Fission of a weakly nonlinear interfacial solitary wave at a step

The transformation of a weakly nonlinear interfacial solitary wave in an ideal two-layer flow over a step is studied. In the vicinity of the step the wave transformation is described in the framework of the linear theory of long interfacial waves, and the coefficients of wave reflection and transmission are calculated. A strong transformation arises for propagation into shallower water, but a weak transformation for propagation into deeper water. Far from the step, the wave dynamics is described by the Korteweg-de Vries equation which is fully integrable. In the vicinity of the step, the reflected and transmitted waves have soliton-like shapes, but their parameters do not satisfy the steady-state soliton solutions. Using the inverse scattering technique it is shown that the reflected wave evolves into a single soliton and dispersing radiation if the wave propagates from deep to shallow water, and only dispersing radiation if the wave propagates from shallow to deep water. The dynamics of the transmitted wave is more complicated. In particular, if the coefficient of the nonlinear quadratic term in the Korteweg-de Vries equation is not changed in sign in the region after the step, the transmitted wave evolves into a group of solitons and radiation, a process similar to soliton fission for surface gravity waves at a step. But if the coefficient of the nonlinear term changes sign, the soliton is destroyed completely and transforms into radiation. The effects of cubic nonlinearity are studied in the framework of the extended Korteweg-de Vries (Gardner) equation which is also integrable. The higher-order nonlinear effects influence the amplitudes of the generated solitons if the amplitude of the transformed wave is comparable with the thickness of lower layer, but otherwise the process of soliton fission is qualitatively the same as in the framework of the Korteweg-de Vries equation.     

Convection in rotating annular channels heated from below. Part 2. Transitions from steady flow to turbulence

We report the results of fully three-dimensional numerical simulations of nonlinear convection in a Boussinesq fluid in an annular channel rotating about a vertical axis with lateral no-slip or stress-free sidewalls, stress-free top and bottom, uniformly heated from below, a problem first studied by Davies-Jones and Gilman (1971 Davies-Jones, RP and Gilman, PA. 1971. Convection in a rotating annulus uniformly heated from below.. J. Fluid Mech., 46: 65–81.  [Google Scholar]) and Gilman (1973 Gilman, PA. 1973. Convection in a rotating annulus uniformly heated from below. Part 2. Nonlinear results. J. Fluid Mech., 57: 381–400.  [Google Scholar]). A substantial range of the Rayleigh number R (R_c≤R≤O(100 R_c)), where R_c denotes the critical value at the onset of convection) is considered. It is found that the wall-localized convection mode, unaffected by the velocity boundary condition imposed on the sidewalls, is nonlinearly robust. Both directions of travelling waves, one propagating against the sense of rotation near the outer sidewall and the other propagating in the same sense as the rotation in the vicinity of the inner sidewall, are always present in the nonlinear solutions. In contrast to nonlinear convection in a rotating Bénard layer, neither convection rolls nor the Küpper–Lortz instability can exist in a rotating annular channel because of the effect of the sidewalls. It is the nonlinear interaction between the wall-localized modes and the internal mode that plays an essential role in determining the nonlinear properties of convection in a rotating annular channel. Our studies reveal systematically the various nonlinear phenomena, from steady travelling waves trapped in the vicinities of the sidewalls to convective turbulence exhibiting columnar structure.     

Nonlinearities and Linearities in Internal Gravity Waves of the Atmosphere and Oceans

The spectrum of internal gravity waves in the atmosphere and oceans is sufficiently intense that nonlinear interactions must occur, if these waves are analyzed in Eulerian coordinates as is usually done. As it happens, however, if these waves are analyzed in Lagrangian coordinates the most important nonlinearity can be entirely avoided: it is an Eulerian mathematical construct only, not a physical process. The mathematical basis for this assertion is developed here, and some of its consequences are discussed. Among the latter is a questioning of the validity of standard Eulerian eikonal methods of calculating ray paths and related functions in a multiwave environment, discussed in an appendix.     

Nonlinear pulsations of horizontal jets

A shallow-water model with horizontally nonuniform density is used to study the dynamics of jet flows that arise under the influence of buoyancy and the Coriolis force. Within this approach, the jet is described by a self-similar compactly-localized solution and interpreted as a band of shear flow having a temperature contrast with the ambient fluid. In addition to stationary states, the dynamics of such jets admit cyclonic rotation with a constant angular velocity and transverse nonlinear pulsations. The phase portrait corresponding to this model shows that regimes with pulsating jets develop along closed trajectories bounded by the separatrix loop. The theory predicts that the period for warm jet pulsations is longer than the inertial oscillation period caused by the Earth’s rotation, while for cold jet pulsations, it is shorter. Thus, only warm jets can have a noticeable effect on the atmospheric dynamics in the synoptic range. In particular, they may well be responsible for additional spectral peaks that appear in this range of wind speed fluctuations.     

