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Development of an evaluation method to determine cavitation inception speed with aft hull vibration using kurtosis of the DEMON spectrum
Ocean Engineering 152 (2018) 167–180
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Development of an evaluation method to determine cavitation inception speed with aft hull vibration using kurtosis of the DEMON spectrum Hyung Suk Han *, Chang Noh Lee, Soo Hong Jeon, Kyoung Hyun Lee, Sung Ho Park Naval System Research Team, Busan Center, Defense Agency for Technology and Quality, 525-2,Gwangan 1 dong, Busan, Republic of Korea
A R T I C L E I N F O
A B S T R A C T
Keywords: Cavitation inception speed(CIS) Kurtosis Detection of envelope modulation on noise(DEMON) Acceleration
The cavitation inception speed (CIS) is very important capability for a navy vessel because the underwater radiated noise dramatically increases when cavitation occurs. Therefore, the CIS has been severely restricted and evaluated at sea trial test steps in the shipbuilding process. After a naval vessel is delivered to the navy, CIS can vary according to the propeller and sea state conditions. However, inspection of CIS cannot be performed frequently due to time and expense limitation. Therefore, real time monitoring of CIS is urgently required by navy personnel. In order to inspect CIS with real time monitoring technique, the measuring method detecting fluctuation pressure of the propeller in a ship at sailing state should be developed. In this research, CIS monitoring techniques using acceleration on the hull above the propeller that can measure the fluctuation pressure of the propeller indirectly are suggested and verified experimentally. In addition, the evaluation algorithm of CIS using the kurtosis of the Detection of Envelope Modulation on Noise (DEMON) spectrum is used to determine the CIS quantitatively.
1. Introduction Of the naval vessels, submarines and surface ships use sonar in order to detect and identify adversary ships. In order to increase the detection ability of the sonar, the self radiated sound of the naval vessel should be controlled and reduced. The main sources of underwater radiated noise of a naval vessel can be classified into either machinery noise such as from an engine, generator, and pump or fluid dynamically induced noise such as propeller cavitation noise. Usually, machinery noise is dominant at the low speed range. However, as the speed increases and cavitation occurs, the cavitation sound becomes dominant and its level is much higher than the machinery noise. In order to improve the capability of antisubmarine warfare of the naval vessel, cavitation inception speed (CIS) should be increased as much as possible. Many studies on the propeller with respect to the CIS as well as the sound of cavitation bubbles have been performed. Generally, the cavitation sound is produced when cavitation cavities are created and collapse. Minnaert (1933) derived the natural frequency of a rising bubble in water using energy conservation for the bubble motion. Strasberg (1956) assumed that the bubble could be modeled asa spring-mass-damper model with 1 degree of freedom, and determined
the radiated sound of a rising bubble in the water. Han (Han et al., 2011) found that the natural frequency of the refrigerant bubble in the pipe was varied according to its shape. In the case of cavitation of the propeller, tip vortex cavitation occurs when a bubble is created at the end of the propeller tip; the cavitation then develops to sheet and cloud cavitation. The sound of the tip cavitation occurs from the vibratory wave of the vortex cavity and differs from the sound of the spherical bubble. Thomson (1880) determined the natural frequency of the vortex bubble using the dispersion relation of the inviscid cavitating vortex. Morozov (1974) extended Thomson's theory to the sound field of the compressive fluid. Bosschers (2007) modified Morzov's theory to determine the natural frequency of the vortex cavity at the axial free stream velocity. Pennings et al. (2016). measured the sound of the vortex cavity produced by the model propeller in the cavitation tunnel and verified that the measured natural frequency of the vortex cavity concurred with the calculated natural frequency by theory. Even though the propeller cavitation is usually monitored by experiment at the cavitation tunnel with the model propeller, this differs to the cavitation occurring in a real ship condition. Therefore, CIS is evaluated by the measurement of underwater radiated noise at sea when the ship is sailing. In addition, CIS is monitored through the cavitation window
* Corresponding author. Defense Agency of Technology and Quality, 525-2, Gwangan 1-dong, Suyeong-gu, Busan, 613-808, Republic of Korea. Tel.: þ82 51 2822 4245; fax: þ82 51 758 3992. E-mail address: [email protected] (H.S. Han). https://doi.org/10.1016/j.oceaneng.2018.01.075 Received 1 February 2017; Received in revised form 28 August 2017; Accepted 16 January 2018 0029-8018/© 2018 Elsevier Ltd. All rights reserved.
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frequency of the cavity can be derived as shown in Eq. (1). The higher order modes have been suggested by Lamb (1945) as shown in Eq. (2).
installed at the aft side hull near the propeller. However, CIS cannot be monitored continuously when the naval vessel is on a voyage since the underwater radiated noise does not occur frequently and the cavitation window is usually removed after the evaluation of the sea trial test with CIS has finished. In this research, a new method of real time monitoring using CIS in the voyage state is suggested using accelerations on the aft hull near the propeller based on the phenomenon that the cavitation bubble induces an alternating pressure on the aft hull. In section 2, the sound characteristics of the spherical and vortex cavity related to the bubble dynamics are described. In section 3, the experiment setup for defining CIS using acceleration on the aft hull and the signal processing technique such as Detection of Envelope Modulation on Noise (DEMON) to identify CIS effectively are described. In section 4, the CIS estimation method using acceleration on the hull is described.
f0 ¼
fn ¼
1 1 ¼ T 2π r0
sffiffiffiffiffiffiffiffi 3κp
(1)
ρ
2 1=2 . ð2π r0 Þ n 1 ðn þ 2ÞT=ρr0
(2)
here, n is the order, T is the surface tension, r0 is the mean radius of the cavity, ρ is the density of the surrounding fluid of the cavity, κ is the specific heat ratio, and p is the pressure of the surrounding fluid of the cavity. Since the restoring force does not exist for the translating mode, the 1st mode has no resonance frequency. The sound from the cavity is dominantly related to the volume variation. In Fig. 2, the 0th mode is the mode at which only the volume variation occurs. However, for the higher order mode, the shape variation only occurs without volume variation. Therefore, the sound of the higher order modes is much smaller than that of the 0th mode. In this research, the 0th natural frequency is only considered and the high order modes are ignored.
2. Theoretical background The cavitation cavity generally produces a sound before the cavity can be visually inspected. When the cavity is initiated at the tip of the propeller, the cavity size is too small to inspect visually. Since the cavity size is so small, the shape of the cavity should be spherical due to the surface tension of the surrounding water. After the cavity is sufficiently developed, the tip vortex cavity is created and visualized. The general sonar equipment for the detection of objects in waterinstalling submarines and military surface ships can detect the sound up to approximately 10 kHz. In addition, since the sound of the small spherical bubble is usually lower than that of the well-developed vortex bubble, the sonar equipment can detect the sound from the vortex bubble rather than that from the small spherical one. Therefore, in the current state, the navy is interested in the sound from the vortex bubble rather than that from the small spherical bubble in terms of the detection of the ship CIS. However, since sonar-detection technology is being developed continuously, the frequency-detection range can be wider. Therefore, in this section, the sound characteristics of both the spherical and vortex cavities are described.
2.2. Line vortex cavity Observing the cavity in the tip vortex cavity as shown in Fig. 3, a wave can be seen in the cavity. Bosschers (2007) determined the 0th natural frequency in the vortex cavity with constant axial stream velocity as shown in Eqs. (3)–(5). "
ω1;2 ¼ Wkx þ Ω n
Ω¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # kr rc H 1'n ðkr rc Þ Tw H 1n ðkr rc Þ
Vc rc
(3)
(4)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 Tw ¼ 1 þ n þ k 2z r 2c 1 ρrc V 2c
(5)
here, ω is the natural frequency (rad/sec), W is the fluid velocity at the axial direction flow (m/s), kz is the axial wave number (1/m), kr is the radial wave number (1/m), n is the azimuthal wave number, Hn1 is the Hankel function of the 1st kind, rc is the radius of the cavity(m), Tis the surface tension, ρ is the density of the fluid (kg/m3), and Vc is the azimuthal velocity at the cavity radius. When the mean axial velocity is sufficiently slower than the sound speed, the radial wave number can be determined as shown in ) (Bosschers, 2007).
2.1. Spherical cavity The sound of the spherical cavity was first investigated by Minnaert (1933) in 1933. He assumed that the cavity vibrated in the radial direction as shown in Fig. 1. When the cavity size is at its maximum, its potential energy reaches the maximum and the kinetic energy becomes zero. When the oscillating radius of cavity(x) is “r”, the velocity of the oscillating radius of the cavity becomes maximum and the potential energy of the cavity becomes zero. When the energy of these two conditions are equal through the energy conservation law, the 0 mode's natural
kr2 ¼
1 ðω Wkz Þ2 kz2 c2
(6)
When the square of the radial wave number is smaller than 0, it becomes an imaginary value and the wave decreases with time. Therefore, the Hankel function of the 1st kind can be written with the Macdonald function (modified Hankel function of the 2nd kind) (K). In addition, when the azimuthal velocity at the cavity radius is represented with a cavitation number (σ n ), Eq. (3) can be approximately represented as ) (Bosschers, 2007).
