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Higher-harmonic focused-wave forces on a vertical cylinder
ARTICLE IN PRESS Ocean Engineering 36 (2009) 595–604
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Higher-harmonic focused-wave forces on a vertical cylinder Yuxiang Ma a,, Guohai Dong a,, Marc Perlin b, Shuxue Liu a, Jun Zang c, Yiyan Sun a a b c
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023, PR China Department of Naval Architecture & Marine Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA Department of Architecture and Civil Engineering, University of Bath, Bath BA2 7AY, UK
a r t i c l e in fo
abstract
Article history: Received 1 September 2008 Accepted 18 February 2009 Available online 13 March 2009
This paper considers higher-harmonic forces due to wave focusing on a vertical circular cylinder. A series of experiments has been conducted in a wave flume. The first six-harmonic components of the measured wave forces are analyzed using the scale-averaged wavelet spectrum. It is noted that due to the transient nature of focused (freak) waves, Fourier analysis would not provide equivalent information to that gleaned from the analyses used herein. The results for the experiments with very steep wave crests show significant amplitudes at the fourth and fifth harmonics. These harmonics exhibit amplitudes that are the same order as the second harmonic, but much larger than those of the third harmonic. The wavelet-based bicoherence is used to detect the quadratic nonlinearity of the measured forces. And the bicoherence spectra reveal the primary mathematical reason for the existence of the striking amplitudes of the fourth and fifth harmonics: the interaction between the lower-harmonic components couple more strongly with the fourth and fifth harmonics, thus the fourth and fifth harmonics glean more energy than those of the third-harmonic components. However, the physical explanation for this remains elusive. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Higher-harmonic forces Vertical cylinder Nonlinear interactions Wave transformation Scale-averaged wavelet energy Wavelet-based bicoherence
1. Introduction To obtain predictions of the total wave force on offshore structures, in recent years the research community has considered necessarily higher-harmonic forces. One important higher-order response is so-called ‘‘ringing’’, which is a vibration at the resonant frequency of a structure, and is a concern with regard to extreme loading. The generation mechanism of the higherharmonic wave loads leading to ringing of offshore structures is not fully understood. The ringing phenomenon has been detected in large and extremely steep waves (Chaplin et al., 1997), but Grue and Huseby (2002) show that it may occur in moderately-steep transient waves. The theoretical solutions of Faltinsen et al. (1995) and Malenica and Molin (1995), hereafter referred to as M&M, with the assumption of incoming Stokes waves, can capture the first three harmonic components of wave loads. A fully nonlinear model has also been developed to analyze this problem (Ferrant, 1998; Ferrant et al., 1999) and higher-harmonic wave loads were analyzed through an experimental approach. Huseby and Grue (2000) experimentally investigated the first seven harmonic components of the horizontal forces on vertical cylinders in
Corresponding authors. Tel.: +86 41184706213; fax: +86 41184708526.
E-mail addresses: [email protected] (Y. Ma), [email protected] (G. Dong). 0029-8018/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2009.02.009
Stokes waves. Boo (2006) studied the first five harmonic wave forces on a small scale truncated vertical cylinder in Stokes waves. However, an investigation of the higher-harmonic forces on a vertical cylinder during focused-wave events is lacking; this is the motivation for the study herein. The present effort considers the higher-harmonic loads of focused waves on a vertical cylinder in both the horizontal, longitudinal, and transverse directions. Moderate and steep slopes of focused waves are considered. Additionally the question of how these harmonic components appear in the wave forces is examined and explained through the wavelet-based bicoherence. The wavelet transform is a signal-processing technique that can generate localized time–frequency information from time series. In recent years, this technique has been applied successfully in wave analysis as well as other ocean engineering applications such as wave groupiness (Dong et al., 2008a; Liu, 2000), non-stationary processes and noise identification (Huang, 2004; Massel, 2001). Wavelet-based bicoherence, introduced by Milligen et al. (1995), is a powerful tool to detect quadratic phase coupling associated with nonlinear wave–wave interactions. Chung and Powers (1998) studied the statistical properties of the wavelet bicoherence and pointed out that the wavelet-based bicoherence estimate has a larger number of effective degrees of freedom than the Fourier-based bicoherence estimate. Thus, it can be used to detect quadratic phase coupling in a short time series. (Here short is used in comparison to that required for the usual bispectrum of Fourier analysis.) Wavelet-based bicoherence has
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been used successfully to study nonlinear wave–wave interactions by Elsayed (2006) in deep water, and by Dong et al. (2008b) in the coastal zone. This paper is organized as follows. Following the Introduction, a brief overview of the wavelet transform and wavelet-based bicoherence is presented in Section 2. The experimental set-up is described in Section 3. A discussion of the experimental results is presented in Section 4. Lastly, the conclusions of this research are given in Section 5.
2. Wavelet transform and wavelet-based bicoherence 2.1. Continuous wavelet transform The continuous wavelet transform of a time function x(t) can be defined as Z 1 xðtÞca;t ðtÞ dt; (1) WTða; tÞ ¼ 1
2.3. Wavelet-based bicoherence The wavelet-based cross bispectrum is defined as follows: Z WT x ða1 ; tÞWT x ða2 ; tÞWT y ða; tÞ dt, (6) Bxxy ða1 ; a2 Þ ¼ T
where a1, a2 and a must satisfy the frequency sum rule: 1 1 1 þ . ¼ a a1 a2
(7)
The wavelet-based cross bispectrum measures the amount of quadratic phase coupling in the time interval T between wavelet components of scale lengths a1, a2 of x(t) and a of y(t) in a manner such that the sum rule is satisfied (Milligen et al., 1995). From the relationship between scale and frequency, we can interpret the wavelet-based cross bispectrum as the coupling between waves of frequencies f ¼ f1+f2. Hence we will graph the wavelet bispectrum as a function of frequency rather than scale. Likewise the auto bispectrum is defined as Bx ða1 ; a2 Þ ¼ Bxxx ða1 ; a2 Þ.
(8)
where the asterisk denotes the complex conjugate, and ca,t(t) represents a family of functions called wavelets that are constructed by translating in time, t and dilation with scale, a, of a mother wavelet function c(t). The scale a can be interpreted as the reciprocal of frequency, fE1/a. The ca,t(t) expression is defined as tt . (2) ca;t ðtÞ ¼ jaj0:5 c a
Here the third x subscript and its use in Eq. (6) to obtain Eq. (8) should be obvious. To directly measure the degree of phase coupling, the squared wavelet cross-bicoherence can be defined as the normalized squared wavelet bispectrum
One of the most extensively used mother wavelets in continuous wavelet analysis is the Morlet wavelet (Torrence and Compo, 1998); it is a plane wave modulated by a Gaussian envelope and is defined as
which can assume values between 0 and 1. Similarly, the squared auto-bicoherence is defined as
cðtÞ ¼ p1=4 exp
t2 expðio0 tÞ, 2
(3)
where o0 is the peak frequency of the wavelet, usually chosen to be 6.0 to meet the admissible condition (Farge, 1992). Thus, analogous to the Fourier energy spectrum, the wavelet energy spectrum can be defined as Wða; tÞ ¼
WTða; tÞWT ða; tÞ . a
(4)
jBxxy ða1 ; a2 Þj2 R , 2 2 T jWT x ða1 ; tÞWT x ða2 ; tÞj dt T jWT y ða; tÞj dt
½bxxy ða1 ; a2 Þ2 ¼ R
(9)
½bx ða1 ; a2 Þ2 ¼ ½bxxx ða1 ; a2 Þ2 .
