A constantly accelerating circular plate was investigated towards understanding the effect of non-stationarity on shear-layer entrainment and topology. Dye visualizations and time-resolved particle image velocimetry measurements were collected for normalized accelerations spanning three orders of magnitude. Increasing acceleration acts to organize shear-layer topology. Specifically, the Kelvin–Helmholtz instabilities within the shear layer better adhered to a circular path and exhibited consistent and repeatable spacing. Normalized starting-vortex circulation was observed to collapse with increasing acceleration, which one might not expect due to increased levels of mixing at higher instantaneous Reynolds numbers. The entrainment rate was shown to increase nonlinearly with increasing acceleration. This was attributed to closer spacing between instabilities, which better facilitates the roll-up of fluid between the shear layer and vortex core. The shear-layer organization observed at higher accelerations was associated with smaller spacings between instabilities. Specifically, analogous point-vortex simulations demonstrated that decreasing the spacing between instabilities acts to localize and dampen perturbations within an accelerating shear layer.

We examine the wave-induced flow of small-amplitude, quasi-monochromatic, three-dimensional, Boussinesq internal gravity wavepackets in a uniformly stratified ambient. It has been known since Bretherton (J. Fluid Mech., vol. 36 (4), 1969, pp. 785–803) that one-, two- and three-dimensional wavepackets induce qualitatively different flows. Whereas the wave-induced mean flow for compact three-dimensional wavepackets consists of a purely horizontal localized circulation that translates with and around the wavepacket, known as the Bretherton flow, such a flow is prohibited for a two-dimensional wavepacket of infinite spanwise extent, which instead induces a non-local internal wave response that is long compared with the streamwise extent of the wavepacket. One-dimensional (horizontally periodic) wavepackets induce a horizontal, non-divergent unidirectional flow. Through perturbation theory for quasi-monochromatic wavepackets of arbitrary aspect ratio, we predict for which aspect ratios which type of induced mean flow dominates. We compose a regime diagram that delineates whether the induced flow is comparable to that of one-, two- or compact three-dimensional wavepackets. The predictions agree well with the results of fully nonlinear three-dimensional numerical simulations.

Large-scale atmospheric flow shares certain attributes with two-dimensional turbulence. In this paper, we study the effect of spherical geometry on two-dimensional turbulence.

Energy transfer is multi-component in spherical geometry in contrast to energy transfer among triads of wave vectors in Cartesian geometry. It follows that energy transfer is more local in spherical than in Cartesian geometry. Enstrophy transfer to higher wavenumbers in spherical geometry is less than enstrophy transfer to higher wavenumbers in Cartesian geometry. Since both energy and enstrophy are inviscid constants of motion, the back transfer of energy is also less in spherical than in Cartesian geometry. Therefore, with a finite viscosity, enstrophy decays more slowly in spherical geometry than in Cartesian geometry. Here these conjectures are tested numerically by spectral methods. The numerical results agree well with the conjectures.

The instability of Poiseuille flow in a pipe is considered for small disturbances. An asymptotic analysis is used which is similar to that found successful in plane Poiseuille flow. The disturbance is taken to travel in a spiral fashion, and comparison of the radial velocity component with the transverse component in the plane case shows a high degree of similarity, particularly near the critical point where the disturbance and primary flow travel with the same speed. Instability is found for azimuthal wave-numbers of 2 or greater, although the corresponding minimum Reynolds numbers are too small to compare favourably with either experiments or the initial restrictions on the magnitude of the wave-number.

The stability of spiral flow between rotating and sliding cylinders is considered. In the limit of narrow gap, a’ modified’ energy theory is constructed. This theory exploits the consequences of assuming the existence of a preferred spiral direction along which disturbances do not vary. The flow is also analyzed from the viewpoint of linearized theory. Both problems depend strongly on the sign of Rayleigh's discriminant, – 2Ωζ. Here Ω is the component of angular velocity, and ζ is the component of total vorticity of the basic flow in the direction perpendicular to the spiral ribbons on which the disturbance is constant. When the discriminant is negative, there is evidently no instability to infinitesimal disturbances, and the spiral disturbance whose energy increases at the smallest R is a roll whose axis is perpendicular to the stream. This restores and generalizes Orr's non-linear results for disturbances having a preferred spiral direction. When the discriminant is positive, the critical disturbances of linear theory and the modified energy theory are spiral vortices. The differences between the energy and linear limits can be made smaller in the restricted class of disturbances with coincidence achieved for axisymmetric disturbances in the rotating cylinder problem in the limit of narrow gap. For the sliding-rotating case, the critical disturbance of the linear theory appears as a periodic wave in a co-ordinate system fixed on the outer cylinder. This wave has a dimensionless frequency equal to - ½ a sin (χ-ψ), where a is the wave-number, χ is the angle between the pipe axis and the direction of motion of the inner cylinder relative to the outer one, and ψ is the disturbance spiral angle.

