research into time-frequency characteristics of cavitation noise using wavelet scalogram

Experimental research into time–frequency characteristics of cavitation noise using wavelet scalogramYongyong He ⇑,Yuan LiuThe State Key Laboratory of Tribology,Tsinghua University,Beijing 100084,PR Chinaa r t i c l e i n f o Article history:Received 17September 2010Received in revised form 23January 2011Accepted 29March 2011Available online 4May 2011Keywords:Cavitation detectionTime–frequency analysis Wavelet scalograma b s t r a c tThe cavitation has become the main cause of the damage to the hydraulic machine.Cavitation detection is very important to guarantee the safe running of the hydraulic machine.The sound,especially the audi-ble sound,based methods are becoming attractive due to their simplicity and logicality in the application.However,the cavitation noise is easy to be contaminated by the background noise.In order to understand the characteristics of the cavitation noise deeply,using the wavelet scalogram analysis,this paper pre-sents an experimental study to investigate the time–frequency characteristics of the cavitation noise of various cavitation states and the relation between the cavitation noise and the cavitation process.In addi-tion,the method of parameters optimization for the wavelet is used to improve the transform perfor-mance of the wavelet scalogram.The results show that:the cavitation noise is composed of the components with wide band frequency and obvious impulse feature;but the cavitation noise of different cavitation stages has different time–frequency characteristics and compositions;in addition,the cavita-tion noise can be distinguished from the background noise because they have totally different frequency characteristic.This study validates that the cavitation noise can be used to detect the cavitation state and monitor the cavitation process.Ó2011Elsevier Ltd.All rights reserved.1.IntroductionThe cavitation is an important and unwanted phenomenon in the flowing liquid,which is defined by Knapp et al.[1].This occurs when a liquid reaches a state at which vapor cavities or bubbles are formed and then grow due to a decrease in local pressure to the va-por pressure of the liquid.These formed and growing cavities will collapse implosively and disappear as soon as they reach higher pressure regions in the flowing liquid [2].The violent process of cavity collapse takes place always in a very short time of about sev-eral nanoseconds and results in the emission of large amplitude shock-waves [3].And a high-speed reentrant liquid micro-jet directed towards the boundary can also occur when cavities col-lapsing close to a solid surface [4].This process has the character of water hammer blows [5].Therefore,the cavitation phenomenon will cause some undesirable effects,even serious damage,to the hydraulic machine,such as the deterioration of the hydraulic per-formance,possible pitting and material erosion (i.e.cavitation ero-sion)to the machine structures in the vicinity of the collapsing cavities,and excessive vibration and noise [6].Actually,the cavita-tion and cavitation erosion have become the main causes of the damage to the hydraulic machine.However,it seems difficult,even impossible,to eliminate the cavitation totally when the existing hydraulic machine running.Real-time monitoring and timely detection of the cavitation during running to avoid further harmful situations seem the best solution.The cavitation will cause a significant drop in the efficiency of the hydraulic machine.Thus,the most commonly used method of cavitation detection is based on the observation of the efficiency drop [7].However,this method needs a special test stand and a set of measurement results to determine the ‘‘critical’’point of the efficiency drop at different flow rates.Therefore,it is not suitable for the cavitation detection in some onsite situation.Visualization of the flow is the second most popular technique for the cavitation monitoring and detection,which uses the underwater image equipment to visualize and photograph the cavitation states in the flowing liquid [8].This method has its limitation in the appli-cation because it only provides qualitative observation and analy-sis of the cavitation and it is also difficult to fix the image equipment to the existing hydraulic machine in some cases.The other techniques,such as the paint erosion testing,the static pres-sure measurement based method,the vibration analysis based method and the sound pressure measurement based method,have been proposed over recent years [6].Especially,the sound pressure measurement based method is attracting