(10)
2
bx indicates the relative degree of quadratic phase coupling 2 between waves, with bx ¼ 0 for random phase relationships, and 2 2 bx ¼ 1 for maximum coupling. For a three-wave system, bx represents the fraction of power at scale a due to quadratic phase coupling of the 3 modes (a1, a2 and a). Detailed discussions on the statistical properties of the wavelet-based bicoherence are presented by Chung and Powers (1998) who have shown that the wavelet-based bicoherence can detect the quadratic phase coupling in a short duration series.
3. Experimental set-up 2.2. Scale-averaged wavelet energy
3.1. Wave flume
Another important spectrum based on the wavelet transform is the scale-averaged wavelet energy (SAWE), which is defined as
The measurements are conducted in the wave flume at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, PR China. This flume is 50.0 m long, 3.0 m wide and is used with a water depth of 0.5 m. The flume is equipped with a hydraulically driven piston-type wave maker, and in the following x ¼ 0 m is defined as the mean position of the wave maker. At the end of the flume, wave absorbers are installed to help mitigate wave reflection. The experimental set-up is shown in Fig. 1.
W n ðtÞ ¼
j¼j2 Wðaj ; tÞ dDt X
Cd
j¼j1
aj
,
(5)
in which d is the scale factor, Dt is the sample interval of the time series, and Cd is the reconstruction factor. The factor Cd is a constant for each wavelet function; for the Morlet wavelet, Cd is taken as 0.776 (Torrence and Compo, 1998). The SAWE is a time series of the average energy variance in a certain scale band (aj1 through aj2). Thus, it can be used to examine the modulation of one series by another, or modulation of one frequency band by another within the same series. In the present investigation, it is used to examine the energy at the scale bands which corresponds to the nth (n ¼ 1, 2, 3, y) harmonic components for both the focused waves and the wave forces.
3.2. Wave generation To generate the required focused waves at the specified point in space and time, a linear solution is used to provide a first approximation to the dispersive characteristics of the free waves generated at the wave paddle. This method is similar to the approaches adopted by Rapp and Melville (1990) and Baldock
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Fig. 1. Plan view of the wave flume experimental set-up.
et al. (1996). Assuming waves are focused at a specified point xf at time tf, linear wave theory defines the wave elevation at an arbitrary point as
Zðx; tÞ ¼
N X
ai cos½ki ðx xf Þ 2pf i ðt t f Þ,
(11)
Table 1 Regular wave conditions. f (Hz) kr max(A/r)
0.62 0.2525 0.3968
0.83 0.3811 0.3552
i¼1
where N is the total number of wave components, and ai, ki and fi represent the wave amplitude, the wave number and the wave frequency of the ith wave component, respectively. The dispersion relation establishes the relation between space and time, i.e. between ki and fi. In the analysis ai is determined from a chosen energy frequency spectrum and the phases of the wave components are such that they are all zero at the focal point. In this study, the focused wave-amplitude AF is defined as the input parameter, hence ai and AF satisfy AF ¼
N X
ai ,
(12)
i¼1
and ai can be calculated using AF S ðf ÞDf i a i ¼ PN i , i¼1 Si ðf ÞDf i
Case
Frequency range (Hz)
Frequency components, N
Input focused wave amplitude, AF (cm)
J1 J2 J3 J4 J5 J6 J7 J8 J9
0.6–1.06 0.5–1.16 0.5–1.36 0.6–1.06 0.5–1.16 0.5–1.36 0.6–1.06 0.5–1.16 0.5–1.36
29 29 50 29 29 50 29 29 50
3 3 3 6 6 6 8 8 8
(13)
where Si(f) is the desired frequency spectrum. In addition, the discrete frequency fi is spaced uniformly over the frequency band [fmin, fmax], and the frequency bandwidth is defined as
Df ¼ f max f min .
Table 2 Focusing wave conditions.
case. In this study, all experimental wave trains are adjusted to focus at x ¼ 11.4 m (without the cylinder present).
(14) 3.4. Model set-up
3.3. Wave conditions Before the experiment of forces on cylinders by focusing waves is conducted, the forces induced by regular waves should be measured to verify the accuracy of the experimental equipment. Incident regular waves with the wave slope Ak (A is the airy wave amplitude) ranging from 0.03 to 0.14 are considered. Two wave frequencies, 0.62 and 0.83 Hz, are used. The detailed information about regular waves is shown in Table 1. For focusing waves, the present investigation considers three different frequency bandwidths of wave simulations based on JONSWAP spectra with the same peak frequency, fp ¼ 0.83 Hz. For each frequency bandwidth, three different focused wave-amplitudes AF are considered. Wave groups with the largest waveamplitude are very close to an extreme event, as waves with larger input amplitude are found to be breaking in the primary test. Detailed information regarding the wave conditions is listed in Table 2 along with a designation assigned to each experimental
Once the focused waves at the desired location (x ¼ 11.4 m) are achieved in the wave flume, a cylinder that extends through the entire water depth is installed at x ¼ 11.625 m (see Fig. 1). The cylinder is 79 cm long with a diameter of 25 cm, and is constructed of plastic tubing with natural frequency (9.52 Hz) larger than the super-harmonics of the wave forces studied. Hence, the slenderness, kpr (kp is the linear dispersion wave number corresponding to fp and r is the cylinder radius), is about 0.383. The ratio of the width of the flume to the cylinder diameter is 12 in this investigation; therefore, the width of the flume has little effect on the results. The water surface elevations at the focal location are measured by a capacitance gauge with absolute accuracy of order 71 mm. For the regular wave cases, the wave elevations are measured in the absence of the cylinder, at the location where the centre of the cylinder is located. Prior to use, the integrity of the wave-elevation gauge is verified, and then it is calibrated. A force transducer, which can measure force in the longitudinal direction of the wave flume, is attached to the top of the cantilevered cylinder and mounted rigidly to a carriage to
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restrain the motions of the cylinder (Fig. 2). The waves and the wave forces are recorded at a sampling rate of 25 Hz.
between the measured first-harmonic forces and the M&M model are also small. However, the measured third-harmonic forces are larger than the theoretical values obtained by the M&M model.
4. Results and discussion
4.2. Nonlinear focusing waves and forces
4.1. Forces by regular waves
The measured water surface elevations at the focal point with and without the vertical cylinder are shown in Fig. 4 for Cases J1, J4 and J7, which have the same input frequency bandwidths (Df ¼ 0.46 Hz), but different focused forcing amplitudes. (Other cases give similar results, which are not shown.) It is noted that for each experiment without the cylinder, the left and right trough elevations of the maximum crest are almost equal at the focal point. The maximum crest elevation is enhanced additionally by augmentation of the input wave-focused amplitude, suggesting that the nonlinearity of the waves is increasing with increased amplitude forcing. For the condition with the cylinder present, however, the left and right trough elevations of the maximum crest elevation are no longer equal, and the water surface elevations are much larger (about 50%) than in the absence of the cylinder. The amplification of the wave crest elevations with the cylinder present is due primarily to the run-up of the waves on the cylinder, indicating as expected that the cylinder changes the local wave field significantly. The maximum crest elevations, without and with the cylinder present, are shown in Fig. 5. The vertical axis is the crest elevation measured at the focus location; the horizontal axis is the amplitude (based on the linear solution). It can be seen that nonlinearity is increasing with the amplitude forcing and that the presence of the cylinder enhances the nonlinearity of waves. It is evident also that the bandwidth has little influence on the nonlinearity of waves. The measured wave forces for Cases J1, J4 and J7 are shown in Fig. 6. There are obvious secondary load cycles (Grue and Huseby, 2002) around the maximum elevations of the loads for Case J7, suggesting that the higher harmonics of the forces are present. It is difficult to analyze the wave forces at each harmonic using Fourier transforms as adopted by Boo (2006) and Huseby and Grue (2000) owing to the transient nature of focused waves. As mentioned previously, due to the transient nature of focused (freak) waves, Fourier analysis simply cannot provide equivalent information to that gleaned from the analyses used herein.
