A multiscale analysis of nonlinear rolling

A new analysis of nonlinear rolling carried out by the multiscale perturbation method is herewith presented. The behaviour of a ship in a regular beam sea is considered and approximate analytical solutions in three nonlinear resonance regions are obtained. These concern the transient and the steady state roll oscillations. The latter fits in well with a previous one obtained through the averaging method and with the results of the numerical simulation.The obtained results appear to be particularly convenient due to their major mathematical simplicity. Moreover, they allow a simple estimation of the maximum roll amplitudes predictable for a given excitation intensity.     

The effect of nonlinear damping and restoring in ship rolling

Many researchers have studied a wide range of nonlinear equations of motion describing a ship rolling in waves. In this study, a form of nonlinear equation governing the motion of a rolling ship subjected to synchronous beam waves is suggested and solved by the generalized Duffing's method in the frequency domain. Various representations of damping and restoring terms found in the literature are investigated and their solutions are analyzed by the above-mentioned method. Comparative results of nonlinear roll responses are obtained for four distinct vessel types at resonance conditions which constitute the worst situation. The results indicate the importance of roll damping and restoring, when constructing a nonlinear roll model. An inappropriate selection of damping and restoring terms may lead to serious discrepancies with reality, especially in peak roll amplitudes.     

Application of extended state space to nonlinear ship rolling

It is well known in the field of marine hydrodynamics that the added mass, damping and wave exciting forces are functions of frequency (Newman, 1977. Marine Hydrodynamics. MIT Press, Cambridge). Although most previous studies of nonlinear ship rolling motion have assumed that these forces do not vary with frequency, in this study the frequency dependent added mass and damping coefficients are approximated in the time domain with extended state space variables. Using numerical time simulation (integration), the extended state space model is compared to the constant coefficient model with a constant frequency forcing and the results for two constant value approximations of the added mass and damping are compared to the extended state space model with a multiple component pseudo random forcing.     

Application of extended state space to nonlinear ship rolling

It is well known in the field of marine hydrodynamics that the added mass, damping and wave exciting forces are functions of frequency (Newman, 1977. Marine Hydrodynamics. MIT Press, Cambridge). Although most previous studies of nonlinear ship rolling motion have assumed that these forces do not vary with frequency, in this study the frequency dependent added mass and damping coefficients are approximated in the time domain with extended state space variables. Using numerical time simulation (integration), the extended state space model is compared to the constant coefficient model with a constant frequency forcing and the results for two constant value approximations of the added mass and damping are compared to the extended state space model with a multiple component pseudo random forcing.     

A nonlinear model of parametric rolling stabilization by anti-roll tanks

The present paper describes a mathematical model in which the fluid motion inside a U-tank is nonlinearly coupled to the heave, roll and pitch motions of the ship. The main purpose of the investigation is centred on the control of roll motion in the case of parametric resonance in longitudinal waves. A transom stern small vessel, known to be quite prone to parametric amplification, is employed in the study. Four tank designs are employed in order to study the influence of tank mass, tank natural frequency and tank internal damping on the control of parametric rolling at different head seas conditions. Additionally, the influence of the vertical position of the tank is also investigated. The main results are presented in the form of limits of stability, with encounter frequency and wave amplitudes as parameters. Distinct dynamical characteristics are discussed and conclusions are drawn on the relevant parameters for the efficient control of the roll amplifications in head seas.     

Cyclostrophic adjustment and nonlinear oscillations in the core of an intense atmospheric vortex

Intense atmospheric vortices are characterized by a regime of cyclostrophic balance, i.e., the balance between the pressure gradient and centrifugal force. To describe motions in the core of an axisymmetrical vortex, a class of exact solutions to the equations of gas dynamics with a linear dependence on radius is derived for the velocity components and with a quadratic dependence for temperature. It is shown that small deviations from the balance state give rise to oscillations of the hydrothermodynamic fields in the vortex core with a frequency proportional to the angular velocity of the rotation of the core. For fairly large initial deviations, oscillations are clearly anharmonic and, under the conditions of the prevailing centrifugal force, result in a significant temperature decrease on the vortex axis. The application of this class of solutions to describing the Ranque vortex effect (the intense cooling of gas during rapid rotations) and the acoustic radiation from tornadoes is discussed.     

