#### Year

1996

#### Degree Name

Doctor of Philosophy

#### Department

Department of Mathematics

#### Recommended Citation

Yinglong, Zhang, On steady and unsteady free-surface flows, Doctor of Philosophy thesis, Department of Mathematics, University of Wollongong, 1996. https://ro.uow.edu.au/theses/1557

#### Abstract

Various problems on steady and unsteady (including quasi-steady) free-surface flows over an uneven bottom topography or submerged or surface-piercing objects are concerned. Our primary interest is on the generation of surface gravity waves on the free surface.

The first part "Steady Problems" deals with motions that can be treated as steady (i.e., time-independent) flows by a proper choice of reference frame. A perturbation method consistent to the second-order is proposed and applied to a semicircular trench. Also introduced is an integral-equation method for the solution of the fully nonlinear problem for an arbitrary bottom topography. This part culminates in a comparison study of various solutions for a semi-circular trench. Some interesting features associated with the generated nonlinear waves are discussed.

The second part deals with unsteady problems, i.e., motions that are timedependent in any frame of reference. Our primary interest is to examine the possible existence of a steady state. That is, whether or not a steady state will eventually emerge from transient motions and if the answer is yes, what does it look like? We begin with a quasi-steady (i.e., time-periodic) problem of a uniform current past an oscillating object. For this problem, we are mainly concerned with the resonant frequency at which the classical linear model fails. A quasi-linear model is thus proposed to remedy this failure.

Then a two-dimensional Neumann-Kelvin problem associated with a surfacepiercing flat-ship-like object is solved analytically. This problem is difficult as it is an initial-boundary value problem with mixed boundary conditions. With this analytical solution, we can prove the existence of a steady state for either bow or stern flows and derive the so-called "radiation condition" for the steady problem.

In the last two chapters of Part II, we study free-surface flows over two types of peculiar topographies, i.e., a "step" and a "wavy" bed (Djordjevic and Redekopp 1992). Chapter 6 is dedicated to the study of a step-like topography with both resonant and non-resonant cases being considered. With the transient non-resonant model, we identify an "upstream influence" and derive the "radiation condition" for a step-like topography. The resonant problem is inherently nonlinear and possesses no steady state in general, but we find that a nonlinear steady state exists if the is properly forced. Our solution also strongly suggests the stability of this nonlinear steady state.

In the last chapter, we show that for a "wavy bed" of finite extent, steady state is attainable even within the resonant transcritical regime, the only remnant of the resonance being an "upstream influence". If the bed extends semi-infinitely to far downstream, the linear model breaks down in the near field, where the motion is governed by a series of forced KdV equations, which are matched to the far-field linear solutions.