Tainan,Taiwan,R.O.C., -3 December 3 Mode Decomposition Analysis Applied to Study the Low-Frequency Embedded in the Vortex Shedding Process Chin-Tsan Wang Department of Electrical Engineering Kau Yuan Institute of Technology Kaushiung, Taiwan 8, R.O.C. Jiun-Jih Miau Institute of Aeronautics and Astronautics National Cheng Kung University Tainan, Taiwan 7, R.O.C. Abstract A new method of the mode decomposition method developed by Huang et al. (998) was applied to decompose the raw signal into a range of fluctuating components from high to low-frequencies in the study. Experimental results of base pressure signals were obtained at Re = 76 and 36, respectively. Results show that decomposition method is useful for further studying the low-frequency variations and other fluctuating components in rang of high to low frequencies structures. At the flow condition of Re = 76, the vortex-shedding structures were exactly referred to the mode C and the residue component of R5 mainly contained the component of the low-frequency variations. Similarly, results obtained at Re = 36 show that the vortex-shedding structure can be identified with the mode C and the residue component of R4 also contained a substantial component of low frequency variations. These results show that all the flow structures could be exactly decomposed into intrinsic oscillatory modes of flow structure and residue components of other flow structure by this technique. Here, the low-frequency variations naturally belong to the residue component but not the mono flow structure. That is to say, the appearance of low-frequency variations embedded in the vortex shedding process would be induced during the modulation process of energy transmission between all the mono flow structures. Key words: low-frequency, mode decomposition analysis. Introduction The present work was motivated by the concerns of signal quality of vortex flowmeters. Miau et al. (993a) reported that pressure fluctuation signals measured at the surface of a vortex shedder contained a substantial component of low-frequency variations at the Reynolds number larger than 3. The energy of low-frequency variations was increased with Reynolds numbers, which might degrade the quality of the vortex shedding signals significantly. Miau et al. (993a) further found that the vortex formation length will change with time, which can be realized from the flow visualization. Hence, Miau et al. (993b) postulated that the low-frequency variations in vortex shedding process relate to unsteady variations of the vortex formation length. This speculation was confirmed later by Miau et al. (999). One of the earliest evidence of low-frequency unsteadiness mentioned in the literature can be traced to the paper by Bloor (964). Bloor (964) noted that the
Tainan,Taiwan,R.O.C., -3 December 3 low frequency unsteadiness appeared in the velocity signals measured in the near wake region of a circular cylinder for Reynolds numbers ranging from 3 to 5. Many studies on the low-frequency variations embedded in the vortex shedding process have been investigated. In spite of the above investigations, many aspects of low-frequency behavior remain unexplained. In this study, the results obtained by the empirical mode decomposition method (Hung et al. 998) were applied to study the characters of low-frequency variation imbedded in the vortex shedding process. Figure The experimental arrangement of the trapezoidal cylinder 3. Experimental Method. Experimental Apparatus The close-type and low-speed wind tunnel with a square test section 5 mm by 5 mm was employed to obtain the base pressure signals of the cylinder whose aspect ratio, named AR, is 4.7 at Re = 76 and 36, respectively. The turbulent intensity of the wind tunnel is about.6 % of the mean velocity measured at the centerline of test section. The two-dimensional bluff body employed was a trapezoidal cylinder installed with the wider side facing the incoming flow. The maximum width is 3 mm, denoted as D. Therefore, the blockage ratio based on the frontal area of the bluff body was about %. The coordinate system shown in figure and employed in the present study is described as follows: X represents the streamwise axis with the positive direction pointing downstream. Y represents the vertical axis with the positive direction pointing downward and Z represents the spanwise axis. The origin of X =, Y =, and Z = is located at the geometrical center of the trapezoidal cylinder. In this study an empirical mode decomposition method developed by Huang et al. (998) was adopted to analyze the signal traces measured. Because this method is intuitive, direct, a posteriori and adaptive, with the basis of the decomposition based on, and derived from, the