The first three harmonic components of the measured forces (F1, F2, F3) exerted on the cylinder by regular waves are evaluated by Fourier analysis. The normalized components of the measured first three harmonics are shown in Fig. 3. The first-harmonic forces obtained from the Morison equation and the theoretical results of the M&M model (second and third order) given by Malenica and Molin (1995) are also presented in Fig. 3 for comparison. Similar to Huseby and Grue (2000), the differences between the measured first-harmonic forces and the values evaluated by the Morison equation are small. The differences
Fig. 2. Elevation view of the wave flume longitudinal section. The force transducers measure integrated horizontal longitudinal force only on the cantilevered cylinder.
Fig. 3. Solid circles: measured amplitudes of the first three components of forces for the regular wave cases. Dotted line: Morison equation. Dashed line: M&M model. (a) kr ¼ 0.318, (b) kr ¼ 0.381.
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Fig. 4. The measured water elevations at the focus location without (left panels) and with (right panels) the vertical cylinder present for the experiments (from top to bottom) J1, J4 and J7. (a) Case J1. (b) Case J4. (c) Case J7.
Fig. 5. Measured crest elevations: (a) without the cylinder; (b) with the cylinder. (??) linear solutions; (’) Df ¼ 0.46 Hz; (K) Df ¼ 0.66 Hz; (m) Df ¼ 0.86 Hz.
Fig. 6. The measured wave forces for Cases J1, J4 and J7.
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Fig. 7. The scale-averaged wavelet energy (SAWE) for the first six harmonics of the measured force for Case J1. (a) Case J1. (b) Case J4. (c) Case J7.
Fig. 8. The scale-averaged wavelet energy (SAWE) for the first six harmonics of the measured force for Case J4.
As mentioned in the preceding section, the scale-averaged wavelet energy is a time series of the average variance in a certain band. Thus, we can use it to examine the energy variance of the wave forces at the scale bands which corresponds to the nth (n ¼ 1, 2, 3, y) harmonic components of the wave forces. The SAWE of the first six super-harmonics of the wave forces for Cases J1, J4 and J7 are shown, respectively, in Figs. 7–9. It can be seen that the higher-harmonic forces are significantly smaller than the first-harmonic force, as one would expect. And, the higherharmonic components tend to appear simultaneously with the maximum wave forces, i.e. tE10.2 s. For Case J1, which has the smallest forcing amplitude, only the second-harmonic components are evident. For the next-largest focused-amplitude experiment (Case J4), except for the second harmonic, there is energy evident at the other higher harmonics, but it is ordered as
expected. It is interesting to notice that the fourth and fifth harmonics’ peak values of Case J7 are comparable to the peak value of the second harmonic. Indeed the sixth harmonic is the same order as the third harmonic, but both are much smaller than the fourth- and fifth-harmonic components. The peak values of the SAWE at the higher-harmonic components (from second to sixth) for all the measured wave forces are shown in Fig. 10. It is noted that only the peak values of the SAWE at the fourth and fifth harmonics for the largest input focused-amplitude cases (AF ¼ 8 cm, Fig. 10c) are comparable to the values for the second harmonics. In addition the amplitudes for the experiments with small input amplitude (AF ¼ 3 cm and AF ¼ 6 cm) of the third, fourth and fifth harmonics are much smaller than those of the second-harmonic components. At the same time, the peak values of the ith (i ¼ 3, 4, 5, 6) harmonic
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Fig. 9. The scale-averaged wavelet energy (SAWE) for the first six harmonics of the measured force for Case J7.
Fig. 10. The peak values of the scale-averaged wavelet energy (SAWE) for the higher harmonics of the measured forces for the experimental cases: (a) cases with A ¼ 3 cm; (b) cases with A ¼ 6 cm; (c) cases with A ¼ 8 cm.
forces for wave conditions with the same input AF but different Df are almost equal, suggesting that the input frequency bandwidth has little influence on the forces at these higher harmonics. The above discussions indicate that the considerably larger higherharmonic forces tend to appear in the waves with transient character and very steep slope.
Fig. 11 depicts the peak values of the SAWE at the higher harmonics (from second to sixth) for the measured water elevations at the focal location for all the experiments. The results shown in Fig. 11 indicate that the amplitudes of the components at the fourth, fifth and sixth harmonics are negligible compared to those at the second and third harmonics, even for the cases with
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Fig. 11. The peak values of the scale-averaged wavelet energy (SAWE) for the higher harmonics of the wave elevations at the focal location: (a) cases with A ¼ 3 cm; (b) cases with A ¼ 6 cm; (c) cases with A ¼ 8 cm.
Fig. 12. Wavelet-based auto-bicoherence of the measured forces for the cases with A ¼ 3 cm. (a) Case J1. (b) Case J2. (c) Case J3.
Fig. 13. Wavelet-based auto-bicoherence of the measured forces for the cases with A ¼ 6 cm. (a) Case J4. (b) Case J5. (c) Case J6.
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very steep slope. These results are distinct from the results of the wave forces discussed above.