more and more attention and becoming a potential method due to its simplicity and logical-ity in the application.It has been known that the appearance of the0003-682X/$-see front matter Ó2011Elsevier Ltd.All rights reserved.doi:10.1016/j.apacoust.2011.03.008Corresponding author.Tel./fax:+861062787932.E-mail address:heyy@ (Y.He).cavitation is clearly heard by normal listening.Therefore,the cav-itation noise can be used to monitor and detect the onset and development of the cavitation.However,the cavitation noise is easy to be contaminated by the surrounding environmental noise,i.e.the background noise.It means that:for better application of this method,it is necessary and crucial to deeply analyze and understand the characteristics of the cavitation noise,the differ-ence between the cavitation noise and the background noise,and the relation between the cavitation noise and the cavitation pro-cess.The noise spectrum analysis method has been extensively used to analyze the characteristics of the cavitation noise.How-ever,the noise spectrum analysis can only provide the frequency information.Especially,it is known that the cavitation noise is always time-variant and non-stationary.The Fast Fourier Trans-form (FFT)based noise spectrum analysis is not appropriate for such signal.Therefore,to understand the characteristics of the cav-itation noise more deeply,some time–frequency analysis methods,which can simultaneously provide the time domain and the fre-quency domain information of the signal,should be introduced to analyze the cavitation noise.A particular time–frequency analy-sis method,the wavelet transform (WT)[9,10],has become an attractive and popular analysis method of the non-stationary signal due to its evident pared with other time–frequency analysis methods,such as the Gabor transform (win-dowed Fourier transform)[11]and the bilinear time–frequency representation [12],the WT can be used for multi-scale analysis of the signal through dilation and translation and has better time–frequency characteristics,and therefore has been utilized widely for the time–frequency analysis of non-stationary signals in various areas.Corresponding to the spectrums of the Gabor transform,the wavelet scalogram provides another alternative time–frequency analysis method for non-stationary signals,which can extract more effective time–frequency information of the sig-nal and can display periodic,quasi-periodic,and even randomly occurring signals.Therefore,it seems that the wavelet scalogram can provide a strong tool for analyzing and investigating the char-acteristics of the cavitation noise.In this paper,using the wavelet scalogram method,an experi-mental study is presented to investigate the time–frequency char-acteristics and compositions of the cavitation noise of different cavitation states.The aim of this study is to further understand the characteristics of the cavitation noise and reveal the relation between its time–frequency characteristics and the cavitation pro-cess.This study should provide some experimental support and base for the better application of the cavitation noise based moni-toring and detection method of the cavitation.2.Continuous wavelet transform and Its scalogram [13]Assume that x (t )is a finite-energy function,i.e.x (t )2L 2(R ).Then the continuous wavelet transform (CWT)of x (t )is defined asW x ða ;b ;w Þ¼x ðt Þ;w a ;b ðt Þ¼a À1=2Zx ðt Þw Ãa ;b ðt Þdtð1Þwhere a >0,the superscript ⁄denotes the complex conjugation andh i denotes the inner product.w a ,b (t )is generated by dilation and translation from the mother wavelet w (t ),byw a ;b ðt Þ¼aÀ1=2wt Àb að2Þwhere a is the scale parameter and b is the time parameter.The fac-tor a À1/2is used to ensure energy conservation for the transform.Many functions may be used as the mother wavelet,but they must belong to L 2(R )and satisfy the admissible conditions,which ensures the mother wavelet function to be oscillatory and with suitabletime localization.From this definition of the wavelet transform,the wavelet transform does not lose any information and energy.Thus,x ðt Þ;x ðt Þh i ¼Z1À1x ðt Þj j 2dt ¼1w Z1À1aÀ2Z1À1W x ða ;b ;w Þj j 2dbdað3Þfor some constant C w .Thus |W x (a ,b ;w )|2/C w a 2can be considered asthe energy density function of the time-scale plane (a ,b )and j W x ða ;b ;w Þj 2D a D b =C w a 2represents the total energy of a domain centered at (a ,b )with scale interval D a and time interval D b .The wavelet scalogram is defined as the spectrum of the Gabor trans-form,SG x ða ;b ;w Þ¼j W x ða ;b ;w Þj 2,which can be seen as a spectrum with constant relative bandwidth [13].Fig.1gives out a simulation example to