ω1;2 ¼
ω1;2 rc W
¼κþ
κ ¼ kz rc ¼ ikr rc ¼ Fig. 1. Spherical shape bubble. 168
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # κK 'n ðκÞ σn n Kn ðκÞ
pffiffiffiffiffi
2π rc λz
"
(7)
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Fig. 2. Mode shapes of spherical bubble.
considering that their frequencies are in the range of 10 kHz–25.6 kHz based on the general sound spectrum at CIS, the size of the cavitation cavities at CIS are in the range of 0.15 mm–0.39 mm. When the cavities are sufficiently developed and after the formation of the vortex cavity that is visually inspected, it can be estimated that the natural frequency of the vortex cavity is lower than that of the spherical cavity with a small radius. From Maines and Arndt (1997) experiment results, the natural frequencies of the vortex cavities that are visually inspected are in the range of 0.4 kHz–1.1 kHz. When calculating the radius of the vortex cavities, observing Maines and Arndt's experiment using Eq. (7), the radius of the vortex is in the ranges of 1.7 mm–4.0 mm at W ¼ 5 m/s and 2.2 mm–10.0 mm at W ¼ 8 m/s. By comparing these natural frequencies of the vortex cavities to those of the spherical cavities, it can be estimated that the natural frequency of the vortex cavity by visual inspection is much lower than that of the spherical cavities with a small radius. If the CIS is determined with the sound produced by small size spherical cavities, signal processing should be performed with the sound of high frequency range considering the natural frequency of the incepted spherical cavities. However, if the CIS is determined with the sound produced by tip vortex cavities, it should be performed with the sound of the frequency range including the natural frequency of the tip vortex cavities.
Fig. 3. Observation of the singing vortex, Maines and Arndt (1997).
here, λz is the wave length (m). Junger and Feit (1986) stated that the high order mode does not contribute to the radiated sound of the cavity, and only the 0th mode contributes to the radiated sound of the cavity because the maximum amplitude of the alternating pressure of the cavity is scaled to ðkrc Þn , and the high order mode is low ðkrc Þn , where n is the order number. The mode shapes of each mode are shown in Fig. 4 (Bosschers, 2007). Fig. 5 shows the natural frequencies of the spherical and vortex cavity when W ¼ 5 m/s, 8 m/s, and λ ¼ 30 mm. When the axial velocity is 5 m/ s, the natural frequency of the spherical cavity is higher than that of the vortex cavity, the radius of which is equal to that of the spherical cavity as shown in Fig. 5(a). However, as the axial velocity increases to 8.0 m/s, the gap of the natural frequency between spherical and vortex cavity decreases as shown in Fig. 5(b). In Fig. 5, the natural frequency is calculated at the point where the density of the sea water is 1025 kg/m3, the specific heat ratio is 1.0, and the distance between the free surface of the sea and the propeller is 10 m. Assuming that the shape of cavities is spherical at the CIS step and
2.3. DEMON analysis The cavitation begins at one of the propeller blades when the speed of rotation is over the critical speed, such that the static pressure of the surrounding water approaches close to the vapor pressure. When the cavitation continuously develops according to the increasing speed, cavitation occurs at all of the blades. When the cavitation occurs, the sound of the cavities modulates synchronously to the shaft rate and the blade passing frequency. When applying DEMON analysis, the noise component at the shaft rate as well as the blade passing frequency is revealed in the DEMON spectrum. The analytic signal of DEMON analysis is obtained from Eq. (8) and the enveloped signal is the amplitude of the analytic signal as shown in ) (Randall, 1987). b hðtÞ ¼ hðtÞ þ j~hðtÞ
(8)
0:5 b hðtÞ ¼ hðtÞ2 þ ~hðtÞ2
(9)
~ here, h(t) is the time signal of the raw data and hðtÞ is the Hilbert transform of h(t), as shown in Eq. (10). ~hðtÞ ¼ HðhðtÞÞ ¼ 1 ∫ ∞ hðtÞdt π ∞ τ t
(10)
From the fast Fourier transform (FFT) spectrum of Eq. (9), CIS inspects whether the DEMON spectrum reveals the sound of the rotating shaft or blade passing frequency. In DEMON analysis, the selection of the bandwidth and center frequency of the band-pass filter is very important. They must be selected considering the frequency range where the
Fig. 4. Main vortex cavity oscillation modes (reproduced from Bosschers (2007)). Top: monopole breathing mode (n ¼ 0); middle: dipole serpentine centerline displacement mode (n ¼ 1); and bottom: quadrupole helical mode (n ¼ 2). 169
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Fig. 5. Resonance frequency of the bubble according to its size. (a) W ¼ 5 m/s, λ ¼ 30 mm (b) W ¼ 8 m/s, λ ¼ 30 mm.
According to the measurement results of the underwater-radiated noise with the hydrophone array that was performed prior to the present research, CIS was preliminary estimated and determined as VCIS knots. Therefore, the acceleration on the aft hull was measured as from “VCIS-8” knots to “VCISþ4” knots as shown in Table 1 in order to compare the CIS estimated with accelerations to that with underwater radiated noise.
modulation coincides with the shaft rotating and blade passing frequency. This proper selection can be performed with spectral kurtosis of the short time FFT described in the next section. In addition, CIS can be identified when the overall noise level is increased to over 2dB/knots as the ship speed increases. In this research, CIS is determined with the vibration signal on the aft hull located close to the propeller instead of the sound signal radiated to the under water. Since the pressure wave produced by the propeller cavitation induces the adjacent hull of the propeller, the acceleration on the hull is the same pattern to the cavitation sound of the propeller. Therefore, in the following section, for a specific naval vessel, CIS is evaluated using a 1/3 octave band spectrum, the overall level, and DEMON spectrum of the hull vibration adjacent to the propeller.
3.1. Test setup The tested ship is a battleship with a combined diesel-engine/gasturbine-propulsion system. The propeller type is the controllable pitched propeller, and two of each propeller are installed at the port and the starboard sides. To inspect the propeller CIS, three accelerometers were installed at the aft hull above the propeller as shown in Fig. 6. Since the space wherein the accelerometers were installed is void space, equipment is not necessary. The No.1 accelerometer was installed on the hull at the port side, No.3 was installed on the hull at the starboard side, and No.2 was installed between the No.1 and No.3 accelerometers as shown in Fig. 6.
3. Experiment In this section, the acceleration on the aft hull above the propeller is measured according to the ship speed in order to estimate CIS based on the acoustic characteristics of the cavitation cavity described in section 2.
Table 1 Speed of the test ship. Test No.
t1
t2
t3
t4
t5
t6
t7
t8
t9
t10
Speed [knots] RPM Pitch %
VCIS-8 N0 0%
VCIS-5 N0 33%
VCIS-3 N0 66%
VCIS-2 N0 100%
VCIS-1 N0 þ 5
VCIS N0 þ 10
VCIS þ 1 N0 þ 14
VCIS þ 2 N0 þ 21
VCIS þ 3 N0 þ 38
VCIS þ 4 N0 þ 45
170
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Fig. 6. Test setup for CIS measurement with accelerometers.
- Since 2 propellers are installed at the port and starboard sides of the ship and their blades rotate outward, the cavitation cavities of the 2 propellers would be created below the center position installing No. 2 accelerometers where the static pressure of the water near the propeller is the least. The water can be vaporized easily at this position, when the propeller speed is critical for the cavitation, because the static pressure is at its lowest value. Therefore, the pressure wave from the cavity may be transferred to the hull more easily where installing the No.2 accelerometer rather than where installing the No.1 and 3 accelerometers.
A PCB Type 352A60 accelerometer and a B&K3053-B12/0 data acquisition system were used to collect the signal. The acceleration was recorded for 2 min for each ship speed, as shown in Table 1, and the sampling rate was set at 65,536samples/sec. 3.2. Test results Fig. 7 shows the 1/3 octave band spectrum of the acceleration for each ship speed to be tested. The acceleration levels at the high frequency range of over 5 kHz for the No.1 and 3 accelerometers increased rapidly when the ship speed was t6–t7. Even though the distances from the No. 1 and No. 3 accelerometers to the center line are identical, the hull structures (for example, the distribution of the stiffeners) are different from each other because the hulls are not quite bilaterally symmetrical. In addition, the dynamic pressure-induced acceleration on the hull can be different between the port and starboard sides, because the dynamic pressure of the propeller blade for the port and starboard sides can be nonlinearly different in accordance with the sailing conditions. Therefore, the difference between the acceleration spectra of No. 1 and No. 3 can exist, as shown in Fig. 7. Even though the spectra shape of the No. 1 and No. 3 accelerations are different from each other, they are increased rapidly and identically when the ship speed is from t6–t7. The acceleration levels at the high frequency range of over 5 kHz for the No.2 accelerometer increased rapidly at t5 ~ t6, earlier than those for No.1 and 3 accelerometers. This increasing acceleration may be caused by the cavitation. Therefore, CIS can be estimated to t5–t7. The reason for why this increasing acceleration for the No. 2 accelerometer started earlier than those of the No. 1 and No. 3 accelerometers was not clearly identified. We could simply guess that the reason is as follows:
Even though the reason for this difference was not clearly identified, the CIS that was estimated from the acceleration can be varied in accordance with the accelerometer-installation position. In this research, it is assumed that the frequency range of the cavitation sound caused by the vortex and large spherical cavities is set to 3 kHz–10 kHz and that caused by the spherical cavities with a very small radius is set to 10 kHz–25.6 kHz. Fig. 8 shows the overall acceleration level in the ranges of 3 kHz–10 kHz and 10 kHz–25.6 kHz. Generally, CIS is determined when the overall sound pressure level is increased by 2 dB per 1 knot increase in speed. In order to identify CIS using the overall acceleration level, the same criterion for identifying CIS with sound is adopted. In Fig. 8, the overall acceleration levels on the hull in which the frequency range is from 3 kHz to 10 kHz are increased by 2 dB per 1 knot increase in speed, similar to the sound pressure when the ship speed varies from t6 to t7 for the No.1 and 3 accelerometers and from t4 to t5 for the No.2 accelerometer. However, the overall acceleration levels on the hull in which the frequency range is from 10 kHz to 25.6 kHz are increased by 2 dB per 1knot increase in speed, similar to the sound pressure when the ship
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Fig. 7. 1/3 octave band spectra of the acceleration of the hull above the propeller. (a) accelerometer No.1. (b) accelerometer No.2. (c) accelerometer No.3. Fig. 8. Overall rms acceleration of the hull above the propeller. (a) accelerometer No.1. (b) accelerometer No.2. (c) accelerometer No.3.
speed varies from t4 to t5 for the No.1 and 2 accelerometers but the No. 3 accelerometer does not sufficiently vary.