Instability limits, frequencies and wave-numbers are computed numerically when the cylinder gap is not narrow. These are in even closer agreement with Ludwieg's experimental results than the approximate results which were given in part 1.

The structure and two- and three-dimensional stability properties of a linear array of compressible Stuart vortices (CSV; Stuart 1967; Meiron et al. 2000) are investigated both analytically and numerically. The CSV is a family of steady, homentropic, two-dimensional solutions to the compressible Euler equations, parameterized by the free-stream Mach number $M_{\infty}$, and the mass flux $\epsilon$ inside a single vortex core. Known solutions have $0 \,{<}\,M_{\infty}\,{<}\,1$. To investigate the normal-mode stability of the generally spatially non-uniform CSV solutions, the linear partial-differential equations describing the time evolution of small perturbations to the CSV base state are solved numerically using a normal-mode analysis in conjunction with a spectral method. The effect of increasing $M_{\infty}$ on the two main classes of instabilities found by Pierrehumbert & Widnall (1982) for the incompressible limit $M_{\infty} \,{\rightarrow}\, 0$ is studied. It is found that both two- and three-dimensional subharmonic instabilities cease to promote pairing events even at moderate $M_{\infty}$. The fundamental mode becomes dominant at higher Mach numbers, although it ceases to peak strongly at a single spanwise wavenumber. We also find, over the range of $\epsilon$ investigated, a new instability corresponding to an instability on a parallel shear layer. The significance of these instabilities to experimental observations of growth in the compressible mixing layer is discussed. In an Appendix, we study the CSV equations when $\epsilon$ is small and $M_{\infty}$ is finite using a perturbation expansion in powers of $\epsilon$. An eigenvalue determining the structure of the perturbed vorticity and density fields is obtained from a singular Sturm–Liouville problem for the stream-function perturbation at $O(\epsilon)$. The resulting small-amplitude steady CSV solutions are shown to represent a bifurcation from the neutral point in the stability of a parallel shear layer with a tanh-velocity profile in a compressible inviscid perfect gas at uniform temperature.

The temporal and spatiotemporal stability of thermal plumes is investigated for laminar velocity and temperature profiles, under the Boussinesq approximation, in the far self-similar region as well as in the region close to a finite-size inlet. In the self-similar case, Prandtl and Grashof numbers are systematically varied, and azimuthal wavenumbers $m=0$, 1 and 2 are considered. In the temporal analysis, helical modes of $m=1$ are found to be dominant throughout the unstable parameter space, with few exceptions. Axisymmetric modes typically present smaller growth rates, but they may dominate at very low Prandtl and Grashof numbers. Double-helical modes of $m=2$ are unstable over a very restricted range of parameters. Only the helical $m=1$ mode is found to ever become absolutely unstable, whereas $m=0$ and $m=2$ modes are at most convectively unstable. In a temporal setting, an analysis of the perturbation energy growth identifies buoyancy- and shear-related mechanisms as the two potentially destabilizing flow ingredients. Buoyancy is demonstrated to be important at low Grashof numbers and long wavelengths, whereas classical shear mechanisms are dominant at high Grashof numbers and shorter wavelengths. The physical mechanism of destabilization through the effect of buoyancy is investigated, and an interpretation is proposed. In the near-source region, both axisymmetric and helical modes may be unstable in a temporal sense over a significant range of wavenumbers. However, absolute instability is again only found for helical $m=1$ modes.

This research presents a new theory that explains analytically the behaviour of any fully developed incompressible turbulent pipe flow, steady or unsteady. We propose the name of theory of underlying laminar flow (TULF), because its main consequence is the description of any turbulent pipe flow as the sum of two components: the underlying laminar flow (ULF) and the purely turbulent component (PTC). We use the framework of the TULF to explain analytically most of the important and interesting phenomena reported in He & Jackson (J. Fluid Mech., vol. 408, 2000, pp. 1–38). To do so, we develop a simple model for the pressure gradient and Reynolds shear stress that could be applied to the linearly accelerated pipe flow described by He & Jackson (2000). The following features of the unsteady flow are explained: the deformation undergone by the mean velocity profiles during the transient, the velocity overshoot observed in the more rapid excursions, the dual deformation of mean velocity profiles when overshoots are present, the laminarisation effects described during acceleration, the rapid decrease of the Reynolds shear stress upon approaching the wall that brings forth the laminar sublayer, and some other minor effects. A new field is defined to characterise the degree of turbulence within the flow, directly calculable from the theory itself. Arguably, this new field would describe the degree of turbulence in a pipe flow more accurately than the familiar turbulence intensity parameter. Finally, a paradox is found in the deformation of the unsteady mean velocity profiles with respect to equal-Reynolds-number steady profiles, which is duly explained. The research also predicts the occurrence of mean velocity undershoots if the flow is decreased rapidly enough.