4.3. Wavelet-based bicoherence of the measured nonlinear wave forces As mentioned, the measured wave forces at the fourth, fifth and sixth harmonics cannot be neglected, especially for the highly nonlinear wave conditions in which the components at the fourth and fifth harmonics are comparable to those of the secondharmonic components. Perplexingly, the results of the wave elevations measured at the leading edge of the cylinder show that the fourth-, fifth- and sixth-harmonic components of the waves are negligible, and that all harmonics are ordered. This phenomenon cannot be explained by any existing linear spectral approach. Thus, in this section, the wavelet-based bicoherence which has been proven to detect quadratic nonlinear phase coupling in a short duration time series is used to measure the quadratic nonlinear characteristics of the wave forces. The wavelet-based auto-bicoherence spectra for the cases with the smallest input amplitude (Cases J1, J2 and J3) are shown in Fig. 12; the bicoherence spectra indicate that the quadratic wave interactions are primarily amongst the first-harmonic compo2 nents. For example, bx (0.83, 0.83) ¼ 0.61 in the bicoherence spectrum of force for Case J1 (Fig. 12a) indicates a self–self wave interaction at the energy-frequency peak (f ¼ 0.83 Hz) coupled with the energy at twice the peak frequency (f ¼ 1.66 Hz). That is why there are only the second-harmonic components of the wave forces evident. It can be seen that the self–self interaction within the primary components of the force for Case J2 is larger than those of Cases J1 and J3 (Fig. 12b), suggesting that the secondharmonic force of that test obtains more energy from the primary components. This is consistent with the second-harmonic force of Case J2 which is the largest of the tests with AF ¼ 3 cm. Fig. 13 presents the auto-bicoherence spectra of the measured wave forces for the cases with AF ¼ 6 cm (Cases J4, J5 and J6). Compared to the runs with the smaller AF, the degree of quadratic phase coupling among the harmonic components of the forces are stronger, and more harmonic components participated in the nonlinear process, indicating that the quadratic nonlinearity of
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the forces for these tests are stronger than those of cases with smaller AF. The auto-bicoherence spectra of the wave forces for the largest input wave amplitudes are presented in Fig. 14. In these spectra, not only do the nonlinear interactions involve more components of the forces at higher harmonics than those of cases with lower input wave amplitudes, but the degree of nonlinear interaction between those higher-harmonic components is very strong. For 2 2 2 example, bx (2.49, 0.83) ¼ 0.72, bx (3.32, 0.83) ¼ 0.59 and bx (1.66, 1.66) ¼ 0.70 for Case J7 (Fig. 14a). All these indicate that the fourth- and fifth-harmonic components obtain significant energy from the first three harmonic components of the wave forces. The physical explanation of why this occurs remains a mystery. Meanwhile, we notice that the degree of quadratic phase coupling between the first-harmonic components and the second-harmonic components is smaller than those of the interactions between the first harmonic and other higher harmonics for almost all cases; this suggests that the third-harmonic components obtain less energy from the lower harmonics than do the fourth and fifth harmonics. For example, the value of bicoherence between 2 f ¼ 0.83 Hz and f ¼ 1.66 Hz is 0.49, i.e. bx (0.83, 1.66) ¼ 0.49, for the force of Case J8, but the value of bicoherence between f ¼ 0.83 Hz and f ¼ 2.49 Hz is 0.72 (Fig. 14b). In addition, the fourth- and fifth-harmonic components have more avenues to obtain energy than those of the third-harmonic components. For example, the fourth-harmonic components can obtain energy through the phase coupling between the harmonic components of the first and the third or through the phase coupling within the second-harmonic components. Whereas, the third-harmonic components obtain energy only through the phase coupling between the first two harmonic components, and at the same time, the third-harmonic components may transfer energy to the higher-harmonic components through nonlinear phase coupling with other harmonic components. Thus, the results obtained by the bicoherence spectra afford a reasonable explanation of the phenomenon (that there are large forces at the fourth and fifth harmonics, and that their amplitudes are significantly higher than that of the third harmonic). Furthermore, the bicoherence spectra of the forces with the same AF but different Df suggest that the input frequency bandwidth has little influence on the nonlinearity of the wave forces.
Fig. 14. Wavelet-based auto-bicoherence of the measured forces for the cases with A ¼ 8 cm. (a) Case J7. (b) Case J8. (c) Case J9.
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5. Concluding remarks This study concerns the higher-harmonic wave forces acting on a vertical circular cylinder, especially the forces induced by focused waves. Before conducting the experiment with focusing waves, the forces exerted on a cylinder by regular waves are studied to compare with theoretical models and to verify the experimental set-up. The results show that the first-harmonic forces can well be estimated by the Morison equation. The M&M (Malenica and Molin, 1995) model can predict approximately the second-harmonic forces, but under-predicts the third-harmonic forces. To study the characteristics of the higher-harmonic wave forces induced by focused waves, the horizontal longitudinal forces acting on a vertical circular cylinder in focused waves (based on JONSWAP spectra with different focused amplitudes and different input frequency bandwidths) are measured in a wave flume. The first six harmonic components for both the forces and the wave elevations at the focal location are analyzed using the scale-averaged wavelet energy (SAWE) which provides a time series of the average energy variance in a certain frequency band. The results demonstrate that SAWE is a good choice to address harmonic analysis for transient and short duration time–space series. Through the SAWE of the measured time series, it is revealed that the higher-harmonic forces are significantly smaller than the primary harmonic force for all the test cases. And, for the cases with very steep waves at the leading edge of the cylinder, the amplitudes of the forces for the harmonic components at the fourth and fifth are almost equal to that of the second-harmonic components, but much larger than those of the third-harmonic components, even though the amplitudes of the corresponding harmonic component for the wave elevations are negligible. To reveal the reasons why there are small forces at the thirdharmonic component but remarkable contributions at the fourthand fifth-harmonic components for those cases with highest input amplitude, the nonlinearity of the measured forces is studied through the wavelet-based bicoherence which can detect the quadratic nonlinear phase coupling in a short duration series. The phenomenon of small forces at the third harmonic, but noteworthy forces at the fourth and fifth harmonics for those cases with the highest input amplitude is substantiated through the wavelet-based bicoherence. The bicoherence spectra of the forces for these cases with high wave steepness show that the phase couplings between the harmonic components of the first and third, and the first and fourth are larger than between the first two harmonic components, indicating that the components at the fourth and fifth harmonics obtain more energy from lower harmonics than does the third harmonic. In addition, the fourthand fifth-harmonic components have more paths to obtain energy than those of the third-harmonic components. This is the reason that there are noteworthy forces at the fourth- and fifth-harmonic components, and why their amplitudes are significantly larger than those of the third-harmonic components. Furthermore, from
the bicoherence spectra, it is noticed that the input frequency bandwidth of the waves seems to have little influence on the nonlinearity of the wave forces.