data. Briefly speaking, this method is able to identify the intrinsic oscillatory modes through a sifting process on the data, regardless of stationary or non-stationary. The decomposition is based on the assumptions:() the signal has at least two extrema-one maximum and one minimum; () the characteristic time scale is defined by time lapse between the extrema; (3) if the data were totally devoid of extrema but contained only inflection points, then it can be differentiated once or more times to reveal the extrema. Final results can be obtained by integration(s) of the components. The essence of the method is to identify the intrinsic oscillatory modes by their characteristic time scales in the data empirically, and then decompose the data accordingly. According to Drazin (99), the first step of data analysis is to examine the data by eye. From
Tainan,Taiwan,R.O.C., -3 December 3 this examination, one can immediately identify the different scales directly in two ways: by the time lapse between the successive alternations of local maxima and minima; and by the time lapse between the successive zero crossings. The interlaced local extra and zero crossings give us the complicated data: one undulation is riding on top of another, and they, in turn, are riding on still other undulations, and so on. Each of these undulations defines a characteristic scale of the data; it is intrinsic to the process. We have decided to adopt the time lapse between successive extrema as the definition of the time scale for the intrinsic oscillatory mode, because it not only gives a much finer resolution of the oscillatory modes, but also can be applied to data with non-zero mean, either all positive or all negative values, without zero crossings. According to the method (Huang et al. 998), a systematic way to extract them, designated as the sifting process, is described as follows. By virtue of the intrinsic mode function (Huang et al. 998), the decomposition method can simply use the envelopes defined by the local maxima and minima separately. Once the extra are identified, all the local maxima are connected by a cubic spine line as the upper envelope. Repeat the procedure for the local minima to produce the lower envelope. The upper and lower envelopes should cover all the data between them. Their mean is designated as C, and the difference between the data, called as X(t), and C is the first component, dedicated as R, i.e. X(t) C = R () Hence, C should contain the finest scales or the shortest period component of the signal and R contains the components of longer time scales, comparatively speaking. This procedure can be repeated, hence the subsequent Rj components are reduced as follows. R C = R,, Rn- Cn = Rn () As described above, the process is indeed like sifting: to separate the finest local mode from the data first based only on the characteristic time scale. The sifting process, however, has two effects: (i) to eliminate riding waves; and (ii) to smooth uneven amplitudes. While the first condition is absolutely necessary for the instantaneous frequency to be meaningful, the second condition is also necessary in case the neighboring wave amplitudes have too large a disparity. Unfortunately, the second effect, when carried to the extreme, could obliterate the physically meaningful amplitude fluctuations. Therefore, the sifting process should be applied with care, for carrying the process to an extreme could make the resulting a pure frequency modulated signal of constant amplitude. To guarantee that the IMF (Huang et al. 998) components retain enough physical sense of both amplitude and frequency modulations, we have to determine a criterion for the sifting process to stop. This can be accomplished by limiting the size of the standard deviation, SD, computed from the two consecutive sifting results as T SD = = R ( k ) ( t) R R t k ( t) (Huang et al. 998) (3) A typical value for SD can be set between. and.3. As a comparison, the two Fourier spectra, computed by shifting only five out of 4 points from the same data, can have an equivalent SD of.-.3 calculated point-by-point. Therefore, a SD value of.-.3 for the sifting procedure is a very rigorous limitation for the difference between siftings. The sifting process of the equation () is repeated until either of the following criteria is reached: (a) when the component, Cn, or the residue Rn, become so small that it is less than the value of the standard deviation suggested by Huang et al. (998), which is set between the normalized value of. to.3, or (b) when the k ( t)
