Towed body measurements of flow noise from a turbulent boundary layer under sea conditions J. Abshagena) and V. Nejedl Research Department for Underwater Acoustics and Marine Geophysics,b) Bundeswehr Technical Center WTD71-FWG, Berliner Straße 115, 24340 Eckernf€ orde, Germany

(Received 6 February 2013; revised 18 November 2013; accepted 3 December 2013) Results from an underwater experiment under sea conditions on flow noise beneath a flat-plate turbulent boundary layer are presented. The measurements were performed with a towed body at towing speeds U ¼ 2:3; …; 6:1 m/s and depths h ¼ 150; …; 100 m. Flow noise is measured with a linear array of equally spaced hydrophones (Dx ¼ 70 mm) that is orientated in streamwise direction and embedded within a laterally attached flat plate. In order to separate flow noise from ocean ambient noise and other acoustical noise sources wavenumber-frequency filtering is applied. The (nondimensionalized) spectral power density of flow noise UðxÞ hU1 i= ðhd i ð1=2q hU1 iÞ2 Þ is found to scale like ðxhd i=hU1 iÞ4:3 in a wide frequency range at higher towing speeds. Here, x, hd i, and hU1 i denote frequency, boundary layer displacement thickness, and potential flow velocity in the array region, respectively. Potential flow velocity is estimated from numerical simulations around a symmetrical, two-dimensional body with a semi-elliptical nose. Evidence is given that a v2 -(Tsallis) superstatistics provides a reasonable representation of the probability distribution C 2014 Acoustical Society of America. function of flow noise at higher towing speeds. V [http://dx.doi.org/10.1121/1.4861238] PACS number(s): 43.30.Wi, 43.30.Xm, 43.28.Ra, 43.60.Gk [JIA] I. INTRODUCTION

Pressure fluctuations in turbulent boundary layers are of fundamental scientific interest with large relevance for many applications.1–4 In incompressible flows, pressure couples nonlocally with different spatial regions, while in compressible flows, turbulent pressure fluctuations can give rise to sound production.4–6 Sound can be generated in the near-wall region of a body in motion by a surrounding turbulent boundary layer and transmitted through the outer (thin) wall structure into the (quiescent) interior of the body as flow noise. Here, the interaction of the turbulent wall-pressure fluctuations with the outer wall structure plays an important role for flow noise production.3,7 Flow noise is of relevance, for instance, in underwater applications,3,8 but also in automotive and aircraft technology.9,10 In incompressible, isotropic turbulence, the probability distribution function (PDF) of pressure fluctuations is found to be non-Gaussian, i.e., it is negatively skewed due to an intermittent occurrence of tube-like, low-pressure filaments. This behavior has been revealed analytically11 and is supported by direct numerical simulations (DNS) of turbulence in periodic boxes.12,13 Negatively skewed PDF of turbulent pressure has been observed in swirling flows,14,15 jet flows,16 and boundary layer experiments.17 Near the wall, the PDF of turbulent pressure fluctuations becomes more symmetric, as it is shown, e.g., in channel flow DNS18 and wind-tunnel experiments.17 The PDF of wall-pressure fluctuations resembles, therefore, those of other turbulent quantities such as the

a)

Author to whom correspondence should be addressed. Electronic mail: [email protected] b) Formerly FWG: Bundeswehr Research Institute for Underwater Acoustics and Marine Geophysics. J. Acoust. Soc. Am. 135 (2), February 2014

