The Hydrodynamic Behavior of Vortex Shedding behind Circular Cylinder in the Presence of Group Focused Waves

: Vortex shedding behind an elastically mounted circular cylinder in the presence of group focused waves propagating upstream was investigated using a classical approach (time series and FFT) and nonclassical approach (complex 2D Morlet wavelets). Wavelet analysis emerged as a novel solution in this regard. Our results include wave trains with different nonlinearities propagating in different water depths and derived from three types of spectra (Pierson–Moskowitz, JONSWAP ( γ = 3.3 or γ = 7)). It was found that the generated wave trains could modify regimes of shedding behind the cylinder, and subharmonic frequency lock-in could arise in particular situations. The occurrence of a lock-in regime in the case of wave trains propagating in intermediate water locations was shown experimentally even for small nonlinearities. Moreover, the application of time-localized wavelet analysis was found to be a powerful approach. In fact, the frequency lock-in regime and its duration could be readily identiﬁed from the wavelet-based energy and its corresponding ridges.


Introduction
Vortex shedding behind cylindrical structures has been the focus of many experimental and numerical studies over recent decades, given its practical importance in the maritime industry [1][2][3][4]. When exposed to highly nonlinear perturbations, vortex shedding provides modifications particularly in the wake around the cylinder. Therefore, these modifications should be considered when evaluating vortex shedding's disadvantages. The frequency of vortex shedding f sh in the wake depends on the frequency of the perturbation f w . More precisely, there is a fundamental synchronization region in which f sh is controlled by the imposed perturbation, which is referred to as the vortex shedding lock-in regime [5,6]. This regime may occur at frequencies of perturbation f w equal to f sh (harmonic lock-in), 2f sh (subharmonic lock-in), or f sh /n with n = 2, 3, . . . (superharmonic lock-in) [7]. In the present work, we only focused on subharmonic lock-in regime, i.e., f w~2 f sh .
In the global context of climate change evolving the expansion of marine renewable energy and the impacts of storminess on man-induced frames, cylindrical structures should be designed to remain functional in increasingly intermediate and deep-water locations during extreme events. For that reason, it is essential to study vortex shedding around the cylinder in similar harsh environments. In previous studies, several physical mechanisms have been suggested as causing the formation of extreme waves, including dispersive focusing in shallow or intermediate water depth [8,9] and modulation instability in deep-water depth, i.e., kh > 1.363, where k and h are the wavenumber and water depth, respectively [10]. In the present work, we have decided to limit our focus to extreme waves generated by the spatiotemporal wave focusing as a consequence of the dispersive nature of water waves. Dispersive focusing is traditionally used in laboratory modeling [11][12][13] to represent the expected wave profile based on the NewWave theory [14], which describes the wave velocity components and the free surface elevation of a focused group of localized waves. The

Materials and Methods
The tests were conducted at the Morphodynamique Continentale et Côtière de Caen, France. The wave flume was 16 m long, 0.5 m wide, and filled with tap water to a depth of h 0 = 0.25 m (Figure 1). A piston-type wave maker was located at one end of the flume to generate wave trains, and a foam absorber beach was at the other end. Two honeycombs of about 20 cm were installed in the flume, as shown in Figure 1a, in order to reduce wave reflection and to minimize the intensity of turbulent fluctuations generated by the pump. Temporal variations of free surface elevation measured at x = 2 m from the wave maker, were recorded by a wave gauge with a sampling rate of 100 Hz. These measurements were used to identify input wave train parameters.