Acknowledgements This research is supported financially by the National Natural Science Foundation (50679010, 50379002), Program for New Century Excellent Talents in Universities of China (NCET-04-0267, NCET-05-0282), and Program for Changjiang Scholars and Innovative Research Teams of Colleges and Universities of China (IRT0420). References Baldock, T.E., Swan, C., Taylor, P.H., 1996. A laboratory study of non-linear surface waves on water. Philosophical Transactions of the Royal Society of London, Series A—Mathematical Physical and Engineering Sciences 354, 1–28. Boo, S.Y., 2006. Measurements of higher harmonic wave forces on a vertical truncated circular cylinder. Ocean Engineering 33 (2), 219–233. Chaplin, J.R., Rainey, R.C.T., Yemm, R.W., 1997. Ringing of a vertical cylinder in waves. Journal of Fluid Mechanics 350, 119–147. Chung, J., Powers, J., 1998. The statistics of wavelet-based bicoherence. In: Proceedings of the IEEE-SP International Symposium on Time–Frequency and Time–Scale Analysis, Pittsburgh, Pennsylvania, USA, pp. 141–144. Dong, G.H., Ma, Y.X., Ma, X.Z., 2008a. Cross-shore variations of wave groupiness by wavelet transform. Ocean Engineering 35, 676–684. Dong, G.H., Ma, Y.X., Perlin, M., Ma, X.Z., Yu, B., Xu, J.W., 2008b. Experimental study of wave–wave nonlinear interactions using the wavelet-based bicoherence. Coastal Engineering 55 (9), 741–752. Elsayed, M.A.K., 2006. A novel technique in analyzing non-linear wave–wave interaction. Ocean Engineering 33 (2), 168–180. Faltinsen, O.M., Newman, J.N., Vinje, T., 1995. Nonlinear-wave loads on a slender vertical cylinder. Journal of Fluid Mechanics 289, 179–198. Farge, M., 1992. Wavelet transforms and their applications to turbulence. Annual Review of Fluid Mechanics 24, 395–457. Ferrant, P., 1998. Fully non-linear interactions of long-crest wave packets with a three-dimensional body. In: Proceeding of the 22nd ONR Symposium on Naval Hydrodynamics. Washington, DC, pp. 403–415. Ferrant, P., Malenica, S., Molin, B., 1999. Nonlinear wave loads and runup on a vertical cylinder. In: Mahrenholtz, O., Markiewicz, M. (Eds.), Advances in Fluid Mechanics (non-linear water wave interaction), Vol. 24(3). WIT Press, Computational Mechanics Publications, Southampton, UK, pp. 101–136. Grue, J., Huseby, M., 2002. Higher-harmonic wave forces and ringing of vertical cylinders. Applied Ocean Research 24 (4), 203–214. Huang, M.C., 2004. Wave parameters and functions in wavelet analysis with filtering. Ocean Engineering 31 (7), 813–831. Huseby, M., Grue, J., 2000. An experimental investigation of higher-harmonic wave forces on a vertical cylinder. Journal of Fluid Mechanics 414, 75–103. Liu, P.C., 2000. Wave grouping characteristics in nearshore Great Lakes. Ocean Engineering 27 (11), 1221–1230. Malenica, S., Molin, B., 1995. Third-harmonic wave diffraction by a vertical cylinder. Journal of Fluid Mechanics 302, 203–229. Massel, S.R., 2001. Wavelet analysis for processing of ocean surface wave records. Ocean Engineering 28 (8), 957–987. Milligen, B.P.V., Sanchez, E., Estrada, T., Hidalgo, C., Branas, B., Carrersa, B., Garcia, L., 1995. Wavelet bicoherence: a new turbulence analysis tool. Physics of Plasmas 8, 3017–3032. Rapp, R.J., Melville, W.K., 1990. Laboratory measurements of deep-water breaking waves. Philosophical Transactions of the Royal Society of London, Series A—Mathematical Physical and Engineering Sciences 331 (1622), 735–800. Torrence, C., Compo, G.P., 1998. A practical guide to wavelet analysis. Bulletin of the American Meteorological Society 79 (1), 61–78.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Higher-harmonic focused-wave forces on a vertical cylinder Yuxiang Ma a,, Guohai Dong a,, Marc Perlin b, Shuxue Liu a, Jun Zang c, Yiyan Sun a a b c
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023, PR China Department of Naval Architecture & Marine Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA Department of Architecture and Civil Engineering, University of Bath, Bath BA2 7AY, UK
a r t i c l e in fo
abstract
Article history: Received 1 September 2008 Accepted 18 February 2009 Available online 13 March 2009
This paper considers higher-harmonic forces due to wave focusing on a vertical circular cylinder. A series of experiments has been conducted in a wave flume. The first six-harmonic components of the measured wave forces are analyzed using the scale-averaged wavelet spectrum. It is noted that due to the transient nature of focused (freak) waves, Fourier analysis would not provide equivalent information to that gleaned from the analyses used herein. The results for the experiments with very steep wave crests show significant amplitudes at the fourth and fifth harmonics. These harmonics exhibit amplitudes that are the same order as the second harmonic, but much larger than those of the third harmonic. The wavelet-based bicoherence is used to detect the quadratic nonlinearity of the measured forces. And the bicoherence spectra reveal the primary mathematical reason for the existence of the striking amplitudes of the fourth and fifth harmonics: the interaction between the lower-harmonic components couple more strongly with the fourth and fifth harmonics, thus the fourth and fifth harmonics glean more energy than those of the third-harmonic components. However, the physical explanation for this remains elusive. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Higher-harmonic forces Vertical cylinder Nonlinear interactions Wave transformation Scale-averaged wavelet energy Wavelet-based bicoherence
1. Introduction To obtain predictions of the total wave force on offshore structures, in recent years the research community has considered necessarily higher-harmonic forces. One important higher-order response is so-called ‘‘ringing’’, which is a vibration at the resonant frequency of a structure, and is a concern with regard to extreme loading. The generation mechanism of the higherharmonic wave loads leading to ringing of offshore structures is not fully understood. The ringing phenomenon has been detected in large and extremely steep waves (Chaplin et al., 1997), but Grue and Huseby (2002) show that it may occur in moderately-steep transient waves. The theoretical solutions of Faltinsen et al. (1995) and Malenica and Molin (1995), hereafter referred to as M&M, with the assumption of incoming Stokes waves, can capture the first three harmonic components of wave loads. A fully nonlinear model has also been developed to analyze this problem (Ferrant, 1998; Ferrant et al., 1999) and higher-harmonic wave loads were analyzed through an experimental approach. Huseby and Grue (2000) experimentally investigated the first seven harmonic components of the horizontal forces on vertical cylinders in
Corresponding authors. Tel.: +86 41184706213; fax: +86 41184708526.
E-mail addresses: [email protected] (Y. Ma), [email protected] (G. Dong). 0029-8018/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2009.02.009
Stokes waves. Boo (2006) studied the first five harmonic wave forces on a small scale truncated vertical cylinder in Stokes waves. However, an investigation of the higher-harmonic forces on a vertical cylinder during focused-wave events is lacking; this is the motivation for the study herein. The present effort considers the higher-harmonic loads of focused waves on a vertical cylinder in both the horizontal, longitudinal, and transverse directions. Moderate and steep slopes of focused waves are considered. Additionally the question of how these harmonic components appear in the wave forces is examined and explained through the wavelet-based bicoherence. The wavelet transform is a signal-processing technique that can generate localized time–frequency information from time series. In recent years, this technique has been applied successfully in wave analysis as well as other ocean engineering applications such as wave groupiness (Dong et al., 2008a; Liu, 2000), non-stationary processes and noise identification (Huang, 2004; Massel, 2001). Wavelet-based bicoherence, introduced by Milligen et al. (1995), is a powerful tool to detect quadratic phase coupling associated with nonlinear wave–wave interactions. Chung and Powers (1998) studied the statistical properties of the wavelet bicoherence and pointed out that the wavelet-based bicoherence estimate has a larger number of effective degrees of freedom than the Fourier-based bicoherence estimate. Thus, it can be used to detect quadratic phase coupling in a short time series. (Here short is used in comparison to that required for the usual bispectrum of Fourier analysis.) Wavelet-based bicoherence has
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been used successfully to study nonlinear wave–wave interactions by Elsayed (2006) in deep water, and by Dong et al. (2008b) in the coastal zone. This paper is organized as follows. Following the Introduction, a brief overview of the wavelet transform and wavelet-based bicoherence is presented in Section 2. The experimental set-up is described in Section 3. A discussion of the experimental results is presented in Section 4. Lastly, the conclusions of this research are given in Section 5.
2. Wavelet transform and wavelet-based bicoherence 2.1. Continuous wavelet transform The continuous wavelet transform of a time function x(t) can be defined as Z 1 xðtÞca;t ðtÞ dt; (1) WTða; tÞ ¼ 1
2.3. Wavelet-based bicoherence The wavelet-based cross bispectrum is defined as follows: Z WT x ða1 ; tÞWT x ða2 ; tÞWT y ða; tÞ dt, (6) Bxxy ða1 ; a2 Þ ¼ T
where a1, a2 and a must satisfy the frequency sum rule: 1 1 1 þ . ¼ a a1 a2
(7)
The wavelet-based cross bispectrum measures the amount of quadratic phase coupling in the time interval T between wavelet components of scale lengths a1, a2 of x(t) and a of y(t) in a manner such that the sum rule is satisfied (Milligen et al., 1995). From the relationship between scale and frequency, we can interpret the wavelet-based cross bispectrum as the coupling between waves of frequencies f ¼ f1+f2. Hence we will graph the wavelet bispectrum as a function of frequency rather than scale. Likewise the auto bispectrum is defined as Bx ða1 ; a2 Þ ¼ Bxxx ða1 ; a2 Þ.