demonstrate the advan-tage of the wavelet scalogram analysis compared with FFT.The simulation signal is composed of an amplitude-modulated signal and a frequency-modulated &amplitude-modulated signal.From Fig.1,it can be seen that the FFT spectrum cannot give almost any amplitude-modulated or frequency-modulated information of the signal.On the other hand,the time domain and the fre-quency domain information are manifested detailedly in the time–frequency plane of the wavelet scalogram.The amplitude-modulated and the frequency-modulated information can be ob-served clearly.3.Experimental study and results 3.1.Experimental system for cavitationFig.2shows the schematic diagram of the experimental rig,the cavitation generating segment of which includes a Venturi tube.When the liquid flows over the Venturi tube,increasing flow rate of the liquid causes the decrease of the pressure at the throat of the Venturi tube and thus leads to cavitation generation.Fig.3shows the particular structure of the cavitation generating segment.The blocks with different thicknesses can be placed at the corre-sponding location in the Venturi tube (see Fig.3)to make the cavi-tation generate easily.The flow rate of the liquid is controlled by the pump and an asynchronous motor.The asynchronous motor is ad-justed by the frequency converter.The regulating range of the con-verter is from 8Hz to 50Hz.The flow rate of the liquid is measured by the flowmeter.Fig.4gives out the calibration curve between the frequency of the converter and the flow rate of the liquid.We find that the cavitation can be observed always when the frequency of the converter reaches about 17Hz.Therefore,in the experiment of this study,the regulating range of the frequency from 14Hz to 40Hz is selected.For the convenience of description,the frequency of the converter is used to denote the flow rate of the liquid in this paper.A Bruel &Kjær (B &K)high frequency hydrophone of type 8103is used to collect the cavitation noise signal at the corresponding location (see Fig.3),the frequency range of which is from 0.1Hz to 180kHz.The B &K PULSE 3560c system with highest sampling frequency of 65kHz is used to sample the cavitation noise signal and collect the data.In this study,the audible noise is considered.Therefore,the sampling frequency is set to 32kHz.3.2.Experimental schemeIn order to investigate the characteristics of the background noise and its influence on the cavitation noise,we first perform the experiment without the cavitation.In this experiment,a thin-ner block is placed at the corresponding location (see Fig.3)to make its surface accordant to the inner surface of the Venturi tube of the cavitation generating segment.It means that the whole inner722Y.He,Y.Liu /Applied Acoustics 72(2011)721–731surface of the Venturi tube is smooth and there is no bulge (denoted0mm bulge in the following description).Under this experimental condition,through the observation window,we found no cavitation emerged when the frequency of the converter was adjusted from14Hz to even25Hz.Therefore,the collected noise signal should be regarded as the background noise.The noise signals under theflow rates of14Hz,15Hz,16Hz,17Hz,19Hz, 21Hz,23Hz and25Hz respectively are sampled and collected. Fig.5shows two examples of the noise signals under theflow rates of15Hz and19Hz.From Fig.5,it can be seen that the amplitude of the background noise increases from about2000Pa to about4000Pa with the increase of the liquidflow rate from15Hz to19Hz.However, any other information cannot be observed just from these original waveforms.In the next section,the wavelet scalogram is used to investigate its time–frequency characteristics.Then,the cavitation experiments are carried out.In these exper-iments,to make the cavitation generate easily,a thicker block is placed at the corresponding location(see Fig.3).The block bulges over the inner surface of the Venturi tube about3mm.It means that there exists a3mm bulge in the inner surface of the Venturi tube(denoted3mm bulge in the following description).Adjust the frequency of the converter from14Hz to50Hz,the noise sig-nals under theflow rates of14Hz,15Hz,16Hz,17Hz,19Hz, 21Hz,23Hz,28.5Hz,35Hz,40Hz and50Hz are sampled and col-lected respectively.In the process of these experiments,through the observation window,the cavitation inception is observed when the liquidflow rate reaches about17Hz.After that,the cavitation phenomenon and its development process from the inception to the supper cavitation state are observed clearly.From Fig.6,it can be seen that the amplitude of the noise increases sharply with the emergence and development of the