Boundary-layer receptivity in the leading-edge region of a cambered thin airfoil is analysed for the case of a low-Mach-number flow. Acoustic free-stream disturbances are considered. Asymptotic results based on large Reynolds number ($U^2 / \omega \nu \,{\gg}\, 1$) are presented, supplemented by numerical solutions. The influence of mean aerodynamic loading enters the theory through a parameter $\mu$, which provides a measure of the flow speed variations in the leading-edge region, due to flow around the leading edge from the lower surface to the upper. A Strouhal number based on airfoil nose radius, $S\,{=}\,\omega r_n/U$, also enters the theory. The variation of the receptivity level as a function of $\mu $ and $S$ is analysed. Modest levels of aerodynamic loading are found to decrease the receptivity level for the upper surface of the airfoil, while the receptivity is increased for the lower surface. For larger angles of attack close to the critical angle for boundary layer separation, a local rise in the receptivity occurs for the upper surface, while on the lower surface the receptivity decreases. These effects are more pronounced at larger values of $S$. While the Tollmien–Schlichting wave does not emerge until a downstream distance of $O((U^2 / \omega \nu)^{1/3} U / \omega)$, the amplitude of the Tollmien–Schlichting wave is influenced by the acoustic free-stream disturbances only in a relatively small region near the leading edge, of length approximately $4 U/\omega$. The numerical receptivity coefficients calculated, together with the asymptotic eigenfunctions presented, provide all the necessary information for transition analysis from the interaction of acoustic disturbances with leading-edge geometry.

Strong near-shore earthquakes are the most frequent sources of tsunamis in many oceans of the world. In the framework of the nonlinear shallow-water theory, the initial sea-surface tsunami elevation is assumed to equal the sea-floor co-seismic displacement produced by the seismic event. This is quantified by means of the analytical formulas due to Okada (1985, 1992), dealing with seismic faults buried in an elastic medium. In this work the propagation of tsunamis is studied along two-dimensional profiles on an idealized constant-slope sea bed, an approximation that allows one to reduce the governing nonlinear equations to a linear problem by means of the classical Carrier & Greenspan (1958) approach. We introduce an analytical solution that is sufficiently general to account for initial conditions associated with paradigmatic cases of sea-bottom deformations produced by near-shore earthquakes, such as subsidence or uplift of the coastal area, and can be also used to treat more complex deformations. The main result is that the amplification of the tsunami height at the coast is found to range between approximately 1 and 2. The amplification is around 1 for tsunamis induced by earthquakes with their epicentre inland and tends to grow as the fault moves seaward. We restrict our analysis to earthquakes that dislocate the shore region. Within the class of sources that we consider, the tsunamis that are most amplified are the ones having initial profiles with a crest–trough–crest system or conversely with a trough–crest–trough system. The bottom slope is found to have no effect on tsunami run-ups and run-downs, but to influence tsunami periods and tsunami speed remarkably. Breaking analysis shows that wave breaking does not occur if the initial wave height is less than 8–9 m, and that the simplest sea-level profiles, which are associated with earthquakes with their epicentre on land, are not expected to break even if their initial height exceeds 19 m.

The similarity of the coherent structures (streaks and hairpin vortices) naturally occurring in different fully developed bounded turbulent shear flows as well as in transitional flows suggests the existence of a basic mechanism responsible for the formation of these structures, under various base flow conditions. The common elements for all such flows are the shear of the base flow and the presence of a localized vortical disturbance. The objective of the present numerical study is to examine the capability of a simple model of interaction, between a localized vortical disturbance and laminar uniform unbounded shear flow, to reproduce the generation mechanism and characteristics of the coherent structures that naturally occur in turbulent bounded shear flows. The effects of the disturbance ‘localized character’ in the stream-wise and spanwise directions as well as its initial orientation relative to the base flow are investigated by using several geometries of the initial disturbance. The results demonstrate that a small-amplitude initial disturbance (linear case) eventually evolves into a streaky structure independent of its initial geometry and orientation, whereas, a large-amplitude disturbance (strongly nonlinear case) evolves into a hairpin vortex (or a packet of hairpin vortices) independent of its geometry over a wide range of the initial disturbance orientations. The main nonlinear effects are: (i) self-induced motion, which results in the movement of the vortical structure relative to the base flow and the destruction of its streamwise symmetry, and (ii) the alignment of the vortical structure with the vorticity lines. This is unlike the linear case, where there is a strong deviation of the vorticity vector from the direction of the vortical structure. Qualitatively, the disturbance evolution is sufficiently independent of its initial geometry, whereas the associated quantitative characteristics, i.e. inclination angle, centre and strength (which is governed by the transient growth mechanism), strongly depend on the disturbance geometry. The Reynolds number is found to have a negligible effect on the kinematics of the vortical structure, but does have a significant effect on its transient growth. Finally, the formation of the asymmetric hairpin vortex, due to minor spanwise asymmetries of the initial disturbance, is demonstrated.