Using wavelet transforms to analyze nonlinear ship rolling and heave-roll coupling

The use of wavelet transforms is explored to investigate the nonlinear dynamical characteristics of ship roll and coupled heave-roll motion. The harmonic character, double period character and chaotic character are observed via a time–frequency window of the wavelet transform. Typical wave parameters in different stability regions are considered. Features such as restoring rolling, divergence rolling, steady state and chaotic responses of ship roll are obtained as well. The investigation in this paper not only highlights the feasibility of using wavelet transforms in the analysis of nonlinear dynamic characteristics of ship rolling in waves, but also shows how it could enhance the analysis abilities.     

Stability and capsizing analysis of nonlinear ship rolling in wind and stochastic beam seas

Considering the actual seaway condition, stability and capsizing of nonlinear ship rolling system in stochastic beam seas is of significant importance for voyage safety. Safe zone are defined in the phase space plan of the unperturbed Hamilton system to qualitatively distinguish ship motions as capsize and noncapsize. Capsize events are defined by solutions passing out of the safe zone. The probability of such an occurrence is studied by virtue of the random Melnikov function and the concept of phase space flux. In this paper, besides conventional wave excitation, the effect of wind load is also taken into account. The introduction of wind load will lead to asymmetry, in other words, it transforms the symmetric heteroclinic orbits into asymmetric homoclinic orbits. For asymmetric dynamical system, the orbital analytic solutions and its power spectrum are not readily available, and the technique of discrete time Fourier transformation (DTFT) is used. In the end, as verification of theoretical critical significant wave height, capsizing probability contour diagram is generated by means of numerical simulation. The contour diagram shows that these analytical methods provide reliable and predictive results about the likelihood of a vessel capsizing in a given seaway condition.     

Thermohydrodynamics of the ocean decay of nonlinear oscillations in a bounded basin of variable depth

We study the time decay of surges of a liquid in a round shallow-water basin of variable depth. The dependence of the logarithmic decrement of oscillations on the bottom topography and wind velocity is analyzed. The role of convective acceleration and bottom friction in the formation of both the level of vertical displacement of the surface of the basin and the velocity field of horizontal wave currents is estimated. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 2, pp. 3–11, March–April, 2006.     

The influence of nonlinear hysteretic seabed interaction on slug-induced stress oscillations in steel catenary risers

Steel catenary risers (SCRs) are usually cost-effective solutions in the development of offshore fields and the transferring of the hydrocarbons from the seabed to the floating facilities. These elements are subjected to the fatigue loads particularly in the touchdown zone (TDZ), where the oscillating SCR is exposed to cyclic contact with the seabed. The slug-induced oscillation is a significant contributor to the fatigue loads in the TDZ. The cyclic seabed soil softening under the wave-induced riser oscillations and the gradual penetration of the SCR into the seabed are widely accepted to have a significant influence on SCR fatigue performance. However, this has never been investigated for slug-induced oscillations due to the lack of integrated access to comprehensive numerical models enabling the simulation of the riser slugging and nonlinear hysteretic riser-seabed interaction at the same time. In this paper, an advanced interface was developed and verified using the multi-point moving tie constraint in order to examine the influence of cyclic seabed soil softening on slug-induced oscillations of SCR. The interface was integrated with a pre-developed user subroutine for modeling of the nonlinear hysteretic riser-seabed interaction and incorporated into a global SCR model in ABAQUS. A comprehensive parametric study was conducted to investigate the influence of slug characteristics and nonlinear seabed soil model on slug-induced, wave-induced, and combined wave/slug induced oscillations of SCR in the TDZ. It was observed that the nonlinear seabed model could significantly affect the embedment of the SCR into the seabed under the slug-induced oscillations and consequently improve the fatigue life. The developed user interface was found to be a strong framework for modeling riser slugging.     

The nonlinear rolling response of a vessel including chaotic motions leading to capsize in regular seas

The rolling motion of a ship has been successfully modelled using a semi-empirical nonlinear differential equation by a number of researchers. Experimental data has been used to model nonlinear damping and righting lever characteristics and comparison with observed behaviour has been reasonably good. The present article describes a numerical, phenomenological approach to analyse this type of behaviour. The stability of the periodic motion, and in particular the possibility of capsize, is explored with reference to qualitative prediction techniques. The appearance of chaotic motions in regular beam seas is a new feature which should be of interest to designers. The inability of traditional quantitative methods, such as the perturbation technique, to detect chaos is a further justification for using numerical simulation guided by dynamical systems theory.     

