Laminar hypersonic flow past a sphere is examined on the basis of the constant-density approximation. The Navier-Stokes equations governing the flow are reduced to a nearly parabolic form so that backward influence is essentially eliminated. Two methods of solution are then used on the resulting equations. The first method is the so-called series-truncation method (local similarity), and the second method is an implicit finite-difference method. The solutions from the two methods are compared for various values of the shock Reynolds number. These solutions are also compared with Lighthill's inviscid constant-density solution for high-shock Reynolds number.

We consider non-geostrophic homogeneous baroclinic turbulence without solid boundaries, and we focus on its energetics and dynamics. The homogeneous turbulent flow is therefore submitted to both uniform vertical shear $S$ and stable vertical stratification, parametrized by the Brunt–Väisälä frequency $N$, and placed in a rotating frame with Coriolis frequency $f$. Direct numerical simulations show that the threshold of baroclinic instability growth depends mostly on two dimensionless numbers, the gradient Richardson number $\mathit{Ri}= {N}^{2} / {S}^{2} $ and the Rossby number $\mathit{Ro}= S/ f$, whereas linear theory predicts a threshold that depends only on $\mathit{Ri}$. At high Rossby numbers the nonlinear limit is found to be $\mathit{Ri}= 0. 2$, while in the limit of low $\mathit{Ro}$ the linear stability bound $\mathit{Ri}= 1$ is recovered. We also express the stability results in terms of background potential vorticity, which is an important quantity in baroclinic flows. We show that the linear symmetric instability occurs from the presence of negative background potential vorticity. The possibility of simultaneous existence of symmetric and baroclinic instabilities is also investigated. The dominance of symmetric instability over baroclinic instability for $\mathit{Ri}\ll 1$ is confirmed by our direct numerical simulations, and we provide an improved understanding of the dynamics of the flow by exploring the details of energy transfers for moderate Richardson numbers.

The motion of puffs which move in the direction of the buoyancy force was considered in a previous paper (Richards 1965). We now consider cases in which the directions of motion and of buoyancy are inclined or opposed. The main assumption is that the distributions of velocity inside the puffs are similar throughout the motion. Although this assumption is thought to be accurate only when the surroundings are neutral, we have reason to think that the same assumption may sometimes be used as an approximation to the internal distribution of velocity in other types of environment.

The particular case of motion when the buoyancy force is constant is calculated in detail. The calculations are supported by observations from new laboratory experiments.

New large-eddy simulations of flow over a flexible plant canopy by Dupont et al. (J. Fluid Mech., 2010, this issue, vol. 652, pp. 5–44) have produced apparently paradoxical results. Work over the last three decades had suggested that turbulent eddies could ‘lock onto’ to the waving frequency of uniform cereal canopies. Their new simulations contradict this view, although a resolution may lie in the essentially three-dimensional nature of the instability process that generates the dominant eddies above plant canopies.

This paper deals with the mass transport induced by sea waves propagating over a sloping beach and fully reflected from a wall. It is shown that for moderate slopes the classical recirculation cell structure holds for small Reynolds numbers only. When the Reynolds number is large, the cells interact among themselves giving rise to the merging of the negative cells and the confinement of the positive ones near the bottom. Under such circumstances the fluid moves onshore near the bottom and offshore near the free surface. The seaward decrease of the vorticity produced at the bottom appears to be the reason for the merging phenomenon.

A complete Hamiltonian formalism is suggested for inertial waves in rotating incompressible fluids. Resonance three-wave interaction processes – decay instability and confluence of two waves – are shown to play a key role in the weakly nonlinear dynamics and statistics of inertial waves in the rapid rotation case. Future applications of the Hamiltonian approach to inertial wave theory are investigated and discussed.

Simultaneous steady forced and free convective flow past a heated or cooled body in a semi-infinite porous medium subject to the Darcy–Boussinesq approximation is treated analytically and numerically. A correction term is derived in terms of the Rayleigh (Ra) and Péclet (Pe) numbers for the velocity and temperature fields far from the body; this is subsequently implemented in a numerical treatment, using finite-difference techniques in elliptical coordinates, for 1[les ]Pe[les ]100 and [mid ]Ra[mid ][les ]103. Flow separation is observed for both heating and cooling, perhaps surprisingly so for the former case since the flow near the plate is being accelerated by comparison with the forced convection case. A simple analogy with inviscid flow theory serves to illustrate the manner in which separation eddies are formed for both heating and cooling cases.