The experimental setup is shown in Figure 1. The rigid circular cylinder model, situated at x = 7 m from the wave maker, was made from Plexiglas with diameter D = 0.04 m and had an immersed length of L = 0.248 m. An acoustic doppler velocimeter (ADV), placed at a distance d~D from the cylinder, was used to measure the longitudinal wave velocity V x upstream the structure. Wave trains propagated on the background of a steady current U = 0.16 m/s, corresponding to a Reynolds number Re = UD ν = 6400 (ν is the kinematic viscosity) and a Froude number Fr = U √ gh 0 = 0.1. The definition of the flow velocity and the Reynolds and Froude numbers were the same as in Hans et al. (2016). The cylinder was vertically clamped by its upper end to an elastic support, which was fixed on the flume's structure. The lower end of the cylinder was at approximately 2 mm (5% D) from the flume's floor (Figure 1b,c). Focused wave trains were generated with a technique similar to that in [13]. The resulting wave trains were derived from Pierson-Moskowitz, JONSWAP (γ = 3.3), and JONSWAP (γ = 7) spectra. Figure 2 shows three sets of time series of Pierson-Moskowitz, JONSWAP (γ = 3.3), and JONSWAP (γ = 7) wave trains and their corresponding dimensionless frequency spectra.        By adjusting the nonlinearity S 0 , breaking wave trains with different amplitudes were generated. The focus point was set to x = 12 m > x = 7 m and therefore, all of the generated wave trains broke downstream the cylinder. The nonlinearity parameter S 0 = k S 0 ∑ n i = 1 a i , also called local wave steepness, was the same as S 0 in [13]. Here, k S 0 was the characteristic wavenumber, and a i was the amplitude of the ith wave component. The nonlinearity parameter S 0 was calculated at x = 2 m from the wave maker. The generated wave trains propagated in intermediate water depth (0.9 < k w h 0 < 2.04). Tables 1 to 3 present some key parameters of the generated wave trains, which were categorized via their spectra, wave nonlinearity S 0 , and the water depth k w h 0 . One hundred and twenty crest-focused wave groups were generated in this study.

Fourier Analysis
The longitudinal velocity V x measured downstream the cylinder and FFT analysis were used in order to obtain the velocity energy spectrum S (V x (t)) = 2 |F(V x (t))| 2 . Here, F was the Fourier transform of the flow velocity V x . The sampling time for ADV measurements was truncated to 187.5 s, corresponding to 37,500 points. The lower and the upper cut-off frequencies were set, respectively, to 0.5 Hz and 2f w . Outside this frequency range, the spectral energy was very low (less than 2% of the total energy) and could be neglected. The data for this paper are available in the Supplementary Materials. Figure 3 exhibits a typical set of velocity spectra for twelve wave trains having approximately the same nonlinearity S 0 , derived from the three spectra and propagating in four different water depths. For each spectrum, the amplitude was normalized by the maximum of the spectral energy, and the shedding frequency f sh was normalized by the wave train peak frequency f w . The ratio f sh /f w and the spectrum width were used as indicators for detecting the subharmonic synchronization regime. In Hans et al. (2016), a spectral width of less than 3% of f sh /f w = 0.5 was used, in the case of regular waves, as a criterion by which it is possible to know if the frequency lock-in occurred or not. In this study, the presence of a subharmonic-like regime was characterized by f sh /f w = 0.5 and a spectrum width ∆f = 10% of f sh /f w = 0.5, in which more than 90% of the total energy was present.
Fluids 2022, 7, x FOR PEER REVIEW 6 of 16 width of less than 3% of fsh/fw = 0.5 was used, in the case of regular waves, as a criterion by which it is possible to know if the frequency lock-in occurred or not. In this study, the presence of a subharmonic-like regime was characterized by fsh/fw = 0.5 and a spectrum width Δf = 10% of fsh/fw = 0.5, in which more than 90% of the total energy was present. As the water depth increased, decreasing shedding frequency was featured, and the energy accumulation near f = fw/2 was evident. The existence of the subharmonic-like regime narrowed the high and low amplitude range of the velocity spectra. The shedding frequency and the wave peak frequency subsynchronized at kwh0 > 2.04, while the FFT spectra of the ADV signal were broad-banded with peak frequencies higher than fsh/fw = 0.5 in shallower water depths kwh0 < 2.04.
The dominant frequency of longitudinal velocity fluctuations in the wake fsh measured for all the studied wave trains is summarized in Figure 4. One can observe that the wave nonlinearity S0 and the spectrum type had no noticeable impact on the behavior of the shedding frequency evolution versus the steepness. As the water depth increased, decreasing shedding frequency was featured, and the energy accumulation near f = f w /2 was evident. The existence of the subharmonic-like regime narrowed the high and low amplitude range of the velocity spectra. The shedding frequency and the wave peak frequency subsynchronized at k w h 0 > 2.04, while the FFT spectra of the ADV signal were broad-banded with peak frequencies higher than f sh /f w = 0.5 in shallower water depths k w h 0 < 2.04.
