Wavelet-based Bicoherence Research Articles

Study on suppression of violent sloshing in a multiple tuned liquid column damper by porous media layers

Multiple tuned liquid multi-column damper (MTLCD) is a passive damping device that is used to suppress the motion of semisubmersible floating offshore wind turbine (FOWT). However, the violent sloshing within the MTLCD under extreme sea environments reduces the anti-pitch motion effect. Thus, this study employed an innovative MTLCD with porous media layers, which can increase the flow energy dissipation, suppress the violent sloshing and enhance the damping moment (Mdamping). The OpenFOAM is used to simulate the hydrodynamic force in MTLCD, where the numerical method is validated by the decay test of a scaled model. The numerical natural period of the MTLCD shows good agreement with the experimental and analytical results. The sloshing characteristics, including pressure, damping moment impulse and work of the MTLCD without porous media, are analyzed firstly under simple harmonic rotation amplitudes (θ0) of 2°–20°. Subsequently, porous media layers are installed in the MTLCD with different heights (hp), including immersion and suspension, in order to improve the MTLCD's damping moment. Wavelet transform and wavelet-based bicoherence are utilized to analyze the nonlinear behaviors of the damping moment under violent sloshing. The breaking, surging and converging in the MTLCD are captured by the numerical method. The analyses show that the porous layers can suppress the violent sloshing by dissipating the fluid kinetic energy, and the damping moment work can be significantly improved when θ0 >6°. The liquid surging with bubbles leads to the nonlinear behaviors in the MTLCD, the sloshing energy transfer to the frequency doubling by three-wave interaction.

Experimental study on the interactions between wave groups in double-wave-group focusing

Nonlinear interactions in wave-group focusing are regarded as one of the main mechanisms in the generation of destructive extreme waves. In real seas, the wideband bimodal state is a typical configuration, containing interactions within a single wave group and between different wave groups. The former has been well uncovered under the assumption of narrow bandwidth, but the latter is poorly understood. In this paper, physical experiments are conducted to reveal the physics of double-wave-group focusing considering various energy distributions. Superposed wavemaker signals generated by the iteration method are applied to produce a double-wave-group focusing with the interactions being decomposed. Results of the wavelet-based bicoherence spectrum show that double-wave-group focusing is distinguished from the linear superposition of two single-wave-group focusing mainly in the nonlinear interactions induced by the second-order sum harmonics. Under the assumption of equivalent energy, interactions of the second-order sum harmonics between the lower frequency group and higher frequency group cannot be ignored in swell-dominated states, and lesser linear interactions and stronger nonlinear interactions are observed while the spectral distribution of the double-wave-group is more asymmetrical. This work is anticipated to contribute to the understanding of the generation mechanism of extreme waves driven by strong nonlinearity.

Experimental study on the effect of an oyster reef on the nonlinear characteristics of irregular waves

As a “marine ecological engineer”, the oyster reefs not only perform important ecological functions, but also reduce the damage caused by waves to protective structures such as seawalls. However, oyster reefs in shallow water change the nonlinear characteristics of waves and affect sediment transport and coastal evolution. Based on Fourier spectrum and analysis of Wavelet Transform, the influence of artificial bag oyster reefs on the energy and nonlinear phase coupling of irregular waves are studied through physical experiment. The results show that oyster reefs have a substantial effect on the energy of primary harmonic, which transfer to higher harmonics through triad interactions, and a considerable reduction in primary harmonic energy and an increase in higher harmonics energy are reflected in the energy spectra. The transmission spectrum behind the oyster reefs shows three peaks at primary, secondary and third harmonics. The bicoherence spectrum indicates that the peaks at secondary and third harmonics mainly result from the self-coupling of the primary harmonics and phase coupling between the primary and secondary harmonics respectively. As the water depth increases, the degree of nonlinear coupling between wave components decreases, which leads to the energy of wave components at different frequencies increases. With increasing top width, the length of the shoaling region increases, and the growth of triad nonlinear interactions are observed in wavelet-based bicoherence spectra, resulting in the spectral peak energy decreasing while the secondary harmonics energy increasing in the spectrum. Finally, the potential application of an ecological system composed by “oyster reefs + mangroves” is discussed. As the effect of water depth on wave energy is much greater than that of top width, in artificial oyster reef construction, it is recommended that keep the oyster reefs non-submerged in terms of wave dissipation. Further studies should take the dynamic growth effect of oyster reefs into account.