Tainan,Taiwan,R.O.C., -3 December 3 residue, Rn, becomes a monotonic function. obtain By summing up equation () and (), we finally n X(t) = + Rn (4) i= Cpb... 3. 4. 5. - Ci... 3. 4. 5. Where X(t) denotes the original data, Ci denotes the ith fluctuating component with zero mean and a zero crossing occurred between every consecutive maximum and minimum points, and R n denotes a residual component (Huang et al. 998). - -... 3. 4. 5. -... 3. 4. 5. - Re=76; Fs=4.9Hz Original-signals... 3. 4. 5. R R R3 R4 R5 4. Experimental Results and Discussions In this study the results obtained by the empirical mode decomposition method (Hung et al. 998) were reported in the following. A segment of the base pressure signals obtained at Re = 76, for the cylinder of AR = 4.7, was analyzed by this technique. Figure shows the residue components of R, R, R3, R4 and R5 resulted from the sifting process described previously. Thus, the component of R5 corresponds to the lowest frequency component resulted from this process. If the original data and each of the residue components can be regarded as stationary, the Fast Fourier Transform analysis is a relevant tool to obtain the respective spectral results. The results are shown in figure 3, from which one is able to appreciate the effectiveness of this decomposition technique. Furthermore, the results of time-lag auto-correlation with respect to the components of R to R5 are shown in figure 4. The integral time scale of R5 reduced from the correlation results is found to be about two-fold the vortex shedding period, equivalent to the integral time scale of the optimal cut-off frequency shown in the report of Miau et al. (3). Moreover, the spectrum of R5 shown in figure 3 indicates that the frequency components contained in R5 are mainly below 5 Hz. In - 3 4 5 t/ts Figure Empirical mode decomposition of the base-pressure fluctuations: ()original data ()R-R5 denotes five kinds of residue component................... Re=76 ; Fs=4.9Hz Original-spectra (Base-pressure) Spectra of R spectra of R spectra of R3 spectra of R4 spectra of R5 5 5 5 Hz Figure 3 Spectra of the original data and the residue components correspond to Figure. this case, 5Hz is close to the optimal cut-off frequency, Stc =.37, which is about 6 Hz. That is to say, the speculation that the residue component of R5 contained a substantial component of low-frequency variations embedded in the vortex shedding process could be
Tainan,Taiwan,R.O.C., -3 December 3 identified. structure except vortex-shedding structure exit R It: integral time scale It=.87(Ts) simultaneously during the vortex shedding process. Generally speaking, the appearance of the residue - component contained a great deal of low-frequency Auto-correlation coefficient - - R R3 R4 It=.335(Ts) It=.563(Ts) It=.9(Ts) variations component would be induced due to the imbalance of energy transmission of all monotonic flow structures. C - - R5 It=.89(Ts) - C - 4 8 6 Time-lag(t/Ts) - C3 Figure 4 Auto-correlation coefficient and integral time scale of low-frequency unsteadiness (It) of the original data and the residue components corresponding to Figure. - - C4 C5 By the technique of mode decomposition, the signal trace shown in figure can be decomposed into five mono-components of C-C5 in addition to the residue component of R5, which are shown in figure 5. One can immediately see that in each of the C to C5 components fluctuations are varying about zero mean, conformed to the criterion pointed out by Huang et al. (998). The C component appears to contain the fluctuations of highest frequencies and the C5 component shows the fluctuations of much lower frequencies. Therefore, the five kinds of monotonic flow structures, C-C5, and the residue component R5 which posses a great quantity of low frequency variations simultaneously exited at the flow of Re = 76. Furthermore, that the mode C would be the vortex shedding structure at Re = 76 was also confirmed clearly by comparing with figure and figure 4. That is to say, there are different scales of flow 4 8 6 Time (Sec) Figure 5 Empirical mode decomposition of the base-pressure fluctuations: C-C5 denotes five kinds of monotonic component, respectively. The decomposition method was also applied to a segment of the base pressure signal obtained at Re = 36. The residue components of R to R6 are shown in figure 6, and the frequency spectra corresponding to the original signal and the six residue components are given in figure 7. In this case, we find that the energy resided in R5 and R6 components are much less than the component of R4. Moreover, we find that the integral time scale associated with the R4 component, instead of the integral time scales of R5 and R6, is close to the integral time scale of the low-passed signal at the optimal cut-off frequency. Therefore, the component of R4 also contained a substantial component of low-frequency variations at Re = 36.
