An entrainment model for lazy turbulent plumes is proposed, the resulting solutions of the plume conservation equations are developed and the implications for plume behaviour are considered and compared with the available experimental data. Indeed, the applicability of the classic solutions of the conservation equations subject to source conditions that produce lazy plumes, i.e. those with suitably high source Richardson number, contains an essential weakness: the underlying assumption of a constant entrainment coefficient. While entrainment models prescribing the dependence of the entrainment coefficient on the local Richardson number have been proposed for forced plumes, corresponding formulations for lazy plumes have not until now been considered. In the context of saline plumes, the model is applied directly. For hot gaseous plumes, we use a modified definition of buoyancy flux to recover a constant buoyancy flux in a non-stratified environment, despite the specific heat varying with the temperature. After a brief review of existing forced-plume formulations of entrainment, a power-law variation is adopted for the lazy plume. The plume equations are solved for the parameter $0\leqslant \unicode[STIX]{x1D714}<1$, where $\unicode[STIX]{x1D714}$ denotes the exponent of the power law. The cases of pure plumes and lazy plumes are then analysed in more detail; to the best of our knowledge this represents the first modelling of variable entrainment for lazy plumes. Specifically, it is shown that classic plume theory is recovered for $\unicode[STIX]{x1D714}=0$, while for $\unicode[STIX]{x1D714}=1/5$ the plume equations may be solved using usual functions (notably polynomials) only. The results of the models for these cases are very similar, which advocates the idea of selecting systematically $\unicode[STIX]{x1D714}=1/5$, instead of $\unicode[STIX]{x1D714}=0$, for cases where the effect of variation of entrainment is weak, since the new model leads to simple calculations. In the case of very lazy plumes, it is shown that, provided that a relevant value of $\unicode[STIX]{x1D714}$ is chosen, the new model reproduces the available experimental results well.

In this paper, we identify a new type of resonant triad which operates in a parallel or nearly parallel shear flow with a symmetric profile. The triad consists of a planar sinuous mode, an oblique sinuous mode and an oblique varicose mode, but is not of the usual subharmonic-resonance form. The development of the triad is studied in the non-equilibrium critical-layer régime. The equations governing the evolution of the modes are derived. We show that the quadratic resonance can significantly enhance the growth of both the oblique sinuous and varicose modes, and may cause them to grow super-exponentially. This can lead to a subsequent stage in which the oblique sinuous mode produces a back reaction on the oblique varicose mode through a phase-locked interaction, causing both oblique modes to evolve even more rapidly. We suggest that the resonant triad is a viable mechanism for the development of three-dimensional structures and varicose components observed in the later stage of plane wake transition.

An experiment conducted on the Oregon continental shelf has provided measurements of velocity fluctuations in the viscous sublayer and buffer layer of the boundary-layer flow. Spectra from the viscous sublayer collapse considerably when scaled as suggested by Bakewell & Lumley (1967), and buffer-layer spectra collapse reasonably well with laboratory spectra when the scaling customarily used in the logarithmic layer is applied. However, in spite of the usefulness of the spectral scaling, the scaled sublayer and buffer-layer spectra from the ocean floor fall below the scaled laboratory spectra in the energy-containing portion of the spectrum, perhaps because the sea floor is not perfectly planar.

The entire free-surface elevation field of a rotating fluid in the laboratory can be imaged and analysed, by using it as a parabolic Newtonian telescope mirror. This ‘optical altimetry’ readily achieves a precision of better than 1 μm of surface elevation. The surface topography corresponds to the pressure field just beneath the surface. It is the streamfunction for the geostrophic hydrostatic circulation, which can be resolved to better than 0.1 mm s−1. Still and animated images thus produced, of the entire surface elevation field, are of value in themselves, and using a projected image (a speckle pattern), have the promise of providing quantitative slope and height field data recovered by PIV (particle imaging velocimetry) techniques. With homogeneous fluid, geostrophic flow is the same at all depths. Yet of equal interest are sheared stratified rotating flows where the surface pressure is associated with inertial waves, convection, and other motions, geostrophic or ageostrophic.