Dynamics coupling analysis between elastic supporting structure and TLD with and without slat screen

The slat screen (SS) has been widely used in tuned liquid damper (TLD) to increase damping and suppress structural vibration. This study proposes a linked-spring screen that has a movable screen elastically connected to the inner wall. Laboratory experiments of elastic supporting structure with TLD were first conducted on a hexapod table to obtain structural dynamics response under different conditions. Wavelet transform and wavelet-based bicoherence are used to analyze the nonlinear characteristics of sloshing waves. Meanwhile, a two-way coupling numerical model was validated to accurately predict the nonlinear coupling interaction between structure and TLD. Effects of the SS on reducing the structural vibration were numerically evaluated together with the interactions among elastic structure, TLD and SS. Finally, effects of spring stiffness on TLD with linked-spring screen were studied. Findings show that SS can not only enhance the internal dissipation by creating vortexes but also break waves through the perforation. Compared with conventional TLD, screen with Se = 0.75 and D/L = 0.15 can reduce the maximum displacement by 37.86 %. Besides, the linked-spring screen can further improve the damping performance because of the enhanced shear flow and turbulence intensity when it was moving. The optimal damping effect can be achieved for K = 500 N·m−1 where the further reduction ratio is 29.80 %.

Wavelet bispectral analysis and nonlinear characteristics in waves generated by submerged jets

Nonlinear interaction dominates the whole process from wave deformation to breaking, and is one of the significant factors that may cause many unstable changes in waves. This study aims to indicate the nonlinear characteristics of waves generated by submerged jets. The time series of those waves under five dimensionless upstream water depths (l = 3.0, 4.0, 4.5, 5.0, 5.5) are measured experimentally. Moreover, the wave surface asymmetry, skewness, spectrum, and wavelet-based bicoherence are obtained. The results show that the amplitude of wave surface varies with the variation of measuring point position and upstream water depth. From the perspective of asymmetry and skewness, the wave surface varies from tilt forward to tilt backward along the propagation path, and the vertical tilt amplitude decreases gradually; meanwhile, except for few measuring points, it is basically negative skewness. It can be seen from the energy spectrum that the wave energy increases with the increase of upstream water depth and decreases with the propagation of wave surface. In addition, the wavelet-based bicoherence distribution shows that the energy transfer caused by the interaction between waves is weak. The findings can provide an improved understanding of the nonlinearity characteristics of waves generated by submerged jets.

An experiment on reconstruction and analyses of in-situ measured freak waves

Freak waves are destructive to marine and offshore structures, but their generation mechanisms are still unclear. To explore the possible generation mechanism of freak waves, some in-situ measured waves are reconstructed in a wave flume at model scale. To generate high-quality target waves, a modified time reversal (TR) method including a correction step is proposed. Five scaled in-situ measured waves, containing three freak waves and two non-freak waves, are reconstructed successfully.As reconstructed waves provide good agreement with the scaled in-situ measured waves, their formation mechanisms are analyzed by applying various dispersion relations. The results show the key contributions of the dispersive focusing in the freak wave formation. Then, to explore the nonlinear interactions of the quadratic order of these reconstructed waves, the wavelet-based bicoherence and its total bicoherence are presented, and large surface elevations are found to be related to high total bicoherences, indicating inescapable effects of wave-wave interactions in the wave formation. Furthermore, a phase coincidence level F is proposed to evaluate the phase coincidence level of these reconstructed waves. The results show that a low F (high phase focus level) means a high wave height, while a high wave height does not always exhibit a low F.

Experimental investigation of transient harbor resonance induced by solitary waves

Free-surface oscillations within a square harbor subjected to solitary waves are investigated through physical experiments. The effects of various wave heights and water depths on the resonance within the harbor are examined. In this study, the technique of wavelet-based bicoherence spectra is employed for the first time to investigate the nonlinear wave–wave interaction of the transient harbor resonance induced by solitary waves. The results show that the relation between the maximum surface elevation, ηmax, within the harbor and the incident wave height of a solitary wave can be regarded as approximately linear. Moreover, for the same incident solitary wave height, the larger the water depth is, the greater the value of ηmax becomes. The analyses of the wavelet-based bicoherence indicate that the natural modes of the harbor are generated in the following two ways: (1) direct generation from the energy transfer of the solitary waves, and (2) the subsequent generation from the nonlinear interaction between various wave components. The first mechanism plays a decisive role in the generations of low-order natural modes, while the second mechanism also has a vital catalytic role in triggering higher-order modes. Finally, it is examined that the duration of harbor oscillations changes obviously with incident wave heights and water depths.

