The constant-density inviscid rotational flow in the neighbourhood of a general stagnation point on a wall is investigated. In all but very special cases, the solution is non-analytic and the vorticity at the wall is infinite; the stagnation streamline is tangent to the wall at the stagnation point; stagnation points of saddlepoint type cannot exist.

The boundary-layer equations corresponding to the inviscid solutions studied are presented.

We describe new experiments in which particle-laden turbulent fountains with source Froude numbers $20>Fr_{0}>6$ are produced when particle-laden fresh water is injected upwards into a reservoir filled with fresh water. We find that the ratio $U$ of the particle fall speed to the characteristic speed of the fountain determines whether the flow is analogous to a single-phase fountain ($U\ll 1$) or becomes a fully separated flow ($U\geqslant 1$). In the single-phase limit, a fountain with momentum flux $M$ and buoyancy flux $B$ oscillates about the mean height, $h_{m}=(1.56\pm 0.04)M^{3/4}B^{-1/2}$, as fluid periodically cascades from the maximum height, $h_{t}=h_{m}+{\rm\Delta}h$, to the base of the tank. Experimental measurements of the speed $u$ and radius $r$ of the fountain at the mean height $h_{m}$, combined with the conservation of buoyancy, suggest that $Fr(h_{m})=u(g^{\prime }r)^{-1/2}\approx 1$. Using these values, we find that the classical scaling for the frequency of the oscillations, ${\it\omega}\sim BM^{-1}$, is equivalent to the scaling $u(h_{m})/r(h_{m})$ for a fountain supplied at $z=h_{m}$ with $Fr=1$ (Burridge & Hunt, J. Fluid Mech., vol. 728, 2013, pp. 91–119). This suggests that the oscillations are controlled in the upper part of the fountain where $Fr\leqslant 1$, and that they may be understood in terms of a balance between the upward supply of a growing dense particle cloud, at the height where $Fr=1$, and the downward flow of this cloud. In contrast, in the separated flow regime, we find that particles do not reach the height at which $Fr=1$: instead, they are transported to the level at which the upward speed of the fountain fluid equals their fall speed. The particles then continuously sediment while the particle-free fountain fluid continues to rise slowly above the height of particle fallout, carried by its momentum.

The time development of a dense eddy placed on a varying topography in a rotating system is the subject of this paper. It is shown that if the bathymetric variations are comparable with the thickness of the eddy, these will induce a first-order time development in the water mass. Large-amplitude geostrophic theory shows that the water is advected along the depth contours with a speed that is proportional to the slope of the bottom. The results are applied to a laboratory experiment with a rotating parabolic channel. According to theory, an initially circular eddy placed in the centre of the channel becomes elliptical while shifting its axis along the channel. The outcome of the experiment is in accordance with theory, and the importance of topographic advection in the ocean is discussed. It is suggested that the mechanism may be important for moving water from deep-water formation sites and into the global deep-water circulation.

Using the Orr-Sommerfeld equation with the wavenumber as the eigenvalue, a search for higher eigenstates in the stability theory of the Blasius boundary layer has revealed the existence of a number of viscous states in addition to the long established fundamental state. The viscous states are discrete, belong to two series, and are all heavily damped in space. Within the limits of the investigation the number of viscous states existing in the layer increases as the Reynolds number and the angular frequency of the perturbation increase. It is suggested that the viscous eigenstates may be responsible for the excitation of some boundary-layer disturbances by disturbances in the free stream.

We study the propagation of water waves over a ridge structured at the subwavelength scale using homogenization techniques able to account for its finite extent. The calculations are conducted in the time domain considering the full three-dimensional problem to capture the effects of the evanescent field in the water channel over the structured ridge and at its boundaries. This provides an effective two-dimensional wave equation which is a classical result but also non-intuitive transmission conditions between the region of the ridge and the surrounding regions of constant immersion depth. Numerical results provide evidence that the scattering properties of a structured ridge can be strongly influenced by the evanescent fields, a fact which is accurately captured by the homogenized model.