Although the technique is designed for experiments in which Coriolis effects are strong, it is possible to use reflective imaging for flows at such high Rossby number that Coriolis effects are negligible, and hence this becomes a tool of more general interest in non-rotating fluid dynamics (for example, illuminating surface gravity waves).

Examples are given, involving (i) the Taylor–Proudman effect with very slow flows over topography; (ii) quasi-geostrophic and inertial-wave flows over a mountain (f-plane); (iii) inertial waves generated by oscillatory forcing; (iv) Kelvin waves (v) free oscillatory Rossby waves on a polar β-plane; and (vi) stationary waves, blocking, jets and wakes with β-plane zonal flow past a mountain. Movies are available with the online version of the paper.

Thin glass sheets may be manufactured using a two-part process in which a sheet is first cast and then subsequently reheated and drawn to a required thickness. The latter redrawing process typically results in a sheet with non-uniform thickness and with smaller width than the cast glass block. Experiments suggest that the loss of width can be minimized and the non-uniformities can be essentially confined to thickening at the sheet edges if the heater zone through which the glass is drawn is made very short. We present a three-dimensional mathematical model for the redraw process and consider the limits in which (i) the heater zone is short compared with the sheet width, and (ii) the sheet thickness is small compared with both of these length scales. We show that, in the majority of the sheet, the properties vary only in the direction of drawing and the sheet motion is one-dimensional, with two-dimensional behaviour and the corresponding thick edges confined to boundary layers at the sheet extremities. We present numerical solutions to this boundary-layer problem and demonstrate good agreement with experiment, as well as with numerical solutions to the full three-dimensional problem. We show that the final thickness at the sheet edge scales with the inverse square root of the draw ratio, and explore the effect of tapering of the ends to identify a shape for the initial preform that results in a uniform rectangular final product.

For the problem describing steady gravity waves with vorticity on a two-dimensional unidirectional flow of finite depth the following results are obtained. (i) Bounds are found for the free-surface profile and for Bernoulli’s constant. (ii) If only one parallel shear flow exists for a given value of Bernoulli’s constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

A series of initially planar shock waves was allowed to expand from a shock tube into a near half-space. The strength of the wall shock was measured at two positions on the slightly concave front wall. These measurements are compared with shock strengths predicted by a shock-dynamic model based on the cylindrical expansion of a critical shock. Chisnell's (1957) theory is used to account for the effect of the increasing surface area of the expanding wall shock, and Whitham's (1957) treatment to correct for the curvature of the wall. The critical shock strength is obtained from Skews’ (1967) experimental measurements of the Mach number of the self-similar wall shock following two-dimensional diffraction at a 90° edge.

The model predicts the relatively small degree of attenuation observed between the measuring stations, but overestimates the absolute shock strength. The most likely cause is that, in the early stages of expansion, the wall shock experiences further attenuation owing to its interaction with the expanding flow. These effects are shown to be short range and of negligible importance at the first measuring station, 1·86 tube diameters from the axis. Thus, using the experimental results at this station as the starting point, the model predicts accurately the shock strength at 3·76 diameters. It is concluded that Chisnell's theory can be applied to the weakening of the wall shock in ducts with large abrupt changes in cross-section only when the wall shock is some distance from the entrance to the area change.

