Yinglong Zhang

NEW SOLUTIONS FOR THE PROPAGATION OF LONG WATER WAVES OVER VARIABLE DEPTH

Based on the linearized long-wave equation, two new analytical solutions are obtained respectively for the propagation of long surface gravity waves around a conical island and over a paraboloidal shoal. Having been intensively studied during the last two decades, these two problems have practical significance and are physically revealing for wave propagation over variable water depth. The newly derived analytical solutions are compared with several previously obtained numerical solutions and the accuracy of those numerical solutions is discussed. The analytical method has the potential to be used to find solutions for wave propagation over more natural bottom topographies.

On the choice of interpolation functions used in the dual-reciprocity boundary-element method

Abstract The dual-reciprocity boundary-element method is a very powerful technique for solving general elliptic equations of the type ∇ 2 u = b . In this method, a series of interpolation functions is used to approximate b in order to convert the associated domain integral, which it is necessary to evaluate in a traditional boundary-element analysis, into boundary integrals only. Hence the choice of interpolation functions has direct effects on the numerical results. According to Partridge and Brebbia, the adoption of a comparatively simple form of interpolation function gives the best results. Unfortunately, when b contains partial derivatives of the unknown function u ( x , y ), the adoption of such a type of interpolation function inevitably leads to the creation of singularities on all boundary and internal nodes used in a dual-reciprocity boundary-element analysis, as was pointed out by Zhu and Zhang in 1992. To avoid this problem, a functional transformation, which applies only to linear governing equations, can be employed to eliminate these derivative terms and thus to obtain better numerical results. In this paper, two new interpolation functions are proposed and examined; they are proven to be generally applicable and satisfactory.

SCATTERING OF LONG WAVES AROUND A CIRCULAR ISLAND MOUNTED ON A CONICAL SHOAL

Abstract Scattering of simple harmonic long waves by a cylindrical island mounted on a conical shoal in an otherwise open sea of constant depth is solved analytically based on the shallow-water wave (long-wave) theory. The new analytical solution not only confirms some conclusions and conjectures previously drawn from purely numerical studies, such as those showing how the slope of the shoal affects the amplification of the ocean waves around the coastline of islands, but also provides another useful check for numerical model developers.

Open channel flow past a bottom obstruction

A new nonlinear integral-equation model is derived in terms of hodograph variables for free-surface flow past an arbitrary bottom obstruction. A numerical method, carefully chosen to solve the resulting nonlinear algebraic equations and a simple, yet effective radiation condition have led to some very encouraging results. In this paper, results are presented for a semi-circular obstruction and are compared with those of Forbes and Schwartz [1]. It is shown that the wave resistance calculated from our nonlinear model exhibits a good agreement with that predicted by the linear model for a large range of Froude numbers for a small disturbance. The small-Froude-number non-uniformity associated with the linear model is also discussed.

Subcritical, transcritical and supercritical flows over a step

Free-surface flow over a bottom topography with an asymptotic depth change (a ‘step’) is considered for different ranges of Froude numbers varying from subcritical, transcritical, to supercritical. For the subcritical case, a linear model indicates that a train of transient waves propagates upstream and eventually alters the conditions there. This leading-order upstream influence is shown to have profound effects on higher-order perturbation models as well as on the Froude number which has been conventionally defined in terms of the steady-state upstream depth. For the transcritical case, a forced Korteweg–de Vries (fKdV) equation is derived, and the numerical solution of this equation reveals a surprisingly conspicuous distinction between positive and negative forcings. It is shown that for a negative forcing, there exists a physically realistic nonlinear steady state and our preliminary results indicate that this steady state is very likely to be stable. Clearly in contrast to previous findings associated with other types of forcings, such a steady state in the transcritical regime has never been reported before. For transcritical flows with Froude number less than one, the upstream influence discovered for the subcritical case reappears.

Solving general field equations in infinite domains with dual reciprocity boundary element method

Abstract The dual reciprocity boundary element method has been successfully employed to solve general field equations posed in a closed domain, i.e. interior problems. Up to now, however, little effort has been made to extend it to exterior problems (i.e. general field equations posed in an infinite region), which are commonly encountered in engineering practice. In this paper, the interpolation functions associated with exterior problems, which were proposed by Loeffler and Mansur (in Boundary Elements X, Vol. 2 , Springer, 1988), are first examined. We have found that the choice of the arbitrary constant, the inclusion of which is necessary in those interpolation functions, has clear effects on the accuracy of the numerical results. A mapping transformation, through which any exterior problem can be solved by solving an equivalent interior problem, is then proposed. Although there are certain regularity conditions attached to such a mapping, they can be easily satisfied if the unknown function satisfies certain regularity conditions at infinity in the original exterior problem. A successful application of this mapping transformation to a transient heat transfer problem demonstrates the good performance of this approach.

A FLAT SHIP THEORY ON BOW AND STERN FLOWS

An analytical solution of a two-dimensional bow and stern flow model based on a flat ship theory is presented for the first time. The flat ship theory is a counterpart to Michells thin ship theory and leads to a mixed initial-boundary value problem, which is usually difficult to solve analytically. Starting from the transient problem, we shall first show that a steady state is attainable at the large time limit. Then the steady problem is solved in detail by means of the Wiener-Hopf technique and closed-form far-field results are obtained for an arbitrary hull shape. Apart from providing a better understanding of the underlying physics, the newly found analytical solution has shed some light on solving a longtime outstanding problem in the engineering practice of ship building, the optimisation of hull shape.

On nonlinear transient free-surface flows over a bottom obstruction

A fully nonlinear integral-equation model based on the potential theory is used to study transient free-surface flow induced by a bump moving along a flat bottom. Comparison is made between the large-time solution of the transient model and the fully nonlinear steady-state solution in Zhang and Zhu [J. Eng. Math. 30, 487 (1996)] for both subcritical and supercritical flows, and asymptotic agreement between the two is observed. For the transcritical flow, some features associated with the resonant flow motion previously found by approximate theories are confirmed. A diagram of the parameter space is presented showing the regions of different solution categories including wave breaking. Through this diagram, a discrepancy between two previous steady-state models can now be clearly explained. Interestingly, the dividing line separating those solutions with or without wave breaking levels out asymptotically for very large Froude numbers, indicating that there is a limiting dimension of the bump, beyond which ...

RESONANT INTERACTION BETWEEN A UNIFORM CURRENT AND AN OSCILLATING OBJECT

Abstract The interaction between a uniform current and an oscillating, submerged circular cylinder is solved using a quasi-linear model based on Dagan and Milohs single source solution [Dagen, G. & Miloh, T., J. Fluid Mech ., 120 (1982) 139–154]. Being different from those obtained previously, the solution near the resonant frequency is presented and discussed.

COMBINED REFRACTION AND DIFFRACTION OF SHORT WAVES USING THE DUAL-RECIPROCITY BOUNDARY-ELEMENT METHOD

Short waves are those with a wavelength-to-depth ratio of less than approximately 20; they play an important role in coastal sediment transport as well as the harbour oscillations with which coastal engineers are concerned. It is usually much more difficult to numerically model the combined effects of the refraction and diffraction of short waves in coastal regions, due to the fact that a large number of grid points is usually required if the computational domain itself must be discretized. In this paper, the dual-reciprocity boundary-element method model developed by Zhu (1993) is further applied to the case of waves with short wavelengths. The models excellent accuracy and efficiency in modelling short waves is further established after some of the preliminary results are compared with corresponding previously published numerical and experimental data.