(8)
where the asterisk denotes the complex conjugate, and ca,t(t) represents a family of functions called wavelets that are constructed by translating in time, t and dilation with scale, a, of a mother wavelet function c(t). The scale a can be interpreted as the reciprocal of frequency, fE1/a. The ca,t(t) expression is defined as tt . (2) ca;t ðtÞ ¼ jaj0:5 c a
Here the third x subscript and its use in Eq. (6) to obtain Eq. (8) should be obvious. To directly measure the degree of phase coupling, the squared wavelet cross-bicoherence can be defined as the normalized squared wavelet bispectrum
One of the most extensively used mother wavelets in continuous wavelet analysis is the Morlet wavelet (Torrence and Compo, 1998); it is a plane wave modulated by a Gaussian envelope and is defined as
which can assume values between 0 and 1. Similarly, the squared auto-bicoherence is defined as
cðtÞ ¼ p1=4 exp
t2 expðio0 tÞ, 2
(3)
where o0 is the peak frequency of the wavelet, usually chosen to be 6.0 to meet the admissible condition (Farge, 1992). Thus, analogous to the Fourier energy spectrum, the wavelet energy spectrum can be defined as Wða; tÞ ¼
WTða; tÞWT ða; tÞ . a
(4)
jBxxy ða1 ; a2 Þj2 R , 2 2 T jWT x ða1 ; tÞWT x ða2 ; tÞj dt T jWT y ða; tÞj dt
½bxxy ða1 ; a2 Þ2 ¼ R
(9)
½bx ða1 ; a2 Þ2 ¼ ½bxxx ða1 ; a2 Þ2 .
(10)
2
bx indicates the relative degree of quadratic phase coupling 2 between waves, with bx ¼ 0 for random phase relationships, and 2 2 bx ¼ 1 for maximum coupling. For a three-wave system, bx represents the fraction of power at scale a due to quadratic phase coupling of the 3 modes (a1, a2 and a). Detailed discussions on the statistical properties of the wavelet-based bicoherence are presented by Chung and Powers (1998) who have shown that the wavelet-based bicoherence can detect the quadratic phase coupling in a short duration series.
3. Experimental set-up 2.2. Scale-averaged wavelet energy
3.1. Wave flume
Another important spectrum based on the wavelet transform is the scale-averaged wavelet energy (SAWE), which is defined as
The measurements are conducted in the wave flume at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, PR China. This flume is 50.0 m long, 3.0 m wide and is used with a water depth of 0.5 m. The flume is equipped with a hydraulically driven piston-type wave maker, and in the following x ¼ 0 m is defined as the mean position of the wave maker. At the end of the flume, wave absorbers are installed to help mitigate wave reflection. The experimental set-up is shown in Fig. 1.
W n ðtÞ ¼
j¼j2 Wðaj ; tÞ dDt X
Cd
j¼j1
aj
,
(5)
in which d is the scale factor, Dt is the sample interval of the time series, and Cd is the reconstruction factor. The factor Cd is a constant for each wavelet function; for the Morlet wavelet, Cd is taken as 0.776 (Torrence and Compo, 1998). The SAWE is a time series of the average energy variance in a certain scale band (aj1 through aj2). Thus, it can be used to examine the modulation of one series by another, or modulation of one frequency band by another within the same series. In the present investigation, it is used to examine the energy at the scale bands which corresponds to the nth (n ¼ 1, 2, 3, y) harmonic components for both the focused waves and the wave forces.
3.2. Wave generation To generate the required focused waves at the specified point in space and time, a linear solution is used to provide a first approximation to the dispersive characteristics of the free waves generated at the wave paddle. This method is similar to the approaches adopted by Rapp and Melville (1990) and Baldock
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Fig. 1. Plan view of the wave flume experimental set-up.
et al. (1996). Assuming waves are focused at a specified point xf at time tf, linear wave theory defines the wave elevation at an arbitrary point as
Zðx; tÞ ¼
N X
ai cos½ki ðx xf Þ 2pf i ðt t f Þ,
(11)
Table 1 Regular wave conditions. f (Hz) kr max(A/r)
0.62 0.2525 0.3968
0.83 0.3811 0.3552
i¼1
where N is the total number of wave components, and ai, ki and fi represent the wave amplitude, the wave number and the wave frequency of the ith wave component, respectively. The dispersion relation establishes the relation between space and time, i.e. between ki and fi. In the analysis ai is determined from a chosen energy frequency spectrum and the phases of the wave components are such that they are all zero at the focal point. In this study, the focused wave-amplitude AF is defined as the input parameter, hence ai and AF satisfy AF ¼
N X
ai ,
(12)
i¼1
and ai can be calculated using AF S ðf ÞDf i a i ¼ PN i , i¼1 Si ðf ÞDf i
Case
Frequency range (Hz)
Frequency components, N
Input focused wave amplitude, AF (cm)
J1 J2 J3 J4 J5 J6 J7 J8 J9
0.6–1.06 0.5–1.16 0.5–1.36 0.6–1.06 0.5–1.16 0.5–1.36 0.6–1.06 0.5–1.16 0.5–1.36
29 29 50 29 29 50 29 29 50
3 3 3 6 6 6 8 8 8
(13)
where Si(f) is the desired frequency spectrum. In addition, the discrete frequency fi is spaced uniformly over the frequency band [fmin, fmax], and the frequency bandwidth is defined as
Df ¼ f max f min .
Table 2 Focusing wave conditions.
case. In this study, all experimental wave trains are adjusted to focus at x ¼ 11.4 m (without the cylinder present).
(14) 3.4. Model set-up
3.3. Wave conditions Before the experiment of forces on cylinders by focusing waves is conducted, the forces induced by regular waves should be measured to verify the accuracy of the experimental equipment. Incident regular waves with the wave slope Ak (A is the airy wave amplitude) ranging from 0.03 to 0.14 are considered. Two wave frequencies, 0.62 and 0.83 Hz, are used. The detailed information about regular waves is shown in Table 1. For focusing waves, the present investigation considers three different frequency bandwidths of wave simulations based on JONSWAP spectra with the same peak frequency, fp ¼ 0.83 Hz. For each frequency bandwidth, three different focused wave-amplitudes AF are considered. Wave groups with the largest waveamplitude are very close to an extreme event, as waves with larger input amplitude are found to be breaking in the primary test. Detailed information regarding the wave conditions is listed in Table 2 along with a designation assigned to each experimental
Once the focused waves at the desired location (x ¼ 11.4 m) are achieved in the wave flume, a cylinder that extends through the entire water depth is installed at x ¼ 11.625 m (see Fig. 1). The cylinder is 79 cm long with a diameter of 25 cm, and is constructed of plastic tubing with natural frequency (9.52 Hz) larger than the super-harmonics of the wave forces studied. Hence, the slenderness, kpr (kp is the linear dispersion wave number corresponding to fp and r is the cylinder radius), is about 0.383. The ratio of the width of the flume to the cylinder diameter is 12 in this investigation; therefore, the width of the flume has little effect on the results. The water surface elevations at the focal location are measured by a capacitance gauge with absolute accuracy of order 71 mm. For the regular wave cases, the wave elevations are measured in the absence of the cylinder, at the location where the centre of the cylinder is located. Prior to use, the integrity of the wave-elevation gauge is verified, and then it is calibrated. A force transducer, which can measure force in the longitudinal direction of the wave flume, is attached to the top of the cantilevered cylinder and mounted rigidly to a carriage to
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restrain the motions of the cylinder (Fig. 2). The waves and the wave forces are recorded at a sampling rate of 25 Hz.
between the measured first-harmonic forces and the M&M model are also small. However, the measured third-harmonic forces are larger than the theoretical values obtained by the M&M model.
