Nonlinear Wave Model Research Articles

Dynamics of surface and internal long waves generated by atmospheric pressure disturbances

Different from non-dispersive wave models such as the shallow water equations, MCC (Miyata, Choi, and Camassa) type strongly nonlinear dispersive wave model can describe not only the dynamics of surface waves, but also those of internal waves. The MCC model for a two-layer fluid system is applied to study dynamics of both surface waves and internal waves. The resonance mechanisms with atmospheric pressure disturbances are numerically studied not only for surface waves, so-called Proudman resonance, but also for internal waves. Baroclinic internal waves are generated when the speed of the atmospheric pressure disturbance is comparable to the critical linear internal wave speed. We compare the amplitudes of generated internal waves to the resonance theory for surface waves. Although most baroclinic internal wave phenomena can be explained by this linear theory, the relative magnitudes of amplitudes for forced and free waves do not agree with the theory. In addition, sequentially generated surface waves with resonance are observed when the radius of the atmospheric pressure disturbance is small. Finally, the dynamics of both surface and internal waves over non-uniform bottom topography are discussed in terms of Froude numbers to understand how they propagate and interact with each other when the Proudman resonance condition is satisfied while they travel.

Mathematical Modeling of Nonlinear Waves in an Elastic Cylindrical Shell Surrounded by an Elastic Medium and Containing a Viscous Incompressible Liquid

A mathematical model is proposed for a Kirchhoff–Love-type nonlinear elastic cylindrical shell surrounded by an elastic medium and containing a viscous incompressible liquid. The model is used to analyze wave processes both analytically and numerically. On the basis of the proposed computational algorithm, a software package is developed, which makes it possible to plot diagrams and to obtain numerical solutions to Cauchy problems with initial conditions taken in the form of exact solutions to dynamic equations of shells in the absence of the liquid.

A Comparison of Numerical Models with Experimental Data for Stem Waves along a Vertical Wall due to Monochromatic Waves

ABSTRACT Shin, C.H.; Yoon, S.B., and Oh, S.C., 2018. A comparison of numerical models with experimental data for stem waves along a vertical wall due to monochromatic waves. In: Shim, J.-S.; Chun, I., and Lim, H.S. (eds.), Proceedings from the International Coastal Symposium (ICS) 2018 (Busan, Republic of Korea). Journal of Coastal Research, Special Issue No. 85, pp. 1121–1125. Coconut Creek (Florida), ISSN 0749-0208. Recently, vertical-type structures are frequently constructed in coastal area to protect ports and operation areas. When the waves propagate at oblique angles to the vertical-type structures, stem waves are generated along a structure due to the nonlinear interaction of incident and reflected waves. Thus, for the design of vertical structures the stem waves should be accurately evaluated using numerical models. In this study the performance of three nonlinear wave models based on different types of governing equations, REF/DIF 1, COULWAVE, and SWASH, was compared by using the measured data f...

Nonlinear Waves and Dispersive Equations

Nonlinear dispersive equations are models for nonlinear waves in a wide range of physical contexts. Mathematically they display an interplay between linear dispersion and nonlinear interactions, which can result in a wide range of outcomes from finite time blow-up to solitons and scattering. They are linked to many areas of mathematics and physics, ranging from integrable systems and harmonic analysis to fluid dynamics, geometry, general relativity and probability.

Asymptotic Calculation of the Wave Trough Exceedance Probabilities in A Nonlinear Sea

This paper concerns the calculation of the wave trough exceedance probabilities in a nonlinear sea. The calculations have been carried out by incorporating a second order nonlinear wave model into an asymptotic method. This is a new approach for the calculation of the wave trough exceedance probabilities, and, as all of the calculations are performed in the probability domain, avoids the need for long time-domain simulations. The proposed asymptotic method has been applied to calculate the wave trough depth exceedance probabilities of a sea state with the surface elevation data measured at the coast of Yura in the Japan Sea. It is demonstrated that the proposed new method can offer better predictions than the theoretical Rayleigh wave trough depth distribution model. The calculated results by using the proposed new method have been further compared with those obtained by using the Arhan and Plaisted nonlinear distribution model and the Toffoli et al.’s wave trough depth distribution model, and its accuracy has been once again substantiated. The research findings obtained from this study demonstrate that the proposed asymptotic method can be readily utilized in the process of designing various kinds of ocean engineering structures.