KX ¼ 3.3. Spectral kurtosis
hEfX 4 ðtÞgi hEfX 2 ðtÞgi2
2
(11)
here, E{}is the mean function, KX is the spectral kurtosis, and X(t)is the signal value at a particular frequency for time t. If the distribution of the signal is Gaussian, the kurtosis becomes 0, and if the signal has high transient signal, it becomes very high. When the cavitation occurs, the impact signal of the propeller occurs periodically synchronous to the propeller rotation and blade passing frequency. In contrast to the frequency of the bearing and gear mesh fault sound, the frequency of the cavitation sound is not the natural frequency of the system but that of the cavity. Therefore, when DEMON analysis is performed, the natural frequencies of the created cavities on the propeller should be included in the bandwidth of the band-pass filter, considering their shape and size. Fig. 9 shows the kurtogram of the measured acceleration on the hull when the ship speed is t5 ~ t7. The kurtogram is a graph of the spectral kurtosis at each frequency for the short time FFT spectrum performed with 2 k frequency lines. It is used to determine the center frequency and
In the previous section, CIS was estimated to be t5–t7 based on the overall acceleration level and on the 1/3 octave band spectrum analysis. In this section, DEMON analysis described, which is another method used to determine CIS. In order to perform DEMON analysis, the center frequency and bandwidth of the band-pass filter should first be determined. In this research, spectral kurtosis (Antoni and Randall, 2006) is applied to select the center frequency and the bandwidth of the band-pass filter to perform CIS. In the mechanical rotating system, periodic excited sound can be caused by a bearing or gear mesh fault. When the impact force is subject to the mechanical system, the vibration increases at its natural frequency. Therefore, the vibration at the natural frequency of the mechanical system periodically appears and disappears repeatedly. When the signal varies unstably due to the transient signal, the kurtosis of the signal increases. Spectral kurtosis is the kurtosis of the signal for the spectrum at a particular frequency and is defined as ) (Antoni and Randall, 2006). 172
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Fig. 9. Kurtogram of accelerations of the hull above the propeller. (a) accelerometer No.1. (b) accelerometer No.2. (c) accelerometer No.3.
bandwidth of the bandpass filter for DEMON analysis. In the kurtogram, the horizontal axis is the frequency (0–25.6 kHz)
and the vertical axis is k (0–7), and k is defined as NW ¼ 2k (Nw: window length). The spectral-kurtosis level is represented by the color map on the 173
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Fig. 9. (continued).
lower than the frequency of that caused by the spherical cavity. Therefore, when the ship speed is t6–t7, spectral kurtosis increases at the frequency range of 3–10 kHz, which is less than the frequency range when the ship speed is t5. When kurtogram analysis is performed for the mechanical system having mechanical defaults such as bearing and gear faults, the k level should be selected to include the natural frequency of the mechanical system where the response of the system increases significantly from the impact force. However, in cavitation, the response is not the response of the mechanical system but that of the cavities. Therefore, at a specific k level, the frequencies with the highest spectral kurtosis, i.e., natural frequencies of vortex bubbles, are varied according to the different cavity size and environmental conditions. In addition, in the process of the merging and the collapsing of the bubbles, the frequencies with the highest spectral kurtosis can also be varied. Therefore, even though the obtained k level can be from 5 to 7 at speeds from t6–t7, narrow bandwidths such as those from 200 Hz (k ¼ 5)–800 Hz (k ¼ 7) cannot be applied to perform the DEMON analysis for the detection of the CIS. Considering that the frequency band of the DEMON analysis for the cavitation sound at the ship speed of t5 can be obtained at 6.4 k Hz from the kurtogram analysis, and the center frequencies of the vortex bubbles can vary widely according to their size and environmental condition, the bandwidth of the bandpass filter of the DEMON analysis for the acceleration on the hull was selected as 6.4 kHz. In fact, the 6.4-kHz frequency
graph. A high spectral-kurtosis level at a given frequency band means that many transient signals have been caused by the cavitation. Therefore, using the kurtogram, the center frequency and the frequency span that comprise the maximum spectral kurtosis can be defined. In Fig. 9, the spectral kurtosis is at a very high level at a low k (k < 3: bandwidth > 6.4 kHz) and high frequency range (>20 kHz) when the ship speed is t5 for No.1 and 2 accelerometers. It means that, regarding the best-fit bandpass filter for the DEMON analysis at the ship speed of t5, it would be preferable for the center frequency and the frequency span to be more than 20 kHz and 6.4 kHz, respectively. As the ship speed increases to t6 and t7, the frequency range having the highest spectral kurtosis decreases to the lower frequency and the k level having the highest spectral kurtosis increases to 5–7 (200 Hz < bandwidth < 800 Hz) compared to those when the ship speed is t5. Therefore, regarding the center frequency of the best-fit bandpass filter for the DEMON analysis at the ship speeds from t6–t7, it would be preferable for the speed to be lower than t5 and the frequency band to be narrower than its typical t5 value. As described in section 2.2 and Fig. 5, spherical bubbles first appear when the cavitation is incepted and the broadband acceleration of 10 kHz–25.6 kHz which includes the natural frequency of the very small size cavity is modulated with the acceleration at the propeller rotating and blade passing frequency. However, when the cavity is developed and the tip vortex cavity starts to form, the frequency of the acceleration caused by the vortex cavity is 174
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spherical cavities and is set to 6.4 kHz 3.2 kHz with that for the tip vortex cavity.
band is widely used for the bandpass filter that is used to perform the DEMON CIS-detection analysis. Consequently, the center frequencies were defined as 22.4 kHz and 6.4 kHz based on the kurtogram results at the ship speeds t5 and from t6–t7, respectively, and the bandwidth was determined as 6.4 kHz, as previously stated. According to the kurtogram analysis of this chapter, the frequency range of the band-pass filter for DEMON analysis is set to 22.4 kHz 3.2 kHz when CIS is defined with the sound for the very small
3.4. Results from DEMON DEMON analysis for the acceleration on the hull above the propeller is performed according to the ship's speed, at which the bandwidth of the band-pass filter is set to 6.4 kHz and the center frequency of the bandFig. 10. DEMON spectra applying 6.4 3.2 kHz bandpass filter (SR ¼ speed rate frequency, BPF ¼ blade passing frequency, CIS ¼ t6–t7). (a) ship speed ¼ t5. (b) ship speed ¼ t6. (c) ship speed ¼ t7. (d) ship speed ¼ t8.
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Fig. 10. (continued).
pass filter is set to 6.4 kHz and 22.4 kHz based on the kurtogram analysis. Fig. 10 shows the DEMON spectrum when the bandwidth of the bandpass filter is 6.4 kHz and the center frequency is 6.4 kHz. Here, the x axis is the frequency and the y axis is the nominal value that indicates 2 times the measured time duration (2 0.5sec n, n ¼ 0–200). The bandwidth and center frequency of DEMON analysis for the underwater radiated noise were the same as those applied in Fig. 10. When the cavitation is intercepted, the cavity is created at one of the
propeller blades. If the blade dimensions of the propeller are perfectly the same as one another, cavities can occur at all of the blades when they pass through the positions where the distance between end of the blade and the free surface is the smallest. However, due to the variation of the dimensions, a cavity can occur at one of the blades first. And then, as the cavities are developed further, the cavities start to occur at all of the blades. Therefore, in the DEMON spectrum, the peak at the 1st rotating frequency of the propeller shaft is firstly revealed at the CIS, and then its
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Fig. 11. DEMON spectra applying 6.4 3.2 kHz bandpass filter (SR ¼ speed rate frequency, BPF ¼ blade passing frequency, CIS ¼ t5–t6). (a) ship speed ¼ t1. (b) ship speed ¼ t5. (c) ship speed ¼ t6. (d) ship speed ¼ t7. (e) ship speed ¼ t8.
frequency when the ship speed is t7. Fig. 11 shows the DEMON spectrum when the bandwidth of the bandpass filter is 6.4 kHz and the center frequency is 22.4 kHz. The spectrum value at the rotating shaft and blade passing frequency are shown at t5 and t6, respectively, where the ship speed is less than that when using the band-pass filter with a center frequency of 6.4 kHz. Therefore, CIS can be estimated as t5–t6 from Fig. 11. In addition, the peak value at the shaft rate can be found in the DEMON spectrum when the propeller pitch is
2nd and 3rd peaks are revealed according to the rotating frequency of the shaft. When the cavities are sufficiently developed, the peak at the bladepassing frequency is also evident. Therefore, CIS is defined from DEMON analysis when the DEMON spectrum has the peak at the rotating frequency of the shaft. In Fig. 10, CIS can be estimated as t6–t7, similar to CIS estimated with underwater radiated noise since the DEMON spectrum has a peak at the 1st rotating frequency when the ship's speed is t6 and the blade passing 177
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Fig. 11. (continued).
minimum (V ¼ t1). It means that small spherical bubble may occur when the low propeller pitch is applied. As discussed in the previous section, the differences in CIS according to the center frequency of the band-pass filter is due to the different types of cavities (spherical or vortex cavity).
