Coherent flow noise beneath a flat plate in a water tunnel experiment

Journal of Sound and Vibration 340 (2015) 211–220

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Coherent flow noise beneath a flat plate in a water tunnel experiment J. Abshagen a,n, I. Schäfer a, Ch. Will b, G. Pfister b a Research Department for Underwater Acoustics and Marine Geophysics (FWG), Bundeswehr Technical Center WTD71, Berliner Straße 115, 24340 Eckernförde, Germany b Institute of Experimental and Applied Physics, University of Kiel, Olshausenstraße 40, 24098 Kiel, Germany

a r t i c l e i n f o

abstract

Article history: Received 19 November 2013 Received in revised form 25 November 2014 Accepted 26 November 2014 Handling Editor: R.E. Musafir Available online 31 December 2014

Results from a combined experimental and numerical study on the properties of flow noise on the reverse side of a flat plate excited by a turbulent boundary layer flow are reported. Particular focus is given to the coherence between wall pressure fluctuations, plate vibrations, and interior flow noise. The plate was immersed in water and laterally attached to a streamlined model inside the HYKAT cavitation tunnel at HSVA Hamburg (Germany). The flow velocity in the tunnel was U ¼ 7 m/s. Simultaneous measurements of wall pressure fluctuations, structural vibrations, and interior flow noise as well as a numerical response and eigenvalue analysis provide evidence that evanescent plate modes excited by wall pressure fluctuations play a crucial role in flow noise generation. Flow noise in the (quiescent) water inside of the model is found to have large coherence lengths and pronounced amplitudes for distinct frequencies below f  300 Hz. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction Flow induced noise originating from a moving body which is surrounded by a turbulent boundary layer flow is not only of large relevance for many applications, but also of fundamental scientific interest [1,2]. Propagating away from the moving body this flow induced sound may be perceived as unwanted noise in the far field. Examples arise from aircraft noise [3], ship noise [4], and wind turbine noise [5]. If there are fluid filled spaces inside the body, pressure fluctuations can generate additional noise therein. This is known, for instance, as cabin noise of aircrafts [6] and cars [7] as well as in sonar applications [8]. In the latter case flow induced noise is generated by the motion of a sonar antenna through quiescent water and contributes to sonar-self-noise [9]. The fluctuations of velocity and pressure generated inside an (incompressible) turbulent boundary layer flow act on an underlying mechanical wall structure [10]. The fluctuations behave as a random field with a small coherence area and a slow propagation component in the flow direction. Wall pressure fluctuations beneath a turbulent boundary layer are of particular interest due to their dominant role in turbulent sound generation and have therefore been subjected to numerous experimental and theoretical investigations in the past [11–19]. Various semi-empirical models of the temporal and the space–time correlations behaviour of wall pressure fluctuations have been developed in order to understand the underlying physical processes but also to cope with applicational needs [20–25].

n

Corresponding author. E-mail address: [email protected] (J. Abshagen).

http://dx.doi.org/10.1016/j.jsv.2014.11.033 0022-460X/& 2014 Elsevier Ltd. All rights reserved.

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Inside of a fluid-filled (moving) body pressure fluctuations can only propagate as sound waves and the wall structure plays a crucial role in the generation of this interior flow noise [26]. In principle turbulent pressure fluctuations can be transmitted locally through a wall structure into the fluid on the reverse side, but typically the mechanical wall structure responses to excitations from the turbulent boundary layer. This response reflects the modal character of the mechanical wall structure [27]. The vibroacoustical response to turbulent boundary layer excitation has been studied in detail, for instance, for flat plates (see e.g. [1,28,29]). Here, the Finite-Element Method [30–32] has played a crucial role. A particular focus of recent work in vibroacoustic response to turbulent excitations is given to the improvement of predictive methodology [33–36]. Fluid– structure interaction between boundary layer flow and wall structure can furthermore be considered to be of importance for the generation of flow-induced noise [37]. In this work the vibroacoustic response of a flat plate to a turbulent boundary layer excitation and the hydroacoustic pressure field in the vicinity of the vibrating plate are investigated. The plate is entirely immersed in water and flow-induced noise is measured on the reverse side of the plate, i.e. in the (quiescent) interior of a model surrounded by the turbulent boundary layer flow. In order to understand the relevant sources of flow-induced noise in the experiment particular focus is given to the coherence between wall pressure fluctuations, plate vibrations, and interior flow noise. Vibroacoustic properties and hydroacoustic response of the plate are determined by a numerical simulation based on the Finite-Element method.