Pages: 637–645

acceleration of Lagrangian particles,19,20 the pressure gradient in isotropic turbulence,21 and the velocity increments.22,23 The statistical behavior of those turbulent quantities has been modeled by a superstatistics approach.24–26 A Kolmogorov k7=3 spectral decay, as it is found for turbulent pressure fluctuations at sufficiently high Reynolds numbers,16,27 is considered also as an important regime in the wall-pressure power spectra. Various semi-empirical models of the power spectral density have been developed in order to mimic the one-point spectral behavior of turbulent wallpressure fields. In general, a wall-pressure spectrum includes a universal range with a xð0:71:1Þ decay from the spectral peak followed by a viscous subrange with a much stronger decay of x5 or even ex toward higher frequencies (see, e.g., Refs. 28–30 for recent studies). Pressure fluctuations of transitional31,32 and turbulent boundary layers have been preferentially studied in wind tunnel experiments,17,33–36 but measurements were also performed in towing tank facilities.37 Experimental studies have revealed that the meansquare pressure of wall-pressure fluctuations normalized by the square of the shear stress (or, alternatively, the dynamical pressure) is found to be almost constant (with a logarithmic correction) toward higher Reynolds numbers (see, e.g., Refs. 17 and 33, and references therein). Due to the convective nature of turbulent boundary layer flows, not only the spectral, but also the space-time correlation behavior, i.e., the wavenumber-frequency spectra, characterizes a turbulent wall-pressure field.2,3,7,38,39 Semi-empirical models of the wavenumber-frequency spectrum have been developed, for instance, by Corcos,40 Chase,41 and Smol’yakov.42 The convective nature in boundary layer flows also limits resolution of wall-pressure transducers.43 Underwater experiments on turbulent wall-pressure fluctuations and flow noise face particular difficulties since the

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C 2014 Acoustical Society of America V

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sea is a noisy environment and other environmental conditions, such as oceanic turbulence, can also have impact on the measurements.44 Underwater experiments,45 however, provide valuable information about the applicability of results from laboratory experiments and numerical simulation to experiments under sea conditions. Early experiments on wall-pressure fluctuations have been performed with an underwater towed body46 and later measurements were conducted with a free lifting body.47–49 Recently, wavenumberfrequency spectra of underwater wall-pressure fluctuations have been studied with a full scale towed array.50,51 Since flow noise significantly influences the performance of acoustical antennas, various approaches on flow noise reduction have been pursued, such as modifications of the interior structure8 or the outer surface.52 In this paper, we report results from an underwater experiment on flow noise beneath a turbulent boundary layer performed with a towed body under sea conditions. The towed body was designed for flow noise experiments of flat plate turbulent boundary layers (Sec. II A) and measurements were conducted in parameter regimes relevant for underwater applications (Sec. II B). Wavenumber-frequency filtering is applied in order to separate flow noise from other noise sources such as ambient noise (Sec. III A), and statistical and scaling properties of flow noise under sea conditions are studied (Sec. III B). II. EXPERIMENTS A. Underwater towed body

The towed body was designed by the company ATLAS Elektronik (Bremen, Germany) in cooperation with FWG for experiments on sonar self-noise under sea conditions. It is made of a ribbed steal frame surrounded by hard foam and has a length of 5:26 m and a width of 0:935 m. The total width and height (including fins) is 1:353 m and 1:715 m, respectively. The weight in air is 2:8 tons and the total mass of the body flooded with water amounts to 3:5 tons. The body has a net buoyancy of about 0:3 tons and is submerged by deploying the (heavy) towing cable. The length of the towing cable can be adjusted between 400 m and 1000 m in order to achieve the aspired towing depth at given towing speed. Flat plates confine the towed body in the bulk region on each side. On the port side, flat plate conditions are achieved by five rectangular panels ordered in a row. They were encapsulated by an outer layer made of polyurethane and were mounted by elastic damping elements onto the ribbed steel frame of the towed body. All cavities between the panels were smoothed in order to achieve as close as possible flat plate conditions. The panels contain a horizontal line array of M ¼ 30 equally spaced hydrophones having a spacing of Dx ¼ 70 mm, i.e., six hydrophones on each panel, for flow noise measurements. The array is orientated in a streamwise direction and has an acoustical length of L ¼ MDx ¼ 2100 mm. It consists of spherical hydrophones having a radius of r ¼ 14 mm, which are embedded within the outer layer of the panels and located beneath the surface at a distance of y ¼ 21 mm (in normal direction to the surface) with respect to center of the sphere. They are not 638

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directly exposed or connected to the flow, such as in case of flush-mounted or pinhole hydrophones. Phase stability of the array was validated with continuous wave pulses at different frequencies. These pulses were transmitted from a freely drifting buoy system. A schematic plot of the port side of the towed body is shown in Fig. 1(a). Two fins are mounted below the bow, while four fins, two horizontal and two vertically inclined ones, are attached to the tail in order the stabilize the motion of the body during measurements. The two fins mounted horizontally to the tail of the body can be actively controlled, but during data recording no machinery is active in order to avoid additional noise. The position of the linear array is indicated within the flat plate region in Fig. 1(a). A front and top view of the body (fins are omitted) are depicted in Figs. 1(b) and 1(c), respectively. Here, the shape of the leading and trailing edge of the towed body as well as the inclination of the flat plate region can be seen. B. Measurement procedure