4. Discussion When the CIS is estimated with 1/3 octave band spectrum, overall level variation and DEMON spectrum analysis of the acceleration on the hull, it is the same as the CIS estimated with underwater radiated noise. Even though CIS slightly varies according to the location of the installed
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Fig. 11. (continued).
accelerometer, it is verified that CIS can be determined from the acceleration on the hull above the propeller. When the hydrophone and pressure sensor are applied to the hull in order to monitor CIS, a hole needs to be bored to install these sensors. However, this hole is not necessary when the accelerometer is applied to monitor the CIS variation in a real ship. Therefore, for the real time monitoring of CIS, it is possible to use acceleration on the hull simply with the signal processing suggested in this research. Through this monitoring technique, the maintenance duration can be identified exactly with respect to CIS. Even though CIS can be estimated with DEMON spectrum analysis, decoding the DEMON spectrum is very subjective. Therefore, a quantitative index needs to be developed for automatic real time monitoring of CIS. In this section, the kurtosis of the DEMON spectrum is suggested for this quantitative index. Fig. 12 shows the DEMON spectrum at the typical time “t” when the cavitation begins. In Fig. 12, the peak occurs at the blade passing frequency. When the dominant peak forms, as shown in Fig. 12, the kurtosis of the DEMON spectrum (Y) as Eq. (12) becomes very large. Here, xi is the DEMON spectrum level at each frequency, x is the average spectrum level, s is the standard deviation of the spectrum levels, and n is the line numbers of the frequency in the DEMON spectrum. If no dominant peak occurs in the DEMON spectrum, the distribution of the DEMON spectrum level is Gaussian and the kurtosis of the DEMON spectrum should be approximately 0.
Fig. 12. DEMON spectrum when the cavitation occurs.
DEMON spectrum of the acceleration on the hull according to the ship's speed as shown in Table 1. Here, the frequency range of the DEMON spectrum is 4–100 Hz and the number of the data to calculate kurtosis is 192 since the frequency resolution is 0.5 Hz (n¼(100 Hz–4 Hz)/0.5 Hz). The kurtosis is calculated with the DEMON spectrum of the acceleration measured for 0.5sec. The kurtosis is calculated for 100 s and the kurtosis shown in Fig. 13 is the averaged kurtosis for 100 s. In Fig. 13, it is assumed that CIS occurs when the average kurtosis is over than 4.0. Then, CIS can be identified as t5 when the center frequency and bandwidth are set to 22.4 kHz 3.2 kHz as shown in Fig. 13(a). In this case, cavitation also occurs at t1–t2 where the propeller pitch ratio is very small. When the center frequency and bandwidth is set to 6.4 kHz 3.2 kHz, CIS can be estimated as t6, as shown in Fig. 13(b). CIS is estimated with the average kurtosis, as shown in Fig. 13, is the same as that estimated by qualitative analysis performed in section 3. Through this average kurtosis of the DEMON spectrum, CIS can be
"
4 # n X nðn þ 1Þ xi x 3ðn 1Þ2 Y¼ ðn 1Þðn 2Þðn 3Þ i¼1 s ðn 2Þðn 3Þ
(12)
Therefore, the kurtosis of the DEMON spectrum according to the ship's speed can identify CIS. Fig. 13 is the average kurtosis for the
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(1) CIS can be estimated with the acceleration on the hull adjacent to the propeller from the experiment with s specific naval vessel. (2) Through the theoretical investigation, it can be seen that the natural frequency of the spherical cavity is much higher than that of the vortex cavity with the same radius when the axial velocity of the propeller is slow. Therefore, when the spectrum analysis using DEMON is performed to identify CIS, the cavity's shape and size should be considered. (3) When the CIS of a specific naval vessel is estimated from DEMON of the acceleration on the hull, whereby the center frequency of the band-pass filter is 6.4 kHz and 22 kHz, CIS estimated with a center frequency of the band-pass filter of 22.4 kHz is 1 knots faster than that estimated with the center frequency of the bandpass filter of 6.4 kHz. When using 6.4 kHz band-pass filter, CIS is determined to the speed inspecting very small spherical cavities that can't visually be inspected. However, when using 22.4 kHz band-pass filter, CIS is determined to the speed inspecting well developed tip vortex cavities. Therefore, CIS can be different according to the frequency range of the band-pass filter. (4) The estimated CIS differs according to the position at which the accelerometers installed. Therefore, the proper installation position of the accelerometer should be selected. In order to select the best point on the hull when transferring the pressure wave, the CIS estimated from the underwater radiating noise should be compared to that estimated from acceleration. (5) Through the estimation method of CIS using the average kurtosis of the DEMON spectrum, the quantitative evaluation of CIS is possible and the CIS estimated with the average kurtosis of the DEMON spectrum concurs well with the qualitative evaluation of the CIS with 1/3 octave band and DEMON analysis. It is expected that CIS determined using the average kurtosis of the DEMON spectrum can be applied to the real time monitoring of CIS.
Fig. 13. Kurtosis of DEMON spectrum for acceleration on the hull according to the ship's speed. (a) bandpass filter ¼ 22.4 kHz 3.2 kHz (b) bandpass filter ¼ 6.4 kHz 3.2 kHz.
identified quantitatively and can be used in the real time monitoring of CIS in a naval vessel during a voyage. When the center frequency of the band-pass filter is set as 22.4 kHz, CIS is identified earlier at about 1 knots compared to CIS estimated with the band-pass filter using a center frequency of 6.4 kHz. It is widely known that the cavity's sound can be heard ahead of the visual inspection of the bubble. In the investigation of this research study, the sound of the small spherical bubble firstly occurred, and this was followed by the sound of the vortex bubble. After the amplitude of the cavity sound is increased in accordance with the increasing of the cavity size, the cavity can be visualized. Therefore, to accurately identify the CIS, the CISidentification method should be firstly selected. Since the sonar systems that are installed in navy ships detect the CIS with the sound signal, the CIS should be defined not by the visualization of the cavity, but by the sound from the cavity. To define the CIS with the sound signal accurately, the center frequency and bandwidth of the band-pass filter should be determined considering the detection capabilities of the sonar systems for the general naval vessel. Currently, the designs of general CISdetection sonar systems are usually based on the sound detection of the vortex cavity.
In the future, the CIS monitoring and estimating system will be developed, with the acceleration on the hull adjacent to the propeller. In order to develop this system, the installation position of the accelerometer and the type of cavity that most contributes to the underwater radiated noise should be initially identified. References Antoni, J., Randall, R.B., 2006. The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines. Mech. Syst. Signal Process. 20, 308–331. Bosschers, J., 2007. Broadband Hull Pressure Fluctuations and Cavitating Vortices – An Investigation of Resonance Frequencies . Marine Engineering Forum, Propeller Cavitation Workshop, London. Han, H.S., Jeong, W.B., Kim, M.S., Lee, S.Y., Seo, M.Y., 2011. Frequency characteristics of the noise of R600a refrigerant flowing in a pipe with intermittent flow pattern. Int. J. Refrig. 34, 1497–1506. Junger, M.C., Feit, D., 1986. Sound, Structures and Their Interaction, second ed. MIT Press, Cambridge, MA, USA. Lamb, H., 1945. Hydrodynamics. Dover Publication, New York. Maines, B., Arndt, R.E.A., 1997. The case of the singing vortex. J. Fluid Eng. 119, 271–276. Minnaert, W.K., 1933. On musical air bubbles and the sound of running water. Phil. Mag. 16, 235–248. Morozov, V.P., 1974. Theoretical analysis of the acoustic emission from cavitating line vortices. Sov. Phys. Acoust. 19 (No. 5), 468–471. Pennings, P.C., Westerweel, J., Terwisga, T., 2016. Cavitation tunnel analysis of radiated sound from the resonance of a propeller tip vortex cavity. Int. J. Multiph. Flow 83, 1–11. Randall, R.B., 1987. Frequency Analysis. B&K. Strasberg, M., 1956. Gas bubbles as source of sound in liquids. J. Acoust. Soc. Am. 28 (No. 1), 20–27. Thomson, W. (Lord Kelvin), 1880. Vibrations of a columnar vortex. In: Proceedings of the Royal Society of Edinburgh, Scotland, March 1, pp. 152–165. Phil. Mag. X.
5. Conclusion In this research, the CIS estimation method using the signal processing technique from the acceleration on the hull above the propeller is suggested based on the mechanism in which the vibratory cavity transfers the pressure wave to the hull adjacent to the propeller. The detailed conclusions from this research are as follows.