2. Experimental setup

1.6m

900

100

The experiments are performed with a streamlined model in the HYKAT cavitation tunnel at HSVA, Hamburg. The model has a symmetrical outer shape in flow direction and is made of coated wood. Inbetween a curved nose and tail the model contains a flat plate region which is inclined in vertical direction. The shape of the model is based on a towed body which was designed for flow noise measurements at open sea [38]. A schematic drawing of the starboard side of the towed body is depicted in Fig. 1(a). The outer shape of the lower part of wooden model and towed body is identical (below dashed line), except that fins have been omitted in the tunnel model. Furthermore an additional extension of 100 mm in vertical direction has been added to the tunnel model in order to reduce the influence of upper tunnel wall turbulence on the measurement region. The tunnel model (without vertical extension) has a maximal length and width of 5212 mm and 930 mm, respectively, and a height of 900 mm. The width of the model reduces to 393 mm at the bottom which results in a tilt angle within the flat plate region of 16.61 in spanwise direction. The location of the model (front view) inside the cavitation tunnel of height 1600 mm and width 2800 mm is indicated in Fig. 1(b). The inclination of the model as well as the vertical extension (online: green) can be seen. On the port side the tunnel model is closed while on the starboard side a flat plate made of perspex is mounted flush to the model. The plate has a size of 2446 mm  766 mm in streamwise and spanwise direction, respectively, and a thickness of d ¼25 mm. The location of the flat plate can be clearly seen in Fig. 1 (a) (online: blue). Different types of sensors are mounted on and positioned behind the flat plate. The positions of the sensors are concentrated in an area inside the flat plate, as indicated in Fig. 1(a) (online: green). This area is symmetric to the flat plate (dashed line: axis of symmetry) and is located at a distance of 655 mm from the leading edge of the plate. A detailed view of the measurement area and the sensor positions is given in Fig. 2. The accelerometers (ACC, online: red) of Type B&K 5958 are mounted on the inside of the plate. One flush-mounted hydrophone (FMH, online: blue) of type RESON TC4050 is located on the symmetry axis in order to measure wall pressure fluctuations. Inside of the model a linear hydrophone array aligned in streamwise direction is positioned on the symmetry axis at a distance of y¼20 mm in normal direction from the inner wall of the plate. The array consists of 16 equidistant hydrophones of type RESON TC4013 which have a spacing of 11.5 mm. It is mounted by elastic damping elements onto a (heavy) support in order to decouple the array from the wooden model as well as from the plate. The measurement distance between hydrophone array and plate was chosen in order to capture also near-field effects of the hydroacoustic pressure field. Here, a

766

2446

655

393

2.8m

930

Fig. 1. (a) Schematic plot of the towed body (for open sea experiments [38]) with the flat plate region (blue) and position of the measurement region (green) inside of the flat plate area. (b) Schematic front view of the wooden model inside the HYKAT cavitation tunnel of HSVA, Hamburg. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

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213

ACC 1 93

FMH

173

Hyd 1

237

150

400

100

ACC 3

Hyd 16 150

ACC 2

Fig. 2. Measurement region with position of three accelerometers (ACC), one flush-mounted hydrophone (FMH), and 16 equally spaced hydrophones Hydn, n ¼ 1…16 (here only the position of Hyd1 and Hyd16 is plotted). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

