The interaction of nonlinear progressive waves and a uniform current in water of finite depth is investigated analytically by means of the homotopy analysis method (HAM). With HAM, the velocity potential of the flow and the surface elevation are expressed by the Fourier series, and the nonlinear free surface boundary conditions are satisfied by continuous mapping. Unlike a perturbation method, the present approach does not depend on any small parameters; thus, the solutions are suitable for steep waves and strong currents. To verify the HAM solutions, experiments are conducted in the wave–current flume of the Education Ministry Key Laboratory of Hydrodynamics at Shanghai Jiao Tong University (SJTU) in Shanghai, China. It is found that the HAM solutions are in good agreement with experimental measurements. Based on the series solutions of the validated analytical model, the influence of water depth, wave steepness, and current velocity on the physical properties of the coexisting wave–current field are studied in detail. The variation mechanisms of wave characteristics due to wave–current interaction are further discussed in a quantitative manner. The significant advantage of HAM in dealing with strong nonlinear wave–current interactions in the present study is clearly demonstrated in that the solution technique is independent of small parameters. A comparative study on wave characteristics further reveals the great potential of HAM to solve more complex wave–current interaction problems, leading to engineering applications in the offshore industry and the marine renewable energy sector.
The interaction of nonlinear progressive waves and uniform currents in water of finite depth is investigated analytically by means of the homotopy analysis method (HAM). In the HAM, the velocity potential of the wave is expressed by Fourier series and the nonlinear free surface boundary conditions are satisfied by continuous mapping. The present approach does not depend on any small parameters, thus the solutions are suitable for steep waves and strong currents. To verify the HAM solutions, experiments are conducted in wave-current flume of Education Ministry Key Laboratory of Hydrodynamics at SJTU. It is shown that the HAM solutions are in good agreement with experimental measurements. The present study demonstrated that the great potential of the HAM to solve more complex wave-current interaction problems leading to engineering applications in traditional offshore industry and marine renewable energy sector.
Abstract Recent studies of water waves propagating over sloping seabeds have shown that sudden transitions from deeper to shallower depths can produce significant increases in the skewness and kurtosis of the free surface elevation and hence in the probability of rogue wave occurrence. Gramstad et al. (Phys. Fluids 25 (12): 122103, 2013) have shown that the key physics underlying these increases can be captured by a weakly dispersive and weakly nonlinear Boussinesq-type model. In the present paper, a numerical model based on an alternative Boussinesq-type formulation is used to repeat these earlier simulations. Although qualitative agreement is achieved, the present model is found to be unable to reproduce accurately the findings of the earlier study. Model parameter tests are then used to demonstrate that the present Boussinesq-type formulation is not well-suited to modelling the propagation of waves over sudden depth transitions. The present study nonetheless provides useful insight into the complexity encountered when modelling this type of problem and outlines a number of promising avenues for further research.
Recent studies of surface gravity waves propagating over a sloping bottom have shown that an increase in the probability of extreme waves can be triggered by depth variations in sufficiently shallow waters. A boundary element method is used to show that this increase in probability is greatest when the slope is steepest, i.e., for a step. A harmonic separation technique shows that the second-order terms in wave steepness are responsible for the change in the statistical properties near the depth transition.
A fully nonlinear solution for bi-chromatic progressive waves in water of finite depth in the framework of the homotopy analysis method (HAM) is derived. The bi-chromatic wave field is assumed to be obtained by the nonlinear interaction of two monochromatic wave trains that propagate independently in the same direction before encountering. The equations for the mass, momentum, and energy fluxes based on the accurate high-order homotopy series solutions are obtained using a discrete integration and a Fourier series-based fitting. The conservation equations for the mean rates of the mass, momentum, and energy fluxes before and after the interaction of the two nonlinear monochromatic wave trains are proposed to establish the relationship between the steady-state bi-chromatic wave field and the two nonlinear monochromatic wave trains. The parametric analysis on ε1 and ε2, representing the nonlinearity of the bi-chromatic wave field, is performed to obtain a sufficiently small standard deviation Sd, which is applied to describe the deviation from the conservation state (Sd = 0) in terms of the mean rates of the mass, momentum, and energy fluxes before and after the interaction. It is demonstrated that very small standard deviation from the conservation state can be achieved. After the interaction, the amplitude of the primary wave with a lower circular frequency is found to decrease; while the one with a higher circular frequency is found to increase. Moreover, the highest horizontal velocity of the water particles underneath the largest wave crest, which is obtained by the nonlinear interaction between the two monochromatic waves, is found to be significantly higher than the linear superposition value of the corresponding velocity of the two monochromatic waves. The present study is helpful to enrich and deepen the understanding with insight to steady-state wave-wave interactions.
Abstract Many ocean engineering problems involve bound harmonics which are slaved to some underlying assumed close to linear time series. When analyzing signals we often want to remove the bound harmonics so as to “linearise” the data or to extract individual bound harmonic components so that they may be studied. For even moderately broadbanded systems filtering in the frequency domain is not sufficient to separate components as they overlap in frequency. One way to overcome this difficulty is to use input signals with the same linear envelope but with different phases and then use simple addition and subtraction of the resulting signals to extract different harmonics. This approach has been established for the analysis of wave groups. In this paper we examine whether this approach can be used on random time series as well. We analyse random wave time series of wave elevation from the towing tank in Shanghai Jiao Tong University and force measurements on a cylinder taken in the Kelvin tank at the University of Strathclyde.
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