PROCEEDINGS of the 22 nd International Congress on Acoustics Signal Processing in Acoustics (others): Paper ICA2016-111 About Doppler-Fizeau effect on radiated noise from a rotating source in cavitation tunnel Romuald Boucheron (a) (a) DGA Techniques hydrodynamiques, Val-de-Reuil FRANCE, romuald.boucheron@intradef.gouv.fr Abstract The operational requirements for naval and research vessels have seen an increasing demand for quieter ships either to comply the ship operational requirements or to minimize the influence of shipping noise on marine life. To estimate the future radiated noise of a ship, scale measurements are realized in tunnel. DGA Hydrodynamics owns its cavitation tunnel with low background noise which allows such measurements. Among all noise origins, the present paper considers the problem of a rotating blade. Due to flow and pressure load of each blade faces, blades vibrate and ring out at modal resonance frequencies which is a strategic problem for acoustic discretion. The frequency emitted by the blade will be measured by the non-moving hydrophone with a variable shift in frequency due to Doppler effect. The first part of the paper deals with the idealized case of a monopole in rotation for which theoretical equations for the spectra are derived. The classical Bessel-shape spectra obtained for a sine-shape modulation is modified due to non-stationary modulation, imposed by the time dependence of Mach number contribution in Doppler equation. We have developed an algorithm to compute its associated spectra dealing with Bessel function properties. The second part of the paper will consider the more realistic case of a dipole source in tunnel. The algorithm has been adapted to take into account the dipole directivity. A short parametric study on different parameters as rotation velocity, hydrophone position and dipole orientation concludes the proposed communication. Keywords: Doppler Effect, Source in rotation, Bessel-shape, Dipole directivity.
About Doppler-Fizeau Effect on radiated noise from a rotating source in cavitation tunnel 1 Introduction The operational requirements for naval and research vessels have seen an increasing demand for quieter ships either to comply the ship operational requirements or to minimize the influence of shipping noise on marine life. From a general standpoint, three domains of interest [2] can be distinguished: the flow noise which is the wall pressure fluctuations induced by turbulence or by bubbles, the radiated noise in the far field of the ship (typically beyond 100m) which is principally related to the fluctuating hydrodynamic forces on the rotating blades on the propeller, and the radiated noise inside the ship which more concerns the passengers cabin of cruise liners, as schematically explained by figure 1. Figure 1: Main radiated noise origins of ship. To estimate the future radiated noise of a ship, scale measurements are realized in tunnel. DGA Hydrodynamics own a cavitation tunnel with a low background noise which allows proceeding with such measurements. Typically, a model is introduced in the test section of the tunnel as shown by figure 2. Figure 2: Schematic drawing of a classical noise measurement in tunnel. Two main devices can be used for the noise measurement: a streamlined hydrophone, like shown on the left of figure 2 and a hydrophone plug, flush mounted at a window of the tunnel. 2
The flush-mounted hydrophone is perturbed by boundary layer developed on tunnel walls contrary to streamline hydrophone, but does not perturb the flow in tunnel. The noise measurement can be performed by the two devices with good accuracy. Among all noise origins described before, the present paper considers principally the problem of a rotating blade. Due to flow and pressure load of each blade faces, blades can vibrate and ring out at modal resonance frequencies which is a strategic problem for acoustic discretion. This frequency, emitted by the blade, is measured by the non-moving hydrophone with a shift in frequency due to Doppler Effect. Moreover, the particular geometrical configuration and values of rotation speed can affect the spectra of measured hydrophone signal. From a general standpoint, it is well known that a moving source with an emission frequency is measured by a sensor with a modified frequency according to [3,4] = 1 1 M( )cos( θ(t)), (1) with lower index 1 relative to sensor, 0 relative to the source and M the Mach number with parameter notation of the following figure 3. Figure 3: Geometrical configuration and parameter definitions. The Mach number is time dependant like the angle θ (viewed angle of the source by the sensor). In classical case, the relation between these two frequencies can be generally derived taking into account particular geometry and for example a constant velocity with constant direction. In tunnel configuration, the source, generally in the centre of the test section, is rotating according the following experimental arrangement of figure 3. 2 Monopole in rotation In this idealized case of a monopole in rotation, theoretical equations for the spectra can be derived. The source describes a circle shape trajectory which imposes time dependence of the Mach number and of the viewed angle θ. A sine-shape modulation with modulation index β, (defined as the ratio between ω =ω max ω min and the magnitude of pulsation differences 3