This paper extends the analysis of solute dispersion in electrohydrodynamic flows to the case of band broadening in polyelectrolyte-grafted (soft) capillaries by accounting for the effects of ion partitioning, irreversible catalytic reaction and pulsatile flow actuation. In the Debye–Hückel limit, we present the benchmark solutions of electric potential and velocity distribution pertinent to steady and oscillatory mixed electroosmotic–pressure-driven flows in soft capillaries. Afterwards, the mathematical models of band broadening based on the Taylor–Aris theory and generalized dispersion method are presented to investigate the late-time asymptotic state and all-time evolution of hydrodynamic dispersion, respectively. Also, to determine the heterogeneous dispersion behaviour of solute through all spatiotemporal stages and to relax the constraint of small zeta potentials, a full-scale numerical simulation of time-dependent solute transport in soft capillaries is presented by employing the second-order-accurate finite difference method. Then, by inspecting the dispersion of passive tracer particles in Poiseuille flows, we examine the accuracy of two analytical approaches against the simulation results of a custom-built numerical algorithm. Our findings from hydrodynamic dispersion in Poiseuille flows reveal that, compared to rigid capillaries, more time is required to approach the longitudinal normality and transverse uniformity of injected solute in soft capillaries. For the case of dispersion in mixed electrohydrodynamic flows, it is found that the characteristics of the soft interface, including the thickness, permittivity, fixed charge density and friction coefficient of the polymer coating layer, play a significant role in determining the Taylor diffusion coefficient, advection speed and dispersion rate of solutes in soft capillaries.

In this paper we investigate the effect of an inhomogeneous and unsteady velocity field incident on an array of rigid circular cylinders arranged within a circular perimeter (diameter $D_{G}$) of varying solid fraction $\unicode[STIX]{x1D719}$, where the unsteady flow is generated by placing a cylinder (diameter $D_{G}$) upwind of the array. Unsteady two-dimensional viscous simulations at a moderate Reynolds number ($Re=2100$) and also, as a means of extrapolating to a flow with a very high Reynolds number, inviscid rapid distortion theory (RDT) calculations were carried out. These novel RDT calculations required the circulation around each cylinder to be zero which was enforced using an iterative method. The two main differences which were highlighted was that the RDT calculations indicated that the tangential velocity component is amplified, both, at the front and sides of the array. For the unsteady viscous simulations this result did not occur as the two-dimensional vortices (of similar size to the array) are deflected away from the boundary and do not penetrate into the boundary layer. Secondly, the amplification is greater for the RDT calculations as for the unsteady finite Reynolds number calculations. For the two highest solid fraction arrays, the mean flow field has two recirculation regions in the near wake of the array, with closed streamlines that penetrate into the array which will have important implications for scalar transport. The increased bleed through the array at the lower solid fraction results in this recirculation region being displaced further downstream. The effect of inviscid blocking and viscous drag on the upstream streamwise velocity and strain field is investigated as it directly influences the ability of the large coherent structures to penetrate into the array and the subsequent forces exerted on the cylinders in the array. The average total force on the array was found to increase monotonically with increasing solid fraction. For high solid fraction $\unicode[STIX]{x1D719}$, although the fluctuating forces on the individual cylinders is lower than for low $\unicode[STIX]{x1D719}$, these forces are more correlated due to the proximity of the cylinders. The result is that for mid to high solid fraction arrays the fluctuating force on the array is insensitive to $\unicode[STIX]{x1D719}$. For low $\unicode[STIX]{x1D719}$, where the interaction of the cylinders is weak, the force statistics on the individual cylinders can be accurately estimated from the local slip velocity that occurs if the cylinders were removed.