The motion of a light particle in an eccentrically rotating cylinder provides a method for verifying stationary history lift force effects at low but non-zero particle Reynolds numbers. We examine the flow in detail using a Lagrangian equation of motion for constant, non-zero-vorticity flows, and we predict a measurable and stationary contribution of history lift effects that can be verified experimentally with current experimental techniques. Because the history lift contribution is relevant only under certain conditions (which are determined in this work), the present flow configuration also allows one to isolate history drag effects under normal gravitation conditions without resorting to the tethered-particle arrangement used in previous works. We formulate and solve the trajectory problem for light particles that attain stable orbital motion, and we propose an experimental concept that makes possible the study of individual contributions of Lagrangian forces to the motion of small particles in viscous flows.

Numerical simulations are used to study a series of reduced models of homogeneous, rotating flow at moderate Rossby numbers $Ro \,{\approx}\, 0.1$, for which both numerical and physical experiments show the generation of quasi-two-dimensional vortices and symmetry breaking in favour of cyclones. A random force at intermediate scales injects energy at a constant average rate. The nonlinear term of reduced models is restricted to include only a subset of triad interactions in Fourier space. Reduced models of near-resonant, non-resonant and near two-dimensional triad interactions are considered. Only the model of near resonances reproduces all of the important characteristics of the full simulations: (i) efficient energy transfer from three-dimensional forced modes to two-dimensional large-scale modes, (ii) large-scale energy spectra scaling approximately as $k_h^{-3}$, where $k_h$ is the wavenumber in the plane perpendicular to the axis of rotation, and (iii) strong cyclone/anticyclone asymmetry in favour of cyclones. Non-resonances, defined as the complement to near resonances, act to reduce the energy transfer to large scales.

Particle image velocimetry experiments have been performed in a turbulent boundary-layer wind tunnel in order to study the coherent structures taking part in the generation and preservation of wall turbulence. The particular wind tunnel used is suitable for high-resolution experiments ($\delta \gt 0.3$ m) at high Reynolds numbers (up to $R_{\theta} = 19\,000$ in the present results). Eddy structures were identified in instantaneous velocity maps in order to determine their mean characteristics and possible relationships between these structures. In the logarithmic region, the results show that the observed eddy structures appear to organize like elongated vortices, tilted downstream, mainly at an angle of about 45° and having a cane shape. The characteristics of these vortices appear here to be universal in wall units for $R_{\theta}\,{\leq}\,19\,000$. They seem to find their origin at a wall distance of about 25 wall units as quasi-streamwise vortices and to migrate away from the wall while tilting to form a head and a leg. Away from the wall, their radius increases and their vorticity decreases very slowly so that their circulation is nearly constant. Near the wall, the picture obtained is in fair agreement with existing models. The analysis of the results indicates a universality of the buffer-layer mechanism, even at low Reynolds number, and a sensitivity of the logarithmic region to low-Reynolds-number effects.

We analyse the currently popular vortex identification criteria that are based on point-wise analysis of the velocity gradient tensor. A new measure of spiralling compactness of material orbits in vortices is introduced and using this measure a new local vortex identification criterion and requirements for a vortex core are proposed. The inter-relationships between the different criteria are explored analytically and in a few flow examples, using both zero and non-zero thresholds for the identification parameter. These inter-relationships provide a new interpretation of the various criteria in terms of the local flow kinematics. A canonical turbulent flow example is studied, and it is observed that all the criteria, given the proposed usage of threshold, result in remarkably similar looking vortical structures. A unified interpretation based on local flow kinematics is offered for when similarity or differences can be expected in the vortical structures educed using the different criteria.