The dominant frequency of longitudinal velocity fluctuations in the wake f sh measured for all the studied wave trains is summarized in Figure 4. One can observe that the wave nonlinearity S 0 and the spectrum type had no noticeable impact on the behavior of the shedding frequency evolution versus the steepness.

Wavelet Analysis
The spectral method used in the last section did not take into account signals whose constituent frequencies varied over time. Consequently, we needed a tool that has high resolution in the frequency and time domain, allowing us to identify at which frequencies the signal oscillated, and most importantly, at which time they occurred [21]. To efficiently and accurately detect frequency lock-in in the wake of the cylinder, the wavelet transform was proposed. Among all tests, the experimental results of six representative cases are provided in Table 4. It is important to note that similar results were obtained for the remaining wave trains. The continuous wavelet transform WT (a,τ) of the ADV temporal signal Vx (t) was defined as a convolution integral of Vx (t) with dilated and scaled versions of a so-called mother wavelet , * [22,23]: The asterisk indicates the complex conjugate, and ψa,τ represents the so-called mother wavelet function dilated by a factor τ and scaled by a factor a. Informally, Vx(t) × ψa,τ represents the energy of the ADV temporal signal around time t in the frequency band centered at 2 i . Hence, scales can be written as fractional powers of two [23]:

Wavelet Analysis
The spectral method used in the last section did not take into account signals whose constituent frequencies varied over time. Consequently, we needed a tool that has high resolution in the frequency and time domain, allowing us to identify at which frequencies the signal oscillated, and most importantly, at which time they occurred [21]. To efficiently and accurately detect frequency lock-in in the wake of the cylinder, the wavelet transform was proposed. Among all tests, the experimental results of six representative cases are provided in Table 4. It is important to note that similar results were obtained for the remaining wave trains. The continuous wavelet transform WT (a,τ) of the ADV temporal signal V x (t) was defined as a convolution integral of V x (t) with dilated and scaled versions of a so-called mother wavelet ψ * a,τ [22,23]: The asterisk indicates the complex conjugate, and ψ a,τ represents the so-called mother wavelet function dilated by a factor τ and scaled by a factor a. Informally, V x (t) × ψ a,τ represents the energy of the ADV temporal signal around time t in the frequency band centered at 2 i . Hence, scales can be written as fractional powers of two [23]: where a 0 = 0.01 is the smallest resolvable scale, M denotes the largest scale, and δ is the scale factor. The increase of the scale a is equivalent to moving the frequency content of the mother wavelet towards lower frequencies. However, the decrease of a is equivalent to moving its frequency content towards higher frequencies. We opted for a scale factor δ = 0.005, giving a total of 380 frequencies. The time sampling and the number of points were, respectively, ∆t = 0.005 s and N = 37,500. The wavelet transform WT (a,τ) can be displayed on a 2D frequency versus time color plot, with the colors representing the magnitudes of WT (a,τ). Abrupt changes in the temporal signal affect the values of WT (a,τ) at all scales and can be easily detected. As our wavelet transform was complex-valued, it could be expressed in terms of the modulus |WT (a,τ)|, which can be interpreted as the wavelet energy spectrum (i.e., scalogram). The magnitude of WT (a,τ) characterizes the occurrence and the intensity of the counterpart time scale at given time t. Consequently, the energy density of the signal V x (t) in the time scale plan could be estimated by: Wavelet analysis offers a large choice in the form of mother wavelets to represent the temporal signal as compared to cosines and sines used in the Fast Fourier Transform (FFT) and Short-Time Fourier Transform (STFT). The wavelet-based energy of the Mexican hat [24], Daubechies [25], Shannon [26], and Morlet [20,21,27] mother wavelets for the time series of the velocity signal V x resulting from the propagation of a JONSWAP (γ = 7) wave train (Case 120) are presented in Figure 5. The shapes of E (a,τ) were qualitatively similar when using different mother wavelets. Compared to the Morlet wavelet, more irregularities in E (a,τ) based on the Mexican hat, Daubechies, and Shannon mother wavelets were found, and smoothing operations may be needed to investigate frequencies present in the wavelet spectrum. Consequently, the Morlet mother wavelet was selected to be the most suitable for our purpose. The complex Morlet wavelet, also called the Gabor wavelet, can be interpreted as a sine wave tapered by a Gaussian window: where ν 0 is the bandwidth and ω 0 is the center frequency. A wider Morlet provides a weak