Modeling of the nonlinear drift oscillations of moored vessels subject to non-Gaussian random sea-wave excitation

A quadratic system model based on Volterra series representation is utilized to model the nonlinear response of moored vessels subjected to random seas. The key idea is to represent the relationship between the incident sea wave (input) and corresponding sway response of the moored vessel (output) with a parallel combination of linear and quadratic transfer functions, and to estimate them by processing actual input and output data. Compared to previous approaches, we take the important step of removing the restriction that the random input must possess Gaussian statistics. The feasibility and validity of the approach is demonstrated by analyzing experimental data taken in model basin tests. We also describe some of the deleterious consequences of assuming Gaussian sea-wave excitation when in fact the excitation is non-Gaussian.     

Mitigating horizontal divergence “checker-board” oscillations on unstructured triangular C-grids for nonlinear hydrostatic and nonhydrostatic flows

A complication of finite-volume triangular C-grid methods is the numerical emergence of horizontal divergence errors that lead to grid-scale oscillations in vertical velocity. Nonlinear feedback via advection of momentum can lead to numerical instability in velocity modes via positive feedback with spurious vertical velocities induced by horizontal divergence truncation error. Existing strategies to mitigate divergence errors such as direct divergence averaging and increased diffusion do not completely mitigate horizontal vertical velocity oscillations. We present a novel elliptic filtering approach to mitigate this spurious error and more accurately represent vertical velocities via improved calculation of horizontal divergences. These results are applied to laminar curved channel flows, demonstrating the applicability of the method to reproduce secondary flow features.     

Seiche oscillations in Lake Baikal

The variations in the free surface of Lake Baikal at three stations (Bol’shie Koty, Listvyanka, and Baikal’sk) are measured. A modern recording method and an advanced technique of record processing are used. Based on 1-year-long observation data, the amplitudes of seiche oscillations and their seasonal changes are analyzed. It is found, in particular, that 67-min seiches are manifested in different seasons. Numerical calculations of seiches in Lake Baikal are made with the use of up-to-date bathymetric data on one-dimensional, plan, and spherical models. Spatial structures of oscillations with periods of 277, 152, 84, 67, and 59 min, corresponding to the well-expressed peaks of power spectral density, are studied. It is shown that the first four periods correspond to uninodal, binodal, trinodal, and quadrinodal longitudinal seiche modes of Lake Baikal. The periods of three solutions can correspond to the value of 59 min. The first of them is the seiche of the lake’s South Basin, and two others are characterized by significant amplitude growth in the Small Sea and Chivyrkui Bay.     

Sea level oscillations in Wellington Harbour

Periods of oscillations of sea level in Wellington Harbour, New Zealand (41° 17'S, 174° 52’ E), are calculated by spectral analysis of the residual elevations observed in tide gauge records. These periods are compared with those computed by numerically integrating a one‐dimensional linear momentum equation and the continuity equation. The two main oscillations are the first harmonics along and across the harbour with periods of about 27 and 22 min respectively — the second harmonics were also observed. The quarter wavelength oscillation with forcing at the mouth which was excited by the 1960 Chilean Tsunami has a period of about 160 min.     

Tidal oscillations in the Baltic Sea

Long-term hourly data from 35 tide gauge stations, including 15 stations in the Gulf of Finland, were used to examine tidal sea level oscillations of the Baltic Sea. High-resolution spectral analysis revealed the well-defined fine structure of tidal peaks with diurnal peaks at most stations being higher than semidiurnal. At some stations (e.g., Narva, Daugava, and Wladyslawowo), high frequency radiational tidal peaks with periods multiple of the solar day (3, 4, 5, 6, and 8 cpd) were detected; the respective oscillations are supposed to be caused by seabreeze winds. Harmonic analysis of tides for individual yearly sea level series followed by vector averaging over the entire observational period was used to estimate the amplitudes and phases of 16 tidal constituents. The maximum tidal oscillations of 17–19 cm were found to be observed in the Gulf of Finland and, first of all, in Neva Bay (in the head of the gulf). Diurnal or mixed diurnal tides are predominant in almost the entire Baltic Sea. The comparison of the observed tides with those theoretically computed showed that the existing numerical models of the main tidal harmonics generally quite accurately reproduce the structure of the tides in the Baltic Sea except for some regions of the Gulf of Bothnia.     