Impact of turbulence induced loads and wave kinematic models on fatigue reliability estimates of offshore wind turbine monopiles

The cost of offshore wind turbine substructures has a significant impact on competitiveness of the wind energy market and is affected by conservative safety margins adopted in the design phase. This implies that an accurate design load prediction, especially of those resulting in fatigue damage accumulation, may help achieve more cost-effective solutions. In this article, the impact of turbulence and wave loads on fatigue reliability of pile foundations is investigated for a 5-MW offshore wind turbine. Loads obtained by varying turbulence percentiles are compared with those obtained from the full joint probability distribution of wind speed and turbulence through Monte Carlo (MC) simulations, and from the equivalent turbulence level currently adopted by IEC standards. The analyses demonstrate that a lower equivalent turbulence percentile leads to a more realistic and less conservative estimation of fatigue loads. Subsequently, the research focuses on studying the effects of uncertain marine environments on the fatigue load distribution, showing that the latter is insensitive to the random variability of the hydrodynamic coefficients. With respect to the wave kinematic model, a comparison between nonlinear and linear waves clearly suggests that hydrodynamic forces depend significantly on the kinematic model adopted and the operational conditions of the turbine. Furthermore, a term is derived to correct the error introduced by Wheeler stretching at finite water depths. The respective model uncertainties that originate from the nonlinear irregular wave model and Wheeler correction are quantified and employed in a reliability analysis. In a case study, the results are finally compared in terms of estimated probability of failure, with the aim to quantify the influence of environmental models on monopile reliability.

Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face

This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.

Modeling two-dimensional infragravity motions on a fringing reef

This study investigates infragravity motions on a two-dimensional fringing reef with a concave topography using a wave-resolving nonlinear Boussinesq wave model. The effect of bathymetric nonlinearity on spatial variations of infragravity waves is studied from shore-normal incident random waves. The model is first verified from cross-shore/alongshore significant wave height, wave setup and infragravity wave height measured in laboratory experiments. The modeled velocity is further examined to elucidate the infragravity-wave current field. Wavenumber-frequency velocity spectra show that cross-shore infragravity motions are controlled by the leaky waves, while alongshore infragravity motions exhibit edge waves characteristics. Additionally, high-mode edge waves prevail on the entire infragravity frequency band while low-mode edge waves are dominant at very low frequencies. Furthermore, the model results illustrate the spatial and temporal variability of wave-induced nearshore currents, alongshore and cross-shore flows accompanying with circulation cells and rip currents. The infragavity-wave current field is nearly identical to the mean current field and the circulations develop at the same spatial scale as the reef flat.

A conservative numerical scheme for modeling nonlinear acoustic propagation in thermoviscous homogeneous media

A nonlinear system of partial differential equations capable of describing the nonlinear propagation and attenuation of finite amplitude perturbations in thermoviscous media is presented. This system constitutes a full nonlinear wave model that has been formulated in the conservation form. Initially, this model is investigated analytically in the inviscid limit where it has been found that the resulting flux function fulfills the Lax–Wendroff theorem, and the scheme can match the solutions of the Westervelt and Burgers equations numerically. Here, high-order numerical descriptions of strongly nonlinear wave propagations become of great interest. For that matter we consider finite difference formulations of the weighted essentially non-oscillatory (WENO) schemes associated with explicit strong stability preserving Runge–Kutta (SSP-RK) time integration methods. Although this strategy is known to be computationally demanding, it is found to be effective when implemented to be solved in graphical processing units (GPUs). As we consider wave propagations in unbounded domains, perfectly matching layers (PML) have been also considered in this work. The proposed system model is validated and illustrated by using one- and two-dimensional benchmark test cases proposed in the literature for nonlinear acoustic propagation in homogeneous thermoviscous media.

A parametric study of tsunami fission generation in a bay geometry based on numerical simulation

ABSTRACTTsunami fission generates short waves at the front face of tsunami and these short waves may expand the inundation area, increase the hydrodynamic forces exerted on structures, and amplify the damage to coastal communities. It is therefore important to understand to what extent the tsunami fission generates such short wave components under variable conditions for development of better tsunami disaster mitigation strategies and plannings. This study carried out numerical experiments to describe under which conditions the tsunami fission occurs in a bay. The experiments indicated that the amplification of the nearshore tsunami height induced by the narrowing bay geometry and the bay-scale resonance enhance the tsunami fission. Based on the computations, a parametric diagram for the occurrence of tsunami fission was proposed in terms of the tsunami properties, the effect of bay geometry and the bay resonance. The developed diagram was applied to Sendai, Ryori, Yoshihama, and Toni Bays, which were affected by the 2011 Tohoku Tsunami. The proposed diagram consistently explained the occurrence of tsunami fission in Sendai Bay and in Ryori Bay as observed in the video footage. The diagram also suggested that the tsunami fission would have occurred in Yoshihama Bay. A high-resolution numerical computation was performed to simulate the tsunami deformation in the Yoshihama Bay and Toni Bay based on a nonlinear dispersive wave model. The result supported the validity of the diagram in describing the generation of tsunami fission in Yoshihama Bay.