Tainan,Taiwan,R.O.C., -3 December 3 - - - - - - R6 - Figure 6 Empirical mode decomposition of the.. Re=36; Fs=67Hz R e = 3 6 ; F s = 6 7 H z Original-signals base-pressure fluctuations: () Original data () R-R6 4 8 denotes six kinds of residue component, respectively. O r i g i n a l -spectra R R R3 R4 R5 component of highest frequencies whereas the C6 component corresponds to the fluctuating component of lowest frequencies. The speculation that the vortex shedding structure belongs to the mode C also could be confirmed at Re = 36. It is learned from a comparison of figures. 5 and 8 that as the Reynolds number gets higher more components can be resulted from the decomposition method. This is reasonable since turbulence at higher Reynolds number tends to contain wider range of scales. These two cases of flow at different Reynolds numbers also suggest that the lowest-frequency component resolved by the mode decomposition technique does not necessarily correspond to the low-passed signal at the optimal cut-off frequency mentioned earlier. Further verification is necessary by comparing the integral time scales corresponding to different mode components..... s p e c tra o f R s p e c tra o f R - Re=36; Fs=67Hz C..... 4.. 4. s p e c tra o f R 3 s p e c tra o f R 4 s p e ctra of R 5 s p e c tra o f R 6 - - - C C3 C4. 5.. 5.. 5. H z - C5 Figure 7 Spectra of the original data and residue components corresponding to Figure 6. - 4 8 6 Time (Sec) C6 For further information, figure 8 shows the mono-components of C to C6 for the signal trace shown in figure 6. Similarly, C corresponds to the fluctuating Figure 8 Empirical mode decomposition of the base-pressure fluctuations: C-C6 denotes six kinds of monotonic component, respectively.
Tainan,Taiwan,R.O.C., -3 December 3 5. Conclusions 6. Reference The mode decomposition method by Huang et al. (998) is found to be useful in the present study to decompose the raw signal into a range of fluctuating components from high to low frequencies. Some conclusions obtained by the mode decomposition method would be described as following. The vortex shedding structure and low-frequency variations component would be decomposed more exactly by this technology. At flow condition of Re = 76, the vortex shedding structures were referred as the mode C and the low-frequency variations would be included in the component of R5. Similarly, that the vortex shedding structure was corresponding to the mode C and the component of R4 mainly included the component of the low-frequency variations at Re = 36 are also confirmed. According to these analyzed results applied by this technology, it is easily found that different length scales of flow structure, referred as mono mode, and the residue component whose mainly component contained low-frequency variations exit simultaneously in the flow during the energy transmission process. The low-frequency variations should be probably caused by an instantaneous energy imbalance between all different scales of flow structure (Kiya & Sasaki 983). As the Reynolds number gets higher more components of low frequency variations can be found, this phenomenon is similar with turbulence at higher Reynolds number tends to wider range of scale. B LOOR, M. S. 964 The transition to turbulence in the wake of a circular cylinder. J Fluid Mech.., Vol.9, pp.9-33. D RAZIN, P. G. 99 Nonlinear systems. Cambridge University Press. H UANG, N. E., Sheng, Z., Long, S. R., Wu, M. C., Shih. H.H., Zheng, Q., Yen, N. C., Tung, C.C. & Liu, H.H. 998 The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. R. Soc. Lond. Vol.454A, pp.93-995. K IYA, M. & Sasaki, K. 983 Structure of a turbulent separation bubble. J. Fluid Mech., Vol.37, pp.83-3. M IAU, J. J., Yang, C. C., Chou, J. H., & Lee, K. R. 993a Suppression of low-frequency variations in vortex shedding by a splitter plate behind a bluff body. Journal of Fluids and Structures, Vol.7, pp.897-9. M IAU, J. J., Yang, C. C., Chou, J. H., & Lee, K. R. 993b A t-shaped vortex shedder for a vortex flowmeter. Flow Measurement and Instruments 4, No. 4, pp.59-67. M IAU, J. J., Wang, J. T., Chou, J. H., & Wei, C. Y. 999 Characteristics of low-frequency variations embedded in vortex shedding process. Journal of Fluids and Structures, Vol.3, pp.339-359. Miau, J. J., J. T. Wang, J. H. Chou and C. Y. Wei, 3 Low-Frequency Fluctuations in the Near-Wake Region of a trapezoidal Cylinder with low Aspect Ratio Journal of Fluids and Structures Vol.7/5.
Tainan,Taiwan,R.O.C., -3 December 3 Huang et al. (998) (mode decomposition method) Re = 76 36 Huang et al. (998)