The sea trials were conducted with a multipurpose research vessel in the Sognefjord, Norway. Because of its diesel-electric propulsion system, the level of radiated underwater noise of the vessel was relatively low. Thus, the flow noise measurements were not affected by towing vessel noise. The Sognefjord is well suited for towing experiments due to the large depth of more than 1000 m and a sufficient width. The towed body was operated well below the local thermocline at depths h ¼ 150; …; 100 m. Here, the influence from the sea surface as well as from oceanic turbulence is minimized in order to ensure as quiet as possible measurement conditions. The towing speeds varied from U ¼ 2:3; …; 6:1 m/s corresponding to ship speeds 4:5; …; 12 knots. Flow noise measurements were performed after the towed body’s motion was optimally adjusted toward a straight track. During a measurement, roll and pitch angles were stabilized below 65 and 62 . The corresponding

FIG. 1. Towed body in (a) side, (b) front, and (c) top view [note that fins are omitted in (b) and (c) for reasons of visibility]. The flat plate is located in the bulk region of the towed body; the position of the hydrophone array of an acoustic length l ¼ 2100 mm within the flat plate region is indicated. J. Abshagen and V. Nejedl: Towed body measurements of flow noise

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periods were Troll  3 s and Tpitch  5 s to 10 s. Boundary layer parameters were not measured in the experiments under sea conditions, but estimates of these quantities are given in Sec. III B 2. Data were recorded on board the towing vessel with a sampling frequency of fs ¼ 31 250 Hz, i.e., with a sampling time of Dt ¼ 32 ls. Throughout the paper, Na ¼ 10 nonoverlapping time series of length N ¼ 215 obtained from measurements with each of the 30 hydrophones are analyzed for each towing speed. The measurement time of a single time series is therefore T ¼ 1:0485 s and the total measurement time for each towing speed amounts to Na T ¼ 10:485 s. The frequency bandwidth is Df ¼ 1=T ¼ 0:9537 Hz and the wavenumber resolution is Dk ¼ 1=L ¼ 0:4762 1=m. Within each measurement interval, Na T, no significant fluctuations in root-mean-square pressure was observed, but moderate spatial variation of prms along the hydrophone array arose. Spatial Hamming windowing is found to improve wavenumber-frequency analysis qualitatively, but has no significant quantitative effect on the analysis. III. RESULTS A. Estimation of flow noise

Flow noise measurements are performed for the five different towing speeds U 2 f2:3; 3:3; 4:3; 5:4; 6:1g m/s. In Fig. 2, the power spectral density (PSD) Uðf Þ for all towing speeds is depicted. The PSD is calculated from averaging over all 30 single-point PSD of the line array, i.e., from 2   X   2 2pitn fj   pðxm ; tn Þwn e ; (1) Uðfj Þ ¼   2 Dfe n n

na ; m

xm ¼ mDx ðm ¼ 0; …; M  1Þ where pðxm ; tn Þ with denotes the pressure measured with the mth hydrophone at tn ¼ nDt ðn ¼ 0; …; N  1Þ and fj ¼ jDf ðj ¼ 0; …; N=2Þ denotes the jth frequency. The brackets h  ina ; m with

FIG. 2. (Color online) PSD Uðf Þ for each towing speed U 2 f2:3; 3:3; 4:3; 5:4; 6:1g m/s in comparison to the sea state 2 ambient noise level (dashed line; see Refs. 44 and 53 for discussion). Each PSD results from averaging over 30 single-point PSD. At higher frequencies, the spectral behavior is dominated by the ambient noise level. J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014