Dynamical equations for the interaction of high-wavenumber, high-frequency internal waves with a prescribed, linear, large-scale internal-wave field are obtained from the Boussinesq–Euler equations. The relationship of these ‘induced-diffusion’ interactions to the Taylor–Goldstein equation is discussed. Exact equations are derived in the induced-diffusion limit of McComas & Bretherton (1977) for the evolution of the first and second moments of the small-scale flow when the large-scale flow is assumed random. Estimates of corrections to the induced-diffusion approximation for the Garrett–Munk internal-wave model indicate the domain of applicability of these equations. Computations of the autocorrelation function and action transport in wavenumber and physical space are presented. Severe limitations are found on the applicability of two-time perturbation theory and the resonant-interaction approximation. The high transfer rates found by McComas & Bretherton in the induceddiffusion regime are reduced significantly in the present calculations.

A series of experiments is described in which a fully submerged parallel-plate enclosure is subjected to regular incident waves. The resulting wave-interaction effects are defined in terms of a reflexion coefficient derived from wave-height measurements and a pressure coefficient derived from measurements of pressure in the depths of the enclosure. The latter parameter describes the pressure induced in the enclosure by the passage of regular waves above it and is of particular interest, at present, in having a bearing on the operation and performance of a sub-sea wave-energy converter currently under development. The outcome of these experiments is the partial verification of some recent theoretical work, including the demonstration of a pressure intensification at certain wave frequencies.

These experiments form part of a wider study concerned generally with the performance of immersed, pressure-driven, wave-energy converters.

Here we present experimental and theoretical results for how a stratified fluid, initially rotating as a solid body with constant angular velocity, $\Omega $, within a closed cylinder of square cross-section, is spun up when subject to a small, impulsive increase, $ \mrm{\Delta} \Omega $, in the cylinder’s rotation rate. The fluid’s adjustment to the new state of solid rotation can be characterized by: (a) an inviscid, horizontal starting flow which conserves the vorticity of the initial condition; (b) the eruption of Ekman layer fluid from the perimeter region of the cylinder’s base and lid; (c) horizontal-velocity Rayleigh layers that grow into the interior from the container’s sidewalls; and (d) the formation and decay of columnar vortices in the vertical corner regions. Asymptotic results describe the inviscid starting flow, and the subsequent interior spin-up that occurs due to the combined effects of Ekman suction through the base and lid Ekman layers, and the growth of the sidewall Rayleigh layers. Attention is focused on the flow development over the spin-up time scale ${T}_{s} = {E}^{\ensuremath{-} 1/ 2} {\Omega }^{\ensuremath{-} 1} $, where $E$ is the Ekman number. (The spin-up process over the much longer diffusive time scale, ${T}_{d} = {E}^{\ensuremath{-} 1} {\Omega }^{\ensuremath{-} 1} $, is not considered here.) Experiments were performed using particle imaging velocimetry (PIV) to measure horizontal velocity components at fixed heights within the flow interior and at regular stages during the spin-up period. The velocity data obtained are shown to be in excellent agreement with the asymptotic theory.

We present an experimental and numerical study of the upstream internal wavefield in a channel generated by constant density intrusions propagating into a linearly stratified ambient fluid during the initial phase of translation. Using synthetic schlieren imaging and two-dimensional direct numerical simulations, we quantify this wave motion within the ambient stratified fluid ahead of the advancing front. We show that the height of the neutral buoyancy surface in the ambient fluid determines the vertical modal response with the predominant waves being mode 2 for intrusions near the mid-depth of the channel and mode 1 waves being produced by intrusions nearer the top or bottom of the domain. All higher vertical modes travel slower than the intrusion and so do not appear upstream ahead of the intrusion front. We find the energy flux into this upstream wavefield to be approximately constant, and to be between 10 and 30 % of the rate of available potential energy transfer into the flow.

Here we show that there exist internal gravity waves that are inherently unstable, that is, they cannot exist in nature for a long time. The instability mechanism is a one-way (irreversible) harmonic-generation resonance that permanently transfers the energy of an internal wave to its higher harmonics. We show that, in fact, there are a countably infinite number of such unstable waves. For the harmonic-generation resonance to take place, the nonlinear terms in the free surface boundary condition play a pivotal role, and the instability does not occur in a linearly stratified fluid if a simplified boundary condition, such as a rigid lid or a linearized boundary condition, is employed. Harmonic-generation resonance presented here provides a mechanism for the transfer of internal wave energy to the higher-frequency part of the spectrum hence affecting, potentially significantly, the evolution of the internal waves spectrum.