temporal precision but good spectral precision, and vice versa for a narrower Morlet [28]. Obviously, the wavelet energy of the signal in the time scale plan was calculated for wavelets with different center frequencies ω 0 (namely 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, and 6). For the signal decomposition, the maximum of the wavelet energy was used as an indicator for selecting the optimal center frequency ω 0 [29]. When ω 0 = 3 for most of the studied wave signals, the maximum of energy was obtained. Thus, the number three is the optimum frequency of the mother wavelet for signal decomposition. Concerning the bandwidth ν 0 , it was fixed to 1 for real-time requirement and simplicity. Additionally, it can be useful to plot local maxima in the two-dimensional representation of wavelet information. These maxima, also called wavelet ridges, represent those locations in the time-frequency plan and are useful for pinpointing important frequencies at a given time t [21,28]. In addition to the energy wavelet density E (a,τ), to simplify its visualization, the dynamics of the local maxima (i.e., the time evolution of the wavelet ridges) can be calculated as follows: ( , ) = 0 (6) Additionally, it can be useful to plot local maxima in the two-dimensional representation of wavelet information. These maxima, also called wavelet ridges, represent those locations in the time-frequency plan and are useful for pinpointing important frequencies at a given time t [21,28]. In addition to the energy wavelet density E (a,τ), to simplify its visualization, the dynamics of the local maxima (i.e., the time evolution of the wavelet ridges) can be calculated as follows: Figure 6 exhibits a simple illustration of the wavelet-based spectrum of the longitudinal velocity V x in the presence of a sine wave with a frequency f = 1.35 Hz. Hans et al. [7] demonstrated that this wave belongs to the subharmonic lock-in region. Figure 6b shows a noticeable energetic band centered at f sh /f w = 0.5, which means that the subharmonic lock-in was continuously present. Figure 6c shows that the associated wavelet ridges were almost constant, i.e., dE~0, and deviated marginally from 0.5 to 0.5 ± 0.01 during the sinusoidal wave propagation. In other words, the effect of this sinusoidal wave on the vortex street led to the appearance of oscillations in the wake with a peak at half the frequency f w /2, and a finite width of the wavelet spectral peak. For that reason, it was necessary to use a criterion by which one could determine if the subharmonic frequency lock-in occurred or not. As a criterion, we chose the width of the wavelet spectral peak. Only if the width of the wavelet spectral peak was less than 5% of f sh /f w = 0.5, this regime was defined as a subharmonic frequency lock-in.  [7] demonstrated that this wave belongs to the subharmonic lock-in region. Figure 6b shows a noticeable energetic band centered at fsh/fw = 0.5, which means that the subharmonic lock-in was continuously present. Figure 6c shows that the associated wavelet ridges were almost constant, i.e., dE~0, and deviated marginally from 0.5 to 0.5 ± 0.01 during the sinusoidal wave propagation. In other words, the effect of this sinusoidal wave on the vortex street led to the appearance of oscillations in the wake with a peak at half the frequency fw/2, and a finite width of the wavelet spectral peak. For that reason, it was necessary to use a criterion by which one could determine if the subharmonic frequency lock-in occurred or not. As a criterion, we chose the width of the wavelet spectral peak. Only if the width of the wavelet spectral peak was less than 5% of fsh/fw = 0.5, this regime was defined as a subharmonic frequency lock-in. Generally, Figures 7-12 show that the complex Morlet power spectrum was found to highlight distinctive signatures in the presence and in the absence of the subharmonic frequency lock-in phenomenon. Figures 7-9 show the time series of the ADV of three wave trains in the presence of subharmonic synchronization, their wavelet energy E (a,τ), and their associated temporal evolution of the wavelet ridges. The respective patterns and graphs looked very similar. The first notable signature in the wavelet pattern was a strip of the domains aligned near fsh/fw = 0.5. This strip was formed by six domains that were red-yellow-shaded, and corresponded well to the trace path of the wave train. Generally, Figures 7-12 show that the complex Morlet power spectrum was found to highlight distinctive signatures in the presence and in the absence of the subharmonic frequency lock-in phenomenon. Figures 7-9 show the time series of the ADV of three wave trains in the presence of subharmonic synchronization, their wavelet energy E (a,τ), and their associated temporal evolution of the wavelet ridges. The respective patterns and graphs looked very similar. The first notable signature in the wavelet pattern was a strip of the domains aligned near f sh /f w = 0.5. This strip was formed by six domains that were red-yellow-shaded, and corresponded well to the trace path of the wave train.  