distance in the same order of magnitude as the plate thickness is found to be appropriate. A check at a smaller distance has not revealed any qualitative changes in the results. In Fig. 2 the position of the two outer hydrophones Hyd1 and Hyd16 (online: green) are depicted (Hyd2⋯15 are omitted for reasons of visibility). During the measurements the wooden model is entirely flooded with water and no evidence for significant fluid motion inside of the model has been found. The tunnel flow velocity is measured with a Prandtl tube located on the port side near the outer tunnel wall at (approximately) the same vertical height as the hydrophone array. The measurements presented in this study have been performed at a tunnel flow velocity of U¼7 m/s. 3. Numerical analysis of plate vibrations In order to understand the vibroacoustic properties and hydroacoustic response of the plate immersed in water a numerical analysis based on the Finite-Element Method (FEM) has been performed. This includes an investigation of the eigenmodes and eigenfrequencies as well as a numerical response analysis. Here, a spatially localised pressure perturbation is applied to the plate in order to mimic the excitation from turbulent wall pressure fluctuations within the coherence area. This approach is aimed to evaluate the relevant modal contribution to the hydroacoustic pressure field at a hydrophone position on the reverse side of the plate. 3.1. Structural modes A numerical eigenvalue analysis of a full scale FEM model of a plate immersed in water has been performed. The shell elements of the plate are of Reissner–Mindlin (RM) type, while the behaviour of the surrounding water is governed by Helmholtz equation. The plate is meshed with triangular elements of size 0.01–0.05 m, while for the surrounding water a tetrahedral mesh with an element size 0.01–0.2 m is used. Near the source region the grid is finer than further away. Perspex (PMMA) of the RM-plate has a density ρPMMA ¼ 1180 kg=m3 , Poisson's ratio ν ¼0.4, Young's modulus E ¼ 3:3 109 N=m2 , and a ratio between loss and storage modulus tan ðδÞ ¼ 0:05, while the density and the speed of sound in the Helmholtz equation is assumed to be ρH2 O ¼ 1000 kg=m3 and c ¼1490 m/s, respectively. The numerical simulation was performed with the COMSOL Multiphysics© packages. Pinning, i.e. vanishing displacement, as boundary conditions of a corresponding plate of size 2350 mm  666 mm has been found to be a sufficient representation of the physical boundaries. This was determined by comparison between the first eigenmodes in experiment and numerics. The reduced size of the corresponding plate in comparison to the original size of 2466 mm  766 mm in streamwise and spanwise direction, respectively, owes to the fact that the perspex plate is screwed to the wooden model. Therefore, the boundary conditions of the original plate are not clearly defined. For reasons of clarity but also due to (unavoidable) uncertainties that are inherent in the modelling of physical boundaries the analysis is restricted to eigenfrequencies below f ¼140 Hz. In spite of this restriction the first 15 eigenmodes are covered by our analysis. In Table 1 the eigenfrequencies f i;j for the eigenmodes Ψ i;j (with i; j ¼ 1; 2; …) of the corresponding flat plate are represented. Note that the numbers ði 1Þ and ðj 1Þ denote the number of nodes in stream- and spanwise direction, respectively. It can be seen that the lowest six eigenmodes which corresponds to eigenfrequencies up to f 6;1 ¼ 60:8 Hz do not have a node in spanwise direction. The spatial displacement pattern of two examples of the eigenmodes, i.e. of (a) Ψ 3;1 and (b) Ψ 3;2 , is depicted in Fig. 3. 3.2. Localised wall pressure excitation In order to shed light on the mechanism of flow noise generation on the reverse side of the plate the excitation of the plate by a localised wall pressure perturbation is studied. The perturbation has the form of a travelling wave in streamwise direction having a (non-moving) modified Gaussian envelope. A travelling wave mimics the wall pressure signature of a localised vortical structure which is advected as part of a frozen turbulent pattern with (a fraction of) the mean flow [1,2,10,11]. Due to the disparity of wavenumber scales between vortical perturbation and plate eigenmodes Ψ ði; jÞ, the calculation is simplified by keeping the streamwise wavenumber of the travelling wave fixed to kx ¼ 40 m  1 . This value results from

