Two-dimensional Wave Trains Research Articles

An experimental study of weakly three-dimensional non-breaking and breaking waves

To investigate the influence of directionality on intermediate- and deep-water (kh=1.09–3.49) wave breaking, an elaborately designed physical experiment was conducted in an ‘X’ configuration, where two identical two-dimensional wave trains were generated and propagated with an approach angle of 8° (i.e. a weakly three-dimensional configuration) and encountered each other in an interaction region. The study involved measuring the wave surface elevations using calibrated wave gauges along the channels as well as recording the variation of the surface elevations spatially using video in the vicinity of the interaction region. In addition, for direct comparison a two-dimensional experiment was also conducted in one of the channels with the ‘X’ configuration essentially removed by a dummy wall in the interaction region. The experimental results show that the weakly three-dimensional experiments have little influence on the profile of wave trains with relatively low initial steepness (S0=0.215. S0=ks∑an, where ks is the wave number corresponding to the spectrally weighted wave frequency, and an is the wave amplitude of the nth wave component), although it is pronounced when breaking occurs. In addition, the frequency spectra of the focusing waves illustrate that the interaction of two wave trains increases the energy in the 0.91<f/fp<1.82 band for non-breaking cases and dissipates the energy from the higher frequency end for breaking cases. Compared with the two-dimensional experiment, it is noted that due to directional wave–wave interaction, breaking events could occur with smaller initial steepness and, more interestingly, with lower energy dissipation. It is somewhat surprising that the breaking dissipation was reduced. We noticed a shift in focusing/breaking location (i.e. premature) due to the increase in crest height using linear focusing theory, which was more pronounced during directional wave–wave interaction.

On analyticity of travelling water waves

In this paper we establish the existence and analyticity of periodic solutions of a classical free-boundary model of the evolution of three-dimensional, capillary–gravity waves on the surface of an ideal fluid. The result is achieved through the application of bifurcation theory to a boundary perturbation formulation of the problem, and it yields analyticity jointly with respect to the perturbation parameter and the spatial variables. The travelling waves we find can be interpreted as resulting from the (nonlinear) interaction of two two-dimensional wavetrains, giving rise to a periodic travelling pattern. Our analyticity theorem extends the most sophisticated results known to date in the absence of resonance; ‘short crested waves’, which result from the interaction of two wavetrains with unit amplitude ratio are realized as a special case. Our method of proof also sheds light on the convergence and conditioning properties of classical boundary perturbation methods for the numerical approximation of travelling surface waves. Indeed, we demonstrate that the rather unstable numerical behaviour of these approaches can be attributed to the strong but subtle cancellations in the formulas underlying their classical implementations. These observations motivate the derivation and use of an alternative, stable, formulation which, in addition to providing our method of proof, suggests new stabilized implementations of boundary perturbation algorithms.

Response of a two-layer compliant slab to inviscid incompressible fluid flow

In this study, the dynamics of two-dimensional wave trains at the interface between a two-layer viscoelastic coating and flow of an inviscid fluid is examined theoretically. The objective is to understand the interaction mechanism between the two media better, and to explore the effects of different geometric and material properties on this mechanism. Each layer of coating is treated as an infinite, elastic, or viscoelastic solid of finite thickness, bonded to each other. The flow is modeled by the potential theory. The two media, two-layer slab and the fluid, are coupled through stress and velocity, normal to the surface of the slab. Two layers of the slab are assumed to be bonded to each other. The proposed mathematical model is used to predict the appearance of unstable surface waves of the coating. A few dimensionless parameters are identified. The model predicts that the instabilities initiate in the form of standing waves for the viscoelastic coating while the speed of the unstable waves at the interface of the elastic coating and the fluid is relatively high.