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Development of an evaluation method to determine cavitation inception speed with aft hull vibration using kurtosis of the DEMON spectrum Hyung Suk Han *, Chang Noh Lee, Soo Hong Jeon, Kyoung Hyun Lee, Sung Ho Park Naval System Research Team, Busan Center, Defense Agency for Technology and Quality, 525-2,Gwangan 1 dong, Busan, Republic of Korea
A R T I C L E I N F O
A B S T R A C T
Keywords: Cavitation inception speed(CIS) Kurtosis Detection of envelope modulation on noise(DEMON) Acceleration
The cavitation inception speed (CIS) is very important capability for a navy vessel because the underwater radiated noise dramatically increases when cavitation occurs. Therefore, the CIS has been severely restricted and evaluated at sea trial test steps in the shipbuilding process. After a naval vessel is delivered to the navy, CIS can vary according to the propeller and sea state conditions. However, inspection of CIS cannot be performed frequently due to time and expense limitation. Therefore, real time monitoring of CIS is urgently required by navy personnel. In order to inspect CIS with real time monitoring technique, the measuring method detecting fluctuation pressure of the propeller in a ship at sailing state should be developed. In this research, CIS monitoring techniques using acceleration on the hull above the propeller that can measure the fluctuation pressure of the propeller indirectly are suggested and verified experimentally. In addition, the evaluation algorithm of CIS using the kurtosis of the Detection of Envelope Modulation on Noise (DEMON) spectrum is used to determine the CIS quantitatively.
1. Introduction Of the naval vessels, submarines and surface ships use sonar in order to detect and identify adversary ships. In order to increase the detection ability of the sonar, the self radiated sound of the naval vessel should be controlled and reduced. The main sources of underwater radiated noise of a naval vessel can be classified into either machinery noise such as from an engine, generator, and pump or fluid dynamically induced noise such as propeller cavitation noise. Usually, machinery noise is dominant at the low speed range. However, as the speed increases and cavitation occurs, the cavitation sound becomes dominant and its level is much higher than the machinery noise. In order to improve the capability of antisubmarine warfare of the naval vessel, cavitation inception speed (CIS) should be increased as much as possible. Many studies on the propeller with respect to the CIS as well as the sound of cavitation bubbles have been performed. Generally, the cavitation sound is produced when cavitation cavities are created and collapse. Minnaert (1933) derived the natural frequency of a rising bubble in water using energy conservation for the bubble motion. Strasberg (1956) assumed that the bubble could be modeled asa spring-mass-damper model with 1 degree of freedom, and determined
the radiated sound of a rising bubble in the water. Han (Han et al., 2011) found that the natural frequency of the refrigerant bubble in the pipe was varied according to its shape. In the case of cavitation of the propeller, tip vortex cavitation occurs when a bubble is created at the end of the propeller tip; the cavitation then develops to sheet and cloud cavitation. The sound of the tip cavitation occurs from the vibratory wave of the vortex cavity and differs from the sound of the spherical bubble. Thomson (1880) determined the natural frequency of the vortex bubble using the dispersion relation of the inviscid cavitating vortex. Morozov (1974) extended Thomson's theory to the sound field of the compressive fluid. Bosschers (2007) modified Morzov's theory to determine the natural frequency of the vortex cavity at the axial free stream velocity. Pennings et al. (2016). measured the sound of the vortex cavity produced by the model propeller in the cavitation tunnel and verified that the measured natural frequency of the vortex cavity concurred with the calculated natural frequency by theory. Even though the propeller cavitation is usually monitored by experiment at the cavitation tunnel with the model propeller, this differs to the cavitation occurring in a real ship condition. Therefore, CIS is evaluated by the measurement of underwater radiated noise at sea when the ship is sailing. In addition, CIS is monitored through the cavitation window
* Corresponding author. Defense Agency of Technology and Quality, 525-2, Gwangan 1-dong, Suyeong-gu, Busan, 613-808, Republic of Korea. Tel.: þ82 51 2822 4245; fax: þ82 51 758 3992. E-mail address: [email protected] (H.S. Han). https://doi.org/10.1016/j.oceaneng.2018.01.075 Received 1 February 2017; Received in revised form 28 August 2017; Accepted 16 January 2018 0029-8018/© 2018 Elsevier Ltd. All rights reserved.
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frequency of the cavity can be derived as shown in Eq. (1). The higher order modes have been suggested by Lamb (1945) as shown in Eq. (2).
installed at the aft side hull near the propeller. However, CIS cannot be monitored continuously when the naval vessel is on a voyage since the underwater radiated noise does not occur frequently and the cavitation window is usually removed after the evaluation of the sea trial test with CIS has finished. In this research, a new method of real time monitoring using CIS in the voyage state is suggested using accelerations on the aft hull near the propeller based on the phenomenon that the cavitation bubble induces an alternating pressure on the aft hull. In section 2, the sound characteristics of the spherical and vortex cavity related to the bubble dynamics are described. In section 3, the experiment setup for defining CIS using acceleration on the aft hull and the signal processing technique such as Detection of Envelope Modulation on Noise (DEMON) to identify CIS effectively are described. In section 4, the CIS estimation method using acceleration on the hull is described.
f0 ¼
fn ¼
1 1 ¼ T 2π r0
sffiffiffiffiffiffiffiffi 3κp
(1)
ρ
2 1=2 . ð2π r0 Þ n 1 ðn þ 2ÞT=ρr0
(2)
here, n is the order, T is the surface tension, r0 is the mean radius of the cavity, ρ is the density of the surrounding fluid of the cavity, κ is the specific heat ratio, and p is the pressure of the surrounding fluid of the cavity. Since the restoring force does not exist for the translating mode, the 1st mode has no resonance frequency. The sound from the cavity is dominantly related to the volume variation. In Fig. 2, the 0th mode is the mode at which only the volume variation occurs. However, for the higher order mode, the shape variation only occurs without volume variation. Therefore, the sound of the higher order modes is much smaller than that of the 0th mode. In this research, the 0th natural frequency is only considered and the high order modes are ignored.
2. Theoretical background The cavitation cavity generally produces a sound before the cavity can be visually inspected. When the cavity is initiated at the tip of the propeller, the cavity size is too small to inspect visually. Since the cavity size is so small, the shape of the cavity should be spherical due to the surface tension of the surrounding water. After the cavity is sufficiently developed, the tip vortex cavity is created and visualized. The general sonar equipment for the detection of objects in waterinstalling submarines and military surface ships can detect the sound up to approximately 10 kHz. In addition, since the sound of the small spherical bubble is usually lower than that of the well-developed vortex bubble, the sonar equipment can detect the sound from the vortex bubble rather than that from the small spherical one. Therefore, in the current state, the navy is interested in the sound from the vortex bubble rather than that from the small spherical bubble in terms of the detection of the ship CIS. However, since sonar-detection technology is being developed continuously, the frequency-detection range can be wider. Therefore, in this section, the sound characteristics of both the spherical and vortex cavities are described.
2.2. Line vortex cavity Observing the cavity in the tip vortex cavity as shown in Fig. 3, a wave can be seen in the cavity. Bosschers (2007) determined the 0th natural frequency in the vortex cavity with constant axial stream velocity as shown in Eqs. (3)–(5). "
ω1;2 ¼ Wkx þ Ω n
Ω¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # kr rc H 1'n ðkr rc Þ Tw H 1n ðkr rc Þ
Vc rc
(3)
(4)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 Tw ¼ 1 þ n þ k 2z r 2c 1 ρrc V 2c
(5)
here, ω is the natural frequency (rad/sec), W is the fluid velocity at the axial direction flow (m/s), kz is the axial wave number (1/m), kr is the radial wave number (1/m), n is the azimuthal wave number, Hn1 is the Hankel function of the 1st kind, rc is the radius of the cavity(m), Tis the surface tension, ρ is the density of the fluid (kg/m3), and Vc is the azimuthal velocity at the cavity radius. When the mean axial velocity is sufficiently slower than the sound speed, the radial wave number can be determined as shown in ) (Bosschers, 2007).
2.1. Spherical cavity The sound of the spherical cavity was first investigated by Minnaert (1933) in 1933. He assumed that the cavity vibrated in the radial direction as shown in Fig. 1. When the cavity size is at its maximum, its potential energy reaches the maximum and the kinetic energy becomes zero. When the oscillating radius of cavity(x) is “r”, the velocity of the oscillating radius of the cavity becomes maximum and the potential energy of the cavity becomes zero. When the energy of these two conditions are equal through the energy conservation law, the 0 mode's natural
kr2 ¼
1 ðω Wkz Þ2 kz2 c2
(6)
When the square of the radial wave number is smaller than 0, it becomes an imaginary value and the wave decreases with time. Therefore, the Hankel function of the 1st kind can be written with the Macdonald function (modified Hankel function of the 2nd kind) (K). In addition, when the azimuthal velocity at the cavity radius is represented with a cavitation number (σ n ), Eq. (3) can be approximately represented as ) (Bosschers, 2007).