Simple Model of Nonlinear Spin Waves in Graphene Structures

A series of theoretical and experimental works is known which investigated the magnetic properties of graphene structures. This is due, among other things, to the prospects of using graphene as a material for the needs of the future nanoelectronics and spintronics. In particular, it is known about the presence of ferromagnetic properties at temperatures up to 200 C and above in a single-layer graphene films that are free from impurities. Previously there was proposed a quantum field theoretical model describing the possible mechanism of ferromagnetism in graphene as a result of spontaneous breaking of spin symmetry of the surface density of valence electrons. The possible spatial configurations of the localized spin density were described. In this paper we investigate such spatially localized nonlinear spin configurations of the valence electron density on the graphene surface such as kinks, and their interactions, as well as quasibound metastable states of the interacting kinks and antikinks, that are breathers. The spectrum of such breathers is investigated. It is shown that under certain conditions, this spectrum has a discrete sector, which, in turn, allows us to speak about the possibility of coherent quantum generation of spin waves in graphene structures, which is important in terms of practical applications in nanoelectronics and spintronics.

Dispersive Shallow Water Wave Modelling. Part I: Model Derivation on a Globally Flat Space

In this paper, we review the history and current state-of-the-art in the modelling of long nonlinear dispersive waves. For the sake of conciseness of this review, we omit the unidirectional models and focus especially on some classical and improved Boussinesq-type and Serre-Green-Naghdi equations. Finally, we propose also a unified modelling framework which incorporates several well-known and some less known dispersive wave models. The present manuscript is the first part of a series of two papers. The second part will be devoted to the numerical discretization of a practically important model on moving adaptive grids.

Нелинейные динамические волновые модели газожидкостных потоков в технических системах

Нелинейные динамические волновые модели газожидкостных потоков в технических системах

An investigation of infrasound propagation over mountain ranges.

Linear theory is used to analyze trapping of infrasound within the lower tropospheric waveguide during propagation above a mountain range. Atmospheric flow produced by the mountains is predicted by a nonlinear mountain gravity wave model. For the infrasound component, this paper solves the wave equation under the effective sound speed approximation using both a finite difference method and a Wentzel-Kramers-Brillouin approach. It is shown that in realistic configurations, the mountain waves can deeply perturb the low-level waveguide, which leads to significant acoustic dispersion. To interpret these results, each acoustic mode is tracked separately as the horizontal distance increases. It is shown that during statically stable situations, situations that are common during night over land in winter, the mountain waves induce a strong Foehn effect downstream, which shrinks the waveguide significantly. This yields a new form of infrasound absorption that can largely outweigh the direct effect the mountain induces on the low-level waveguide. For the opposite case, when the low-level flow is less statically stable (situations that are more common during day in summer), mountain wave dynamics do not produce dramatic responses downstream. It may even favor the passage of infrasound and mitigate the direct effect of the obstacle.

Mean Displacement Near an Interface in a Nonlinear String

Nonlinear waves on a two-part stretched string are considered as a reduced model of nonlinear waves in sea ice near a sudden change in ice properties. Geometric nonlinearity is retained, and the waves are confined to a wave packet. The weakly nonlinear approach results in three coupled nonlinear amplitude equations, one for incident, reflected, and transmitted waves. The waves create a local mean displacement at the interface of the two sides, which provides the primary nonlinear effect. This interfacial mean requires a time-history approach, treated here with Laplace transforms. The results show that the mean is positive when the stiffness decreases across the interface and negative otherwise. The magnitude of the mean displacement near the interface is much larger when the stiffness increases, as it would where ice increases in thickness.