The second feature of the wavelet pattern was a large blue-colored domain, which could be distinguished above and under the red-yellow-shaded domains. These large blue domains indicated that the implication of frequencies f < 0.45 fw and f > 0.55 fw was less important. This means that the energy was mainly concentrated on the map zone, in which the frequency was 0.45 fw ≤ f ≤ 0.55 fw. The third feature of the wavelet pattern seen   The second feature of the wavelet pattern was a large blue-colored domain, which could be distinguished above and under the red-yellow-shaded domains. These large blue domains indicated that the implication of frequencies f < 0.45 fw and f > 0.55 fw was less important. This means that the energy was mainly concentrated on the map zone, in which the frequency was 0.45 fw ≤ f ≤ 0.55 fw. The third feature of the wavelet pattern seen Thus, the wave train spectrum could be a reason why the strip of the domains near fsh/fw = 0.5 was narrow or wide. Moreover, wavelet ridges plots illustrated the presence of several zero slope regions following the passage of the wave train. Consequently, we could distinguish easily the duration of the subharmonic frequency lock-in phenomenon using wavelet ridges plots. It was also noted that in the case of narrow wave train spectra compared to wide wave train spectra, the wavelet ridges' temporal evolution contained less irregularities.   E (a,τ), and their wavelet ridges' temporal evolution. The respective patterns and graphs in the absence of subharmonic synchronization looked very similar. The existence of a wide strip of red-yellow-colored domains allowed us to assume the existence of several implicated frequencies 0.2fw ≤ f ≤ 1.5fw. Over time, additional several frequencies were involved in the wake of the cylinder, and chaotic frequency components appeared. In the absence of subharmonic synchronization, the shedding frequency was irregular in space, and vortex shedding did not occur periodically in time. The temporal evolution of the wavelet ridges demonstrated that local maxima varied randomly between 0 and 1.5 over time.

Conclusions
This paper discusses an experimental study of vortex shedding behind an elastically mounted cylinder in a water flow and in the presence of random wave trains propagating upstream. In the experiments, wave trains were generated via dispersive focusing technique in a two-dimensional wave flume. We investigated one hundred and twenty wave trains derived from Pierson-Moskowitz and JONSWAP (γ = 3.3 or γ = 7) spectra propagating in intermediate water depth. The experimental conditions were selected based on the wave nonlinearity S0, the water depth kwh0, and the spectrum type.
The velocity measurements downstream the cylinder were used to examine the shed- The second feature of the wavelet pattern was a large blue-colored domain, which could be distinguished above and under the red-yellow-shaded domains. These large blue domains indicated that the implication of frequencies f < 0.45 f w and f > 0.55 f w was less important. This means that the energy was mainly concentrated on the map zone, in which the frequency was 0.45 f w ≤ f ≤ 0.55 f w . The third feature of the wavelet pattern seen in Figures 7-9 was a narrower strip of the aligned domains near f sh /f w = 0.5 in the case of narrower wave train spectra. In other words, this strip was narrower in the case of a JONSWAP (γ = 7) wave train when compared to that of a Pierson-Moskowitz wave train. Thus, the wave train spectrum could be a reason why the strip of the domains near f sh /f w = 0.5 was narrow or wide. Moreover, wavelet ridges plots illustrated the presence of several zero slope regions following the passage of the wave train. Consequently, we could distinguish easily the duration of the subharmonic frequency lock-in phenomenon using wavelet ridges plots. It was also noted that in the case of narrow wave train spectra compared to wide wave train spectra, the wavelet ridges' temporal evolution contained less irregularities. Figures 10-12 show the time series of the ADV of three wave trains in the absence of subharmonic synchronization, their wavelet energy E (a,τ), and their wavelet ridges' temporal evolution. The respective patterns and graphs in the absence of subharmonic synchronization looked very similar. The existence of a wide strip of red-yellow-colored domains allowed us to assume the existence of several implicated frequencies 0.2f w ≤ f ≤ 1.5f w . Over time, additional several frequencies were involved in the wake of the cylinder, and chaotic frequency components appeared. In the absence of subharmonic synchronization, the shedding frequency was irregular in space, and vortex shedding did not occur periodically in time. The temporal evolution of the wavelet ridges demonstrated that local maxima varied randomly between 0 and 1.5 over time.