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Table 1 Eigenfrequencies f i;j of eigenmodes Ψ i;j of a flat plate immersed in water with i and j denoting the number of nodes in stream- and spanwise direction, respectively. Ψ i;j f i;j (Hz)

Ψ 1;1 11.6

Ψ 2;1 15.4

Ψ 3;1 21.8

Ψ 4;1 31.3

Ψ 5;1 44.1

Ψ i;j f i;j (Hz)

Ψ 6;1 60.7

Ψ 1;2 62.8

Ψ 2;2 67.6

Ψ 3;2 75.0

Ψ 7;1 81.2

Ψ i;j f i;j (Hz)

Ψ 4;2 86.8

Ψ 5;2 101.7

Ψ 8;1 105.9

Ψ 6;2 120.2

Ψ 9;1 134.8

Fig. 3. Numerically calculated eigenmodes (a) Ψ 3;1 and (b) Ψ 3;2 for the eigenfrequencies f 3;1 ¼ 21:8 Hz and f 3;2 ¼ 75:0 Hz, respectively, of a plate immersed in water (normal displacement is colour coded).

200

100

y=0

2350

Fig. 4. (a) Snapshot of localised wall pressure excitation (real part of pressure is visualised by colour coded displacement), (b) pressure field in the water on both sides resulting from an excitation with 24 Hz (plate is located in horizontal direction at y¼0).

kx ¼ f max =U c and a lower bound for the convective speed U c ¼ U=2 ¼ 3:5 m=s and f max ¼ 140 Hz as an upper limit of the frequency regime considered in the work. Propagation speed of the travelling pressure wave varies therefore between ctw ¼ 0:25 and 7 m/s with 10 Hz being the lowest frequency considered in this work. Though these speeds do not fit exactly to the experimental conditions they are still two to three orders of magnitudes lower than the acoustic wave speed in water of c ¼1485 m/s which has been measured in the tunnel. Therefore travelling pressure waves are far from coincidence. The spatially localised wall pressure excitation reads in complex form: pðx; y; kx Þ ¼ ζ ðxÞe  ðx  xs Þ

2

=2ξx

2

2

e  ðy  ys Þ

=2ξy

2

e  ikx ðx  xs Þ

(1)

with xs and ys representing the position of the perturbation on the plate and ζ ðxÞ the up–downstream asymmetry in the space–time coherence functions. The size of the wave packet in streamwise direction is set to ξx  32 mm which is the estimated thickness of the turbulent boundary layer calculated for the corresponding towed body [38], i.e. the largest scale for a vortical structure. Since the spanwise coherence length is typically considered as significantly smaller than the streamwise [1,2], ξy  12 mm is taken as an upper bound for the size of the wave packet in spanwise direction. Calculations with lower values of ξy have revealed that the results are insensitive to this parameter. In order to mimic up–downstream asymmetry in the space–time coherence functions of turbulent boundary layers [11] an (approximate) step function

ζ ðxÞ ¼ f1  tanh½200ðx xs Þg=2

(2)

is added to the envelope of the wave packet (Eq. (1)). A localised pressure perturbation is visualised in Fig. 4(a) within a rectangular surface of 200 mm  100 mm for an instant point of time (the real part is plotted). For a qualitative analysis the exact shape of the envelope is found not to be crucial, but only the variation of the net pressure with time. Numerically, the wall pressure perturbation is not directly incorporated, but treated as an inhomogeneity of the Helmholtz equation located at the upper fluid–structure interface. It should be stressed that these are therefore purely combined structural and hydroacoustic, but not hydrodynamic, simulations.