cavitation.Likewise,no anyFig.1.A simulation of wavelet scalogram analysis.Fig.2.Schematic diagram of the experimental equipment for cavitation.other detailed information can be observed from these original waveforms.In the next section,the wavelet scalogram is also used to analyze the time–frequency characteristics of the cavitation noise of different cavitation states.4.Experimental analysis and results4.1.Power spectrum analysis of cavitation noiseFor power spectrum analysis,compared with the traditional spectrum estimation methods,such as periodogram method, Welch method,AR spectrum estimation has following two merits:(1)Higher analysis resolution:For the traditional spectrum esti-mation methods,the analysis resolution is equal to the mainlobe width of window function.It means that theanalysis resolution is inversely proportionalof the sampled data.The reason is thattrum estimation methods regard theanalysis length,as zero,which is notin many situations.However,ARnot restricted by this term.The outsidemined by extrapolation.Therefore,thethe restriction of window function width.(2)Better smoothness:AR spectrum estimationmodel,which is continuous derivativeTherefore,it has better smoothness.Fig.7shows the comparison between thetion and the periodogram method with respectsignal which is composed of four sinusoidalure,it can be seen that the power spectrumTherefore,AR spectrum is more appropriate for the power spec-trum analysis of such signal.In this paper,AR spectrum is used to analyze the cavitation noisefirst.Fig.8a shows the AR spectrums of the signals under theflow rates of14Hz,15Hz,16Hz,17Hz,19Hz,21Hz and23Hz with 0mm bulge respectively.As illustrated above,in such situation of0mm bulge,no cavitation is observed.The collected signals are just the background noise.From thisfigure,it can be seen that, such signals have almost similar spectral type and spectrum signa-ture,the energy of which mainly distributes in the low frequency range below1000Hz.This power spectrum analysis also demon-strates that if the bulge is0mm,no cavitation generates even when theflow rate reaches very high value.The collected signal can be regarded as the background noise.Fig.8b shows the AR spectrums of the signals under theflow rates of14Hz,15Hz,16Hz,17Hz,19Hz,21Hz and23Hz with 3mm bulge paring Fig.8b with a,it can be seen that,in this situation of3mm bulge,with increasingflow rate,theFig.3.Structure of the cavitation segment. Fig.4.The calibration curve of frequency vs.flow rate.energy of the signal increases as a whole.Especially,when theflow rate reaches17Hz,the cavitation generation can be observed,and the energy of the cavitation noise increases obviously in the high frequency range over1000Hz.The generating cavitation intro-duces high frequency components into the collected signal.However,the power spectrum analysis provides only frequency information and is a gross description in the frequency domain but has no time domain information relating to the transient informa-tion in the signal.Thus we cannot know if a frequency component exists during the whole signal time span,or whether it is localized,Fig.5.Noise signals underflow rate of15Hz and19Hz with0mm bulge.(a)Noise signal under15Hz.(b)Noise signal under19Hz.Noise signals underflow rate of15Hz,19Hz,28.5Hz and40Hz with3mm bulge.(a)Noise signal under15Hz.(b)Noise signal under19Hz.(c)Noise signal (d)Noise signal under40Hz.because the power spectrum uses transforms the signal globally.Incontrast to the power spectrum,the wavelet scalogram,described in Section 2,reveal the time and frequency information in the sig-nal simultaneously.The time–frequency distribution of the signal is clearly revealed and the duration of each frequency component shown.In the following sections,the wavelet transform and its scalogram are be used to investigate the time frequency character-istics of the cavitation noise.4.2.Parameter optimization of the morlet wavelet [14]The performance of the CWT transform depends greatly upon the selection of the mother wavelet.The mother wavelet is se-lected so that its wave shape is as similar as possible to the ana-with Fourier transformb w ðx Þ¼ffiffiffiffiffiffiffiffiffip =2p exp Àx Àx 0ffiffiffi2p b2"#ð5ÞStrictly speaking,the Morlet wavelet does not satisfy the admis-sibility condition,sinceb wx ¼0ðÞ¼ffiffiffiffiffiffiffiffiffip =2p =b exp Àx 20=2b 2ÀÁ–0ð6ÞHowever,for x 0/b P 5,the condition is approximately satisfied.The parameter,b ,and the central frequency,x 0,control the shape of the basic Morlet wavelet:when x 0is fixed,increasing b will accelerate the attenuation of the basic wavelet,and decrease the support domain