and the modulated pulsation ω ) is used to define the signal. With ω the pulsation generated by the source and using Bessel of first kind J (see [1](9.1.41)) the signal is ( ) = R J (β)!((" # $%" & $) '. (2) This spectra is then easy to represent schematically with the amplitude following the Besselshape according to figure 4. Figure 4: Typical spectra of modulated signal with index of modulation β. In this case of rotating monopole, the time dependence of θ and Mach number drastically complicate the spectra. Actually, starting with geometrical configuration of figure 3, instantaneous frequency received by the hydrophone can be evaluated by [3,4] ( ) = ) * ) * +, -($) with 0( ) = 1 2²% 4²567(" & $ϕ) 1 8²% 2²% 4²%9²%:97;<(" & $ϕ). (3) 0( ) is a periodic function and the phase ϕ is given by ϕ = tan ( =/?)). Taking into account the fact that the time of emission is not the time of reception, we replace it and resampled the new signal according to A = +C( )/D in order to be easily used with classical algorithms with efficiency. Developed in Taylor series thanks to a synchronous detection we get for each mode E its magnitude F GH and its phase ϕ F GH. We stop the calculation of modes at an order I so that the magnitude of the modes is smaller than a tiny value of the maximum of first modes (typically 1 ). The second step consists in introducing this model into the complete signal equation in order to express the amplitude of each component of the spectra. We obtain then, after some mathematical manipulations 4
( ) = R!" #$ JK J Lβ F M S F N N!(F" & $%ϕ O PQ) R with β F = 2π GH F /(EU V ).The summation of Bessel function is also limited to an order P thanks to limiting properties of Bessel functions [1](9.3.1). The product of all sums could be developed in order to together all terms having the same exponential component. Calling the different indexes W F for the i th term of the product, the complex amplitude can be computed by the following equation, associated to its pulsation ', S S F F (4) YJ ZO Lβ F M!Z Oϕ O PQ[ for ω = U^ +U V L W F E M This amplitude will be a contributor of the U^ pulsation peak only if the right sum of equation (4) equals zero. Thereby, to find the magnitude of the U^ peak, we must find all combinations of indexes `W F a that provide the value 0. Hence, we define the domain ϒ b S of indexes combinations so that the sum equals the value q by S `W F a S ϒ b S KE W F = f R (5) F Then, we can express the complex amplitude of the pulsation U^ +fu V as S b = JYJ ZO Lβ F M!Z Oϕ O PQ[ ' h `Z O a ϒ F g (6) The developed algorithm could be decomposed in the following steps: Step 1: Synchronous detection for amplitude and phase of instantaneous frequency Step 2: Determination of maximum order of Bessel functions for the calculus of J (β). b Step 3: Determination of domain ϒ S according to equation (5). Step 4: Calculus of magnitude b of mode q with equation (6). 5
Simulations of propagation of acoustic pressure and peaks estimation have been performed. Spectra are then estimated with the help of Fourier transform (weighted by a Blackman window). Figure 5 presents results of simulation (black lines and results of peak locking by green points) and algorithm by blue points for a monopole in rotation. Figure 5: Simulation of a rotating monopole (black line) with peak (green) and results from algorithm (blue points). Frequency emitted is 744.8Hz and a rotation frequency is 24.6Hz. A good agreement between the blue and green peak for the spectra magnitude and also for the phase values can be observed. 3 Dipole in rotation The configuration of a monopole in rotation is an idealized case. In reality, due to the fact that the blade motion is a transverse translation for main vibration modes, we have to study the more realistic case of a dipole in rotation. We taking into account the directivity pattern via the principal axis of zero efficiency of the dipole (noticed θ^), as shown by the following figure 6 Figure 6: Geometrical configuration for dipole features definition. The amplitude of acoustic pressure radiation toward the hydrophone will be weighted by the directivity pattern of the dipole. This weight can be evaluated by the angle ψ. Performing another synchronous detection (providing mode magnitude γ F GH and phase φ F GH ) on amplitude 6
modulation against time due to dipole directivity leads to a complete model signal measured by hydrophone in the dipole case as 9 PQ ( ) GH!φ = Rγ O F F!F" & $ ' Ri!" #$ F b Fb!F" & $ j (7) We have just to combine the different magnitudes of modes to find coefficient k according to the following equations to find the real magnitude of mode ω^ +lu V N ( ) = R k N!(" # %" & )$ ' with k = J Yγ F GH!φ O PQ [ F 9 F, F nb (8) Figure 7 presents the results of a rotating dipole. We can see a great agreement between simulations and algorithm results for both magnitude and phase. Differences can be observed between the monopole (red crosses) and dipole case (blue points), especially for the peak magnitude shape. The source type is then of important influence on measured spectra. Figure 7: Simulation of a rotating dipole (black line) with peak (green) and results from algorithm (blue points). Frequency emitted is 744.8Hz and a rotation frequency is 24.6Hz. 4 Influence of parameters This section is dedicated to the influence of different parameters. Excepted if specified, all parameter for these simulations are an emitted frequency of 744.8Hz, a 20cm source radius, a rotation frequency of 12Hz and a viewed angle of -30. 7
