We compare two-dimensional vorticity and passive scalar cascades seen as a gradient enhancement process. Our criteria are based on conditional averages of the first and second Lagrangian derivatives of vorticity and passive scalar gradients in relation to the local flow geometry. In order to interpret these criteria, transient properties are derived for random vorticity and scalar fields, showing that the second-order Lagrangian derivatives of vorticity and passive scalar gradients may behave differently. Cascades obtained in numerical simulations of decaying and forced incompressible turbulence are analysed. First-order analysis reveals that the direct cascade in elliptic domains is more efficient than previously suspected. While several first-order diagnostics collapse to a single curve for vorticity and passive scalars, second-order diagnostics consistently show that the vorticity gradient exhibits faster temporal fluctuations than the passive scalar gradient, a property which we anticipate qualitatively in the study of random fields.

We consider the spin-up of a two-layer, stably (density) stratified fluid in a rotating container with an axisymmetric sloping base and cylindrical walls. Details of the spin- up readjustment mechanisms are presented under the assumption of small impulsive changes in the rotation rate of the container. It is shown that the relative positions of the density interface and the discontinuity in wall slope determine the qualitative large-time spin-up response of the fluid. The density interface leads to a spin-up readjustment in each of the fluid layers that is essentially independent. However, when the density interface is below the boundary-slope discontinuity, a sub-region of the upper layer is predicted to readjust in an algebraic rather than exponential manner. A detailed sequence of laboratory experiments have been performed to confirm the predictions of the linear spin-up analysis.

The method of characteristics is used to calculate the flow ahead of an impulsively started piston moving at constant velocity. Particular attention is paid to the development of weak shock waves which are either fully or partly dispersed at very large distances from the piston. It is found that the global features of the flows may be represented in similarity form, and the graphs obtained allow extrapolation to very weak waves.

The present investigation is concerned with natural convection in a wedge-shaped domain induced by constant isothermal heating at the water surface. Complementary to the study of daytime heating by solar radiation relevant to nearshore regions of lakes and reservoirs previously reported by the same authors, this study focuses on sensible heating imposed by the atmosphere when it is warmer than the water body. A semi-analytical approach coupled with scaling analysis and numerical simulation is adopted to resolve the problem. Two flow regimes are identified depending on the comparison between the Rayleigh number and the inverse of the square of the bottom slope. For the lower Rayleigh number regime, the entire flow domain eventually becomes isothermal and stationary. For the higher Rayleigh number regime, the flow domain is composed of two distinct subregions, a conductive subregion near the shore and a convective subregion offshore. Within the conductive subregion, the maximum local flow velocity occurs when the thermal boundary layer reaches the local bottom, and the subregion eventually becomes isothermal and stationary. In the offshore convective subregion, a steady state is reached with a distinct thermal boundary layer below the surface and a steady flow velocity. The dividing position between the two subregions and the major time and velocity scales governing the flow development in both subregions are proposed by the scaling analysis and validated by corresponding numerical simulation.

This paper describes an investigation of the acoustic properties of a rigid plate with a periodic pattern of holes, in a compressible, ideal, inviscid fluid in the absence of mean flow. Leppington & Levine (J. Fluid Mech., vol. 61, 1973, pp. 109–127) obtained an approximation of the reflection and transmission coefficients of a plane wave incident on an infinitely thin plate with a rectangular array of perforations, assuming that a characteristic size of the perforations is negligible relative to that of the unit cell of the grating, itself assumed to be negligible relative to the wavelength. One part of the present study is of methodological interest. It establishes that it is possible to extend their approach to thick plates with a skew grating of perforations, thus confirming recent results of Bendali et al. (SIAM J. Appl. Math., vol. 73 (1), 2013, pp. 438–459), but in a much simpler way without using complex matched asymptotic expansions of the full wave or to a grating of multipoles. As is well-known, effective compliances for the plate can then be derived from the corresponding approximations of the reflection and transmission coefficients. These compliances are expressed in terms of the Rayleigh conductivity of an isolated perforation. Consequently, in one other part of the present study, the methodology recently introduced by Laurens et al. (ESAIM, Math. Model. Numer. Anal., vol. 47 (6), 2013, pp. 1691–1712) to obtain sharp bounds for the Rayleigh conductivity has been extended to include the case for which the openings of the perforations on the upper and lower sides of the plate are elliptical in shape. This not only enables the determination of these bounds and of the associated reflection and transmission coefficients for actual plates with tilted perforations but also yields single expressions covering all usual cases of perforations: straight or tilted with a circular or an elliptical cross-section.