Numerical simulations and experimental visualizations of the vortex characteristics for a solitary wave interacting with a bottom-mounted vertical plate

This study is aimed to investigate numerically and experimentally the interaction of a solitary wave with a bottom-mounted vertical thin plate with focuses on the wave induced fluid kinematics and vortex flow patterns. A streamfunction–vorticity free-surface (SVFS) based fully nonlinear viscous wave model solved in a transient boundary-fitted coordinate system with locally overlaid grids is applied to study this wave-plate interaction problem. Qualitative comparisons between numerically generated flow patterns around the plate and the experimental observations, including images from both the particle-tracking and the laser-dye visualization methods, are presented. The shear-layer velocity profiles are compared quantitatively with the experimental measurements. The detailed flow characteristics and formed vortices including the effect of the shape of the plate top on the vortex formation and development are numerically investigated. For a thin vertical plate, the flow characteristics that are affected by the dimensionless parameters of incident-wave height and plate height are also investigated. The numerical visualizations illustrated by the virtual transport of particle-tracing, streamlines, and equi-vorticity lines are useful to understand the kinematic behaviors of the induced vortical motions. Furthermore, the pressure gradient, shear stress, and maximum wave force are examined to detail the hydrodynamic impacts on the submerged plate.

Stability of electron wave spectra in weakly magnetized plasmas

Analytical models for nonlinear electron plasma waves in weakly magnetized plasmas are developed for single as well as multi-mode conditions, with continuous wave spectra being a limiting case. The conditions for wave decay as well as modulational instabilities are analysed. Our results demonstrate that slow or nearly stationary plasma density variations can be found for weakly magnetized plasmas even for weakly nonlinear electron plasma waves without involving cavitation of large amplitude plasma waves. A reduction of the growth rates for decay as well as modulational instabilities are found when the spectral width of the wave spectrum is increased. Some of our results are relevant for the interpretation of the nonlinearly enhanced ion acoustic lines often observed in non-equilibrium ionospheres.

Fundamental solitons and dynamical analysis in the defocusing Kerr medium and $$\varvec{\mathcal {PT}}$$ PT -symmetric rational potential

We find that a class of parity-time- ( $$\mathcal {PT}$$ -) symmetric rational potentials can support stable solitons in the defocusing Kerr-nonlinear media, though they may not enjoy entirely real linear spectra. Analytical expressions of spatial solitons are elicited at lots of isolated propagation-constant points, around which several families of numerical fundamental solitons can be found to be stable, which is validated by linear stability analysis and nonlinear wave propagation. Many other intriguing properties of nonlinear localized modes are also discussed in detail, including the interactions, excitations, and transverse power flows. The idea of the $$\mathcal {PT}$$ -symmetric rational potentials can also be extended to other types of nonlinear wave models.

Nonlinear and viscous effects on the hydrodynamic performance of a fixed OWC wave energy converter

The hydrodynamic performance of a fixed Oscillating Water Column (OWC) device is experimentally and numerically investigated. Based on the time-domain higher-order boundary element method (HOBEM), by introducing an artificial viscosity term in the dynamic free surface boundary condition, a fully nonlinear numerical wave model is used to simulate the hydrodynamic performance of an OWC device. A set of comprehensive experiments for regular waves is carried out to validate the numerical results as well as to investigate the nonlinear effects on the hydrodynamic performance of OWC. The mechanism of the nonlinear phenomenon is investigated based on the analysis of the experimental and numerical results. The influence of the wave nonlinearity and the viscosity on the hydrodynamic efficiency is quantified by comparing the linear and nonlinear numerical results. It was found that the hydrodynamic efficiency increases with the nonlinearity and viscosity when the incident wave amplitude is small. When the incident wave amplitude is large, the hydrodynamic efficiency is reduced by the weakened transmission of the second-order harmonic wave component due to the strong wave nonlinearity. However, when the wave amplitude is between these two regimes, the wave is weakly nonlinear, the efficiency decreases with the wave amplitude due to the combined effect of the nonlinearity and viscosity.

Fully Nonlinear Model for Simulating Solitary Waves Propagating through a Partially Immersed Rectangular Structure

ABSTRACT Chang, C.-H.; Wang, K.-H., and Hsieh, P.-C., 2017. Fully nonlinear model for simulating solitary waves propagating through a partially immersed rectangular structure. In this study, a two-dimensional fully nonlinear wave model is developed to simulate a solitary wave propagating through the surface of a partially immersed rectangular structure. The analysis includes the transient characteristics of the evolving waveforms throughout the process of wave-structure interaction. The potential-flow–based finite difference model is generalized by solving the transformed equations in grids according to a vertically transient curvilinear coordinate system. The fully nonlinear conditions at the free surface are treated by a mixed explicit-implicit scheme to ensure numerical accuracy and stability. The wave elevations in terms of reflected and transmitted waves under various structural and wave conditions are computed and compared with those from a series of small-scale experimental tests. Good agreement fo...