Analysis of buried pipelines subjected to reverse fault motion

Presently available simplified analytical methods and semi-empirical methods for the analysis of buried pipelines subjected to fault motion are suitable only for the strike-slip and the normal-slip type fault motions, and cannot be used for the reverse fault crossing case. A simple finite element model, which uses beam elements for the pipeline and discrete nonlinear springs for the soil, has been proposed to analyse buried pipeline subjected to reverse fault motion. The material nonlinearities associated with pipe-material and soil, and geometric nonlinearity associated with large deformations were incorporated in the analysis. Complex reverse fault motion was simulated using suitable constraints between pipe-nodes and ground ends of the soil spring. Results of the parametric study suggest that the pipeline's capacity to accommodate reverse fault offset can be increased significantly by choosing a near-parallel orientation in plan with respect to the fault line. Further improvement in the response of the pipeline is possible by adopting loose backfill, smooth and hard surface coating, and shallow burial depth in the fault crossing region. For normal or near normal orientations, pipeline is expected to fail due to beam buckling at very small fault offsets.     

岩石滞后非线性弹性响应的物理模型

介绍了岩石滞后非线性弹性的几个物理模型。赫兹颗粒接触模型是具有多尺度和滞后特性的经典模型,它预测了岩石中强烈的非线性;软的粘结系统几乎决定了岩石的力学性质,粘结系统中的流体对非线性响应的贡献特别显著,但是目前还没有搞清楚粘结系统和孔隙流体究竟是如何影响非线性响应的;GL模型是一个基于金属位错的物理模型,这是滞后动力行为方面一个开拓性的微观模型;PM模型是一个基于岩石细观尺度的唯象模型,它对理解岩石滞后非线性的机制和尺度是很重要的。     

A simplified nonlinear analysis method for piled raft foundation in layered soils under vertical loading

This paper presents a simplified nonlinear solution for piled raft foundations in layered soils under vertical loading. Based on the elastic–plastic analysis of a single pile in a layered soil, the shielding effect between a receiver pile and the soil is taken into account to modify the conventional interaction factor between two piles. An approximate approach with the concept of the interaction factor is employed to study the nonlinear behavior of pile groups with a rigid cap. Considering the variation of soil properties, the solution to multilayered elastic materials is used to calculate the settlement of the soil. The interactions between pile–soil–raft are taken into account to determine the stiffness matrix of the piled raft. By solving the stiffness matrix equations, the settlement and the load shared by the piles and raft could be obtained. Compared with results of the available published literatures, the proposed solution provides reasonable results.     

Empirical modeling of plate load test moduli of soil via gene expression programming

New empirical models were developed to predict the soil deformation moduli using gene expression programming (GEP). The principal soil deformation parameters formulated were secant (E_s) and reloading (E_r) moduli. The proposed models relate E_s and E_r obtained from plate load-settlement curves to the basic soil physical properties. The best GEP models were selected after developing and controlling several models with different combinations of the influencing parameters. The experimental database used for developing the models was established upon a series of plate load tests conducted on different soil types at depths of 1–24 m. To verify the applicability of the derived models, they were employed to estimate the soil moduli of a part of test results that were not included in the analysis. The external validation of the models was further verified using several statistical criteria recommended by researchers. A sensitivity analysis was carried out to determine the contributions of the parameters affecting E_s and E_r. The proposed models give precise estimates of the soil deformation moduli. The E_s prediction model provides considerably better results in comparison with the model developed for E_r. The simplified formulation for E_s significantly outperforms the empirical equations found in the literature. The derived models can reliably be employed for pre-design purposes.     

A hybrid computational approach to formulate soil deformation moduli obtained from PLT

In this study, new empirical equations were developed to predict the soil deformation moduli utilizing a hybrid method coupling genetic programming and simulated annealing, called GP/SA. The proposed models relate secant (E_s), unloading (E_u) and reloading (E_r) moduli obtained from plate load–settlement curves to the basic soil physical properties. Several models with different combinations of the influencing parameters were developed and checked to select the best GP/SA models. The database used for developing the models was established upon a series of plate load tests (PLT) conducted on different soil types at various depths. The validity of the models was tested using parts of the test results that were not included in the analysis. The validation of the models was further verified using several statistical criteria. A traditional GP analysis was performed to benchmark the GP/SA models. The contributions of the parameters affecting E_s, E_u and E_r were analyzed through a sensitivity analysis. The proposed models are able to estimate the soil deformation moduli with an acceptable degree of accuracy. The E_s prediction model has a remarkably better performance than the models developed for predicting E_u and E_r. The simplified formulations for E_s, E_u and E_r provide significantly better results than the GP-based models and empirical models found in the literature.     

Numerical analysis and validation of weakly nonlinear ship motions and structural loads on a modern containership

This paper aims to validate a numerical seakeeping code based on a 3D Rankine panel method by comparing its results with experimental data. Particularly, the motion response and hull-girder loads on a real modern ship, a 6500 TEU containership, are considered in this validation study. The method of solution is a 3D Rankine panel method which adopts B-spline basis function in the time domain. The numerical code is based on the weakly nonlinear scheme which considers nonlinear Froude-Krylov and restoring forces. The main focus of this study is given to investigate the nonlinear characteristics of wave-induced loads, and to validate this present scheme for industrial use in the range of low Froude number. The comparisons show that the nonlinear motions and hull-girder loads, computed by the present numerical code, have good overall agreements with experimental results. It is found that, for the better accuracy of computational results, particularly at extreme waves in oblique seas, the careful treatment of soft-spring (or compatible) system is recommended to the control of non-restoring motions such as surge, sway, and yaw.