ω1;2 ¼
ω1;2 rc W
¼κþ
κ ¼ kz rc ¼ ikr rc ¼ Fig. 1. Spherical shape bubble. 168
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # κK 'n ðκÞ σn n Kn ðκÞ
pffiffiffiffiffi
2π rc λz
"
(7)
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Fig. 2. Mode shapes of spherical bubble.
considering that their frequencies are in the range of 10 kHz–25.6 kHz based on the general sound spectrum at CIS, the size of the cavitation cavities at CIS are in the range of 0.15 mm–0.39 mm. When the cavities are sufficiently developed and after the formation of the vortex cavity that is visually inspected, it can be estimated that the natural frequency of the vortex cavity is lower than that of the spherical cavity with a small radius. From Maines and Arndt (1997) experiment results, the natural frequencies of the vortex cavities that are visually inspected are in the range of 0.4 kHz–1.1 kHz. When calculating the radius of the vortex cavities, observing Maines and Arndt's experiment using Eq. (7), the radius of the vortex is in the ranges of 1.7 mm–4.0 mm at W ¼ 5 m/s and 2.2 mm–10.0 mm at W ¼ 8 m/s. By comparing these natural frequencies of the vortex cavities to those of the spherical cavities, it can be estimated that the natural frequency of the vortex cavity by visual inspection is much lower than that of the spherical cavities with a small radius. If the CIS is determined with the sound produced by small size spherical cavities, signal processing should be performed with the sound of high frequency range considering the natural frequency of the incepted spherical cavities. However, if the CIS is determined with the sound produced by tip vortex cavities, it should be performed with the sound of the frequency range including the natural frequency of the tip vortex cavities.
Fig. 3. Observation of the singing vortex, Maines and Arndt (1997).
here, λz is the wave length (m). Junger and Feit (1986) stated that the high order mode does not contribute to the radiated sound of the cavity, and only the 0th mode contributes to the radiated sound of the cavity because the maximum amplitude of the alternating pressure of the cavity is scaled to ðkrc Þn , and the high order mode is low ðkrc Þn , where n is the order number. The mode shapes of each mode are shown in Fig. 4 (Bosschers, 2007). Fig. 5 shows the natural frequencies of the spherical and vortex cavity when W ¼ 5 m/s, 8 m/s, and λ ¼ 30 mm. When the axial velocity is 5 m/ s, the natural frequency of the spherical cavity is higher than that of the vortex cavity, the radius of which is equal to that of the spherical cavity as shown in Fig. 5(a). However, as the axial velocity increases to 8.0 m/s, the gap of the natural frequency between spherical and vortex cavity decreases as shown in Fig. 5(b). In Fig. 5, the natural frequency is calculated at the point where the density of the sea water is 1025 kg/m3, the specific heat ratio is 1.0, and the distance between the free surface of the sea and the propeller is 10 m. Assuming that the shape of cavities is spherical at the CIS step and
2.3. DEMON analysis The cavitation begins at one of the propeller blades when the speed of rotation is over the critical speed, such that the static pressure of the surrounding water approaches close to the vapor pressure. When the cavitation continuously develops according to the increasing speed, cavitation occurs at all of the blades. When the cavitation occurs, the sound of the cavities modulates synchronously to the shaft rate and the blade passing frequency. When applying DEMON analysis, the noise component at the shaft rate as well as the blade passing frequency is revealed in the DEMON spectrum. The analytic signal of DEMON analysis is obtained from Eq. (8) and the enveloped signal is the amplitude of the analytic signal as shown in ) (Randall, 1987). b hðtÞ ¼ hðtÞ þ j~hðtÞ
(8)
0:5 b hðtÞ ¼ hðtÞ2 þ ~hðtÞ2
(9)
~ here, h(t) is the time signal of the raw data and hðtÞ is the Hilbert transform of h(t), as shown in Eq. (10). ~hðtÞ ¼ HðhðtÞÞ ¼ 1 ∫ ∞ hðtÞdt π ∞ τ t
(10)
From the fast Fourier transform (FFT) spectrum of Eq. (9), CIS inspects whether the DEMON spectrum reveals the sound of the rotating shaft or blade passing frequency. In DEMON analysis, the selection of the bandwidth and center frequency of the band-pass filter is very important. They must be selected considering the frequency range where the
Fig. 4. Main vortex cavity oscillation modes (reproduced from Bosschers (2007)). Top: monopole breathing mode (n ¼ 0); middle: dipole serpentine centerline displacement mode (n ¼ 1); and bottom: quadrupole helical mode (n ¼ 2). 169
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Fig. 5. Resonance frequency of the bubble according to its size. (a) W ¼ 5 m/s, λ ¼ 30 mm (b) W ¼ 8 m/s, λ ¼ 30 mm.
According to the measurement results of the underwater-radiated noise with the hydrophone array that was performed prior to the present research, CIS was preliminary estimated and determined as VCIS knots. Therefore, the acceleration on the aft hull was measured as from “VCIS-8” knots to “VCISþ4” knots as shown in Table 1 in order to compare the CIS estimated with accelerations to that with underwater radiated noise.
modulation coincides with the shaft rotating and blade passing frequency. This proper selection can be performed with spectral kurtosis of the short time FFT described in the next section. In addition, CIS can be identified when the overall noise level is increased to over 2dB/knots as the ship speed increases. In this research, CIS is determined with the vibration signal on the aft hull located close to the propeller instead of the sound signal radiated to the under water. Since the pressure wave produced by the propeller cavitation induces the adjacent hull of the propeller, the acceleration on the hull is the same pattern to the cavitation sound of the propeller. Therefore, in the following section, for a specific naval vessel, CIS is evaluated using a 1/3 octave band spectrum, the overall level, and DEMON spectrum of the hull vibration adjacent to the propeller.
3.1. Test setup The tested ship is a battleship with a combined diesel-engine/gasturbine-propulsion system. The propeller type is the controllable pitched propeller, and two of each propeller are installed at the port and the starboard sides. To inspect the propeller CIS, three accelerometers were installed at the aft hull above the propeller as shown in Fig. 6. Since the space wherein the accelerometers were installed is void space, equipment is not necessary. The No.1 accelerometer was installed on the hull at the port side, No.3 was installed on the hull at the starboard side, and No.2 was installed between the No.1 and No.3 accelerometers as shown in Fig. 6.
3. Experiment In this section, the acceleration on the aft hull above the propeller is measured according to the ship speed in order to estimate CIS based on the acoustic characteristics of the cavitation cavity described in section 2.
Table 1 Speed of the test ship. Test No.
t1
t2
t3
t4
t5
t6
t7
t8
t9
t10
Speed [knots] RPM Pitch %
VCIS-8 N0 0%
VCIS-5 N0 33%
VCIS-3 N0 66%
VCIS-2 N0 100%
VCIS-1 N0 þ 5
VCIS N0 þ 10
VCIS þ 1 N0 þ 14
VCIS þ 2 N0 þ 21
VCIS þ 3 N0 þ 38
VCIS þ 4 N0 þ 45
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Fig. 6. Test setup for CIS measurement with accelerometers.
- Since 2 propellers are installed at the port and starboard sides of the ship and their blades rotate outward, the cavitation cavities of the 2 propellers would be created below the center position installing No. 2 accelerometers where the static pressure of the water near the propeller is the least. The water can be vaporized easily at this position, when the propeller speed is critical for the cavitation, because the static pressure is at its lowest value. Therefore, the pressure wave from the cavity may be transferred to the hull more easily where installing the No.2 accelerometer rather than where installing the No.1 and 3 accelerometers.
A PCB Type 352A60 accelerometer and a B&K3053-B12/0 data acquisition system were used to collect the signal. The acceleration was recorded for 2 min for each ship speed, as shown in Table 1, and the sampling rate was set at 65,536samples/sec. 3.2. Test results Fig. 7 shows the 1/3 octave band spectrum of the acceleration for each ship speed to be tested. The acceleration levels at the high frequency range of over 5 kHz for the No.1 and 3 accelerometers increased rapidly when the ship speed was t6–t7. Even though the distances from the No. 1 and No. 3 accelerometers to the center line are identical, the hull structures (for example, the distribution of the stiffeners) are different from each other because the hulls are not quite bilaterally symmetrical. In addition, the dynamic pressure-induced acceleration on the hull can be different between the port and starboard sides, because the dynamic pressure of the propeller blade for the port and starboard sides can be nonlinearly different in accordance with the sailing conditions. Therefore, the difference between the acceleration spectra of No. 1 and No. 3 can exist, as shown in Fig. 7. Even though the spectra shape of the No. 1 and No. 3 accelerations are different from each other, they are increased rapidly and identically when the ship speed is from t6–t7. The acceleration levels at the high frequency range of over 5 kHz for the No.2 accelerometer increased rapidly at t5 ~ t6, earlier than those for No.1 and 3 accelerometers. This increasing acceleration may be caused by the cavitation. Therefore, CIS can be estimated to t5–t7. The reason for why this increasing acceleration for the No. 2 accelerometer started earlier than those of the No. 1 and No. 3 accelerometers was not clearly identified. We could simply guess that the reason is as follows:
Even though the reason for this difference was not clearly identified, the CIS that was estimated from the acceleration can be varied in accordance with the accelerometer-installation position. In this research, it is assumed that the frequency range of the cavitation sound caused by the vortex and large spherical cavities is set to 3 kHz–10 kHz and that caused by the spherical cavities with a very small radius is set to 10 kHz–25.6 kHz. Fig. 8 shows the overall acceleration level in the ranges of 3 kHz–10 kHz and 10 kHz–25.6 kHz. Generally, CIS is determined when the overall sound pressure level is increased by 2 dB per 1 knot increase in speed. In order to identify CIS using the overall acceleration level, the same criterion for identifying CIS with sound is adopted. In Fig. 8, the overall acceleration levels on the hull in which the frequency range is from 3 kHz to 10 kHz are increased by 2 dB per 1 knot increase in speed, similar to the sound pressure when the ship speed varies from t6 to t7 for the No.1 and 3 accelerometers and from t4 to t5 for the No.2 accelerometer. However, the overall acceleration levels on the hull in which the frequency range is from 10 kHz to 25.6 kHz are increased by 2 dB per 1knot increase in speed, similar to the sound pressure when the ship
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Fig. 7. 1/3 octave band spectra of the acceleration of the hull above the propeller. (a) accelerometer No.1. (b) accelerometer No.2. (c) accelerometer No.3. Fig. 8. Overall rms acceleration of the hull above the propeller. (a) accelerometer No.1. (b) accelerometer No.2. (c) accelerometer No.3.
speed varies from t4 to t5 for the No.1 and 2 accelerometers but the No. 3 accelerometer does not sufficiently vary.