na ¼ 1; …; Na correspond to the average over the non-overlapping time series and all hydrophones of line array. Since Hamming windowing, i.e., a weighting qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi function wn ¼ ða  b cosð2pn=NÞÞ= ð2a2 þ b2 Þ=2 with a ¼ 0:54 and b ¼ 0:46, is applied, the effective noise bandwidth results in Dfe ¼ 1:36Df ¼ 1:2970 Hz. The increase of effective bandwidth by a factor 1:36 is a result from Hamming windowing. It can be seen from Fig. 2 that the PSD increases monotonically with increasing towing speed. For each towing speed, the PSD can be divided qualitatively into two regimes, i.e., a low-frequency regime up to about 103 Hz with a power-law-like spectral decay (i.e., an almost linear decay in the log–log plot) and a high-frequency spectral behavior similar to that of ambient noise. Note that the sea state 2 curve (see Refs. 44 and 53 for discussion) shown in Fig. 2 represents the environmental conditions during the measurements sufficiently. The spectral behavior in the high-frequency regime is dominated by the ambient noise level and, thus, flow noise estimation from PSD is corrupted by ambient noise in this frequency band. In the low-frequency regime of the PSD depicted in Fig. 2, the spectral power level exceeds that of the sea state 2 curve and, therefore, measurements of flowinduced noise are, in principle, possible with a single hydrophone (or with an average over a hydrophone ensemble). Flow noise, however, cannot be discriminated in single-point PSD from other noise source, such as, e.g., far-field noise from the towing vessel or flow-induced structure-borne sound. Wavenumber-frequency (kf) analysis, on the other hand, allows to separate different noise sources by their wave propagation speed. A kf-spectrum obtained from a measurement at towing speed U ¼ 5:4 m/s is depicted in Fig. 3. The kf-spectrum is calculated from

FIG. 3. (Color online) A kf-spectrum 10 logðUðk; f Þ=ðlPa2 m=HzÞÞ measured at towing speed U ¼ 5:4 m/s: Acoustic waves projected onto the line array are limited to the region jkj  f =c with c ¼ 1485 m/s (solid lines), while non-acoustic waves can also exist for jkj > f =c. Acoustic noise dominates the high-frequency regime while the low-frequency regime is governed by flow-induced structure-borne sound. The spectrum is high-pass filtered for reasons of visibility.

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2 Dfe Dke n2 m2 2   X   …  pðtn ; xm Þwn um e2piðtn fj þkl xm Þ  ;

Uðkl ; fj Þ ¼

n; m

na

(2) where kl ¼ lDk ðl ¼ m  M=2Þ denotes the lth wavenumber along the array and um ¼ ða  b cosð2pm=MÞÞ= qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2a2 þ b2 Þ=2 spatial Hamming weighting. The effective wavenumber bandwidth results in Dke ¼ 1:36Dk ¼ 0:6476 1=m. For reasons of visibility, the kf-spectrum is high-pass filtered with a slope of 35 dB/decade asymptotically below 2 kHz. In Fig. 3, a wavenumber, k, at frequency, f , reflects a plane wave propagating with wave speed, cp ¼ f =jkj along the array. Here, k > 0 and k < 0 correspond to propagation in and against the towing direction ex ¼ U=U, respectively. The wave speed c ¼ 1485 m/s of underwater sound is indicated in Fig. 3 (solid lines) and reflects propagating along the array in both directions 6ex . Sound wave with wavevector, k, propagating in directions other than 6ex have a lower projected wavenumber, k ¼ hkjex i, and therefore a higher projected wave speed, cp > c, along the array than underwater sound. Acoustic waves are therefore limited to the triangle jkj  f =c in Fig. 3. Waves that originate from sources other than underwater sound can also have wavenumbers with jkj > f =c, i.e., their wave speed along the array is lower than that of sound waves (cp < c). It can be seen from Fig. 3 that above f  2 kHz, the sound level in the acoustic regime exceeds that in the non-acoustic regime and, therefore, acoustic noise dominates the PSD for those frequencies. At lower frequencies, the sound level in the acoustic regime is governed by non-acoustic sound sources. A curved region of high sound level give raise to low-speed, dispersive structure-borne sound induced by the surrounded turbulent boundary layer as the dominant sound source. Characteristic measures of the sound level distribution in the non-acoustic wavenumber regime, i.e., for jkj > f =c, can be seen in Fig. 4. Here, the frequency dependence of the mean and the modes of the sound level distribution of the non-acoustic wavenumber regime are plotted. Since the values of modes and mean agree above f  2 kHz, the distribution is monomodal and symmetric, while for lower frequencies it becomes bimodal. The monomodal regime in Fig. 4 corresponds to the low-wavenumber regime of the turbulent boundary layer which is assumed to be wavenumber white. Therefore, the (mean) non-acoustic sound level can be extrapolated into the acoustic regime, and the mean sound level in the non-acoustic wavenumber regime provides an estimate for the flow noise level induced by the turbulent boundary layer at higher frequencies. This assumption allows an estimation of the flow noise level even if the PSD is dominated by other noise sources, such as, e.g., ocean ambient noise. It should be noted that below f  2 kHz, the mean sound level in the non-acoustic regime provides a reasonable estimate for the power of flow-induced noise, but it is below the self-noise level in sonar applications. 640