This paper is concerned with the generation of short gravity waves and their radiation from the outer edge of the turbulent boundary layer and wake of a ship. They arise primarily near the ship's stern. The wave spectrum in the direction of wavenumber vector at an angle (90° – δ) to the ship's track is: \[\Phi_{\delta}(\omega) = \Psi\left(\frac{UT_d}{2l},\frac{U\sin\delta}{c_g},\frac{R}{UT_d}\right)\frac{1}{k_0R}\frac{2l\omega^2}{g^2}\gamma\left(0,\frac{\pi}{l};0,\omega \right),\] where Ψ is dimensionless and a function of three dimensionless parameters. γ is the spectrum of the oscillating motion at the boundary, U the ship speed, Td the decay timescale of the oscillating motion, 2l the lengthscale of the eddies, and R the distance away from the boundary along the wavenumber vector. Generally, Φδ has large values near δ = 0 and small values at large δ; it behaves as 1/R at distances not far from the ship, then may vary slower than 1/R at intermediate distances, and finally behaves as 1/R again at distances far from the ship. These are consistent with the pattern found in SAR images of narrow V-like ship wakes. The method developed here is also applicable to various problems of surface wave generation by turbulence in water.

We study granular chute flow using the classic Saint-Venant approach, with the shear stresses within the granular sheet being incorporated via a friction law due to Pouliquen & Forterre (J. Fluid Mech., vol. 453, 2002, pp. 113–151) and with the in-plane stresses (which are ignored in the traditional formulation for normal fluids) being represented by a viscous-like term recently derived by Gray & Edwards (J. Fluid Mech., vol. 755, 2014, pp. 503–534). On the basis of this model, we predict that the granular sheet is able to sustain monoclinal waves, i.e. travelling shock structures that monotonically connect a thick region of uniform flow to a thinner one. We examine the balance of forces that determine the shape of this particular waveform and give the precise window of system parameters for which monoclinal waves are expected to appear in experiments.

An experimental investigation into flow through shaped gauze screens in two-dimensional and axisymmetric situations has shown that there is disagreement between measured velocity profiles and those computed from the method developed by Elder (1958). This disagreement has been shown to be attributable to the retention of a term of second order in the basic linearization and comparisons between experimental and theoretical profiles omitting this term are in good agreement.

The surface flow, pressure distributions, and vortex-shedding properties of the circular cylinder flow in the subcritical regime modulated by a self-excited vibrating rod were experimentally studied in a closed-circuit wind tunnel for Reynolds numbers between $2\,{\times}\,10^4$ and $1.8\,{\times}\,10^5$. The principle of galloping was employed to induce self-vibration of an elastic rod. The natural (uncontrolled) boundary layer showed significantly different flow patterns in two subranges of Reynolds number: $Re_D \,{<}\,0.55\,{\times}\,10^5$ and $Re_D \,{>}\, 0.55\,{\times}\,10^5$. In the low-Reynolds-number subrange, the laminar separation mode, which was similar to the result reported by previous investigators, was observed. While in the high-Reynolds-number subrange, the separation bubble mode, which was rarely discussed previously, was found. The term separation bubble mode was used to denote the existence of secondary recirculation bubbles induced by the reattachment of separated boundary layers. The surface flow modulated by the vibrating rod presented complex variations in these two subranges. Depending on the position of the control rod, the separation bubble mode or the turbulent separation mode might appear because of the increase of the turbulence kinetic energy. Modulating the surface flow patterns could significantly influence the surface pressure distributions on the main cylinder and the vortex shedding in the wake. Forcing the boundary layer at the positions upstream of the natural separation point would drastically lower the values of the minimum pressure coefficient and the base pressure coefficient, and therefore increase the lift. The drag coefficient, however, would not be apparently decreased. The frequency of the vortex shedding in the wake could ‘lock’ to the rod vibration frequency if the position of the vibrating rod was properly adjusted.

A circulatory flow called a ‘backwash vortex’ forms under waves running up a sloping bed. The backwash vortex is classified into two types: the ‘steady backwash vortex’ and the ‘unsteady backwash vortex’. The former is a steady streaming formed when Re2 < 90, and the latter an unsteady separated flow formed when Re2 > 90, where Re2 is a form of Reynolds number. The ratio of the lengthscale of the backwash vortex to the swash length is proportional to Re2−½ when Re2 [Lt ] 30, and becomes constant when Re2 is sufficiently large.