KX ¼ 3.3. Spectral kurtosis
hEfX 4 ðtÞgi hEfX 2 ðtÞgi2
2
(11)
here, E{}is the mean function, KX is the spectral kurtosis, and X(t)is the signal value at a particular frequency for time t. If the distribution of the signal is Gaussian, the kurtosis becomes 0, and if the signal has high transient signal, it becomes very high. When the cavitation occurs, the impact signal of the propeller occurs periodically synchronous to the propeller rotation and blade passing frequency. In contrast to the frequency of the bearing and gear mesh fault sound, the frequency of the cavitation sound is not the natural frequency of the system but that of the cavity. Therefore, when DEMON analysis is performed, the natural frequencies of the created cavities on the propeller should be included in the bandwidth of the band-pass filter, considering their shape and size. Fig. 9 shows the kurtogram of the measured acceleration on the hull when the ship speed is t5 ~ t7. The kurtogram is a graph of the spectral kurtosis at each frequency for the short time FFT spectrum performed with 2 k frequency lines. It is used to determine the center frequency and
In the previous section, CIS was estimated to be t5–t7 based on the overall acceleration level and on the 1/3 octave band spectrum analysis. In this section, DEMON analysis described, which is another method used to determine CIS. In order to perform DEMON analysis, the center frequency and bandwidth of the band-pass filter should first be determined. In this research, spectral kurtosis (Antoni and Randall, 2006) is applied to select the center frequency and the bandwidth of the band-pass filter to perform CIS. In the mechanical rotating system, periodic excited sound can be caused by a bearing or gear mesh fault. When the impact force is subject to the mechanical system, the vibration increases at its natural frequency. Therefore, the vibration at the natural frequency of the mechanical system periodically appears and disappears repeatedly. When the signal varies unstably due to the transient signal, the kurtosis of the signal increases. Spectral kurtosis is the kurtosis of the signal for the spectrum at a particular frequency and is defined as ) (Antoni and Randall, 2006). 172
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Fig. 9. Kurtogram of accelerations of the hull above the propeller. (a) accelerometer No.1. (b) accelerometer No.2. (c) accelerometer No.3.
bandwidth of the bandpass filter for DEMON analysis. In the kurtogram, the horizontal axis is the frequency (0–25.6 kHz)
and the vertical axis is k (0–7), and k is defined as NW ¼ 2k (Nw: window length). The spectral-kurtosis level is represented by the color map on the 173
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Fig. 9. (continued).
lower than the frequency of that caused by the spherical cavity. Therefore, when the ship speed is t6–t7, spectral kurtosis increases at the frequency range of 3–10 kHz, which is less than the frequency range when the ship speed is t5. When kurtogram analysis is performed for the mechanical system having mechanical defaults such as bearing and gear faults, the k level should be selected to include the natural frequency of the mechanical system where the response of the system increases significantly from the impact force. However, in cavitation, the response is not the response of the mechanical system but that of the cavities. Therefore, at a specific k level, the frequencies with the highest spectral kurtosis, i.e., natural frequencies of vortex bubbles, are varied according to the different cavity size and environmental conditions. In addition, in the process of the merging and the collapsing of the bubbles, the frequencies with the highest spectral kurtosis can also be varied. Therefore, even though the obtained k level can be from 5 to 7 at speeds from t6–t7, narrow bandwidths such as those from 200 Hz (k ¼ 5)–800 Hz (k ¼ 7) cannot be applied to perform the DEMON analysis for the detection of the CIS. Considering that the frequency band of the DEMON analysis for the cavitation sound at the ship speed of t5 can be obtained at 6.4 k Hz from the kurtogram analysis, and the center frequencies of the vortex bubbles can vary widely according to their size and environmental condition, the bandwidth of the bandpass filter of the DEMON analysis for the acceleration on the hull was selected as 6.4 kHz. In fact, the 6.4-kHz frequency
graph. A high spectral-kurtosis level at a given frequency band means that many transient signals have been caused by the cavitation. Therefore, using the kurtogram, the center frequency and the frequency span that comprise the maximum spectral kurtosis can be defined. In Fig. 9, the spectral kurtosis is at a very high level at a low k (k < 3: bandwidth > 6.4 kHz) and high frequency range (>20 kHz) when the ship speed is t5 for No.1 and 2 accelerometers. It means that, regarding the best-fit bandpass filter for the DEMON analysis at the ship speed of t5, it would be preferable for the center frequency and the frequency span to be more than 20 kHz and 6.4 kHz, respectively. As the ship speed increases to t6 and t7, the frequency range having the highest spectral kurtosis decreases to the lower frequency and the k level having the highest spectral kurtosis increases to 5–7 (200 Hz < bandwidth < 800 Hz) compared to those when the ship speed is t5. Therefore, regarding the center frequency of the best-fit bandpass filter for the DEMON analysis at the ship speeds from t6–t7, it would be preferable for the speed to be lower than t5 and the frequency band to be narrower than its typical t5 value. As described in section 2.2 and Fig. 5, spherical bubbles first appear when the cavitation is incepted and the broadband acceleration of 10 kHz–25.6 kHz which includes the natural frequency of the very small size cavity is modulated with the acceleration at the propeller rotating and blade passing frequency. However, when the cavity is developed and the tip vortex cavity starts to form, the frequency of the acceleration caused by the vortex cavity is 174
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spherical cavities and is set to 6.4 kHz 3.2 kHz with that for the tip vortex cavity.
band is widely used for the bandpass filter that is used to perform the DEMON CIS-detection analysis. Consequently, the center frequencies were defined as 22.4 kHz and 6.4 kHz based on the kurtogram results at the ship speeds t5 and from t6–t7, respectively, and the bandwidth was determined as 6.4 kHz, as previously stated. According to the kurtogram analysis of this chapter, the frequency range of the band-pass filter for DEMON analysis is set to 22.4 kHz 3.2 kHz when CIS is defined with the sound for the very small
3.4. Results from DEMON DEMON analysis for the acceleration on the hull above the propeller is performed according to the ship's speed, at which the bandwidth of the band-pass filter is set to 6.4 kHz and the center frequency of the bandFig. 10. DEMON spectra applying 6.4 3.2 kHz bandpass filter (SR ¼ speed rate frequency, BPF ¼ blade passing frequency, CIS ¼ t6–t7). (a) ship speed ¼ t5. (b) ship speed ¼ t6. (c) ship speed ¼ t7. (d) ship speed ¼ t8.
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Fig. 10. (continued).
pass filter is set to 6.4 kHz and 22.4 kHz based on the kurtogram analysis. Fig. 10 shows the DEMON spectrum when the bandwidth of the bandpass filter is 6.4 kHz and the center frequency is 6.4 kHz. Here, the x axis is the frequency and the y axis is the nominal value that indicates 2 times the measured time duration (2 0.5sec n, n ¼ 0–200). The bandwidth and center frequency of DEMON analysis for the underwater radiated noise were the same as those applied in Fig. 10. When the cavitation is intercepted, the cavity is created at one of the
propeller blades. If the blade dimensions of the propeller are perfectly the same as one another, cavities can occur at all of the blades when they pass through the positions where the distance between end of the blade and the free surface is the smallest. However, due to the variation of the dimensions, a cavity can occur at one of the blades first. And then, as the cavities are developed further, the cavities start to occur at all of the blades. Therefore, in the DEMON spectrum, the peak at the 1st rotating frequency of the propeller shaft is firstly revealed at the CIS, and then its
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Fig. 11. DEMON spectra applying 6.4 3.2 kHz bandpass filter (SR ¼ speed rate frequency, BPF ¼ blade passing frequency, CIS ¼ t5–t6). (a) ship speed ¼ t1. (b) ship speed ¼ t5. (c) ship speed ¼ t6. (d) ship speed ¼ t7. (e) ship speed ¼ t8.
frequency when the ship speed is t7. Fig. 11 shows the DEMON spectrum when the bandwidth of the bandpass filter is 6.4 kHz and the center frequency is 22.4 kHz. The spectrum value at the rotating shaft and blade passing frequency are shown at t5 and t6, respectively, where the ship speed is less than that when using the band-pass filter with a center frequency of 6.4 kHz. Therefore, CIS can be estimated as t5–t6 from Fig. 11. In addition, the peak value at the shaft rate can be found in the DEMON spectrum when the propeller pitch is
2nd and 3rd peaks are revealed according to the rotating frequency of the shaft. When the cavities are sufficiently developed, the peak at the bladepassing frequency is also evident. Therefore, CIS is defined from DEMON analysis when the DEMON spectrum has the peak at the rotating frequency of the shaft. In Fig. 10, CIS can be estimated as t6–t7, similar to CIS estimated with underwater radiated noise since the DEMON spectrum has a peak at the 1st rotating frequency when the ship's speed is t6 and the blade passing 177
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Fig. 11. (continued).
minimum (V ¼ t1). It means that small spherical bubble may occur when the low propeller pitch is applied. As discussed in the previous section, the differences in CIS according to the center frequency of the band-pass filter is due to the different types of cavities (spherical or vortex cavity).