Tidal oscillations in the Caspian Sea

Long-term hourly data from 12 tide gauge stations were used to examine the character of tidal oscillations in the Caspian Sea. Diurnal and semidiurnal tidal peaks are well-defined in sea level spectra in the Middle and South Caspian basins. High-resolution spectral analysis revealed that the diurnal sea level oscillations in the Middle Caspian Basin have a gravitational origin, while those in the South Caspian Basin are mainly caused by radiational effects: the amplitude of diurnal radiational harmonic S₁ is much higher than those of gravitational harmonics О₁, P₁, and K₁. In the North Caspian Basin, there are no gravitational tides and only weak radiational tides are observed. A semidiurnal type of tide is predominant in the Middle and South Caspian basins. Harmonic analysis of the tides for individual annual series with subsequent vector averaging over the entire observational period was applied to estimate the mean amplitudes and phases of major tidal constituents. The amplitude of the M₂ harmonic reaches 5.4 cm in the South Caspian Basin (at Aladga). A maximum tidal range of 21 cm was found at the Aladga station in the southeastern part of the Caspian Sea, whereas the tidal range in the western part of the South Caspian Basin varies from 5 to 10 cm.     

High-frequency sediment-level oscillations in the swash zone

Sediment-level oscillations with heights of about 6 cm and shore-normal lengths of order 10 m have been measured in the swash zone of a high-energy, coarse-sand beach. Crests of oscillations were shore parallel and continuous alongshore. The oscillations were of such low steepness (height-to-length ratio approximately 0.006) that they were difficult to detect visually. The period of oscillation ranged between 6 and 15 min and decreased landward across the swash zone. The sediment-level oscillations were progressive landward with an average migration rate in the middle to upper swash zone of 0.8 m min⁻¹. Migration was caused mostly by erosion on the seaward flank of the crest of an oscillation during a period of net seaward sediment transport. Thus, the observed migration was a form migration landward rather than a migration involving net landward sediment transport. The observed sediment-level oscillations were different than sand waves or other swash-zone bedforms previously described.     

Tidal oscillations in shallow bays and sediment movement

The present study seeks to understand the interaction of a directly forced, large amplitude, tidal mode with non-directly forced resonant or nearly resonant tidal modes in a coastal bay or basin despite frictional effects. It is shown that in shallow basins a non-directly forced resonant mode may be indirectly activated with a response that may be time dependent, with the amplitude of this mode increasing and then decreasing before increasing again in a cyclical manner associated with amplitude beats. A laboratory experiment and theoretical arguments are used to support these conclusions and to study the effect when the amplitude of the tidal disturbances is comparable to the depth of the basin. It is noted that the presence of such modes, whether directly or indirectly forced, and whether resonant or non-resonant, may be able to move sediment and thereby change the resonance behaviour of the basin by changing its geometry. The changes in the geometry may then change the resonant frequencies of the basin, possibly causing the decay of the mode that caused the sediment motion. The implications of the mechanism for the modelling of the tidal behaviour in other bays with more complicated geometry are considered.     

Controlling spatial oscillations in bed level update schemes

A frequent problem with process-based coastal morphological models is the appearance of high wave number spatial oscillations in the simulated bed levels with time. After a sufficiently long time, these oscillations become dominant and mask the large-scale features of the bed level evolution.The equation for conservation of sediment mass is used to show that the spatial oscillations are generated by the dependence of the bed celerity (celerity of the bed level oscillations) with bed levels, which is due to the non-linear relationship between sediment transport and bed levels. This breeds higher spatial harmonics of the bed level oscillations with time. In this situation, using a Finite Difference (FD) scheme that does not damp oscillations with high wave numbers leads to the generated harmonics being kept in the solution. These generate further harmonics until the entire solution is dominated by high wave number oscillations.In this paper, a finite difference scheme, in combination with a filtering procedure, is used to dissipate high wave number oscillations. Analysis of the amplification portraits show that the filtering procedure in combination with a Lax–Wendroff scheme does not affect oscillations with lower wave numbers (larger scale features resolved with seven or more grid points). Some examples are also presented to illustrate these features.