A wake model is pursued for potential flow past a submerged, finite-length plate that is perpendicular to a uniform, horizontal stream bounded above by a free surface. The effects of gravity are included along the free surface. The approach is to adopt an open-wake model such that the wake boundaries become parallel to the undisturbed stream at some (unknown) point downstream. Boundary integral equations are formed and then discretised along the wake boundaries and free surface in order to obtain a solution numerically. In terms of the dependency of the solution on various parameters, the problem will be formulated in two ways. First, for a given Froude number and ratio of the length of the vertical plate to the draft (the depth of the bottom of the vertical plate relative to the undisturbed free surface), the effect of the wake underpressure coefficient on the size of the wake will be considered. Then, the problem will be discussed where we instead (more naturally) fix the Froude number, draft and length of the vertical, submerged plate. The dependencies of the solution on these parameters will be analysed regarding the effects on several factors, including the size of the wake, the relative lengths of the upper and lower wake boundaries, and the resulting wake underpressure coefficient.

The behaviour of an axisymmetric bubble in a pure liquid forced by an acoustic pressure field is analysed. The bubble is assumed to have a sharp deformable interface, which is subject both to surface tension and to Rayleigh viscosity damping. Two modelling regimes are considered. The first is a linearized solution, based on the assumption of small axisymmetric deformations to an otherwise spherical bubble. The second involves a semi-numerical solution of the fully nonlinear problem, using a novel spectral method of high accuracy. For large-amplitude nonspherical bubble oscillations, the fully nonlinear solutions show that a complicated resonance structure is possible and that curvature singularities may occur at the interface, even in the presence of surface tension. Rayleigh viscosity at the interface prevents singularity formation, but eventually causes the bubble to become purely spherical unless shape-mode resonances occur. An extended analysis is also presented for purely spherical bubbles, which allows for a more detailed study of the effects of resonance and the Rayleigh viscosity at the bubble surface.

A nonlinear evolution equation correct to fourth order is developed for gravity-capillary waves on linear shear currents in finite water depth. Therefore, this equation covers both effects of depth uniform currents and uniform vorticity. Starting from this equation, an instability analysis is then made for narrow banded uniform Stokes waves. The notable feature is that our investigation due to fourth order shows a remarkable improvement compared with the third-order one, and produces an excellent result compatible with the exact result of Longuet-Higgins. We observe that linear shear currents considerably change the modulational instability properties of capillary-gravity waves, such as the growth rate and bandwidth of instability.

We derive a nonlinear Schrödinger equation for the propagation of the three-dimensional broader bandwidth gravity-capillary waves including the effect of depth-uniform current. In this derivation, the restriction of narrow bandwidth constraint is extended, so that this equation will be more appropriate for application to a realistic sea wave spectrum. From this equation, an instability condition is obtained and then instability regions in the perturbed wavenumber space for a uniform wave train are drawn, which are in good agreement with the exact numerical results. As it turns out, the corrections to the stability properties that occur at the fourth-order term arise from an interaction between the mean flow and the frequency-dispersion term. Since the frequency-dispersion term, in the absence of depth-uniform current, for pure capillary waves is of opposite sign for pure gravity waves, so too are the corrections to the instability properties.

A spectral method is developed to study the steady and unsteady flow of fluid into a line sink from a horizontally confined aquifer, and the results are compared to solutions obtained implementing the finite element package COMSOLTM. The aquifer or drain is considered to be confined below so that the solutions are fundamentally unsteady. Comparison is made between the two methods in determining the drawdown of the surface.

A classical problem in free-surface hydrodynamics concerns flow in a channel, when an obstacle is placed on the bottom. Steady-state flows exist and may adopt one of three possible configurations, depending on the fluid speed and the obstacle height; perhaps the best known has an apparently uniform flow upstream of the obstacle, followed by a semiinfinite train of downstream gravity waves. When time-dependent behaviour is taken into account, it is found that conditions upstream of the obstacle are more complicated, however, and can include a train of upstream-advancing solitons. This paper gives a critical overview of these concepts, and also presents a new semianalytical spectral method for the numerical description of unsteady behaviour.

In models of fluid outflows from point or line sources, an interface is present, and it is forced outwards as time progresses. Although various types of fluid instabilities are possible at the interface, it is nevertheless of interest to know the development of its overall shape with time. If the fluids on either side are of nearly equal densities, it is possible to derive a single nonlinear partial differential equation that describes the interfacial shape with time. Although nonlinear, this equation admits a simple transformation that renders it linear, so that closed-form solutions are possible. Two such solutions are illustrated; for a line source in a planar straining flow and a point source in an axisymmetric background flow. Possible applications in astrophysics are discussed.

Fresh water is one of the most significant resources for human activities and survival, and irrigation is among the most important uses of water. The sustainibility and performance of irrigation canals can be greatly affected by sediment transport and deposition. In our previous works, we proposed a Lattice Boltzmann model for simulating a free surface flow in an irrigation canal, as an alternative to more traditional models mainly based on shallow water equations. Here we introduce the sedimentation phenomenon into our model by adding a new algorithm, based on the earlier work by B. Chopard, A. Dupuis and A. Masselot [9,11,12,27]. Transport, erosion, deposition and toppling of sediments are taken into account and enable the global sedimentation algorithm to simulate different transport modes such as bed load and suspended load. In the present work, we study both the behaviour of a sediment deposit located at an underflow submerged gate (depending on the gate opening and the flow discharge) and the influence of the presence of such a deposit on the flow. Both numerical and experimental validations have been performed. The experiments were realized on the micro-canal of the LCIS laboratory at Valence, France. The comparisons between simulations and experiments give good qualitative agreement.

Internal waves arise in a wide array of oceanographic problems of both theoretical and engineering interest. In this contribution we present a new model, valid in the weakly nonlinear regime, for the propagation of disturbances along the interface between two ideal fluid layers of infinite extent and different densities. Additionally, we present a novel high-order/spectral algorithm for its accurate and stable simulation. Numerical validation results and simulations of wave-packet evolution are provided.

Borda's mouthpiece consists of a long straight tube projecting into a large vessel, where fluid enters the tube in a free surface flow that tends to become uniform far downstream in the tube. A two-dimensional approximation to this flow under gravity in the upper part of the tube leads to an evaluation of the contraction coefficient, the ratio of the constant depth of the uniform flow to the width of the tube. The analysis also applies to flow under gravity past a sluice gate, if the semi-infinite wall above the channel is rotated to the vertical. The contraction coefficient depends upon the Froude number F, and is generally less than the zero gravity value of 1/2 that is approached as F → ∞.

We consider the linearization of the three-dimensional water waves equation with surface tension about a flat interface. Using oscillatory integral methods, we prove that solutions of this equation demonstrate dispersive decay at the somewhat surprising rate of ${{t}^{-5/6}}$. This rate is due to competition between surface tension and gravitation at $O(1)$ wave numbers and is connected to the fact that, in the presence of surface tension, there is a so-called “slowest wave”. Additionally, we combine our dispersive estimates with ${{L}^{2}}$ type energy bounds to prove a family of Strichartz estimates.