4. Results and discussion
4.2. Nonlinear focusing waves and forces
4.1. Forces by regular waves
The measured water surface elevations at the focal point with and without the vertical cylinder are shown in Fig. 4 for Cases J1, J4 and J7, which have the same input frequency bandwidths (Df ¼ 0.46 Hz), but different focused forcing amplitudes. (Other cases give similar results, which are not shown.) It is noted that for each experiment without the cylinder, the left and right trough elevations of the maximum crest are almost equal at the focal point. The maximum crest elevation is enhanced additionally by augmentation of the input wave-focused amplitude, suggesting that the nonlinearity of the waves is increasing with increased amplitude forcing. For the condition with the cylinder present, however, the left and right trough elevations of the maximum crest elevation are no longer equal, and the water surface elevations are much larger (about 50%) than in the absence of the cylinder. The amplification of the wave crest elevations with the cylinder present is due primarily to the run-up of the waves on the cylinder, indicating as expected that the cylinder changes the local wave field significantly. The maximum crest elevations, without and with the cylinder present, are shown in Fig. 5. The vertical axis is the crest elevation measured at the focus location; the horizontal axis is the amplitude (based on the linear solution). It can be seen that nonlinearity is increasing with the amplitude forcing and that the presence of the cylinder enhances the nonlinearity of waves. It is evident also that the bandwidth has little influence on the nonlinearity of waves. The measured wave forces for Cases J1, J4 and J7 are shown in Fig. 6. There are obvious secondary load cycles (Grue and Huseby, 2002) around the maximum elevations of the loads for Case J7, suggesting that the higher harmonics of the forces are present. It is difficult to analyze the wave forces at each harmonic using Fourier transforms as adopted by Boo (2006) and Huseby and Grue (2000) owing to the transient nature of focused waves. As mentioned previously, due to the transient nature of focused (freak) waves, Fourier analysis simply cannot provide equivalent information to that gleaned from the analyses used herein.
The first three harmonic components of the measured forces (F1, F2, F3) exerted on the cylinder by regular waves are evaluated by Fourier analysis. The normalized components of the measured first three harmonics are shown in Fig. 3. The first-harmonic forces obtained from the Morison equation and the theoretical results of the M&M model (second and third order) given by Malenica and Molin (1995) are also presented in Fig. 3 for comparison. Similar to Huseby and Grue (2000), the differences between the measured first-harmonic forces and the values evaluated by the Morison equation are small. The differences
Fig. 2. Elevation view of the wave flume longitudinal section. The force transducers measure integrated horizontal longitudinal force only on the cantilevered cylinder.
Fig. 3. Solid circles: measured amplitudes of the first three components of forces for the regular wave cases. Dotted line: Morison equation. Dashed line: M&M model. (a) kr ¼ 0.318, (b) kr ¼ 0.381.
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Fig. 4. The measured water elevations at the focus location without (left panels) and with (right panels) the vertical cylinder present for the experiments (from top to bottom) J1, J4 and J7. (a) Case J1. (b) Case J4. (c) Case J7.
Fig. 5. Measured crest elevations: (a) without the cylinder; (b) with the cylinder. (??) linear solutions; (’) Df ¼ 0.46 Hz; (K) Df ¼ 0.66 Hz; (m) Df ¼ 0.86 Hz.
Fig. 6. The measured wave forces for Cases J1, J4 and J7.
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Fig. 7. The scale-averaged wavelet energy (SAWE) for the first six harmonics of the measured force for Case J1. (a) Case J1. (b) Case J4. (c) Case J7.
Fig. 8. The scale-averaged wavelet energy (SAWE) for the first six harmonics of the measured force for Case J4.
As mentioned in the preceding section, the scale-averaged wavelet energy is a time series of the average variance in a certain band. Thus, we can use it to examine the energy variance of the wave forces at the scale bands which corresponds to the nth (n ¼ 1, 2, 3, y) harmonic components of the wave forces. The SAWE of the first six super-harmonics of the wave forces for Cases J1, J4 and J7 are shown, respectively, in Figs. 7–9. It can be seen that the higher-harmonic forces are significantly smaller than the first-harmonic force, as one would expect. And, the higherharmonic components tend to appear simultaneously with the maximum wave forces, i.e. tE10.2 s. For Case J1, which has the smallest forcing amplitude, only the second-harmonic components are evident. For the next-largest focused-amplitude experiment (Case J4), except for the second harmonic, there is energy evident at the other higher harmonics, but it is ordered as
expected. It is interesting to notice that the fourth and fifth harmonics’ peak values of Case J7 are comparable to the peak value of the second harmonic. Indeed the sixth harmonic is the same order as the third harmonic, but both are much smaller than the fourth- and fifth-harmonic components. The peak values of the SAWE at the higher-harmonic components (from second to sixth) for all the measured wave forces are shown in Fig. 10. It is noted that only the peak values of the SAWE at the fourth and fifth harmonics for the largest input focused-amplitude cases (AF ¼ 8 cm, Fig. 10c) are comparable to the values for the second harmonics. In addition the amplitudes for the experiments with small input amplitude (AF ¼ 3 cm and AF ¼ 6 cm) of the third, fourth and fifth harmonics are much smaller than those of the second-harmonic components. At the same time, the peak values of the ith (i ¼ 3, 4, 5, 6) harmonic
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Fig. 9. The scale-averaged wavelet energy (SAWE) for the first six harmonics of the measured force for Case J7.
Fig. 10. The peak values of the scale-averaged wavelet energy (SAWE) for the higher harmonics of the measured forces for the experimental cases: (a) cases with A ¼ 3 cm; (b) cases with A ¼ 6 cm; (c) cases with A ¼ 8 cm.
forces for wave conditions with the same input AF but different Df are almost equal, suggesting that the input frequency bandwidth has little influence on the forces at these higher harmonics. The above discussions indicate that the considerably larger higherharmonic forces tend to appear in the waves with transient character and very steep slope.
Fig. 11 depicts the peak values of the SAWE at the higher harmonics (from second to sixth) for the measured water elevations at the focal location for all the experiments. The results shown in Fig. 11 indicate that the amplitudes of the components at the fourth, fifth and sixth harmonics are negligible compared to those at the second and third harmonics, even for the cases with
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Fig. 11. The peak values of the scale-averaged wavelet energy (SAWE) for the higher harmonics of the wave elevations at the focal location: (a) cases with A ¼ 3 cm; (b) cases with A ¼ 6 cm; (c) cases with A ¼ 8 cm.