Linear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich & Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.

A numerical solution of the Navier-Stokes equations of motion by means of finite-difference forms of the vorticity and continuity equations is presented. This is applied to the study of the flow of an incompressible fluid produced by the motion from rest of a piston in a cylindrical tube of circular cross-section.

Experiments at high Reynolds number indicated the presence in the starting flow of a ring vortex which was not reproduced by computation. Iteration to determine the stream function was not found to be necessary to achieve 1% accuracy. Omitting iteration is equivalent to only slightly disturbing the flow. An additional random disturbance applied to the flow at each time step was found to result in the production of the ring vortex.

The initial development of turbulence in the shear layer of a circular jet is observed to show disturbances of increasing scale at discrete frequencies in an approximate 4:2:1 sequence. These are completely confined within the laminar shear layer and convect at 0·59 of the jet velocity. Such relatively regular behaviour was not observed once the disturbances became larger in transverse dimension than the laminar shear layer.

In the development of the subsequent turbulent shear layer it was found that the fluctuating turbulent shear did not scale with local mean shear, but rather scaled more closely with the minus half power of distance from the apparent origin of the laminar shear layer emanating from the nozzle. The scale of the shear fluctuations departed from growth in proportion to shear layer thickness, remaining almost constant. In these aspects it appears that the turbulent shear layer is not well described by a similar growth with axial distance.

The shear fluctuations convected at a speed closer to the local mean velocity than do velocity fluctuations and showed a relatively more patchy distribution with a distinct rotational sense and no reversal of skewness across the layer. Velocity fluctuations showed phase lags approaching $\frac{1}{3}\pi$ relative to shear fluctuations suggesting that dominant velocity disturbances are those associated with entrainment behind concentrations of rotation.

The application of turbulent plume theory in describing the dynamics of emptying filling boxes, control volumes connected to an infinite exterior through a series of openings along the upper and lower boundaries, has yielded novel strategies for the natural ventilation of buildings. Making the plume laminar and having it fall through a porous medium yields a problem of fundamental significance in its own right, insights from which may be applied, for example, in minimizing the contamination of drinking water by geologically sequestered $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\mathrm{CO}}_2$ or the chemicals leached from waste piles. After reviewing the theory appropriate to rectilinear and axisymmetric plumes in porous media, we demonstrate how the model equations may be adapted to the case of an emptying filling box. In this circumstance, the long-time solution consists of two ambient layers, each of which has a uniform density. The lower and upper layers comprise fluid that is respectively discharged by the plume and advected into the box through the upper opening. Our theory provides an estimate for both the height and thickness of the associated interface in terms of, for example, the source volume and buoyancy fluxes, the outlet area and permeability, and the depth-average solute dispersion coefficient, which is itself a function of the far-field horizontal flow speed. Complementary laboratory experiments are provided for the case of a line source plume and show very good agreement with model predictions. Our measurements also indicate that the permeability, $k_f$, of the lower opening (or fissure) decreases with the density of the fluid being discharged, a fact that has been overlooked in some previous studies, wherein $k_f$ is assumed to depend only on the fissure geometry.

Oceanic large- and meso-scale flows are nearly balanced in forces between Earth’s rotation and density stratification effects (i.e. geostrophic, hydrostatic balance associated with small Rossby and Froude numbers). In this regime advective cross-scale interactions mostly drive energy toward larger scales (i.e. inverse cascade). However, viscous energy dissipation occurs at small scales. So how does the energy reservoir at larger scales leak toward small-scale dissipation to arrive at climate equilibrium? Here we solve the linear instability problem of a balanced flow in a rotating and continuously stratified fluid far away from any boundaries (i.e. an interior jet). The basic flow is unstable not only to geostrophic baroclinic and barotropic instabilities, but also to ageostrophic instabilities, leading to the growth of small-scale motions that we hypothesize are less constrained by geostrophic cascade behaviours in a nonlinear regime and thus could escape from the inverse energy cascade. This instability is investigated in the parameter regime of moderate Rossby and Froude numbers, below the well-known regimes of gravitational, centrifugal, and Kelvin–Helmholtz instability. The ageostrophic instability modes arise with increasing Rossby number through a near-degeneracy of two unstable modes with coincident phase speeds. The near-degeneracy occurs in the neighbourhood of an identified criterion for the non-integrability of the ‘isentropic balance equations’ (namely with the absolute vertical vorticity and the horizontal strain rate associated with the basic flow), beyond which development of an unbalanced component of the flow is expected. These modes extract energy from the basic state with large vertical Reynolds stress work (unlike geostrophic instabilities) and act locally to modify the basic flow by reducing the isopycnal Ertel potential vorticity gradient near both its zero surface and its critical surface (phase speed equal to basic flow speed).