Experiments were conducted to examine the development and establishment of the withdrawal current in the selective withdrawal of a stratified fluid from a line sink. The experiments were performed in a Plexiglas channel 30·5 ft long 14 in. wide, and filled with a linearly stratified salt solution to a depth of 18 in. The line sink was located at mid-depth. The flow was symmetric about the middepth. Velocity measurements and flow visualization were obtained by neutrally buoyant liquid droplets and vertical dye lines. Density measurements were made by a salinity probe. The development of the upstream velocity field was found by measurement to be brought about by the successive arrival of ‘columnar disturbance modes’, discussed in part 1. Agreement with the theoretical predictions is excellent. A similarity profile of the steady-state horizontal velocity is obtained, and the Froude number based on the thickness of the flowing zone is found to approach a constant value in the essentially inviscid region. The results are compared with field data from the Tennessee Valley Authority (TVA) reservoirs.

In this investigation we study extreme vortex states defined as incompressible velocity fields with prescribed enstrophy ${\mathcal{E}}_{0}$ which maximize the instantaneous rate of growth of enstrophy $\text{d}{\mathcal{E}}/\text{d}t$. We provide an analytic characterization of these extreme vortex states in the limit of vanishing enstrophy ${\mathcal{E}}_{0}$ and, in particular, show that the Taylor–Green vortex is in fact a local maximizer of $\text{d}{\mathcal{E}}/\text{d}t$ in this limit. For finite values of enstrophy, the extreme vortex states are computed numerically by solving a constrained variational optimization problem using a suitable gradient method. In combination with a continuation approach, this allows us to construct an entire family of maximizing vortex states parameterized by their enstrophy. We also confirm the findings of the seminal study by Lu & Doering (Indiana Univ. Math. J., vol. 57, 2008, pp. 2693–2727) that these extreme vortex states saturate (up to a numerical prefactor) the fundamental bound $\text{d}{\mathcal{E}}/\text{d}t<C{\mathcal{E}}^{3}$, for some constant $C>0$. The time evolution corresponding to these extreme vortex states leads to a larger growth of enstrophy than the growth achieved by any of the commonly used initial conditions with the same enstrophy ${\mathcal{E}}_{0}$. However, based on several different diagnostics, there is no evidence of any tendency towards singularity formation in finite time. Finally, we discuss possible physical reasons why the initially large growth of enstrophy is not sustained for longer times.

This paper uses matched asymptotic expansions to study the non-localized (which we refer to as global) boundary layer instabilities generated by free-stream acoustic and vortical disturbances at moderate supersonic Mach numbers. The vortical disturbances produce an unsteady boundary layer flow that develops into oblique instability waves with a viscous triple-deck structure in the downstream region. The acoustic disturbances (which for reasons given herein are assumed to have obliqueness angles that are close to a certain critical angle) generate slow boundary layer disturbances which eventually develop into oblique stable disturbances with an inviscid triple-deck structure in a region that lies downstream of the viscous triple-deck region. The paper shows that both the vortically generated instabilities and the acoustically generated oblique disturbances ultimately develop into modified Rayleigh-type instabilities (which can either grow or decay) further downstream.

A nonlinear analysis is presented of waves propagating around the free surface of water contained in a circular basin of finite uniform depth. The property that distinguishes these waves from unidirectional gravity waves is the occurrence of low-order resonant interactions between the components composing them. Steady waves in the neighbourhood of resonance are calculated in the fully nonlinear problem, when it is shown that multiple families of free-wave solutions, each family having a different set of resonating wave components, are associated with each of the water depths at which resonance occurs. Linear stability calculations indicate that most steady waves dominated by resonating wave components are linearly unstable. The nonlinear time evolution of perturbed waves is calculated as a check on the linear stability results, leading to doubts about the relevance of the linear prediction when marginal instability is said to occur.