Laboratory study of non-linear wave–wave interactions of extreme focused waves in the nearshore zone

Abstract. Extreme waves play a crucial role in marine inundation hazards and coastal erosion. Prediction of non-linear wave–wave interactions is crucial in assessing the propagation of shallow water extreme waves in coastal regions. In this article, we experimentally study non-linear wave–wave interactions of large-amplitude focused wave groups propagating in a two-dimensional wave flume over a mild slope (β=1:25). The influence of the frequency spectrum and the steepness on the non-linear interactions of focused waves are examined. The generated wave trains correspond to Pierson–Moskowitz and JONSWAP (γ=3.3 or γ=7) spectra. Subsequently, we experimentally approach this problem by the use of a bispectral analysis applied on short time series, via the wavelet-based bicoherence parameter, which identifies and quantifies the phase coupling resulting from non-resonant or bound triad interactions with the peak frequency. The bispectral analysis shows that the phase coupling increases gradually and approaches 1 just prior to breaking, accordingly with the spectrum broadening and the energy increase in high-frequency components. Downstream breaking, the values of phase coupling between the peak frequency and its higher harmonics decrease drastically, and the bicoherence spectrum becomes less structured.

A numerical investigation on nonlinear transformation of obliquely incident random waves on plane sloping bottoms

The statistical properties of obliquely incident irregular waves over a planar sloping bottom were investigated numerically by the well-known numerical wave model FUNWAVE 2.0. Irregular waves based on the averaged JONSWAP spectra with various wave heights and peak periods were simulated to propagate over planar bottoms. A wide range of incident angles from 0° though 75° were considered to study the influence of incident angles. It was found that incident angles have a significant influence on the wave nonlinearity. The wavelet-based bicoherence revealed that the degree of triad wave interactions of primary waves and the higher harmonics was apparently weakened by increasing wave incident angles. Importantly, using the simulated data, the Klopman wave height distribution was improved by incorporating the influence of obliquely incident angles. It was found that the improved Klopman distribution shows better performance for describing wave height distributions in shallow water depth. Moreover, two empirical formulae are recommended to reflect the relationship between the skewness and the asymmetry of waves and the Ursell number for obliquely incident waves on plane sloping bottoms.

Laboratory study of the nonlinear transformation of irregular waves over a mild slope

This paper considers the nonlinear transformation of irregular waves propagating over a mild slope (1:40). Two cases of irregular waves, which are mechanically generated based on JONSWAP spectra, are used for this purpose. The results indicate that the wave heights obey the Rayleigh distribution at the offshore location; however, in the shoaling region, the heights of the largest waves are underestimated by the theoretical distributions. In the surf zone, the wave heights can be approximated by the composite Weibull distribution. In addition, the nonlinear phase coupling within the irregular waves is investigated by the wavelet-based bicoherence. The bicoherence spectra reflect that the number of frequency modes participating in the phase coupling increases with the decreasing water depth, as does the degree of phase coupling. After the incipient breaking, even though the degree of phase coupling decreases, a great number of higher harmonic wave modes are also involved in nonlinear interactions. Moreover, the summed bicoherence indicates that the frequency mode related to the strongest local nonlinear interactions shifts to higher harmonics with the decreasing water depth.

Evidence for four- and three-wave interactions in solar type III radio emissions

Abstract. The high time resolution observations obtained by the STEREO/WAVES experiment show that in the source regions of solar type III radio bursts, Langmuir waves often occur as intense localized wave packets with short durations of only few ms. One of these wave packets shows that it is a three-dimensional field structure with WLneTe ~ 10−3, where WL is the peak energy density, and ne and Te are the electron density and temperature, respectively. For this wave packet, the conditions of the oscillating two-stream instability (OTSI) and supersonic collapse are satisfied within the error range of determination of main parameters. The density cavity, observed during this wave packet indicates that its depth, width and temporal coincidence are consistent with those of a caviton, generated by the ponderomotive force of the collapsing wave packet. The spectrum of each of the parallel and perpendicular components of the wave packet contains a primary peak at fpe, two secondary peaks at fpe ± fS and a low-frequency enhancement below fS, which, as indicated by the frequency and wave number resonance conditions, and the fast Fourier transform (FFT)-based tricoherence spectral peak at (fpe, fpe, fpe + fS, fpe − fS), are coupled to each other by the OTSI type of four-wave interaction (fpe is the local electron plasma frequency and fS is the frequency of ion sound waves). In addition to the primary peak at fpe, each of these spectra also contains a peak at 2fpe, which as indicated by the frequency and wave number resonance conditions, and the wavelet-based bicoherence spectral peak at (fpe, fpe), appears to correspond to the second harmonic electromagnetic waves generated as a result of coalescence of oppositely propagating sidebands excited by the OTSI. Thus, these observations for the first time provide combined evidence that (1) the OTSI and related strong turbulence processes play a significant role in the stabilization of the electron beam, (2) the coalescence of the oppositely propagating up- and down-shifted daughter Langmuir waves excited by the OTSI probably is the emission mechanism of the second harmonic radiation, and (3) the Langmuir collapse follows the route of OTSI in some of the type III radio bursts.