We present a theoretical and experimental study of the propagation of gravity currents in porous media with variations in the topography over which they flow, motivated in part by the sequestration of carbon dioxide in saline aquifers. We consider cases where the height of the topography slopes upwards in the direction of the flow and is proportional to the $n\text{th} $ power of the horizontal distance from a line or point source of a constant volumetric flux. In two-dimensional cases with $n\gt 1/ 2$, the current evolves from a self-similar form at early times, when the effects of variations in topography are negligible, towards a late-time regime that has an approximately horizontal upper surface and whose evolution is dictated entirely by the geometry of the topography. For $n\lt 1/ 2$, the transition between these flow regimes is reversed. We compare our theoretical results in the case $n= 1$ with data from a series of laboratory experiments in which viscous glycerine is injected into an inclined Hele-Shaw cell, obtaining good agreement between the theoretical results and the experimental data. In the case of axisymmetric topography, all topographic exponents $n\gt 0$ result in a transition from an early-time similarity solution towards a topographically controlled regime that has an approximately horizontal free surface. We also analyse the evolution over topography that can vary with different curvatures and topographic exponents between the two horizontal dimensions, finding that the flow transitions towards a horizontally topped regime at a rate which depends strongly on the ratio of the curvatures along the principle axes. Finally, we apply our mathematical solutions to the geophysical setting at the Sleipner field, concluding that topographic influence is unlikely to explain the observed non-axisymmetric flow.

The interaction of barotropic tidal currents and baroclinic geostrophic eddies is considered theoretically and numerically to determine whether energy can be transferred to an internal wave field by this process. The eddy field evolves independently of the tide, suggesting that it acts catalytically in facilitating energy transfer from the barotropic tide to the internal wave field, without exchanging energy with the other flow components. The interaction is identically zero and no waves are generated when the barotropic tidal current is horizontally uniform. Optimal internal wave generation occurs when the scales of tide and eddy fields satisfy resonant conditions. The most efficient generation is found if the tidal current horizontal scale is comparable to that of the eddies, with a weak maximum when the scales differ by a factor of two. Thus, this process is not an effective mechanism for internal wave excitation in the deep ocean, where tidal current scales are much larger than those of eddies, but it may provide an additional source of internal waves in coastal areas where horizontal modulation of the tide by topography can be significant.

Nonlinear triadic interactions are at the heart of our understanding of turbulence. In flows where waves are present, modes must not only be in a triad to interact, but their frequencies must also satisfy an extra condition: the interactions that dominate the energy transfer are expected to be resonant. We derive equations that allow direct measurement of the actual degree of resonance of each triad in a turbulent flow. We then apply the method to the case of rotating turbulence, where eddies coexist with inertial waves. We show that for a range of wavenumbers, resonant and near-resonant triads are dominant, the latter allowing a transfer of net energy towards two-dimensional modes that would be inaccessible otherwise. The results are in good agreement with approximations often done in theories of rotating turbulence, and with the mechanism of parametric instability proposed to explain the development of anisotropy in such flows. We also observe that, at least for the moderate Rossby numbers studied here, marginally near-resonant and non-resonant triads play a non-negligible role in the coupling of modes.

We consider peeling of an elastic sheet away from an elastic substrate through propagation of a fluid-filled crack along the interface between the two. The peeling is driven by a bending moment applied to the sheet and is resisted by viscous flow towards the crack tip and by the toughness of any bonding between the sheet and the substrate. Travelling-wave solutions are determined using lubrication theory coupled to the full equations of elasticity and fracture. The propagation speed $v$ scales like $M^{3}/\unicode[STIX]{x1D707}\bar{E}^{2}d^{5}=Bd\unicode[STIX]{x1D705}^{3}/144\unicode[STIX]{x1D707}$, where $d$ is the sheet’s thickness, $B=\bar{E}d^{3}/12$ its stiffness, $\bar{E}=E/(1-\unicode[STIX]{x1D708}^{2})$ its plane-strain modulus, $\unicode[STIX]{x1D707}$ the fluid viscosity, $M$ the applied bending moment and $\unicode[STIX]{x1D705}=M/B$ the sheet’s curvature due to bending; and the prefactor depends on the dimensionless toughness. If the toughness is small then there is a region of dry shear failure ahead of the fluid-filled region. The expressions for the propagation speed have been used to derive new similarity solutions for the spread of an axisymmetric fluid-filled blister in a variety of regimes: constant-flux injection resisted by elastohydrodynamics in the tip leads to spread proportional to $t^{4/13}$, $t^{4/17}$ and $t^{7/19}$ for peeling-by-bending, gravitational spreading and peeling-by-pulling, respectively.

This article contains numerical and theoretical results on the circular and polygonal hydraulic jumps in the framework of inertial lubrication theory. The free surface and velocity fields are computed along with cross-sections of the vorticity and pressure, in agreement with experimental data. The forces that drive and resist the instability are identified with the radial shear force, the azimuthal surface tension and the hydrostatic azimuthal force, in addition to a nonlinear term in the radial coordinate. Periodic solutions are obtained from the first orders of a perturbation theory by considering azimuthal symmetries. The thresholds of the instability are defined at closed jumps for discontinuous solutions and at one-sided hydraulic jumps for continuous curves that conserve fluid mass density.