A liquid jet falling by gravity ultimately destabilizes by capillary forces. Accelerating as it falls, the jet thins and stretches, causing a capillary instability to develop on a spatially varying substrate. We discuss quantitatively the interplay between instability growth, jet thinning and longitudinal stretching for two kinds of perturbations, either solely introduced at the jet nozzle exit or affecting the jet all along its length. The analysis is conducted for any values of the liquid properties for a sufficiently large flow rate. In all cases, we determine the net gain of the most dangerous perturbation for all downstream distances, thus predicting the jet length, the wavelength at breakup and the resulting droplet size.

The two-dimensional, incompressible flow in a plane sudden expansion is investigated numerically for a systematic variation of the geometry, covering expansion ratios (steps-to-outlet heights) from to . By means of a three-dimensional linear stability analysis global temporal modes are scrutinized. In a symmetric expansion the primary bifurcation is stationary and two-dimensional, breaking the mirror symmetry with respect to the mid-plane. The secondary asymmetric flow experiences a secondary instability to different three-dimensional modes, depending on the expansion ratio. For a moderately asymmetric expansion only one of the two secondary flows (the connected branch) is realized at low Reynolds numbers. Since the perturbed secondary flow does not deviate much from the symmetric secondary flow, both secondary stability boundaries are very close to each other. For very small and very large expansion ratios an asymptotic behaviour is found for suitably scaled critical Reynolds numbers and wavenumbers. Representative instabilities are analysed in detail using an a posteriori energy transfer analysis to reveal the physical nature of the instabilities. Depending on the geometry, pure centrifugal and elliptical amplification processes are identified. We also find that the basic flow can become unstable due to the effects of flow deceleration, streamline convergence and high shear stresses, respectively.

The static pressure-hole problem is investigated both theoretically and experimentally. The influence of all significant dimensionless parameters is brought to light. These parameters represent the effects of the boundary layer, of the velocity gradient and of the wall curvature. A partial linearization makes it possible to propose a formula of correction containing three influence functions which cannot be determined by the theory. A limited number of experiments on appropriate models leads to the determination of these functions in case of practical requirements. So, a method of correction is obtained, but only in incompressible flow. The previous formula has been verified in two complex flows. The importance of the correction on the pressure drag of a slender body is brought to light and the difficulties in the application on the method are emphasized.

Steady potential flow around a two-dimensional bubble with surface tension, either free or attached to a wall, is considered. The results also apply to a liquid drop. The flow and the bubble shape are determined as functions of the contact angle β and the dimensionless pressure ratio γ = (pb − ps)/½ρU2. Here pb is the pressure in the bubble, ps = p∞ + ½ρU2 is the stagnation pressure, p∞ is the pressure at infinity, ρ is the fluid density and U is the velocity at infinity. The surface tension σ determines the dimensions of the bubble, which are proportional to 2σ/ρU2. As γ tends to ∞, the bubble surface tends to a circle or circular arc, and as γ decreases the bubble elongates in the direction normal to the flow. When γ reaches a certain value γ0(β), opposite sides of the bubble touch each other. The problem is formulated as an integrodifferential equation for the bubble surface. This equation is discretized and solved numerically by Newton's method. Bubble profiles, the bubble area, the surface energy and the kinetic energy are presented for various values of β and γ. In addition a perturbation solution is given for γ large when the bubble is nearly a circular arc, and a slender-body approximation is presented for β ∼ ½π and γ ∼ γ0(β), when the bubble is slender.

Numerical integration of the boundary-layer equations associated with flow past a semi-infinite flat plate in the presence of an aligned magnetic field has shown that the solutions are not unique if ε < 1 for certain values of β < 1, where ε is an intrinsic property of the fluid and β a property of conditions at infinity. An analytic explanation of this phenomenon is given here. The main properties as β → 1 of the unique solutions when ε > 1 are elucidated. Further, the equations associated with flow past a solid boundary in which the magnetic field is zero are solved numerically. The solutions appear to be unique but, on the other hand, the maximum value β0, of β, for which they can be found, tends to zero with ε.