Conclusions
This paper discusses an experimental study of vortex shedding behind an elastically mounted cylinder in a water flow and in the presence of random wave trains propagating upstream. In the experiments, wave trains were generated via dispersive focusing technique in a two-dimensional wave flume. We investigated one hundred and twenty wave trains derived from Pierson-Moskowitz and JONSWAP (γ = 3.3 or γ = 7) spectra propagating in intermediate water depth. The experimental conditions were selected based on the wave nonlinearity S 0 , the water depth k w h 0 , and the spectrum type.
The velocity measurements downstream the cylinder were used to examine the shedding frequency. Spectral evolution of the longitudinal velocity V x was examined first. The shedding frequency f sh and the spectral bandwidth were investigated as the nonlinearity and the water depth increased. The spectral bandwidth of the longitudinal velocity V x decreased when the water depth increased. For water depth k w h 0 > 2.04, the spectral peak reached f w /2, and more than 90% of the spectral energy was inside the range 0.45 < f sh /f w < 0.55. Hence, we could conclude the presence of a synchronization-like regime for water depth k w h 0 > 2.04.
The Fourier transform has the assumption of a stationary signal, which means that features of the signal remain constant over time. However, nonstationarities are very important in the case of abrupt changes associated with extreme waves. The assessment over time of the variation of the shedding frequency behind the cylinder assists in easily identifying the subharmonic lock-in and its source. In order to accurately characterize the modes arising in the wake, the wavelet spectrum of the longitudinal velocity V x was studied. Morlet wavelet was chosen as a mother wavelet given its large similarity with the ADV temporal signals.
From the experimental results, it was found that the wavelet energy and the associated ridges of the sampled data could clearly and precisely display locations of each singularity. In the presence of the subharmonic synchronization regime, the dominant frequency components lied in the range 0.45 f w < f sh < 0.55 f w .
Here, the wavelet technique was proven to be useful in identifying the frequency components of a physical system and its multiscale dynamics controlled by a nonstationary behaviour related to the extreme events. In similar contexts, Turki et al. [30,31] have used the spectral wavelet analysis to investigate the dynamics of storms from intermonthly to interannual scales. They found that the extreme energy spectrum at short scales should be largely modulated to increase its impact at larger scales. It will be interesting to test the wavelet technique and the criterion by which one could determine if the subharmonic frequency lock-in occurs or not in the presence of rogue waves in field measurements. This could be achieved using the control procedure employed by [32] to record rogue wave signals and in the presence of a circular cylinder. A full investigation of this multi-timescale dynamics needs a coupling between different approaches of physical (use of experiments in laboratory), numerical (use of mathematical models), and stochastic (use of spectral techniques) which can be an important step to improve our understanding the extreme dynamics in coastal zones. It would also be interesting to investigate these findings in the presence of the cavitation phenomenon [33,34].
This study strengthens the utility of spectral approaches in detecting features that stay hidden in a classical Fourier analysis. Nevertheless, in order to expand results found in this study, efforts should be made by investigating shallower water conditions and higher nonlinearities with the presence of breaking. Finally, an inherent disadvantage of the dispersive focusing mechanism is scaling the results to full scale. Therefore, it will be interesting to compare our findings to field measurements.