Nonlinear Hydroelastic Waves in Deep Water: Propagation and Profile Inversion Captured by Strong CFD-FEA Coupling

Particle trajectories of nonlinear gravity waves in deep water

This paper describes the application of a recently developed analytic approach known as the homotopy analysis method to derive a solution for the classical problem of nonlinear progressive waves in deep water. The method is based on a continuous variation from an initial trial to the exact solution. A Maclaurin series expansion provides a successive approximation of the solution through repeated application of a differential operator with the initial trial as the first term. This approach does not require the use of perturbation parameters and the solution series converges rapidly with the number of terms. In the framework of this approach, a new technique to apply the Pade expansion is implemented to further improve the convergence. As a result, the calculated phase speed at the 20th-order approximation of the solution agrees well with previous perturbation solutions of much higher orders and reproduces the well-known characteristics of being a non-monotonic function of wave steepness near the limiting condition.

The periodic wave solution of a second order nonlinear ordinary differential equation is obtained by the homotopy analysis method, an analytical, totally explicit mathematical technique. By choosing a proper auxiliary parameter, the new series solution converges very fast. The method provides us with a simple way to adjust the convergence region. Furthermore, a significant improvement of the convergence rate and region is achieved by applying Homotopy-Pade approximants. Three examples demonstrate the excellent computation accuracy and efficiency of the present HAM approach. The present method could be extended for more complicated wave equations. References Abbasbandy, S., Homotopy analysis method for generalized Benjamin--Bona--Mahony equation, Z. angew. Math. Phys., 59, 2008, 51--62. doi:10.1007/s00033-007-6115-x Liao, S. J., An approximate solution technique not depending on small parameters: a special example, Int. J. Nonlinear Mech., 30, 1995, 371--380. doi:10.1016/0020-7462(94)00054-E Liao, S. J., Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC, Florida, 2004. Liao, S. J. and Cheung, K. F., Homotopy analysis of nonlinear progressive waves in deep water, J. Eng. Math., 45, 2003, 105--116. doi:10.1023/A:1022189509293 Tao, L., Song, H. and Chakrabarti, S., Nonlinear progressive waves in water of finite depth--óan analytic approximation, Coast. Eng., 54, 2007, 825--834. doi:10.1016/j.coastaleng.2007.05.008 Wang, C., Wu, Y. and Wu, W., Solving the nonlinear periodic wave problems with the Homotopy Analysis Method, Wave Motion, 41, 2005, 329--337. doi:10.1016/j.wavemoti.2004.08.002

The influence of an opposing current on highly non-linear transient breaking waves in deep water is described quantitatively from experiments. 3 new parameters that describe crest front steepness and wave asymmetry are introduced. Further, joint probability density distributions are obtained from an analysis of field data containing nearly 25000 storm waves. Thus, a tool is provided from which estimates for probabilities for occurrences of steep breaking waves in deep water, may be obtained.

This paper examines the signal coherence loss due to internal waves in deep water in terms of the signal coherence time and compare to data reported in the literature over the past 35 years. The coherence time of the early raylike arrivals was previously modeled by Munk and Zachariasen ["Sound propagation through a fluctuating stratified ocean: Theory and observation," J. Acoust. Soc. Am. 59, 818-838 (1976)] using the supereikonal approximation and by Dashen et al. ["Path-integral treatment of acoustic mutual coherence functions for arrays in a sound channel," J. Acoust. Soc. Am. 77, 1716-1722 (1985)] using the path integral approach; a -1 [corrected] power frequency dependence and a -1/2 [corrected] power range dependence were predicted. Recent data in shallow water in downward refractive environments with internal waves suggested that the signal coherence time of the mode arrivals follows a -3/2 power frequency dependence and a -1/2 power range dependence. Since the temporal coherence of the acoustic signal is related to the temporal coherence of the internal waves, based on the observation that the (linear) internal waves in deep and shallow waters have a similar frequency spectrum, it is argued that the modelike arrivals in deep water should exhibit a similar frequency dependence in deep and shallow waters. This argument is supported by a brute-force application of the path integral to mode arrivals based on the WKB relation between the ray and mode. It is found that the data are consistent with the -3/2 power frequency dependence but more data are needed to further test the hypothesis.