The effect of isoflurane anesthesia on the electroencephalogram assessed by harmonic wavelet bicoherence-based indices

Bicoherence quantifies the degree of quadratic phase coupling among different frequency components within a signal. Previous studies, using Fourier-based methods of bicoherence calculation (FBIC), have demonstrated that electroencephalographic bicoherence can be related to the end-tidal concentration of inhaled anesthetic drugs. However, FBIC methods require excessively long sections of the encephalogram. This problem might be overcome by the use of wavelet-based methods. In this study, we compare FBIC and a recently developed wavelet bicoherence (WBIC) method as a tool to quantify the effect of isoflurane on the electroencephalogram. We analyzed a set of previously published electroencephalographic data, obtained from 29 patients who underwent elective abdominal surgery under isoflurane general anesthesia combined with epidural anesthesia. Nine potential indices of the electroencephalographic anesthetic effect were obtained from the WBIC and FBIC techniques. The relationship between each index and end-tidal concentrations of isoflurane was evaluated using correlation coefficients (r), the inter-individual variations (CV) of index values, the coefficient of determination (R2) of the PKPD models and the prediction probability (PK). The WBIC-based indices tracked anesthetic effects better than the traditional FBIC-based ones. The DiagBic_En index (derived from the Shannon entropy of the diagonal bicoherence values) performed best [r = 0.79 (0.66–0.92), CV = 0.08 (0.05–0.12), R2 = 0.80 (0.75–0.85), PK = 0.79 (0.75–0.83)]. Short data segments of ∼10–30 s were sufficient to reliably calculate the indices of WBIC. The wavelet-based bicoherence has advantages over the traditional Fourier-based bicoherence in analyzing volatile anesthetic effects on the electroencephalogram.

Laboratory Study of Unidirectional Focusing Waves in Intermediate Depth Water

The results of laboratory measurements of large focusing wave groups, which were generated using the New Wave theory, are presented. The influences of both the steepness and frequency bandwidth on focused wave characteristics were examined. The influence of frequency bandwidth on focused wave groups with small and moderate steepness was very small. However, for cases with the large steepness, the nonlinearity increased with increasing bandwidth frequency and widened free-wave regimes are identified for those cases with large steepness at the focal location. The underlying nonlinear phase coupling of focused waves was examined using wavelet-based bicoherence and biphase, which can detect nonlinear phase coupling in a short time series. For wave groups with large initial steepness, as wave groups approached the focal location, the values of bicoherence between primary waves and its higher harmonics progressively increased to 1 and the corresponding biphase was gradually close to zero, suggesting that an extreme wave event can be produced by considering Stokes-like nonlinearity to very high-order. Furthermore, the fast change of bicoherence of focused wave groups indicates that the nonlinear energy transfer within focusing waves is faster than that of nonfocusing wave trains.

Higher-harmonic focused-wave forces on a vertical cylinder

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.

Experimental study of wave–wave nonlinear interactions using the wavelet-based bicoherence

The wavelet-based bicoherence, which is a new and powerful tool in the analysis of nonlinear phase coupling, is used to study the nonlinear wave–wave interactions of breaking and non-breaking gravity waves propagating over a sill. Two cases of mechanically generated random waves based on J onswap spectra are used for this purpose. Values of relative depth, k p h ( k p is the wave number of the spectral peak and h is the water depth) for this study range between 0.38 and 1.22. The variations of wavelet-based total bicoherence for the test cases indicate that the degree of quadratic phase coupling increases in the shoaling region consistent with a wave profile that is pitched shoreward, relative to a vertical axis as seen in the experiments, but decreases in the de-shoaling region. For the non-breaking case, the degree of quadratic phase coupling continues to increase until waves reach the top of the sill. Breaking waves, however, achieve their highest level of quadratic phase coupling immediately before incipient breaking and the degree of phase coupling decreases sharply following breaking. In addition the wavelet-based bicoherence spectra provide evidence of the harmonics' growth which is reflected in the energy spectra. The bicoherence spectra also show that quadratic phase coupling between modes within the peak frequency as well as between modes of the peak frequency and its higher harmonics are dominant in the shoaling region, even though there are relatively high levels of quadratic phase coupling occurring between other frequencies. Furthermore, using the temporal resolution property of the wavelet-based bicoherence, we find that the quadratic wave interactions occur more readily during segments of time with large change of wave amplitude, rather than those segments having large wave amplitudes, but small gradients in amplitude.