Six direct numerical simulations of turbulent time-evolving strained plane wakes have been examined to investigate the response of a wake to successive irrotational plane strains of opposite sign. The orientation of the applied strain field has been selected so that the flow is the time-developing analogue of a spatially developing wake evolving in the presence of either a favourable or an adverse streamwise pressure gradient. The magnitude of the applied strain rate $a$ is constant in time $t$ until the total strain e$^{at}$ reaches about 4. At this point, a new simulation is begun with the sign of the applied strain being reversed (the original simulation is continued as well). When the total strain is reduced back to its original value of 1, yet another simulation is begun with the strain again being reversed back to its original sign. This is done for both initially ‘favourable’ and initially ‘adverse’ strains, providing simulations for each of these strain types from three different initial conditions. The evolution of the wake mean velocity deficit and width is found to be similar for all the ‘adversely’ strained cases, with both measures rapidly achieving exponential growth at the rate associated with the cross-stream expansive strain e$^{at}$. In the ‘favourably’ strained cases, the wake widths approach a constant and the velocity deficits ultimately decay rapidly as e$^{-2at}$. Although all three of these cases do exhibit the same asymptotic exponential behaviour, the time required to achieve this is longer for the cases that have been previously adversely strained (by $at \,{\approx}\, 1$). The evolution described above is not consistent with the predictions of classical self-similar analysis; a more general ‘equilibrium similarity solution’ is required to describe the results. Examination of these simulations confirms that the wake width and mean velocity deficit evolutions observed in Rogers (2002) are not a result of the particular initial condition used in that work. At least for the cases considered here, the wake Reynolds number and the ratio of the turbulent kinetic energy to the square of the wake mean velocity deficit are determined nearly entirely by the total strain. For these measures, the order in which the strains are applied does not matter and the changes brought about by the strain are nearly reversible. The wake mean velocity deficit and width, on the other hand, differ by about a factor of 3 when the total strain returns to 1, depending on whether the wake was first ‘favourably’ or ‘adversely’ strained. The strain history is important for predicting the evolution of these quantities.

The effect of the wall-normal diffusion on the spanwise spreading of a steady passive scalar interface is computed for a laminar channel in which the Péclét number, $\Pe$, is high but the velocity profile is parabolic. Two regimes are found according to whether the dimensionless streamwise coordinate $\tilde{x}$ is smaller or larger than $\Pe$. In both cases the mixing layer spreads as $\tilde{x}^{1/2}$ to the lowest approximation in $\Pe^{-1}$, although with different numerical coefficients. When $\tilde{x},{\ll}\,\Pe$ there is a faster growth of order $\tilde{x}^{1/3}$ that is restricted to boundary layers near the wall. The intermediate region between those two limits is universal, and is computed numerically. Quantitative results are given that should be useful to experimentally measure diffusion coefficients. The results are easily generalizable to other velocity profiles.

The heating from above of an initially homogeneous layer of solid crystals, saturated liquid and glass ballotini (an inert matrix filler) is considered both experimentally and theoretically. The heat flux causes crystals at the top of the layer to dissolve, forming liquid which, being more concentrated and dense than the interstitial liquid below, drives convection in the lower layer. Mixing of this high-concentration liquid into the lower layer leads to precipitation, thereby releasing latent heat which raises the temperature of the lower layer. Dissolution of solid crystals from the top leaves behind a closely packed layer of glass ballotini overlain by a layer of clear liquid, both of which deepen with time. The initially homogeneous porous medium thus develops into a three-layer stratified system of (from the top): clear liquid; clear liquid with close-packed ballotini; and the evolved initial assemblage of solid crystals, ballotini and saturated liquid. Data from laboratory experiments compare well with analytical and numerical results from a one-dimensional theoretical model. The model is based on the concept that the heat supplied from above is used entirely for the dissolution of solid crystals at the upper boundary of the lower layer. The resulting compositional convection redistributes the dissolved salt uniformly throughout the lower layer, where it partly recrystallizes to restore chemical equilibrium. The crystallization and associated release of latent heat leads to a gradual and uniform increase of both the solid fraction and temperature of the lower layer. Some geological consequences of the model are presented in the concluding section.