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FIG. 4. (Color online) Mean and the modes of sound level distribution in the non-acoustic wavenumber regime obtained at towing speed U ¼ 5:4 m/s. The distribution is bimodal below f  2 kHz due to the dominance of flow-induced structure-borne sound. Above f  2 kHz, the distribution is unimodal and symmetric. Note that for reasons of visibility, the curves are filtered with a moving-average of width 11Df .

B. Scaling and statistics of flow noise 1. Scaling behavior

In Fig. 5, two estimates of flow noise obtained from kf-analysis are shown, one obtained at low [Fig. 5(a)], i.e., U ¼ 2:3 m/s, and the other at higher [Fig. 5(b)], i.e., U ¼ 5:4 m/s, towing speed. The estimated flow noise level is below the mean sound power level at higher frequencies, but both curves converge toward lower frequencies. In this regime, the PSD is dominated by flow-induced noise. The departure of the flow noise level from the mean sound level at very low frequencies results from the dispersive behavior of structure-borne sound, as can be seen from the kfspectrum in Fig. 4. At very high frequencies, the flow noise level approaches the electronic noise level, which is also indicated in Fig. 5. Between f ¼ 250–2500 Hz, the estimated flow noise level obeys a power-law behavior and the fit results in a spectral decay of Uðf Þ / f 5:1 and Uðf Þ / f 4:5 for U ¼ 2:3 m/s and U ¼ 5:4 m/s, respectively. For all four towing speeds above U ¼ 2:3 m/s, the spectral decay varies only by f 4:560:1 . Therefore the scaling behavior can be considered as universal within the velocity regime U ¼ 3.3–6.1 m/s. In this regime, the flow noise level increases significantly with towing speed and the dependence of the spectral power, ð (3) PF ¼ Uðf Þ df ; F

of flow noise on towing speed in the frequency band F ¼ ½250 : 2500 Hz is depicted in Fig. 6. An increase of towing speed from U ¼ 2:3 m/s to U ¼ 3:3 m/s does not alter the spectral power PF significantly, but from U ¼ 3:3 m/s on the spectral power of flow noise increases with a power-law PF / U 6:9 toward higher towing speeds. J. Abshagen and V. Nejedl: Towed body measurements of flow noise

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FIG. 6. (Color online) Spectral power of flow noise, PF , in the frequency band F ¼ ½250 : 2500 Hz is plotted vs towing speed, U. A power-law increase with PF / U6:9 can be seen from U ¼ 3:3 m/s toward higher towing speeds.

FIG. 5. (Color online) Flow noise estimation for two different towing speeds, i.e., for (a) U ¼ 2:3 m/s and (b) U ¼ 5:4 m/s, kf-analysis allows the estimation of flow noise even below the mean sound level. A spectral decay (dashed line) of (a) f 5:1 and (b) f 4:5 is found in the frequency band F ¼ ½250 : 2500 Hz. It is limited from below by electronic noise (dotted curve) which can be considered as white within this frequency regime. Note that in order to illustrate the difference in spectral decay, the f 5:1 line from (a) is reproduced in (b) (dashed dotted line).