Fig. 12. Wavelet-based auto-bicoherence of the measured forces for the cases with A ¼ 3 cm. (a) Case J1. (b) Case J2. (c) Case J3.
Fig. 13. Wavelet-based auto-bicoherence of the measured forces for the cases with A ¼ 6 cm. (a) Case J4. (b) Case J5. (c) Case J6.
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very steep slope. These results are distinct from the results of the wave forces discussed above.
4.3. Wavelet-based bicoherence of the measured nonlinear wave forces As mentioned, the measured wave forces at the fourth, fifth and sixth harmonics cannot be neglected, especially for the highly nonlinear wave conditions in which the components at the fourth and fifth harmonics are comparable to those of the secondharmonic components. Perplexingly, the results of the wave elevations measured at the leading edge of the cylinder show that the fourth-, fifth- and sixth-harmonic components of the waves are negligible, and that all harmonics are ordered. This phenomenon cannot be explained by any existing linear spectral approach. Thus, in this section, the wavelet-based bicoherence which has been proven to detect quadratic nonlinear phase coupling in a short duration time series is used to measure the quadratic nonlinear characteristics of the wave forces. The wavelet-based auto-bicoherence spectra for the cases with the smallest input amplitude (Cases J1, J2 and J3) are shown in Fig. 12; the bicoherence spectra indicate that the quadratic wave interactions are primarily amongst the first-harmonic compo2 nents. For example, bx (0.83, 0.83) ¼ 0.61 in the bicoherence spectrum of force for Case J1 (Fig. 12a) indicates a self–self wave interaction at the energy-frequency peak (f ¼ 0.83 Hz) coupled with the energy at twice the peak frequency (f ¼ 1.66 Hz). That is why there are only the second-harmonic components of the wave forces evident. It can be seen that the self–self interaction within the primary components of the force for Case J2 is larger than those of Cases J1 and J3 (Fig. 12b), suggesting that the secondharmonic force of that test obtains more energy from the primary components. This is consistent with the second-harmonic force of Case J2 which is the largest of the tests with AF ¼ 3 cm. Fig. 13 presents the auto-bicoherence spectra of the measured wave forces for the cases with AF ¼ 6 cm (Cases J4, J5 and J6). Compared to the runs with the smaller AF, the degree of quadratic phase coupling among the harmonic components of the forces are stronger, and more harmonic components participated in the nonlinear process, indicating that the quadratic nonlinearity of
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the forces for these tests are stronger than those of cases with smaller AF. The auto-bicoherence spectra of the wave forces for the largest input wave amplitudes are presented in Fig. 14. In these spectra, not only do the nonlinear interactions involve more components of the forces at higher harmonics than those of cases with lower input wave amplitudes, but the degree of nonlinear interaction between those higher-harmonic components is very strong. For 2 2 2 example, bx (2.49, 0.83) ¼ 0.72, bx (3.32, 0.83) ¼ 0.59 and bx (1.66, 1.66) ¼ 0.70 for Case J7 (Fig. 14a). All these indicate that the fourth- and fifth-harmonic components obtain significant energy from the first three harmonic components of the wave forces. The physical explanation of why this occurs remains a mystery. Meanwhile, we notice that the degree of quadratic phase coupling between the first-harmonic components and the second-harmonic components is smaller than those of the interactions between the first harmonic and other higher harmonics for almost all cases; this suggests that the third-harmonic components obtain less energy from the lower harmonics than do the fourth and fifth harmonics. For example, the value of bicoherence between 2 f ¼ 0.83 Hz and f ¼ 1.66 Hz is 0.49, i.e. bx (0.83, 1.66) ¼ 0.49, for the force of Case J8, but the value of bicoherence between f ¼ 0.83 Hz and f ¼ 2.49 Hz is 0.72 (Fig. 14b). In addition, the fourth- and fifth-harmonic components have more avenues to obtain energy than those of the third-harmonic components. For example, the fourth-harmonic components can obtain energy through the phase coupling between the harmonic components of the first and the third or through the phase coupling within the second-harmonic components. Whereas, the third-harmonic components obtain energy only through the phase coupling between the first two harmonic components, and at the same time, the third-harmonic components may transfer energy to the higher-harmonic components through nonlinear phase coupling with other harmonic components. Thus, the results obtained by the bicoherence spectra afford a reasonable explanation of the phenomenon (that there are large forces at the fourth and fifth harmonics, and that their amplitudes are significantly higher than that of the third harmonic). Furthermore, the bicoherence spectra of the forces with the same AF but different Df suggest that the input frequency bandwidth has little influence on the nonlinearity of the wave forces.
Fig. 14. Wavelet-based auto-bicoherence of the measured forces for the cases with A ¼ 8 cm. (a) Case J7. (b) Case J8. (c) Case J9.
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5. Concluding remarks This study concerns the higher-harmonic wave forces acting on a vertical circular cylinder, especially the forces induced by focused waves. Before conducting the experiment with focusing waves, the forces exerted on a cylinder by regular waves are studied to compare with theoretical models and to verify the experimental set-up. The results show that the first-harmonic forces can well be estimated by the Morison equation. The M&M (Malenica and Molin, 1995) model can predict approximately the second-harmonic forces, but under-predicts the third-harmonic forces. To study the characteristics of the higher-harmonic wave forces induced by focused waves, the horizontal longitudinal forces acting on a vertical circular cylinder in focused waves (based on JONSWAP spectra with different focused amplitudes and different input frequency bandwidths) are measured in a wave flume. The first six harmonic components for both the forces and the wave elevations at the focal location are analyzed using the scale-averaged wavelet energy (SAWE) which provides a time series of the average energy variance in a certain frequency band. The results demonstrate that SAWE is a good choice to address harmonic analysis for transient and short duration time–space series. Through the SAWE of the measured time series, it is revealed that the higher-harmonic forces are significantly smaller than the primary harmonic force for all the test cases. And, for the cases with very steep waves at the leading edge of the cylinder, the amplitudes of the forces for the harmonic components at the fourth and fifth are almost equal to that of the second-harmonic components, but much larger than those of the third-harmonic components, even though the amplitudes of the corresponding harmonic component for the wave elevations are negligible. To reveal the reasons why there are small forces at the thirdharmonic component but remarkable contributions at the fourthand fifth-harmonic components for those cases with highest input amplitude, the nonlinearity of the measured forces is studied through the wavelet-based bicoherence which can detect the quadratic nonlinear phase coupling in a short duration series. The phenomenon of small forces at the third harmonic, but noteworthy forces at the fourth and fifth harmonics for those cases with the highest input amplitude is substantiated through the wavelet-based bicoherence. The bicoherence spectra of the forces for these cases with high wave steepness show that the phase couplings between the harmonic components of the first and third, and the first and fourth are larger than between the first two harmonic components, indicating that the components at the fourth and fifth harmonics obtain more energy from lower harmonics than does the third harmonic. In addition, the fourthand fifth-harmonic components have more paths to obtain energy than those of the third-harmonic components. This is the reason that there are noteworthy forces at the fourth- and fifth-harmonic components, and why their amplitudes are significantly larger than those of the third-harmonic components. Furthermore, from
the bicoherence spectra, it is noticed that the input frequency bandwidth of the waves seems to have little influence on the nonlinearity of the wave forces.
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