of the wave shape.As b tends to infinity,the basic wavelet becomes a Dirac function,which has the finest time resolution possible,but no frequency resolution.As b tends to 0,the basic wavelet becomes a cosine function,which has the finest frequency resolution possible but no time resolution.When b is fixed,x 0controls the oscillating frequency on the time support do-main;increasing x 0will accelerate the oscillation of the basic wavelet.When used to analyze the same frequency component,increasing (decreasing)the Morlet wavelet central frequency x 0will make the frequency resolution higher (lower),while time res-olution lower (higher).From the above discussion,it can be seen that b and x 0balance the time–frequency resolution.For different signals,different parameters should be selected for better performance.The half-power bandwidth of the Morlet wavelet is b ffiffiffiffiffiffiffiffiln 2p =p Hz,and the central frequency in Hz is f 0=x 0/2p .The quality factor Q (central frequency /bandwidth)may be calculated as x 0=2ffiffiffiffiffiffiffiffiln 2p b ,and hence b and x 0determine the quality factor of the Morlet wavelet.Thus,the wavelet analysis performance is optimized simulta-Fig.7.Power spectrum comparison between two spectrum estimations.Power spectrum analysis of collected noise signals under various conditions.(a)Power spectrum analysis of noise signals under various flow rates with 0mm Power spectrum analysis of noise signals under various flow rates with 3mm bulge.72(2011)721–731and also demonstrates the effect of this method.From Figs.9and 10,it can be seen that parameters optimization of the wavelet depresses the intersection problem and improves the transform performance of the wavelet scalogram.In the following time–frequency analysis,this method is used and optimized Morletselected for the wavelet scalogram analysis of the cavi-Time–frequency analysis of cavitation noise using waveletsection,the wavelet scalogram is used to analyze the characteristics and the composition of the cavita-signal under various conditions.Fig.11shows the analysis results of the noise signals under various0mm bulge.It has been said above that no cavita-observed under this condition.These signals just are the background noise.From Fig.11,it can be seen that the noise level increases with the increase of theflow rate gently.However,from 15Hz to even25Hz,the frequency ranges of the noise signals almost seem similar,which are basically within1000Hz.Their time–frequency distribution and characteristics also look similar. It means that increasingflow rate of the liquid doesthe time–frequency characteristics of the background sources if no cavitation is generated.In addition,thesenoises should be produced by some noise sources frequency,such as mechanical noise and liquidflowing Fig.12shows the time–frequency analysis resultssignals under variousflow rates with3mm bulge.Fromit can be seen that the noise level increases with theflow rate sharply.Especially,their time–frequencyand the composition have obvious change with thetheflow rate.Under theflow rate of14Hz,noobserved.Thus,the time–frequency distributionFig.9.Performance comparison of wavelet scalogram by simulation.Fig.10.Performance comparison of wavelet scalogram.(a)Wavelet scalogram with optimization.(b)Wavelet scalogram without optimization.similar to that of the noises in Fig.11(comparing Fig.12a with Fig.11).When the flow rate increases to 15Hz,some new fre-quency components over 1000Hz emerge in the time–frequency plane of the noise (see Fig.12b).The time–frequency characteris-tics of these components are also obviously different from that of the background noise (comparing Fig.12b with Figs.12a and 11).They are of obvious impulse feature.However,no cavities or bub-bles are observed under this flow rate.As we know,at the incep-tion stage of the cavitation,travelling bubbles will be generated,which usually appear around the bodies of micro-sized nuclei [2].Such bubbles always travel with the flow and are too small in size to be observed,thus relatively weak in energy when collaps-ing and imploding.The new frequency components over 1000Hz in Fig.12b should be induced by such travelling bubbles.When the flow rate increases to 17Hz,some sporadic bubbles with macro size can be observed.When collapsing and imploding,these bubbles produce shock wave and introduce some scattered frequency components with impulse feature in thetime–frequencyTime–frequency analysis of noise signals under various flow rates with 0mm bulge.(a)Wavelet scalogram under 15Hz.(b)Wavelet scalogram under scalogram under 19Hz.(d)Wavelet scalogram under 21Hz.(e)Wavelet scalogram under 23Hz.