The long-time limit of the response of incompressible three-dimensional boundary layer flows on infinite swept wedges and infinite swept wings to impulsive forcing is examined using causal linear stability theory. Following the discovery by Lingwood (1995) of the presence of absolute instabilities caused by pinch points occurring in the radial direction in the boundary layer flow of a rotating disk, we search for pinch points in the cross flow direction for both the model Falkner–Skan–Cooke profile of a swept wedge and for a genuine swept-wing configuration. It is shown in both cases that, within a particular range of the parameter space, the boundary layer does indeed support pinch points in the wavenumber plane corresponding to the crossflow direction. These crossflow-induced pinch points do not constitute an absolute instability, as there is no simultaneous pinch occurring in the streamwise wavenumber plane, but nevertheless we show here how they can be used to find the maximum local growth rate contained in a wavepacket travelling in any given direction. Lingwood (1997) also found pinch points in the chordwise wavenumber plane in the boundary layer of the leading-edge region of a swept wing (i.e. at very high flow angles). The results presented in this paper, however, demonstrate the presence of pinch points for a much larger range of flow angles and pressure gradients than was found by Lingwood, and indeed describe the flow over a much greater, and practically significant, portion of the wing.

As shown in Foster & Smith (1989), at large flow force M, Long's self-similar vortex is in the form of a swirling ring-jet, whose axial velocity profile is of sech2 form. At azimuthal wavenumber n of comparable order to the axial wavenumber, linear helical modes of instability are essentially those of the Bickley jet varicose and sinuous modes. However, at small axial wavenumbers, the three-dimensionality of the vortex is important, and the instabilities depend heavily on the effects of the swirl. We explore here the effects of finite Reynolds number Re on these long-wave inertial modes. It is shown that, because the radial velocity scales with Re−1M, the non-parallelism of the flow is more important than the viscous terms in determining the finite-Re behaviour. The three-layer structure of the parallel-flow instability modes remains, but with a critical layer considerably modified by radial velocity. In investigating the critical range Re = O(M3), we find the following: for n > 1, the non-parallelism stabilizes the unstable inertial modes, leading to determination of neutral curves; for n < − 1, the non-parallel effects always destabilize the vortex to these helical modes. Determination of the unstable modes and neutral curves for the n > 1 case requires a computational scheme that accounts for the presence of viscosity. It turns out that the n < 1 (n > − 1) modes are prograde (retrograde) with respect to the rotation of the main vortex.

When two particles collide in a liquid, the impulsive acceleration due to the rebound produces a pressure pulse that is transmitted through the fluid. Detailed measurements were made of the pressure pulse and the motion of the particles by generating controlled collisions with an immersed dual pendulum. The experiments were performed for a range of impact velocities, angles of incidence, and distances between the wall and the pairs of particles. The radiated fluid pressure was measured using a high-frequency-response pressure transducer, and the motion of the particles was recorded using a high-speed digital camera. The magnitude of the impulse pressure was found to scale with the particle velocity, the particle diameter and the density of the fluid. Additionally, a model is proposed to predict the impulse field in the fluid based on the impulse pressure theory. The model agrees well with the experimental measurements.

Leaves falling in air and marine larvae settling in water are examples of unsteady descents due to complex interactions between gravitational and aerodynamic forces. Understanding passive flight is relevant to many branches of engineering and science, ranging from estimating the behaviour of re-entry space vehicles to analysing the biomechanics of seed dispersion. The motion of regularly shaped objects falling freely in homogenous fluids is relatively well understood. However, less is known about how density stratification of the fluid medium affects passive flight. In this paper, we experimentally investigate the descent of heavy discs in stably stratified fluids for Froude numbers of order 1 and Reynolds numbers of order 1000. We specifically consider fluttering descents, where the disc oscillates as it falls. In comparison with pure water and homogeneous saltwater fluid, we find that density stratification significantly enhances the radial dispersion of the disc, while simultaneously decreasing the vertical descent speed, fluttering amplitude and inclination angle of the disc during descent. We explain the physical mechanisms underlying these observations in the context of a quasi-steady force and torque model. These findings could have significant impact on the design of unpowered vehicles and on the understanding of geological and biological transport where density and temperature variations may occur.