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In Fig. 4(b) the streamwise and wall normal component of the pressure field in water as a response to a localised pressure perturbation applied to the plate with an excitation frequency fe ¼24 Hz is shown. The perturbation is located at ðxs1 ; ys1 Þ with xs1 ¼ x0 þ250 mm and ys1 ¼ y0 (x0 : ¼ lx =2 ¼ 1175 mm and y0 : ¼ ly =2 ¼ 333 mm being the centre of the corresponding plate) and it can be seen that the response is non-local in streamwise direction, i.e. eigenmodes are excited. A non-local response occurs also in the spanwise direction, but this is not shown here. Since the excitation frequency fe is not an eigenfrequency, the plate responds with a superposition of eigenmodes which yield a spatially (inhomogeneous) pattern shown in Fig. 4(b). Note that the perturbation is not shown in Fig. 4(b) but only the modal response. It can be seen in Fig. 4(b) that the hydroacoustic sound field in the surrounding water decays with normal distance to the wall, and therefore the excited hydroacoustic modes are evanescent and do not radiate in wall normal direction. Nevertheless, they are of crucial importance for the flow noise generation in the direct vicinity of the plate. The hydroacoustic pressure level of the modal response dominates by far the contribution of the local transmission of the localised pressure perturbation through the plate to the reverse side. The localised part of the hydroacoustic sound field has therefore been neglected in our analysis. In particular the modal response to a localised perturbation is non-local and contributions from perturbations at all positions on the plate have to be taken into account in order to determine the pressure field at a single measurement position. In our experiment it can be assumed that wall pressure fluctuations in the turbulent boundary layer flow have a coherence area that is much smaller than the size of the plate. Therefore, in our model the modal response from localised pressure perturbations with a distance larger than the coherence lengths has to be summed up incoherently, i.e. the sum over the square magnitudes of the pressure has to be taken. In order to save computational cost the reciprocity principle is used [41]. Instead of summing up the modal response from various perturbations distributed over the plate at a single measurement point, only a single localised wall pressure perturbation is applied at the measurement position and average the square magnitude of the pressure over the plate area, i.e. N 〈p2 ðf Þ〉A ¼ ∬A jpðf Þj2 dA A

(3)

Here, A ¼ lx ly is the total area of the corresponding plate with lx ¼2350 mm and ly ¼ 666 mm in stream- and spanwise direction, respectively. The pressure field at the measurement position is yielded by the sum over N incoherent sources pffiffiffiffiffiffiffiffiffiffi weighted with the (spatially) averaged mean-square pressure. If half-width at half maximum, i.e. 2ln2ξx;y , is assumed as a typical length scale of the localised wall pressure perturbation in stream- and spanwise direction then the area can be estimated as Ae ¼ π ln2ξy ξx  836 mm2 (note that only half of an elliptical area is considered due to the step function ζ ðxÞ). The entire plate can therefore be covered by N ¼ A=Ae  1872 incoherent sources. Note that in case of a δ-function the result for 〈p2 ðf Þ〉A would be exact due to reciprocity principle. Because of the strong spatial localisation of the applied pressure perturbation, however, our procedure still provides a good approximation. The calculation of the averaged pressure response 〈p2 ðf Þ〉A is done separately for each frequency f. Two different positions of the perturbation on the plate have been analysed, one located at ðxs1 ; ys1 Þ and the second at xs2 ¼ x0 þ350 mm ys2 ¼ y0 . Measurement position ðxs2 ; ys2 Þ agrees roughly with hydrophone position Hyd1 in streamwise direction. This is located at a distance of dðHyd1 2446=2 mmÞ ¼ 342 mm from the centre of the perspex plate. The result of the calculations for two different measurement positions ðxs1;2 ; ys1;2 Þ is given in Fig. 5.

Fig. 5. Averaged pressure response 〈p2 ðf Þ〉A (Eq. (3)) numerically calculated for measurement positions ðxs1 ; ys1 Þ ¼ ðx0 þ 250 mm; y0 Þ ( ) and ðxs2 ; ys2 Þ ¼ ðx0 þ 350 mm; y0 Þ ( ). Eigenfrequencies f i;j are given for comparison.

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Fig. 6. (a) Power spectral density Φðf Þ of wall pressure fluctuations ( ), interior flow noise ( ), and the incoherent part of spectral power, i.e. Φðf Þinc ðÞ. (b) Coherence γ2 of interior flow noise along the hydrophone array with respect to Hyd1 at d ¼ 0 mm.