The effect of phase variations and cross-shear on vortical structures in a plane mixing layer

The evolution of spanwise phase variations of nominally two-dimensional instability modes in a plane shear layer is studied in a closed-return water facility using time-harmonic excitation having spanwise-non-uniform phase or frequency distributions. The excitation waveform is synthesized by a linear array of 32 surface film heaters flush-mounted on the flow partition. A span wise-linear phase distribution leads to the excitation of oblique waves and to the rollup of oblique primary vortices. When the prescribed phase distribution is piecewise-constant and spanwise-periodic, the flow is excited with a linear combination of a two-dimensional wavetrain and pairs of equal and opposite oblique waves, the amplitudes of which depend on the magnitude of the phase variation ΔΦ. As a result of the excitation, the primary vortices undergo spanwise-non-uniform rollup and develop spanwise-periodic deformations that induce cross-shear and secondary vortices in the braid region. The amplitude of the deformations of the primary vortices and the shape and strength of the secondary vortices depend on the magnitude of ΔΦ. When ΔΦ is small, the secondary vortices are counter-rotating vortex pairs. As ΔΦ increases, cross-shear induced by oblique segments of the primary vortices in the braid region results in the formation of single secondary vortex strands. The flow is not receptive to spanwise phase variations with wavelengths shorter than the streamwise wavelength of the Kelvin–Helmholtz instability. When the phase variation is ΔΦ = ϕ, the flow is excited with pairs of oblique waves only and undergoes a double rollup, resulting in the formation of spanwise-deformed vortices at twice the excitation frequency. Measurements of the streamwise velocity component show that the excitation leads to a substantial increase in the cross-stream spreading of the shear layer and that distortions of transverse velocity profiles are accompanied by an increase in the high-frequency content of velocity power spectra. Detailed schlieren visualizations shed light on the nature of ‘vortex dislocations’ previously observed by other investigators. Complex spanwise-non-uniform pairing interactions between the spanwise vortices are forced farther downstream by spanwise-amplitude or phase variations of subharmonic excitation wavetrains.

Dynamics and morphology of sustained two-dimensional wavetrains

Self-sustained trains of excitation waves have been produced in an open annular gel reactor in which the Belousov-Zhabotinsky (BZ) reaction was operated. The detailed morphology of the crescent-shaped waves which rotate around the annulus (“excyclons”) depends both on the value of the constraints and on the number of waves in the circular wavetrains. Equal spacing between waves is asymptotically reached and a monotonic dispersion relation is found. However, relaxation to the asymptotic state can be oscillatory. A multistability phenomenon between various morphological states is evidenced as a function of constraints (temperature). The variability of the morphology of excyclons seems fundamentally linked to two-dimensional interactions in the wavetrains.

Two-dimensional modelling of synthetic-aperture-radar imaging of ocean waves using the subwindowing technique

Abstract The subwindowing method of modelling synthetic-aperture-radar (SAR) imaging of ocean waves is extended to simulate the imaging of two-dimensional wave trains. Simulated images show that the azimuthal smearing induced by random wind waves reduces the image response to swells that are primarily imaged by velocity bunching. The magnitude of this effect is not accurately predicted by degrading the SAR resolution using a scene coherence lime. The smearing does not depend on the wind direction when imaging swell, and does not affect the imaging of range travelling swell. Theoretical analysis shows that adjusting the focus of the SAR processor introduces an offset in the mapping of the surface scatterers that compensates the incoherent azimuthal propagation of the wave being imaged, leading to a higher image contrast. The focus adjustment yielding the maximum contrast is approximately one half the azimuthal phase velocity of the wave when it is propagating within 45° of the SAR flight direction. The dep...

Resonant interaction of wave packets in a boundary layer

The space-time evolution of resonance-coupled triads of wave packets in a Blasius boundary layer is studied within the framework of weakly nonlinear stability theory. The amplitude behavior of the packet envelopes is determined in relation to their initial shape, the carrier frequency and the region of propagation. As in the case of triads with a discrete spectrum, interaction leads to parametric pumping of the low-frequency fluctuations and explosive nonlinear growth of the packet maxima. The space-time evolution characteristics are expressed in the deformation of the shape and the spectra of the disturbance. Parts of the envelopes are amplified, depending on the local values of the parameters. This leads to sharp discrimination of the peaks and the equalization of their propagation velocities. These effects make it possible to explain the broadening of the spectrum, the stable distribution of the visualization pattern, and the appearance of irregularities in the oscillograms observed in the S transition. In order to analyze the nonlinear evolution of a disturbance initiated by an instantaneous point source, the interaction of a two-dimensional wave train with variable carrier frequency and pairs of three-dimensional low-frequency packets is examined. (The train frequency corresponds to the local maximum of the linear growth rate with respect to R.) The possibility of the progressive parametric excitation of fluctuations over the entire band of frequency parameters is established. This may explain the acceleration of the transition process in the presence of an impulsive disturbance of the boundary layer.