4. Discussion When the CIS is estimated with 1/3 octave band spectrum, overall level variation and DEMON spectrum analysis of the acceleration on the hull, it is the same as the CIS estimated with underwater radiated noise. Even though CIS slightly varies according to the location of the installed
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Fig. 11. (continued).
accelerometer, it is verified that CIS can be determined from the acceleration on the hull above the propeller. When the hydrophone and pressure sensor are applied to the hull in order to monitor CIS, a hole needs to be bored to install these sensors. However, this hole is not necessary when the accelerometer is applied to monitor the CIS variation in a real ship. Therefore, for the real time monitoring of CIS, it is possible to use acceleration on the hull simply with the signal processing suggested in this research. Through this monitoring technique, the maintenance duration can be identified exactly with respect to CIS. Even though CIS can be estimated with DEMON spectrum analysis, decoding the DEMON spectrum is very subjective. Therefore, a quantitative index needs to be developed for automatic real time monitoring of CIS. In this section, the kurtosis of the DEMON spectrum is suggested for this quantitative index. Fig. 12 shows the DEMON spectrum at the typical time “t” when the cavitation begins. In Fig. 12, the peak occurs at the blade passing frequency. When the dominant peak forms, as shown in Fig. 12, the kurtosis of the DEMON spectrum (Y) as Eq. (12) becomes very large. Here, xi is the DEMON spectrum level at each frequency, x is the average spectrum level, s is the standard deviation of the spectrum levels, and n is the line numbers of the frequency in the DEMON spectrum. If no dominant peak occurs in the DEMON spectrum, the distribution of the DEMON spectrum level is Gaussian and the kurtosis of the DEMON spectrum should be approximately 0.
Fig. 12. DEMON spectrum when the cavitation occurs.
DEMON spectrum of the acceleration on the hull according to the ship's speed as shown in Table 1. Here, the frequency range of the DEMON spectrum is 4–100 Hz and the number of the data to calculate kurtosis is 192 since the frequency resolution is 0.5 Hz (n¼(100 Hz–4 Hz)/0.5 Hz). The kurtosis is calculated with the DEMON spectrum of the acceleration measured for 0.5sec. The kurtosis is calculated for 100 s and the kurtosis shown in Fig. 13 is the averaged kurtosis for 100 s. In Fig. 13, it is assumed that CIS occurs when the average kurtosis is over than 4.0. Then, CIS can be identified as t5 when the center frequency and bandwidth are set to 22.4 kHz 3.2 kHz as shown in Fig. 13(a). In this case, cavitation also occurs at t1–t2 where the propeller pitch ratio is very small. When the center frequency and bandwidth is set to 6.4 kHz 3.2 kHz, CIS can be estimated as t6, as shown in Fig. 13(b). CIS is estimated with the average kurtosis, as shown in Fig. 13, is the same as that estimated by qualitative analysis performed in section 3. Through this average kurtosis of the DEMON spectrum, CIS can be
"
4 # n X nðn þ 1Þ xi x 3ðn 1Þ2 Y¼ ðn 1Þðn 2Þðn 3Þ i¼1 s ðn 2Þðn 3Þ
(12)
Therefore, the kurtosis of the DEMON spectrum according to the ship's speed can identify CIS. Fig. 13 is the average kurtosis for the
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(1) CIS can be estimated with the acceleration on the hull adjacent to the propeller from the experiment with s specific naval vessel. (2) Through the theoretical investigation, it can be seen that the natural frequency of the spherical cavity is much higher than that of the vortex cavity with the same radius when the axial velocity of the propeller is slow. Therefore, when the spectrum analysis using DEMON is performed to identify CIS, the cavity's shape and size should be considered. (3) When the CIS of a specific naval vessel is estimated from DEMON of the acceleration on the hull, whereby the center frequency of the band-pass filter is 6.4 kHz and 22 kHz, CIS estimated with a center frequency of the band-pass filter of 22.4 kHz is 1 knots faster than that estimated with the center frequency of the bandpass filter of 6.4 kHz. When using 6.4 kHz band-pass filter, CIS is determined to the speed inspecting very small spherical cavities that can't visually be inspected. However, when using 22.4 kHz band-pass filter, CIS is determined to the speed inspecting well developed tip vortex cavities. Therefore, CIS can be different according to the frequency range of the band-pass filter. (4) The estimated CIS differs according to the position at which the accelerometers installed. Therefore, the proper installation position of the accelerometer should be selected. In order to select the best point on the hull when transferring the pressure wave, the CIS estimated from the underwater radiating noise should be compared to that estimated from acceleration. (5) Through the estimation method of CIS using the average kurtosis of the DEMON spectrum, the quantitative evaluation of CIS is possible and the CIS estimated with the average kurtosis of the DEMON spectrum concurs well with the qualitative evaluation of the CIS with 1/3 octave band and DEMON analysis. It is expected that CIS determined using the average kurtosis of the DEMON spectrum can be applied to the real time monitoring of CIS.
Fig. 13. Kurtosis of DEMON spectrum for acceleration on the hull according to the ship's speed. (a) bandpass filter ¼ 22.4 kHz 3.2 kHz (b) bandpass filter ¼ 6.4 kHz 3.2 kHz.
identified quantitatively and can be used in the real time monitoring of CIS in a naval vessel during a voyage. When the center frequency of the band-pass filter is set as 22.4 kHz, CIS is identified earlier at about 1 knots compared to CIS estimated with the band-pass filter using a center frequency of 6.4 kHz. It is widely known that the cavity's sound can be heard ahead of the visual inspection of the bubble. In the investigation of this research study, the sound of the small spherical bubble firstly occurred, and this was followed by the sound of the vortex bubble. After the amplitude of the cavity sound is increased in accordance with the increasing of the cavity size, the cavity can be visualized. Therefore, to accurately identify the CIS, the CISidentification method should be firstly selected. Since the sonar systems that are installed in navy ships detect the CIS with the sound signal, the CIS should be defined not by the visualization of the cavity, but by the sound from the cavity. To define the CIS with the sound signal accurately, the center frequency and bandwidth of the band-pass filter should be determined considering the detection capabilities of the sonar systems for the general naval vessel. Currently, the designs of general CISdetection sonar systems are usually based on the sound detection of the vortex cavity.
In the future, the CIS monitoring and estimating system will be developed, with the acceleration on the hull adjacent to the propeller. In order to develop this system, the installation position of the accelerometer and the type of cavity that most contributes to the underwater radiated noise should be initially identified. References Antoni, J., Randall, R.B., 2006. The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines. Mech. Syst. Signal Process. 20, 308–331. Bosschers, J., 2007. Broadband Hull Pressure Fluctuations and Cavitating Vortices – An Investigation of Resonance Frequencies . Marine Engineering Forum, Propeller Cavitation Workshop, London. Han, H.S., Jeong, W.B., Kim, M.S., Lee, S.Y., Seo, M.Y., 2011. Frequency characteristics of the noise of R600a refrigerant flowing in a pipe with intermittent flow pattern. Int. J. Refrig. 34, 1497–1506. Junger, M.C., Feit, D., 1986. Sound, Structures and Their Interaction, second ed. MIT Press, Cambridge, MA, USA. Lamb, H., 1945. Hydrodynamics. Dover Publication, New York. Maines, B., Arndt, R.E.A., 1997. The case of the singing vortex. J. Fluid Eng. 119, 271–276. Minnaert, W.K., 1933. On musical air bubbles and the sound of running water. Phil. Mag. 16, 235–248. Morozov, V.P., 1974. Theoretical analysis of the acoustic emission from cavitating line vortices. Sov. Phys. Acoust. 19 (No. 5), 468–471. Pennings, P.C., Westerweel, J., Terwisga, T., 2016. Cavitation tunnel analysis of radiated sound from the resonance of a propeller tip vortex cavity. Int. J. Multiph. Flow 83, 1–11. Randall, R.B., 1987. Frequency Analysis. B&K. Strasberg, M., 1956. Gas bubbles as source of sound in liquids. J. Acoust. Soc. Am. 28 (No. 1), 20–27. Thomson, W. (Lord Kelvin), 1880. Vibrations of a columnar vortex. In: Proceedings of the Royal Society of Edinburgh, Scotland, March 1, pp. 152–165. Phil. Mag. X.
5. Conclusion In this research, the CIS estimation method using the signal processing technique from the acceleration on the hull above the propeller is suggested based on the mechanism in which the vibratory cavity transfers the pressure wave to the hull adjacent to the propeller. The detailed conclusions from this research are as follows.
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