The current work analyses the onset characteristics of Rayleigh–Bénard convection in confined two-dimensional two-layer systems. Owing to the interfacial interactions and the possibilities of convection onset in the individual layers, the two-layer systems typically exhibit diverse excitation modes. While the attributes of these modes range from the non-oscillatory mechanical/thermal couplings to the oscillatory standing/travelling waves, their regimes of occurrence are determined by the numerous system parameters and property ratios. In this regard, the current work aims at characterising their respective influence via methodical linear and fully nonlinear analyses, carried out on fluid systems that have been selected using the concept of balanced contrasts. Consequently, the occurrence of oscillatory modes is found to be associated with certain favourable fluid combinations and interfacial heights. The further branching of oscillatory modes into standing and travelling waves seems to additionally rely on the aspect ratio of the confined cavity. Specifically, the modulated travelling waves have been observed to occur (amidst standing wave modes) at discrete aspect ratios for which the onset of oscillatory convection happens at unequal fluid heights. This behaviour corresponds to the typical $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}m$:$n$ resonance where the critical wavenumbers of convection onset in the layers are dissimilar. Based on all of these observations, an attempt has been made in the present work to identify the oscillatory excitation modes with a reduced number of non-dimensional parameters.

We performed numerical experiments on a one-dimensional elastic solid oscillating in a two-dimensional viscous incompressible fluid with the intent of discerning the interplay of vorticity and elastodynamics in flapping wing propulsion. Perhaps for the first time, we have established the role of foil deflection topology and its influence on vorticity generation, through spatially and temporally evolving foil slope and curvature. Though the frequency of oscillation of the foil has a definite role, it is the phase relation between foil slope and pressure that determines thrust or drag. Similarly, the phase difference between flapping velocity, and pressure and inertial forces, determine the power input to the foil, and in turn drives propulsive efficiency. At low frequencies of oscillation, the sympathetic slope and curvature of deformation of the foil allow generation of leading-edge vortices that do not separate; they cause substantial rise in pressure between the leading edge and mid-chord. The circulatory component of pressure is determined primarily by the leading-edge vortex and therefore thrust too is predominantly circulatory in origin at low frequencies. In the intermediate and high-frequency range, thrust and drag on the foil spatially alternate and non-circulatory forces dominate over circulatory and viscous forces. For the mass ratios we simulated, thrust due to flapping varies quadratically as a function of Strouhal number or trailing-edge flapping velocity; further, the trailing edge flapping velocities peak at the same set of frequencies where the thrust is also a maximum. Propulsive efficiency, on the other hand, is roughly a mirror image of the thrust variation with respect to Strouhal number. Given that most instances of flapping propulsion in nature are primarily through distributed muscular actuation that enables precise control of deformation shape, leading to high thrust and efficiency, the results presented here are pointers towards understanding some of the mechanisms that drive thrust and propulsive efficiency.

We consider the interaction of two identical disks freely falling side by side in a fluid at rest for Reynolds numbers ranging from 100 to 300, corresponding to rectilinear and oscillatory paths. For the three aspect ratios of the disks investigated, we observed that the bodies always repel one another when the horizontal distance between their centres of gravity is less than 4.5 diameters. They never come closer for distances spanning between 4.5 and 6 diameters. Beyond the latter distance, the disks appear indifferent to each other. For both rectilinear and periodic paths, the repulsion effect is weak, leading to an overall horizontal drift lower than 3 % of the vertical displacement. We propose a model for the repulsion coefficient Cr, which decreases with the separation distance between the bodies and is inversely proportional to the aspect ratio of the bodies, Cr thus being stronger for the thicker ones. Furthermore, in the case of the oscillatory paths, we show that the effect of the interaction reduces to the repulsion effect, since the characteristics of the oscillatory motion of each disk appear unaffected by the presence of the companion disk and no synchronization is observed between the paths, nor between the wakes, of the two disks.