The occurrence of rogue waves in deep sea waters, and their breaking, is examined with the aid of a pdf model of joint amplitudes and frequencies. New wave breaking considerations allow kinematic, dynamic and maximum average slope concepts to be unified in a single breaking criterion, which allows a more accurate determination of the limiting am- plitudes that rogue waves can reach, including the influence of non-linearity of the wave field and the premature wave breaking concept. The probability of rogue wave occurrence does not significantly depend on the sea spectrum bandwidth. The breaking probability of rogue waves increases with the inverse wave age, but its dependence on the latter parameter weakens, as the limiting crest height criterion is stiffened. The right variation of sea surface kurtosis, and of the Benjamin Feir Index, with the (inverse) wave age and their lower than unity values reconfirms their relation to the rogue wave oc- currence.

A numerical method is developed to study the stability of standing water waves and other time-periodic solutions of the free-surface Euler equations using Floquet theory. A Fourier truncation of the monodromy operator is computed by solving the linearized Euler equations about the standing wave with initial conditions ranging over all Fourier modes up to a given wave number. The eigenvalues of the truncated monodromy operator are computed and ordered by the mean wave number of the corresponding eigenfunctions, which we introduce as a method of retaining only accurately computed Floquet multipliers. The mean wave number matches up with analytical results for the zero-amplitude standing wave and is helpful in identifying which Floquet multipliers collide and leave the unit circle to form unstable eigenmodes or rejoin the unit circle to regain stability. For standing waves in deep water, most waves with crest acceleration below A c = 0.889 A_c=0.889 are found to be linearly stable to harmonic perturbations; however, we find several bubbles of instability at lower values of A c A_c that have not been reported previously in the literature. We also study the stability of several new or recently discovered time-periodic gravity-capillary or gravity waves in deep or shallow water, finding several examples of large-amplitude waves that are stable to harmonic perturbations and others that are not. A new method of matching the Floquet multipliers of two nearby standing waves by solving a linear assignment problem is also proposed to track individual eigenvalues via homotopy from the zero-amplitude state to large-amplitude standing waves.

Laboratory experimental results are presented for nonlinear internal solitary waves (ISW) propagation in ‘deep water’ configuration with miscible fluids. The results are validated against direct numerical simulations and traveling wave exact solutions where the effect of the diffused interface is taken into account. The waves are generated by means of a dam break and their evolution is recorded with laser-induced fluorescence and particle image velocimetry. In particular, data collected in a frame moving with the waves are presented here for the first time. Our results are representative of geophysical applications in the deep ocean where weakly nonlinear theories fail to capture the characteristics of large amplitude ISWs from field observations.