2. Nondimensionalized spectra

Boundary layer parameters and mean flow quantities were not directly accessible in the experiments. Therefore, a detailed analysis of the nature of the boundary layer flow, e.g., with respect to three-dimensional effects, is beyond the scope of this work. In order to nondimensionalize the PSD of flow noise, however, potential flow velocity and boundary layer parameters of the towed body are estimated. For that purpose, the flow around a two-dimensional, symmetric body having a flat plate region located between a semielliptical leading and trailing edge is considered. The flat plate region of the two-dimensional body has the same length and the semi-elliptical nose has the same radii as the leading edge of the towed body taken at the vertical position of the line array. The radii are a ¼ 402 mm and b ¼ 294 mm in streamwise and spanwise direction, respectively. This results in a ratio e ¼ b=a ¼ 0:73 and an arclength s1 ¼ 550 J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014

mm from the stagnation point (at s ¼ 0) along the semielliptical nose of the two-dimensional body toward the entrance of the attached flat plate. Together with the length s2 ¼ 755 mm of the flat plate between semi-elliptical nose and linear array, the total entrance length along the twodimensional body amounts to se ¼ s1 þ s2 ¼ 1305 mm. The two-dimensional potential flow U1 ðsÞ along the boundary of the two-dimensional body is calculated in arclength coordinates (s) from van K arm an’s singularity method (see, e.g., Ref. 54). In the array region behind the flow entrance, the potential flow velocity, U1 , is found to be (almost) constant and it is represented by the spatial average hU1 i ¼ hU1 ðse < s < se þ 2100 mmÞis . The laminar momentum thickness, Hlam , is calculated from U1 ðsÞ along the two-dimensional body from Thwaites’ method (see, e.g., Refs. 1 and 55), vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs u u 5 U1 ds u0:47 t s¼0 ; (4) Hlam ¼ 6 U1 with  ¼ 1:04 106 m2 =s and q ¼ 1026 kg=m3 being the kinematic viscosity and density of sea water, respectively. From Hlam , the laminar boundary layer thickness and the displacement thickness are derived under the assumption of flat and plate conditions, i.e., dlam ¼ 5 Hlam =0:664 dlam ¼ 1:7208 Hlam =0:664, respectively.1 The Reynolds number and the spatial location of laminar-turbulent transition in the boundary layer flow is estimated from an extension of Michel’s method given by Cebeci and Smith (see Ref. 55),   22 400 (5) Re0:46 ReHtr ¼ 1:174 1 þ str ; Restr with ReH ¼ U1 H= and Res ¼ U1 s= being the momentum thickness and arclength Reynolds number, respectively.

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Laminar-turbulent transition is estimated to take place close to the junction between semi-elliptical nose and flat plate region at about str  570 mm for all flow velocities considered. The turbulent boundary layer thickness, d, is estimated from flat plate relations with a virtual transition point s~ ¼ ðs  str  scorr Þ where scorr accounts for the correction due to the laminar boundary layer thickness, i.e., from1  1=5 U1 s~ d ¼ 0:37~ s : 

(6)

The displacement thickness, d , and momentum thickness, H, of the turbulent boundary layer are calculated from d ¼ d=8 and H ¼ 7d=72, respectively.1 The estimated boundary layer thickness, d, and displacement thickness, d , of the array region are shown in Fig. 7 for potential flow velocities between hU1 i ¼ 2:5  6:6 m/s. It can be seen that both the boundary layer thickness and the displacement thickness significantly increase in streamwise direction within the array region. Since the estimated PSD of flow noise results from pressure measurements distributed along the line array, the boundary layer quantities are averaged over the array region in order to perform nondimensionalization of PSD. The average quantities are given in Table I. In Fig. 8, the five nondimensionalized PSD of flow noise are depicted. Here, time is scaled by s ¼ hd i=hU1 i and pressure by the dynamical pressure, q ¼ qðU1 Þ2 =2. Frequency is given in x ¼ 2pf , which yields UðxÞ ¼ Uðf Þ=ð2pÞ. Since hd i is calculated from hU1 i, the (calculated) potential flow velocity instead of the (measured) towing speed, U, is the typical velocity scale for nondimensionalization. It can be seen that four nondimensionalized PSD at the higher (averaged) potential flow velocities hU1 i 2 f3:6; 4:4; 5:9; 6:6g m/s collapse well onto each other. Only the PSD at lowest speed hU1 i ¼ 2:5 m/s depart from the other curves in both the power spectral level and

FIG. 7. (Color online) Estimated values of turbulent boundary layer thickness d (dashed curves) and displacement thickness d (solid curves) for different potential flow velocities from hU1 i ¼ 2:5 m/s (thick line) to hU1 i ¼ 6:6 m/s (medium thick line). 642

