneous geometrical centre does not oscillate (unlike Fig. 3(c–f) or Video 1). Furthermore, since the PTRSL significantly reduces the amount of acoustic flux diverging or propagating away from the source, its implementation results in a near-complete suppression of the sidelobes and enhanced focusing of the acoustic flux at the predicted location resulting in almost doubling of amplitudes at the two focal points as observed from Fig. 5(e). Video 2 (see the Link below) plays the corresponding TR simulation of the spatio-temporal evolution of p~ x; y; t~ field (shown in Fig. 5) with the implementation of PTRSL at the predicted location obtained from the TR first-step.
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Fig. 5. Spatio-temporal evolution of the time-reversed acoustic pressure field p~ x; y; t~ at the Aeolian tone corresponding to U 1 ¼ 32 m s 1 obtained with implementation of the PTRSL damping technique at the predicted location (shown in Fig. 4(a)) using the experimental signals recorded at the top and bottom LAs (filtered in the 1=3rd octave band with f c ¼ 1600 Hz) at the reverse time-instants: (a) t~ ¼ 20Δt; (b) t~ ¼ 60Δt; (c) t~ ¼ 100Δt; (d) t~ ¼ 125Δt; (e) t~ ¼ 400Δt and (f) t~ ¼ 800Δt:
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Video S2. which plays the TR simulation of p~ x; y; t~ field at f a ¼ 1584 Hz corresponding to U 1 ¼ 32 m s 1 with the implementation of PTRSL at the predicted dipole location. A video clip is available online.
Supplementary material related to this article can be found online at doi:10.1016/j.ymssp.2015.09.037.
Fig. 6. Comparison of the TR source maps along y-axis ð 0:3 m r y r 0:3 mÞ; i.e. the grid-line passing through the two focal points of the lift-dipole source obtained without the implementation of PTRSL with that obtained using this damping technique (at the predicted location) at flow speed (a) U 1 ¼ 32 m s 1 ; (b) U 1 ¼ 24 m s 1 and (c) U 1 ¼ 16 m s 1 :
Fig. 4(b) depict the TR source maps obtained with the implementation of PTRSL during the second-TR step at the predicted dipole location generated at f a ¼ 1584 Hz: A comparison of Fig. 4(b) with Fig. 4(a), indicates a significant reduction in the focal spot size, a more compact location of the two focal points and a near-complete suppression of the side-lobes, thereby enhancing the focal-resolution. This improvement is also demonstrated by comparing the source maps along the grid-line
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passing through the two focal points of the dipole (shown in Fig. 6(a)); the width of the focal-lobes obtained with the implementation of PTRSL is significantly smaller than that obtained without its use whilst the peaks of the former are much closer as compared to the latter. A similar improvement in TR focal-resolution obtained on implementing a PTRSL damping during the second-TR step is also demonstrated through a comparison of (1) the TR source maps shown in Fig. 4(d) and (f) with Fig. 4(c) and (e), respectively, and (2) the widths and proximity of dipole focal-lobes obtained with/without the implementation of PTRSL damping shown in Fig. 6(b) and (c). Table 1 quantifies the improvement in the focal-resolution for the flow-speeds considered. The corresponding transverse and longitudinal resolution values of the focal spots shown in the fourth and sixth column (from left) of Table 1, respectively, for U 1 ¼ f32; 24; 16g m s 1 demonstrates that use of PTRSL yields a commensurate reduction in the focal spot size. Indeed, the average values of transverse and longitudinal resolution given by 0:35λa and 0:23λa ; respectively, signify that PTRSL damping overcomes the conventional half-wavelength diffraction limit [2]. Furthermore, a 66% reduction in the transverse size and almost 40% reduction in longitudinal size indicate a much sharper focal-resolution. It is however, noted that since the effectiveness of the PTRSL damping depends on an accurate a-priori estimate of the predicted location of the source and its characteristics [7,8] (obtained from first-TR step or Fig. 4(a), (c) and (e)), in essence, it is a technique/an artefact for enhancing the TR source map in the proximity of the PTRSL centre, i.e. point S (refer to Fig. 1 of Ref. [7]) or the predicted location of the experimental flow-induced lift-dipole source.
6. Conclusions This paper has demonstrated, for the first time, robustness of the numerical PTRSL damping technique to enhance the TR focal-resolution of flow-induced dipole sources using experimental acoustic pressure data (recorded on top/bottom LAs of microphones) due to a classical test-case of the Aeolian tone [17–20] from a full-span cylinder in cross-flow. A PTRSL mimics an acoustic sink or a time-reversed source [2,5] and requires an accurate a-priori knowledge of the predicted source location as well as its characteristics (noted earlier); however, its implementation during aeroacoustic TR is relatively straightforward (by virtue of the PCF [26]) because it directly manipulates the acoustic fluxes, without requiring an estimate of the source strength and phase [7,8]; the technique is robust because it yields the desired improvements regardless of the tonal [7] or broadband [8] nature of the time-reversed signals. As such, the PTRSL damping is expected to have a strong impact in experimental aeroacoustic TR problems because its implementation can enhance the focal-resolution of dipole sources generated by cross-flow over a wall-mounted cylinder [21] or airfoil [24], full-span sharp-edged flat-plate [22,23] and possibly, the noise sources in a free-jet [35].
Acknowledgements The authors would like to thank Mr. Ric Porteous for providing experimental data of the mean flow and acknowledge the support of Australian Research Council (ARC) through grant DP 120102134 “Resolving the mechanics of turbulent noise production.”
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