Fourth order nonlinear evolution equations are derived for two counter-propagating surface gravity wave packets in deep water in the presence of wind flowing over water. The resulting equations are asymptotically exact and nonlocal. Stability analysis is made for a uniform standing surface gravity wave train for longitudinal perturbation on the basis of these equations. Graphs are plotted for maximum growth rate of instability and for wave number at marginal stability against wave steepness for some different values of dimensionless wind velocity. Significant deviations are noticed between the results obtained from third order and fourth order nonlinear evolution equations. This paper has an application in rough waves. References T. Brooke Benjamin and J. E. Feir. The disintegration of wave trains on deep water part 1. theory. Journal of Fluid Mechanics , 27:417–430, 1967. doi:10.1017/S002211206700045X Sudebi Bhattacharyya and K. P. Das. Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline. The ANZIAM Journal , 39:214–229, 1997. doi:10.1017/S033427000000881X K. P. Das. On evolution equations for a three-dimensional surface gravity wave packet in a two-layer fluid. Wave Motion , 8(2):191–204, 1986. doi:10.1016/0165-2125(86)90024-7 A. Davey and K. Stewartson. On three-dimensional packets of surface waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 338(1613):101–110, 1974. doi:10.1098/rspa.1974.0076 Suma Debsarma and K. P. Das. Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water. The ANZIAM Journal , 43:513–524, 2002. doi:10.1017/S1446181100012116 A. K. Dhar and K. P. Das. A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water. Physics of Fluids A: Fluid Dynamics , 2(5), 1990. A. K. Dhar and K. P. Das. Fourth-order nonlinear evolution equation for two Stokes wave trains in deep water. Physics of Fluids A: Fluid Dynamics , 3(12), 1991. A. K. Dhar and K. P. Das. Stability analysis from fourth order evolution equation for small but finite amplitude interfacial waves in the presence of a basic current shear. The ANZIAM Journal , 35:348–365, 1994. doi:10.1017/S0334270000009346 V. D. Djordjevic and L. G. Redekopp. On two-dimensional packets of capillary-gravity waves. Journal of Fluid Mechanics , 79:703–714, 1977. doi:10.1017/S0022112077000408 J. C. Dungey and W. H. Hui. Nonlinear energy transfer in a narrow gravity-wave spectrum. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 368(1733):239–265, 1979. doi:10.1098/rspa.1979.0126 K. B. Dysthe. Note on a modification to the nonlinear schrodinger equation for application to deep water waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 369(1736):105–114, 1979. doi:10.1098/rspa.1979.0154 Tetsu Hara and Chiang C. Mei. Frequency downshift in narrow banded surface waves under the influence of wind. Journal of Fluid Mechanics , 230:429–477, 1991. doi:10.1017/S002211209100085X Tetsu Hara and Chiang C. Mei. Wind effects on the nonlinear evolution of slowly varying gravity-capillary waves. Journal of Fluid Mechanics , 267:221–250, 1994. doi:10.1017/S0022112094001163 S. J. Hogan. The fourth-order evolution equation for deep-water gravity-capillary waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 402(1823):359–372, 1985. doi:10.1098/rspa.1985.0122 Peter A. E. M. Janssen. On a fourth-order envelope equation for deep-water waves. Journal of Fluid Mechanics , 126:1–11, 1983. doi:10.1017/S0022112083000014 M. S. Longuet-Higgins. The instabilities of gravity waves of finite amplitude in deep water. I. superharmonics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 360(1703):471–488, 1978. doi:10.1098/rspa.1978.0080 M. S. Longuet-Higgins. The instabilities of gravity waves of finite amplitude in deep water II. subharmonics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 360(1703):489–505, 1978. doi:10.1098/rspa.1978.0081 R. D. Pierce and E. Knobloch. On the modulational stability of traveling and standing water waves. Physics of Fluids , 6(3), 1994. Michael Stiassnie. Note on the modified nonlinear Schrodinger equation for deep water waves. Wave Motion , 6(4):431–433, 1984. doi:10.1016/0165-2125(84)90043-X M. A. Weissman. Nonlinear wave packets in the Kelvin–Helmholtz instability. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 290(1377):639–681, 1979. doi:10.1098/rsta.1979.0019 V. E. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics , 9(2):190–194, 1968. doi:10.1007/BF00913182

This paper presents a new spectral model for solving the fully nonlinear potential flow problem for water waves in a single horizontal dimension. At the heart of the numerical method is the solution to the Laplace equation which is solved using a variant of the ‐transform. The method discretizes the spatial part of the governing equations using the Galerkin method and the temporal part using the classical fourth‐order Runge‐Kutta method. A careful investigation of the numerical method's stability properties is carried out, and it is shown that the method is stable up to a certain threshold steepness when applied to nonlinear monochromatic waves in deep water. Above this threshold artificial damping may be employed to obtain stable solutions. The accuracy of the model is tested for: (i) highly nonlinear progressive wave trains, (ii) solitary wave reflection, and (iii) deep water wave focusing events. In all cases it is demonstrated that the model is capable of obtaining excellent results, essentially up to very near breaking.