Errors in static pressure measurements caused by taps which protrude beyond the wall and disturb the boundary layer locally are investigated experimentally. The resulting error is presented as a function of the wall shearing stress, the protrusion height and the probe diameter in a representation similar to the calibration curve of Preston tubes. If suitably plotted, the present results can be made to fit this calibration curve to within the experimental accuracy, in spite of the differences between the geometries being compared.

Experimental work on mixing in microfluidic devices has been of growing importance in recent years. Interest in probing reaction kinetics faster than the minute or hour time scale has intensified research in designing microchannel devices that would allow the reactants to be mixed on a time scale faster than that of the reaction. Particular attention has been paid to the design of microchannels in order to enhance the advection phenomena in these devices. Ultimately, in vitro studies of biological reactions can now be performed in conditions that reflect their native intracellular environments. Liau et al. (Anal. Chem., vol. 77, 2005, p. 7618) have demonstrated a droplet-based microfluidic mixer that induces improved chaotic mixing of crowded solutions in milliseconds due to protrusions (‘bumps’) on the microchannel walls. Liau et al. (2005) have shown it to be possible to mix rapidly plugs of highly concentrated protein solutions such as bovine hemoglobin and bovine serum albumin. The present work concerns an analysis of the underlying mechanisms of shear stress transfer at liquid–liquid interfaces and associated enhanced mixing arising from the protrusions along the channel walls. The role of non-Newtonian rheology and surfactants is also considered within the mixing framework developed by Aref, Ottino and Wiggins in several publications. Specifically, we show that proportional thinning of the carrier fluid lubrication layer at the bumps leads to greater advection velocities within the plugs, which enhances mixing. When the fluid within the plugs is Newtonian, mixing will be enhanced by the bumps if they are sufficiently close to one another. Changing either the rheology of the fluid within the plugs (from Newtonian to non-Newtonian) or modifying the mechanics of the carrier fluid-plug interface (by populating it with insoluble surfactants) alters the mixing enhancement.

In a previous paper (Grimshaw 1979) the resonant over-reflection of internal gravity waves from a vortex sheet was considered in the weakly nonlinear regime. It was shown there that the time evolution of the amplitude of the vortex sheet displacement was balanced by a cubic nonlinearity. For one vortex sheet mode, symmetrical with respect to the interface, it was shown that a steady finite-amplitude wave was possible. For the other, asymmetric modes, a singularity develops in a finite time. In the present paper, that analysis is extended by replacing the vortex sheet with a thin shear layer of thickness α2, where α is the amplitude of the shear layer displacement. The effect of this extension is to introduce a linear growth rate term in the amplitude equation, which is otherwise unaltered. The linear growth rate can be computed from a formula due to Drazin & Howard (1966, p. 67). The effect on the modes is that the symmetric mode is linearly damped and requires sustained forcing to be observed, while the asymmetric modes are slightly destabilized by the linear term and, as in the vortex-sheet model, develop a singularity in finite time.

Experimental results are presented for two turbulent boundary-layer experiments conducted at a free-stream Mach number of 4 with wall cooling. The first experiment examines a constant-temperature cold-wall boundary layer subjected to adverse and favourable pressure gradients. It is shown that the boundary-layer data display good agreement with Coles’ general composite boundary-layer profile using Van Driest's transformation. Further, the pressuregradient parameter βK found in previous studies to correlate adiabatic highspeed data with low-speed data also correlates the present cooled-wall high-speed data. The second experiment treats the response of a constant-pressure highspeed boundary layer to a near step change in wall temperature. It is found that the growth rate of the thermal boundary layer within the existing turbulent boundary layer varies considerably depending upon the direction of the wall temperature change. For the case of an initially cooled boundary layer flowing onto a wall near the recovery temperature, it is found that δT ∼ x whereas the case of an adiabatic boundary layer flowing onto a cooled wall gives δT ∼ x½. The apparent origin of the thermal boundary layer also changes considerably, which is accounted for by the variation in sublayer thicknesses and growth rates within the sublayer.