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TABLE I. Boundary layer thickness hdi, displacement thickness hd i, and momentum thickness hhi calculated for different potential flow velocities hU1 i. The quantities are spatially averaged (h  i) over the array region and the corresponding standard deviation (stdð  Þ) of the boundary layer quantities is given. hU1 i ðm=sÞ hdi ðmmÞ stdðdÞ ðmmÞ hd i ðmmÞ stdðd Þ ðmmÞ hhi ðmmÞ stdðhÞ ðmmÞ

2.5 31.6 8.8 3.9 1.1 2.7 0.9

3.6 29.4 8.2 3.7 1.0 3.1 0.8

4.7 27.9 7.8 3.5 1.0 2.9 0.8

5.9 26.4 7.5 3.3 0.9 2.7 0.7

6.6 25.8 7.3 3.2 0.9 2.5 0.7

the spectral decay. No evidence is found that the departure goes along with a significant change in the flow regime, such as, for instance, the appearance of laminar-turbulent transition within the array region or with a qualitative change in the motion of the towed body. This gives rise to the conclusion that this departure indicates the lower limit of the scaling regime. Note that a collapse of PSD normalized by outer scaling variables s and q implies a connection between spectral decay xa and the dependence of spectral power on flow velocity like /ðxÞ / d 1a hU1 i3þa xa , with d being a function of hU1 i (see Ref. 47). C. PDF

Wall-pressure fluctuations in turbulent boundary layers have a non-Gaussian PDF, while the PDF of ocean ambient noise is known to be Gaussian. Since the pressure signal measured at the hydrophones results from a mixture of different noise sources, such as underwater ambient noise, flow-induced structure-borne sound, and flow noise, two different types of filtering, a high-pass filter and a kf-filter, are applied in order the determine the PDF of flow noise. In the low-frequency regime, flow-induced structure-borne sound

FIG. 8. (Color online) Flow noise spectra nondimensionalized by (averaged) displacement thickness hd i for different potential flow velocity hU1 i. All curves collapse onto a single spectrum with a scaling behavior ððxhd iÞ=hU1 iÞ4:3 except for the one for lowest towing speed (corresponding to hU1 i ¼ 2:5 m/s). Here, both spectral decay and (normalized) sound level depart from the other curves. J. Abshagen and V. Nejedl: Towed body measurements of flow noise

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dominates the PSD. Therefore, a high-pass filter having an asymptotic slope of 70 dB/decade and an edge frequency of 2 kHz is used. Furthermore, the amplitude in the acoustic wavenumber regime, i.e., the region within the triangle (jkj < f =c0 ), is reduced by an additional 60 dB. Since the width of the non-acoustic wavenumber regime ðjkna j > f =c0 Þ is frequency dependent, the amplitude of flow noise is corrected by the inverse of the relative width, i.e., by Mdk=kka . Note, that this correction has only a very weak quantitative effect and does not change the behavior of the PDF qualitatively. In Fig. 9(a), the relative frequency of the filtered pressure time series is depicted for two different towing speeds. For the towed body being at rest, i.e., for U ¼ 0 m/s, a Gaussian PDF (straight line) is an excellent representation of the filtered pressure data (open circles). At high towing speed, i.e., at U ¼ 6:1 m/s, the distribution of the filtered pressure data (filled circles) becomes non-Gaussian. A

reasonable representation is given by a v2 (Tsallis)superstatistics,26   n þ 1 rffiffiffiffiffiffi ðnþ1Þ=2 C b b 2 2a  P¼ : (7) 1 þ jp=rj n pn n C 2 The fit to the filtered pressure data results in b ¼ 1:0306, n ¼ 61:7788, and a ¼ 1:0005. The difference in frequency between filtered pressure data and the Gaussian PDF is depicted in Fig. 9(b). The condition a ¼ 2  q, with q ¼ 1 þ 2=ðn þ 1Þ, which has been found in studies22,25,26 on other turbulence quantities is not exactly, but only approximately, met. Note that without filtering, the estimated PDF of the pressure time series is found to be Gaussian for all towing speeds. Evidence is given that non-Gaussian PDF of flow noise can be represented by a v2 (Tsallis)-superstatistics, which is in accordance with studies on other turbulent quantities, such as on the velocity difference or the pressure gradient.22,24–26 Departures, in particular, in the tail of the distributions may become relevant which are not accessible in this study. Therefore, other possible statistical representations, such as a log-normal superstatistic,26 may also be possible. It can be concluded, however, that concepts from nonGaussian statistics are applicable and play a role in flow noise measurements under sea conditions. IV. CONCLUSION