4.1 Hydrophone position The hydrophone position is of great influence on magnitude of different peaks. In order to appreciate this influence, the differences (in decibel) between the first peak and main peak are computed (peak at ω^ +U V minus ω^ ). A map of these differences is represented in colour and is confined in figure 8 both for monopole and dipole with the same colour scale. a) Monopole b) Dipole Figure 8: Side map of decibel peak differences of mode 1 for monopole and dipole. We observe a symmetrical mapping for the monopole with high differences in the plane of the propeller (x=0) which is an expected result. The dipole map is non symmetrical, due to dipole directivity. We can observe in near field an area where the peak of first mode is higher than the principal mode (represented with hot colours) as already found in figure 7. Figure 9 presents the same kind of results for mode 1 in a horizontal plane downstream the propeller. We can see that the first peak is small in the propeller axis (due to symmetrical aspect and no Doppler modulation because viewed angle is not time depending in this direction). The dipole map presents also this symmetry around the propeller axis but area with positive values of the peak differences can be observed in near-field. a) Monopole b) Dipole Figure 9: Front map of decibel peak differences of mode 1 for both monopole and dipole. 8
4.2 Source radius influence The rotation radius of the source will also influence the spectra measured by the hydrophone. The same kind of map for the dipole is shown by figure 10 a) R=0.1m b) R=0.25m Figure 10: Map of decibel peak differences of mode 1 for the dipole for two different radius. This kind of map shows that position of hydrophones is a key point to measure or not this Doppler-Fizeau effect. Differences between the spectra measured by two hydrophones located at different positions can occur merely due to these Doppler-Fizeau variations. Huge variations of the spectra occur in near field. 4.3 Dipole directivity direction Obviously, the directivity of the dipole plays an important role by interferences. As example, two side maps with different directivity direction are presented by figure 11. a) θ^= -50 b) θ^ =30 Figure 11: Vertical side map of decibel peak differences of mode 1 for different dipole directivity. As expected, the different spatial patterns which can be observed on figure 13 are highly dependent on directivity angle. These patterns could be similar to regular bump or circle with different diameter and thickness. 9
4.4 Evolution from a given point In this last part, we focus on a given hydrophone position. Experimentally, relative position of hydrophone to centre of propeller is known contrary to position of source, kind of source and directivity. If measurement is enough efficient estimating the peaks magnitude, source feature could be estimated by comparison between results and typical curves associated to blade shape like shown by figure 12. For each radius, viewed angle is known. Peak differences could be computed and an equivalent dipole source position could be estimated for different order. Figure 12: Peak difference evolution for a given hydrophone position. Order 1,Order 2. 5 Conclusion In this communication, we discuss about Doppler Effect from a rotating source in tunnel. Equations for magnitude and phase spectra are derived for both monopole and dipole source models. An algorithm for simulations of such spectra has been developed allowing quick estimation. This study is focused on near field and shows that the spectral aspect of received signal contains information about source position, directivity angle of the dipole. This study is limited by the fact that acoustic field is considered as a free field which is not the case in the reality of tunnel measurements. However, differences that could be observed on experiments can be analysed with the filter of the present study to explain (or not) the kind of source and its positon. Even if this effect seems not to be linear with radius, position either dipole direction, it could be considered as a signature of all these parameters. Future works will tend to make the reverse analysis of the present work that is to say to start with measurements to estimate parameters of sources (which can only be done presently by empirical way) and to combine different kind of sources to be more realistic that only considering monopole and dipole. References [1] Abramovitz M.; Stegun I.A. Handbook of mathematical functions, Dover publications 1970. [2] Aucher M. Hélices marines, Techniques de l'ingénieur n B(4-360), 1996. [3] Rossi M. Audio, Presses polytechniques et Universitaires romandes, 2007. [4] Morse P.M. and Ingard K.U. Theoretical acoustics, Princeton University Press, 1968. 10