Hilbert-Huang transf..

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42 H. Ding et al. / Flow Measurement and Instrumentation 18 (2007) 37–46 Fig. 3. Comparison between Hilbert spectra and wavelet spectra of the differential pressure fluctuation signals for the 15 mm-diameter pipeline: (a)–(d) Hilbert spectra; (e)–(h) wavelet spectra. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) The above results imply that the energy characteristic H j changes with the flow pattern and can therefore be regarded as an indicator for the identification of the flow pattern of gas–liquid two-phase flow. The results also demonstrate that the HHT is suitable for analyzing the differential pressure fluctuation signal. Moreover, the influence of the pipe diameter on the energy characteristic is not significant. Thus, the extracted energy characteristics can be used as an objective indicator for flow pattern monitoring of gas–liquid twophase flow. Though WT method can also be used to study gas–liquid two-phase flow, it is difficult to demonstrate the local characteristics of the signal in both time and frequency
domains because of the non-adaptive feature of the form of elementary wavelet function in a practical application [4]. In the next section, detailed analysis results for both HHT and WT will be given. 5.2. Comparison between the HHT and WT methods The experimental results presented above indicate that the differential pressure fluctuation signals from different sized pipelines have similar spectral characteristics. So we take the experimental results for the 15 mm-diameter pipeline as an example to analyze the Hilbert spectrum and the WT spectrum of the signal. The Hilbert spectra and wavelet spectra of the typical flow patterns in the 15 mm-diameter pipeline are plotted in Fig. 3. The Hilbert spectra as the greyscale coded maps are shown on the left, while the wavelet spectra of the signals are illustrated on the right. The colour bars beside each Hilbert spectrum denote the time and frequency dependent energy. The energy is normalized and represented on a logarithmic scale. The colour bars next to each wavelet spectrum represent the absolute wavelet coefficient, i.e., the amplitude of the signal [3]. Fig. 3(a) shows the Hilbert spectrum of bubble flow. The fluctuations of differential pressure are stable and the amplitude of the signals is small. The main fluctuation locates at the higher frequency, consequently the energy concentrates mainly on the high-frequency components and the total energy is low. The energy is evenly distributed in the high-frequency plane over 25 Hz in Fig. 3(a) and there is very little energy below 25 Hz. The wavelet spectrum shows a similar energy distribution as the Hilbert spectrum, as shown in Fig. 3(e). It can be seen that the energy concentrates mostly in the smaller scale range. Fig. 3(b) shows the Hilbert spectrum of the plug flow. Compared to the bubble flow, the ratio of energy embedded in different frequency bands is changed. The ratio of energy embedded in the high-frequency range has reduced rapidly whereas the ratio of energy embedded in the middle–lowfrequency range has increased. As there still exist many tiny bubbles between two successive plugs, there is low energy in H. Ding et al. / Flow Measurement and Instrumentation 18 (2007) 37–46 43 Fig. 3. (continued) the high-frequency range. Fig. 3(f) shows the results derived from WT. It is clear that the peak amplitude is greater than that of the bubble flow. The Hilbert spectrum of the wavy flow (Fig. 3(c)) indicates that the ratio of energy embedded in the higher-frequency range is greater than that in other frequency ranges. Compared to the plug flow, the ratio of energy embedded in the middlefrequency range begins to decrease and that in the lowerfrequency range begins to increase slowly. As shown in Fig. 3(g), the ratio of energy embedded in smaller scales begins increasing. Fig. 3(d) depicts the Hilbert spectrum of the slug flow. The size of gas slug is bigger than the bubble and the fluctuation of the signal is more drastic than other three flow patterns. As a result, the ratio of energy embedded in the lower-frequency range is the greatest, which is consistent with the analysis result from the wavelet spectrum shown in Fig. 3(h). As shown in Fig. 3, the variation in the ratio of energy embedded in different frequency bands is distinctive in the Hilbert spectrum and is in agreement with the wavelet spectrum. However, the frequency and time localization of the Hilbert spectrum is superior to that of the wavelet spectrum. The localizations of WT are accomplished by modifying the scale of the elementary wavelet function. If a local event occurs in the low-frequency range, it is necessary to modify the scale for characterizing the signal continuously [4]. WT indeed improves the time resolution as compared to the FT, but HHT gives much finer local properties than WT. Although Fig. 3 gives a comparison between the Hilbert and the wavelet spectra for the same set of data, the differences between the two methods are not so obvious according to the counter-intuitive figures. From a theoretical point of view, the choice of the elementary wavelet function will have a significant impact on the analysis results because the elementary wavelet function is fixed during analysis. Once the elementary function is determined, it will be used to analyze all the data [2,4]. Since WT is Fourier based, the wavelet spectrum possesses some fundamental characteristics of a Fourier spectrum. WT can analyze a non-stationary signal
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