We consider the solution in the time domain of the two-dimensional water-wave scattering by fixed bodies, which may or may not intersect with the free surface. We show how the problem with arbitrary initial conditions can be found from the single-frequency solutions using a generalized eigenfunction expansion, required because the operator has a continuous spectrum. From this expansion we derive simple formulas for the evolution in time of the initial surface conditions, and we present some examples of numerical calculations.

This paper deals with the motion of a single helical vortex in an unbounded inviscid incompressible fluid. The vortex is an infinite tube whose centreline is a helix and whose cross-section is a small circle where the vorticity is uniform and parallel to the centreline. Ever since Joukowsky (Trudy Otd. Fiz. Nauk Mosk. Obshch. Lyub. Estest., vol. 16, 1912, pp. 1–31) deduced that this vortex translates and rotates steadily without change of form, numerous attempts have been made to compute the velocities. Here, Hardin’s (Phys. Fluids, vol. 25, 1982, pp. 1949–1952) solution for the velocity field is used to find new expressions for the linear and angular velocities of the vortex. The theoretical results are verified by numerically computing the velocity at a single point using the Helmholtz integral and the Rosenhead–Moore approximation to the Biot–Savart law, and by numerically simulating the vortex evolution, under the Euler equations, in a triple-periodic cube. The new formulae are also shown to be more accurate than previous results over the whole range of values of the vortex pitch and cross-section.

Results from a series of pipe-flow experiments using a range of water-soluble drag-reducing polymers are presented. Degradation has been investigated by means of multiple passes of the solutions through a pipe. A theory predicting drag reduction in pipe flow has been devised which agrees with the experimental results. Changes in polymer molecular weight due to degradation are taken into account. The analysis is then applied to a turbulent boundary layer with polymer injection.

We consider the nonlinear spin-up/down of a rotating stratified fluid in a conical container. An analysis of axisymmetric similarity-type solutions to the relevant boundary-layer problem, Duck, Foster & Hewitt (1997), has revealed three types of behaviour for this geometry. In general, the boundary layer evolves to either a steady state, or a gradually thickening boundary layer, or a finite-time singularity depending on the Schmidt number, the ratio of initial to final rotation rates, and the relative importance of rotation and stratification.

In this paper we emphasize the experimental aspects of an investigation into the initial readjustment process. We make comparisons with the previously presented boundary-layer theory, showing good quantitative agreement for positive changes in the rotation rate of the container (relative to the initial rotation sense). The boundary-layer analysis is shown to be less successful in predicting the flow evolution for nonlinear decelerations of the container. We discuss the qualitative features of the spin-down experiments, which, in general, are dominated by non-axisymmetric effects. The experiments are conducted using salt-stratified solutions, which have a Schmidt number of approximately 700.

The latter sections of the paper present some stability results for the steady boundary-layer states. A high degree of non-uniqueness is possible for the system of steady governing equations; however the experimental results are repeatable and stability calculations suggest that ‘higher branch’ solutions are, in general, unstable. The eigenvalue spectrum arising from the linear stability analysis is shown to have both continuous and discrete components. Some analytical results concerning the continuous spectrum are presented in an appendix.

A brief appendix completes the previous analysis of Duck, Foster & Hewitt (1997), presenting numerical evidence of a different form of finite-time singularity available for a more general boundary-layer problem.

Up-current-rotated, shoreface-connected ridges are found in various coastal areas around the world. An often-quoted conjecture is that these ridges form during storm conditions through free instabilities in the erodible bed. Under these conditions both waves and currents are expected to play a significant role in the hydrodynamics. Although some existing models have included the effects of waves parametrically in their bottom friction terms and sediment equations, the dynamical effects of wave–current interaction have not been explored. In this paper we begin to rectify this by considering the effects of wave–current interaction on the bed-form instabilities of a simple model. This raises the possibility of unsteady alongshore flows and questions about the roles of wave parameters and boundary conditions, which we address here. We show that the flow is stable under the wave forcing; however the waves do affect the bed-form instability. The main dynamical effect of the waves is in altering the shapes of the unstable modes. Under various conditions, however, waves may enhance or suppress the instability or introduce new unstable modes. They also affect the celerity of the ridges. In addition, we investigate the mechanisms whereby the waves affect the instability. We also show a potential problem with the parameterization in terms of wave orbital velocity.