Measurements are reported of the fluctuating lift acting on a sphere and the moment acting about the centre of a sphere at supercritical Reynolds numbers (R > 4 × 105). The lift and moment fluctuations are random functions of time which scale with the free-stream dynamic pressure and sphere dimensions. The power spectra of the lift and moment also scale with the above parameters and with the Strouhal number, nd/U. The spectra contain a maximum spectral density at very low frequencies (nd/U < 0·0003) and do not reveal appreciable effects of vortex shedding at discrete frequencies.

Hot wire anemometers were placed near the surface but outside the boundary layer along a great circle in the meridian plane in which the lift was measured. The fluctuating velocity component near the surface on the upstream hemisphere in this meridian plane is highly correlated with the fluctuating lift in the same meridian plane. The correlation betweeen the lift and tangential velocity near the surface suggests that the fluctuating lift is produced by the component of fluctuating bound vorticity about the sphere that is normal to the meridian plane in which the lift force is measured. The fluctuating moment measured about an axis passing through the centre of the sphere and perpendicular to the above meridian plane is almost perfectly correlated with the fluctuating lift (the measured correlation coefficients were 0·99 and 1·00). The fluctuating moment coefficient is very small ($\sqrt{C^2_m}\simeq 5\times 10^{-4}$) compared to the fluctuating lift coefficient ($\sqrt{C^2_L}\simeq 6\times 10^{-2}$). The exceptional correlation between the random lift and moment suggests that the unsteady moment about the sphere centre (which can be produced only by shear stress fluctuations) is caused by the fluctuations of bound vorticity (residing in the boundary layer and wake) that are responsible for the unsteady lift.

We discuss instabilities of fluid films of nanoscale thickness, with a particular focus on films where the destabilising mechanism allows for linear instability, metastability, and absolute stability, depending on the mean film thickness. Our study is motivated by nematic liquid crystal films; however, we note that similar instability mechanisms, and forms of the effective disjoining pressure, appear in other contexts, such as the well-studied problem of polymeric films on two-layered substrates. The analysis is carried out within the framework of the long-wave approximation, which leads to a fourth-order nonlinear partial differential equation for the film thickness. Within the considered formulation, the nematic character of the film leads to an additional contribution to the disjoining pressure, changing its functional form. This effective disjoining pressure is characterised by the presence of a local maximum for non-vanishing film thickness. Such a form leads to complicated instability evolution that we study by analytical means, including the application of marginal stability criteria, and by extensive numerical simulations that help us develop a better understanding of instability evolution in the nonlinear regime. This combination of analytical and computational techniques allows us to reach novel understanding of relevant instability mechanisms, and of their influence on transient and fully developed fluid film morphologies. In particular, we discuss in detail the patterns of drops that form as a result of instability, and how the properties of these patterns are related to the instability mechanisms.

The collision of liquids of different densities is studied theoretically for the case of liquids having wedge-shaped configuration before the impact. Both liquids are assumed to be ideal and incompressible, and the velocity potential theory is used for the flow of each liquid. Surface tension and gravity effects are neglected. The problem is decomposed into two self-similar problems, one for each liquid. Across the interface between the liquids, continuity of the pressure and the normal component of the velocity is enforced through iteration. This determines the shape of the interface and other flow parameters. The integral hodograph method is employed to derive the solution consisting of analytical expressions for the complex-velocity potential, the complex-conjugate velocity, and the mapping function. They are all defined in the first quadrant of a parameter plane, in which the original boundary-value problem is reduced to a system of integro-differential equations in terms of the velocity magnitude and the velocity angle relative to the flow boundary. They are solved numerically using the method of successive approximations. The results are presented through streamlines, interface and free-surface shapes, the pressure and velocity distributions. Special attention is given to the structure of the splash jet rising as a result of the impact.