FIG. 9. (Color online) (a) Relative frequency obtained from filtered pressure time series for two different towing speeds, U ¼ 0; 6:1 m/s, Gaussian PDF, and fit with v2 -superstatistics, (b) frequency difference to Gaussian PDF. The pressure is centralized by the mean and normalized by the standard deviation, r. J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014

Measurements of underwater flow noise beneath a flatplate turbulent boundary have been performed with an acoustical array inside a towed body at various towing speeds U ¼ 2:3  6:3 m/s and depths of about h ¼ 150; …; 100 m. Properties of the turbulent boundary layer, such as, e.g., boundary layer thickness, were not directly accessible in the experiments, but estimates have been obtained from numerical simulations and semi-empirical turbulent laws.1,55 Accurate estimates of flow noise spectra could be obtained and a physically reasonable scaling behavior ðxhd i=hU1 iÞ4:3 has been revealed from kf-analysis of the experimental data (except for the lowest towing speed U ¼ 2:3 m=s). kf-analysis allows to focus on the contribution of non-acoustic noise sources to the PSD and separate those from hydroacoustic sources, such as ocean ambient noise and towing vessel noise. In particular, the (nondimensionalized)  2 power spectral densities UðxÞhU1 i= ðhd i 1=2qhU1 i2 Þ obtained at higher towing speeds collapse onto each other with the appropriate scaling parameters x, hd i, and hU1 i, i.e., (angular) frequency, (averaged) boundary layer displacement thickness, and (averaged) potential flow velocity, respectively. A power law decay ðxhd i=hU1 iÞ4:3 over a wide frequency range differs qualitatively from wall-pressure fluctuation spectra. We observe neither a universal range with xð0:71:1Þ decay nor a strong viscous decay of x5 , or even ex . The spectral behavior found in our experiment resembles those observed in previous underwater towed body46

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and laboratory47 experiments. Particular agreement is found to buoyant body and wind-tunnel data (collected in Ref. 56). Here, a spectral decay of ðxd =U1 Þ4:5 is reported. A steep increase of spectral power with flow velocity, such as, e.g., P / U 6:9 , is known from turbulent dipole (/ U 6 ) or quadrupole (/ U 8 ) sources. Wall-pressure fluctuations, on the other hand, increase approximately to the fourth power of the flow velocity (with logarithmic corrections).33 In our experiments, the hydrophones are not exposed directly to the flow, but are embedded in the panels. The contribution of flow-induced structure-borne sound to flow noise generation, in particular, in the low-frequency regime can be seen from the kf-spectrum depicted in Fig. 3. In this regime, it differs qualitatively from kf-spectra of wall-pressure fluctuations measured, e.g., with a towed array.50 In this work, we have provided experimental evidence that underwater flow noise beneath a flat plate turbulent boundary layer exhibits a non-Gaussian PDF in the lowwavenumber, high-frequency regime, i.e., for wavenumbers far away from the convective ridge. A v2 (Tsallis)-superstatistics is found to provide a representation of the filtered pressure data, which is in accordance with studies on other turbulent quantities. Under sea conditions, the PSD from a single-point hydrophone measurement (or from an average over a hydrophon ensemble) is dominated by structure-borne sound and ocean ambient noise in the low- and high-frequency regime, respectively. It is therefore difficult to obtain a reliable estimate of the flow noise level from such a measurement. kf-analysis, on the other hand, can provide an appropriate mean for flow noise estimation even if the flow noise level is below those of the other noise sources. ACKNOWLEDGMENTS

We thank the team from ATLAS Electronics, T. Richter (WTD71-FWG), and the crew of Multi Purpose Vessel Schwedeneck for excellent technical support during the sea trial. 1

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Towed body measurements of flow noise from a turbulent boundary layer under sea conditions.

Results from an underwater experiment under sea conditions on flow noise beneath a flat-plate turbulent boundary layer are presented. The measurements...
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