It can be seen that 〈p2 ðf ; xs1;2 Þ〉A has pronounced peaks at the eigenfrequencies corresponding to the even eigenmodes Ψ i;1 (dotted lines). The odd eigenmodes Ψ i;2 (dashed lines) do not contribute substantially to the sound field at the measurement position ys1;2 due to the node being located on the up–down symmetry axis. Furthermore it is evident from Fig. 5 that the peak levels of 〈p2 ðf Þ〉A depend significantly on the exact measurement positions. For instance, point ðxs1 ; ys1 Þ corresponds to position Hyd9 of the hydrophone array and is located in the vicinity of a nodal point of Ψ 6;1 . Therefore the peak 〈p2 ðf 6;1 Þ〉A is substantially reduced. 4. Coherence of interior flow noise In this section the spectral and coherence properties of flow noise in the interior of the streamline model inside the water tunnel are investigated. Power spectral density (PSD) of wall pressure fluctuations (þ) and interior flow noise (solid line) measured at positions FMH and Hyd1 are depicted in Fig. 6(a). The flow speed in the water tunnel was U¼7 m/s and a bandwidth of Δf ¼ 1 Hz was used. Below f¼10 Hz disturbances which originate from the experimental setup contaminates the spectra and therefore this low frequency regime is not considered here. The wall pressure fluctuations display a broadband spectral behaviour showing a strong decay for frequencies above f  300 Hz. Note that the kink in the spectrum at about f  650 Hz is related to wavenumber filtering due to the finite transducer size of the flush-mounted hydrophone [39]. The spectral behaviour of interior flow noise deviated significantly from that of wall pressure fluctuations. The spectral level is substantially lower and the spectral decay differs qualitatively from that of wall pressure fluctuations. Interior flow 3 noise is found to decay with a power law Φðf Þ p f in the frequency regime above f  300 Hz. For lower frequencies pronounced spectral peaks can be observed in the power spectral density depicted in Fig. 6(a). Along with the appearance of spectral peaks goes an increase of the spatial coherence length of interior flow noise. The incoherent contribution Φinc ðf Þ of the power spectral density Φðf Þ at Hyd1 is shown in Fig. 6(a). Φinc ðf Þ is calculated from

Φinc ðf Þ ¼ f1  γ 2 ðHyd1 ; Hyd9 ÞgΦðf Þ

(4)

Here, Hyd9 is taken as a reference hydrophone. Since γ 2 ðHyd1 ; Hyd9 Þ ¼ 1 would correspond to a noise-free, linear relation between the two hydrophone signals, Φinc ðf Þ can be interpreted as the noise spectrum (see e.g. [40] for details). The spatial coherence as a function of frequency and distance d from Hyd1 is given in Fig. 6(b). Above f  300 Hz the coherence decays strongly with d and has almost vanished for d≳60 mm. Below f  300 Hz a substantial increase of spatial coherence superimposed by scattered (narrow) bands of low coherence length can be seen in Fig. 6(b). The increase of spatial coherence is reflected by a reduced spectral level of Φinc ðf Þ below f  300 Hz, as shown in Fig. 6(a). The power spectral density Φðf Þ as well as the (spatially) coherent part of interior flow noise at position Hyd1 is depicted in Fig. 7(a) and (b), respectively. The latter is represented by the ratio 10log10 ðΦðf Þ=Φðf ÞÞinc Þ (ref. Hyd9). The (spatial) coherence of interior flow noise determined at three different distances from Hyd1 along the array, i.e. at d ¼46, 115, 184 mm, is shown in Fig. 7(c). These curves correspond to the colour coded values of coherence depicted in Fig. 6(b). The bandwidth used in Fig. 7(a) and (b) is Δf ¼ 0:5 Hz instead of Δf ¼ 1 Hz. This smaller bandwidth allows an accurate comparison between power spectral density and numerically calculated eigenfrequencies. Coherence, however, is typically found to be better represented with a bandwidth of Δf ¼ 1 Hz, such as in Fig. 7(c). The lowest five eigenfrequencies (corresponding to the even eigenmodes Ψ ð1⋯5Þ;1 ) agree (almost) quantitatively with spectral peaks of interior flow noise represented in (a). This goes along with a high degree of spatial coherence along the hydrophone array which can be seen from Fig. 7(b) and (c). The signature of the seventh and the ninth even eigenmode, i.e. of Ψ 7;1 and Ψ 9;1 , is less pronounced (though still noticeable) in the power spectral density, but the level of coherent flow

