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.

We derive an evolution equation for the free-surface dynamics of a thin film of a second-grade fluid over an unsteady stretching sheet using long-wave theory. For the numerical investigation of the viscoelastic effect on the thin-film dynamics, a finite-volume approach on a uniform grid with implicit flux discretization is applied. The present results are in excellent agreement with results available in the literature for a Newtonian fluid. We observe that the fluid thins faster with the rapid stretching rate of the sheet, but the second-grade parameter delays the thinning behaviour of the liquid film.

This paper re-examines the problem of the flow of a fluid of finite depth over two Gaussian-shaped obstructions on the stream bed. A weakly nonlinear analysis in the form of the Korteweg–de Vries equation is used to compare with the results of the fully nonlinear problem. The main focus is to find waveless subcritical solutions, and contours showing the obstruction height and separation values that result in waveless solutions are found for different Froude numbers and different obstruction widths.

We describe an experimental study of the forces acting on a square cylinder (of width $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}b$) which occupies 10–40 % of a channel (of width $w$), fixed in a free-surface channel flow. The force experienced by the obstacle depends critically on the Froude number upstream of the obstacle, ${\mathit{Fr}}_1$ (depth $h_1$), which sets the downstream Froude number, ${\mathit{Fr}}_2$ (depth $h_2$). When ${\mathit{Fr}}_1<{\mathit{Fr}}_{1c}$, where ${\mathit{Fr}}_{1c}$ is a critical Froude number, the flow is subcritical upstream and downstream of the obstacle. The drag effect tends to decrease or increase the water depth downstream or upstream of the obstacle, respectively. The force is form drag caused by an attached wake and scales as $\overline{F_{D}}\simeq C_D \rho b u_1^2 h_1/2$, where $C_D$ is a drag coefficient and $u_1$ is the upstream flow speed. The empirically determined drag coefficient is strongly influenced by blocking, and its variation follows the trend $C_D=C_{D0}(1+C_{D0}b/2w)^2$, where $C_{D0}=1.9$ corresponds to the drag coefficient of a square cylinder in an unblocked turbulent flow. The r.m.s. lift force is approximately 10–40 % of the mean drag force and is generated by vortex shedding from the obstacle. When ${\mathit{Fr}}_1={\mathit{Fr}}_{1c}\, (<1)$, the flow is choked and adjusts by generating a hydraulic jump downstream of the obstacle. The drag force scales as $\overline{F}_D\simeq C_K \rho b g (h_1^2-h_2^2)/2$, where experimentally we find $C_K\simeq 1$. The r.m.s. lift force is significantly smaller than the mean drag force. A consistent model is developed to explain the transitional behaviour by using a semi-empirical form of the drag force that combines form and hydrostatic components. The mean drag force scales as $\overline{F_{D}}\simeq \lambda \rho b g^{1/3} u_1^{4/3} h_1^{4/3}$, where $\lambda $ is a function of $b/w$ and ${\mathit{Fr}}_1$. For a choked flow, $\lambda =\lambda _c$ is a function of blocking ($b/w$). For small blocking fractions, $\lambda _c= C_{D0}/2$. In the choked flow regime, the largest contribution to the total drag force comes from the form-drag component.

The subcritical flow of a stream over a bottom obstruction or depression is considered with particular interest in obtaining solutions with no downstream waves. In the linearised problem this can always be achieved by superposition of multiple obstructions, but it is not clear whether this is possible in a full nonlinear problem. Solutions computed here indicate that there is an effective nonlinear superposition principle at work as no special shape modifications were required to obtain wave-cancelling solutions. Waveless solutions corresponding to one or more trapped waves are computed at a range of different Froude numbers and are shown to provide a rather elaborate mosaic of solution curves in parameter space when both negative and positive obstruction heights are included.

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 → ∞.