The response of an incompressible, viscoelastic coating to pressure fluctuations in a turbulent boundary layer

The response of the interface between a compliant surface and a turbulent boundary-layer flow is examined theoretically. This response is forced by transient, convected, interfacial pressure pulses that represent the footprints of turbulent flow structures in the boundary layer. Calculations are presented for coatings with a wide range of damping and densities equal to the density of the flow. For coatings with moderate damping, three regimes of response are found. When the flow speed U∞ is less than about 1.2 (non-dimensionalized by the shear wave speed of the coating), the response is stable and primarily localized under the pressure pulse. For flow speeds from 1.2 to as high as 2.8, depending on the damping, the response is also stable, but it includes a wave pattern behind the pressure pulses. For flow speeds above 2.8, the response is unstable and eventually forms a two-dimensional wavetrain moving in the flow direction. At the highest stable flow velocity, the amplitude of the surface displacements reaches 4.0% of the boundary-layer displacement thickness δ* and the energy transfer from the pressure pulse reaches 5.0 × 10−4 U∞2 (δ*)2. For high damping, the coating response is again stable when the flow speed is below 2.8. However, there is no wavelike response regime; the path that is traversed by the pressure pulse is covered by a scar that heals according to the viscous relaxation properties of the material. The amplitude of the response is at most 0.01δ*.

Experiments on nonlinear instabilities and evolution of steep gravity-wave trains

A series of experiments on strong nonlinear instabilities of gravity-wave trains of large steepness 0·25 ≤ ak ≤ 0·34 in deep water in a long tow tank and a wide basin were performed. These experiments clarified several phenomena such as subharmonic instabilities, wave breaking, evolution of power spectra and directional energy spreading. It was found that an initial two-dimensional wavetrain of large steepness evolved into a series of three-dimensional spilling breakers, and was followed by a transition to a two-dimensional modulated wave train during which a series of oblique wave groups were radiated. The final stage of evolution was a series of modulated, two-dimensional wave groups with lower steepness and frequency. The three most frequent normal modes of two-dimensional subharmonic instabilities underlying the frequency downshifting, in the order of their relative occurrence, are found to be about 4/3, 5/4 and 3/2 in terms of the ratio of the fundamental to perturbed wave frequencies. The Bimescale for the phenomenon of the frequency downshifting is from 40 to 60 fundamental wave periods. Comparisons of the experimental results with existing theories showed that observed processes have several qualitative similarities with theoretical computations.

Three-dimensional deep-water waves. Part 1. Experimental measurement of skew and symmetric wave patterns

The three-dimensional structures of symmetric and skew wave patterns in deep water observed in a tow tank and a wide basin are described. These symmetric waves are the result of three-dimensional subharmonic bifurcation of two-dimensional wave-trains with steepness a0k0 ≥ 0·25, where a0, and k0, are the wave amplitude and wave-number respectively. The wave profiles, local surface slopes and amplitude spectra at various cross-sections of the crescent-shaped symmetric waves are presented and discussed.The skew waves are another type of three-dimensional bifurcation from a uniform wavetrain, and occur most clearly when 0·16 [siml ] a0k0 [siml ] 0·18. These skew wave patterns are found to interact between two sets of themselves propagating from different directions, and to be subject to the Benjamin–Feir type modulations. The interactions cause compact three-dimensional wave packets. Agreement between experiment and theory is found for the case of symmetric wave patterns. The bifurcation of uniform Stokes waves into symmetric wave patterns provide a new physical process for directional energy spreading.

A new type of three-dimensional deep-water wave of permanent form

A new class of three-dimensional, deep-water gravity waves of permanent form has been found using an equation valid for weakly nonlinear waves due to Zakharov (1968). These solutions appear as bifurcations from the uniform two-dimensional wave train. The critical wave heights are given as functions of the modulation wave vector. The three-dimensional patterns may be skewed or symmetrical. An example of the skewed wave pattern is given and shown to be stable. The results become exact in the limit of very oblique modulations.