A numerical simulation is performed to study the velocity, streamlines, vorticity and shear stress distributions in viscous water waves with different wave steepness in intermediate and deep water depth when the average wind velocity is zero. The numerical results present evidence of ‘clockwise’ and ‘anticlockwise’ rotation of the fluid at the trough and crest of the water waves. These results show thicker vorticity layers near the surface of water wave than that predicted by the theories of inviscid rotational flow and the low Reynolds number viscous flow. Moreover, the magnitude of vorticity near the free surface is much larger than that predicted by these theories. The analysis of the shear stress under water waves show a thick shear layer near the water surface where large shear stress exists. Negative and positive shear stresses are observed near the surface below the crest and trough of the waves, while the maximum positive shear stress is inside the water and below the crest of the water wave. Comparison of wave energy decay rate in intermediate depth and deep water waves with laboratory and theoretical results are also presented.

Nonlinear diffraction of low-amplitude gravity waves in deep water due to a slightly submerged obstacle is studied experimentally in a wave channel and theoretically. The obstacle is either a circular cylinder or a rectangular shelf. The incoming waves (with wavelength λ) undergo strong nonlinear deformations at the obstacle when the wave amplitude is finite. An infinite number of superharmonic waves are then introduced to the flow. Their wavelengths far away from the obstacle are λ/4, λ/9, λ/16,…, due to the dispersion relation being quadratic in the wave frequency. The superharmonic wave amplitudes grow with increasing incoming wave amplitude up to saturation values. They are found to be prominent at the obstacle's lee side and vanishingly small at the weather side. The second- and third-harmonic wave amplitudes are, surprisingly, in several examples found to be comparable to the incoming wave amplitude. Up to 25% of the incoming energy flux may be transferred to the shorter waves. The theoretical model accounts for nonlinearity by the Boussinesq equations in the shallow region above the obstacle, with patching to linearized potential theory in the deep water. The theory explains both qualitatively and quantitatively the trends observed in the experiments up to breaking.

<p>Long-living coherent wave patterns embedded into the irregular wave fields are studied using the data of extensive numerical simulations of the Euler equations in deep water. The distributions of the rogue wave lifetimes according to the numerical simulations of JONSWAP waves with narrow and broad angle spectra are discussed. The observation of a wave group persisting for more than 200 periods in the direct numerical simulation of nonlinear unidirectional irregular water waves is discussed. Through solution of the associated scattering problem for the nonlinear Schrodinger equation, the persisting group is identified as the intense envelope soliton with remarkably stable parameters. Most of extreme waves occur on top of this group, resulting in higher and longer rogue wave events. It is shown that the persisting wave structure survives under the conditions of directional waves with moderate spread of directions. The survivability of coherent wave patterns is expected to further increase when the waves are guided by currents or the topography.</p><p> </p><p>The research is supported by the RSF grant No. 19-12-00253; the study of trapped waves is performed for the RFBR grant No. 21-55-15008.</p>

This paper aims to investigate the accuracy and computational efficiency of three CFD-based numerical codes to accurately simulate extremely large regular and irregular waves of different steepness in deep and shallow water conditions. The work assesses the performance of three numerical techniques with different formulations of the fluid dynamic equations. Firstly, an open-sourced smoothed particle hydrodynamics (SPH) code; secondly, a finite difference method (FDM) based 3D numerical model with the assumption of inviscid and incompressible fluid flow; and thirdly, a commercial CFD code that uses a finite volume method (FVM) to solve the Reynolds-averaged Navier-Stokes (RANS) equations. A suite of metrics and methodologies, considering three key performance parameters: accuracy, computational requirements and available features for providing a consistent framework for the quantitative assessment of different techniques, has been presented. Numerically simulated free surface elevations, wave periods, and spectrum (for irregular waves only) are compared with experimental data previously acquired at an Offshore Engineering Basin (OEB) facility. Extensive convergence studies were carried out for each numerical tool for a selected large wave before predictions were model for all waves. All three models reproduced waves with an accuracy comparable to physical wave makers in the wave basin experiments for the deep-water regular and irregular waves; however, the SPH model performed better than the other two models for the shallow water waves. The challenge remains for wave basins to reduce unwanted basin effects and numerical facilities to accurately model waves with proper account for boundary effects and numerical diffusions. In addition, only flat-bottom domains were considered in the investigation, leaving the wave modelling for uneven bottom for future studies.

Integrating short- and long-term statistics for short-crested waves in deep and intermediate waters