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Fig. 7. (a) Power spectral density Φðf Þ ( ) of interior flow measured at position Hyd1, (b) ratio ( ) between Φðf Þ and Φinc ðf Þ (ref. Hyd9) at position Hyd1, (c) coherence γ2 determined at three positions along the hydrophone array with respect to position Hyd1 ðΔf ¼ 1 HzÞ. Eigenfrequencies f i;j are given for comparison.

noise as well as their coherence function represented by Fig. 7(b) and (c), respectively, has high values near those eigenfrequencies. In contrast to that the spectral contribution of the sixth and eighth eigenmodes, i.e. to Ψ 6;1 and Ψ 8;1 , is not significant in the experiments. The additional peaks in the vicinity of the eigenfrequency corresponding to Ψ 6;1 are not covered by the numerical simulation. It should be stressed that the quantitative agreement in frequency declines towards higher frequencies. In a perfect up–down-symmetric system odd eigenmodes would not contribute to the flow noise measured at a point located directly on the up–down symmetry axis. This behaviour is reflected in the numerical calculation of 〈p2 ðf Þ〉A for different measurement positions ðxs1;2 ; ys1;2 Þ, as can be seen from Fig. 5(a). Indeed, no significant spectral signature can be found for the lowest odd eigenmodes, i.e. Ψ ð1;3Þ;2 , in the experiments. From the second lowest odd eigenmodes on, however, signatures in the power spectral density can be found. This holds particularly for the 14th eigenmode Ψ 6;2 . 5. Structural vibrations and flow noise In this section the behaviour for structural vibrations and its origin as a source for interior flow noise in the experiment is investigated. Experimentally plate vibrations and spatial coherence are measured with three accelerometer positioned as represented in Fig. 2. The accelerometer ACC3 is located on the up–down symmetry axis at a sufficient distance from the hydrophone array while both the accelerometers ACC1 and ACC2 are located off-axis on either side of the symmetry axis. The power spectral density of plate acceleration measured at position ACC3 is depicted in Fig. 8(a). Here, pronounced spectral peaks can be seen which correspond predominately to the numerically calculated eigenfrequencies of the plate. The eigenfrequencies are represented by vertical lines in Fig. 8. This agreement holds in particular for lower frequencies. Note that in Fig. 8(a) a bandwidth of Δf ¼ 0:5 Hz is used instead of Δf ¼ 1 Hz, as in (b) and (c). In order to determine the contribution of different eigenmodes to plate vibrations the coherence in spanwise direction is analysed. This allows in particular to distinguish the contribution of eigenmodes having even and odd number of nodes. In Fig. 8(b) the coherence γ2 and in (c) the phase difference ΔΦ between ACC1 and ACC2 is shown. As represented in Table 1 only eigenmodes having zero (Ψ i;1 ; i ¼ 1⋯9) and one ðΨ i;2 ; i ¼ 1⋯6Þ node in spanwise direction exist for f r 140 Hz. The corresponding eigenfrequencies to Ψ i;1 and Ψ i;2 are represented by dotted (online: blue) and dashed (online: green) vertical lines, respectively, in Fig. 8. It can be seen in (b) that the coherence γ2 of plate vibrations is pronounced in most cases in the vicinity of the numerically calculated eigenfrequencies. The agreement is significant in particular for the lowest

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Fig. 8. (a) Power spectral density (Fig:5}= >Þ3ð1Þ; ðÞγ2ðÞðÞΔϕðÞð1–2Þ:; :ð; :ÞÞðÞΔϕðÞð1–2Þ:; :ð; :Þ <:¼ }8Continued: vibrations in spanwise direction (ACC1–ACC2). Eigenfrequencies f i;j are given for comparison.