(f)Wavelet scalogram under 25Hz.plane of the noise (see Fig.12c).Such frequency components have broader frequency range than that in Fig.12b (comparing Fig.12c with b).With further increasing of the flow rate,more and more bubbles can be observed and the cavitation develops to the sheet cavitation state and the cloud cavitation state.From Fig.12d–f ,it can be seen that the amplitude of the cavitation noise increases sharply.Especially,more and more components with higher fre-quency are produced,the frequency range of which extends to 8000Hz.These frequency components are still of obvious impulse feature.The cavitation noise becomes stable gradually in the time–frequency distribution and the composition.The cavitation is called attached cavitation at this stage,the bubbles of which al-ways attach the solid body in the flow.With further increasing of the flow rate,an interesting phenom-enon is observed.From Fig.12g–j,it can be seen that the cavitation noise increases in the amplitude more sharply and becomesdenserTime–frequency analysis of noise signals under various flow rates with 3mm bulge.(a)Wavelet scalogram under 14Hz.(b)Wavelet scalogram under scalogram under 17Hz.(d)Wavelet scalogram under 19Hz.(e)Wavelet scalogram under 21Hz.(f)Wavelet scalogram under 23Hz.(g)Wavelet scalogram Wavelet scalogram under 35Hz.(i)Wavelet scalogram under 40Hz.(j)Wavelet scalogram under 50Hz.in the time–frequency distribution but shrinks in the frequency to a lower frequency paring Fig.12f and h with g,the cav-itation state under28.5Hz seems the transition state of this pro-cess.After this transition,the cavitation noise becomes more stable and denser in the time–frequency distribution(comparing Fig.12h and i with j)but still is of obvious impulse feature.At this stage,much denser cavities and bubbles are generated and stable cavitation is establishedfinally.In addition,we conjecture that denser and bigger bubbles maybe produce coupling effect in the noise when imploding and consequently induce shrinking in the frequency range of the cavitation noise.Actually,this experiment phenomenon and analysis result verify the research results of the bubble dynamics[16–18],i.e.the bigger bubble will induce more low-frequency acoustic emitting.The cavitation of this stage is called super cavitation state,which overruns the range of theflow around a body and generates a stable cavitation tail.The super cavitation is most harmful to the hydraulic machine because the water hammer blows are very dense and powerful when bubbles imploding.From above time–frequency analysis,the cavitation phenome-non and process can be understood better from the view of the cav-itation noise.The results indicate that:with the increasing of the flow rate,the cavitation develops from the inception to the at-tached cavitation state andfinally the super cavitation state.The amplitude and level of the cavitation noise increase with the devel-opment of the cavitation.But the time–frequency distribution and characteristics of the cavitation noises from different cavitation states are different:travelling bubbles with micro-size cause some weak components with less broad frequency range from1000Hz to2000Hz in the noise;with the emergence and the development of the attached cavitation,more and more components with higher frequency are introduced into the cavitation noise,the frequency range of which extends to8000Hz gradually;when the cavitation develops to the super cavitation state,the impulse components be-come much denser in the time–frequency plane of the cavitation noise and the time–frequency distribution becomes stable.How-ever,the frequency range of the cavitation noise of this stage shrinks to lower frequency of5000Hz.Generally,the cavitation noise from whatever state is composed of broad frequency compo-nents with obvious impulse feature,the characteristic frequency range of which should be from about1000Hz to about4000Hz. In addition,the background noise is composed of much lower frequency components,the frequency range of which is basically below about1000Hz.Therefore,the background noise cannot influence the monitoring and detection of the cavitation noise. 5.ConclusionsIn this experimental study,the wavelet scalogram method is used to investigate the time–frequency characteristics and the composition of the background noise and the cavitation noise un-der variousflow rates and different cavitation states.The results reveal that:the cavitation noise from whatever stage is composed of broad frequency components with obvious impulse feature,the characteristic frequency range of which should be fromabout Fig.12(continued)。