) and (c) phase difference Δϕ (

) of plate

eigenfrequencies. Here, the phase difference ΔΦ is (almost) zero, as can be seen from Fig. 8(c). At higher frequencies the agreement is also reasonable though not comprehensive. For instance, a phase shift of ΔΦ  π appears for f ≳60 Hz which corresponds to the appearance of eigenmodes Ψ i;2 having a node in spanwise direction. The identification of spectral peak near the plate eigenfrequencies gives rise to the conclusion that the response of the plate to wall pressure fluctuation is of crucial importance for generation of interior flow noise. In Fig. 9 the coherence between (a) wall pressure fluctuations and (b,c) structural vibration on the one hand and interior flow noise on the other is shown. It can be clearly seen from (a) that effectively no coherence between wall pressure fluctuations and interior flow noise exists. Here, flow noise is represented not only by a single-point measurement at positions Hyd1 but also by the spatial average over all 16 pressure time-series 〈Hyd〉. The latter represents a time-delay beamforming with a wall-normal beam direction and allows the detection of a plane wave response. In contrast to that pronounced peaks of significant level of coherence between structural vibrations and flow noise can be found, as demonstrated for example in Fig. 9(b). Here, structural vibration and flow noise are represented by the wallnormal acceleration at position ACC3 and pressure fluctuations at position Hyd1, respectively. Though single point measurements cannot represent the vibrational and the flow noise behaviour comprehensively, a reasonable agreement of the peaks of the coherence function with several numerically calculated eigenfrequencies of the plate can be found. For instance, eigenfrequencies corresponding to the even eigenmodes agree quantitatively for Ψ ð1;3Þ;1 and qualitatively for Ψ ð4;5;7;9Þ;1 while the eigenmodes Ψ ð2;6;8Þ;1 are insignificant. This behaviour is similar to that of interior flow noise represented in Fig. 7. Also the odd eigenmodes, in particular Ψ ð3;5Þ;2 , display a significant level of coherence. The plane wave response in flow noise represented in Fig. 9(c) generally display a lower level of coherence than the single-point measurement shown in (b), except of few peaks such as for instance for Ψ 7;1 . This reflects the strong spatial dependence of flow noise response along the array, which is supported by the numerical simulations of 〈p2 ðf Þ〉A for different measurement positions (see Fig. 5). 6. Conclusion Experiments on flow noise generation in the interior of a streamlined model surrounded by a turbulent boundary layer have been performed in the HYKAT cavitation tunnel at HSVA, Hamburg. Flow noise near the wall is found to be substantially generated by evanescent eigenmodes of the plate which are excited by the wall pressure fluctuations beneath

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Fig. 9. Coherence γ2 between interior flow noise and (a) wall pressure fluctuations measured at position FMH ðΔf ¼ 0:5 HzÞ and (b,c) structural vibrations measured at position ACC3 ðΔf ¼ 1 HzÞ. Flow noise is detected (a,b) at position Hyd1 (Fig:5}= >Þð; Þ16ð〈〉ðÞðÞ; Þ:; :ð; :Þ <:¼ }9Continued: ) and (a,c) averaged over all 16 hydrophones (〈Hyd〉 ( ) and ( ), respectively). Eigenfrequencies f i;j are given for comparison.

the turbulent boundary layer on the reverse side of the plate. Evidence is provided that the non-local, modal response of the plate to spatially distributed, localised wall pressure fluctuations is the dominant source of interior flow noise in the experiment. No significant contribution of pressure fluctuations transmitted locally into the interior could be found. This is supported by experimental spectral and also coherence analysis and by numerical FEM simulations of plate response and eigenvalue analysis. In the simulations the wall pressure field is mimicked by incoherent, localised wave packets and flow noise is calculated on the basis of the reciprocity principle.

Acknowledgements We thank H. Bretschneider from HSVA, Hamburg, and R. Kühl and S. Osburg from FWG for excellent technical support. Discussions with V. Nejedl from FWG are gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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