The intrusion of a fixed volume of fluid which is released from rest and then propagates horizontally at the neutral buoyancy level in a vertically stratified ambient fluid is investigated. The density change is linear, in a restricted layer or over the full depth of the container, and locks of both rectangular and cylindrical shapes are considered. A closed one-layer shallow-water inviscid formulation is used to obtain solutions of the initial-value problem. Similarity solutions for the large-time developed motion and an approximate box model are also presented. The results are corroborated by numerical solutions of the full two-dimensional Navier–Stokes equations and comparisons with previously published experiments. It is shown that the model is a versatile predictive tool which clarifies essential features of the flow field. Accurate insights are provided concerning the governing dimensionless parameters and the major features of the motion. In particular, the theory predicts and explains: ($a$) the fact that the initial propagation is with constant speed for intrusions released from a rectangular lock; ($b$) the effect of the shape of the lock on the motion; ($c$) the spread with time at some power in the developed stage; and ($d$) the sub-critical (compared to the mode 2 linear waves) speed in a full-depth stratified container configuration. The main deficiency of the shallow-water model is that internal gravity waves are not incorporated, but some insight into this effect is provided by the comparisons with the Navier–Stokes simulations and experiments.

Equations relating the energy flux, energy dissipation rate, and pressure within a three-dimensional vibrofluidized bed are derived and solved numerically, using only observable system properties, such as particle number, size, mass and coefficient of restitution, to give the granular temperature and packing fraction distributions within the bed. These are compared with results obtained from positron emission particle tracking experiments and the two are found to be in good agreement, without using fitting parameters, except at high altitudes when using a modified heat law including a packing fraction gradient term. Criteria for the onset of the Knudsen regime are proposed and the resulting temperature profiles are found to agree more closely with the experimental distributions. The model is then used to predict the scaling relationship between the height of the centre of mass and mean weighted bed temperature with the number of particles in the system and the excitation level.

Steady incompressible laminar flow of an electrically conducting fluid down a helically symmetric pipe is investigated with regard to possible dynamo action. Both the fluid motion and the magnetic field are assumed to be helically symmetric, with the same pitch. Such a velocity field can be represented by its down-pipe component, $v$, and a streamfunction $\Psi$ defining the secondary cross-pipe flow.

The helical geometry automatically links the cross-pipe and down-pipe field components and permits laminar dynamo action. It is found that the relatively weak secondary motion, which is always present in real pipe flows, has an inhibitory effect on the magnetic field growth and frequently suppresses dynamo action completely. In such a case for large magnetic Reynolds number ($R_m{\to}\infty$) the asymptotic structure of the neutral mode is analysed using a streamline integral approach.

Kinematic velocity fields, without the cross-pipe flow ($\Psi{=}0$), usually generate a dynamo even for perfectly conducting walls. For large $R_m$ the growing modes are shown to have a two-layer structure with rapid tangential variation.

For appropriate pipe geometry, steady pressure-driven pipe flow is found to drive a dynamo for moderate values (${\sim}1000$) of the magnetic Reynolds number.

Grease ice is an agglomeration of disk-shaped ice crystals, named frazil ice, which forms in turbulent waters of the Polar Oceans and in rivers as well. It has been recognized that the property of grease ice that it damps surface gravity waves could be explained in terms of the effective viscosity of the ice slurry. This paper is devoted to the study of the dynamics of a suspension of disk-shaped particles in a gravity wave field. For dilute suspensions, depending on the strength and frequency of the external wave flow, two orientation regimes of the particles are predicted: a preferential orientation regime with the particles rotating in coherent fashion with the wave field, and a random orientation regime in which the particles oscillate around their initial orientation while diffusing under the effect of Brownian motion. For both motion regimes, the effective viscosity has been derived as a function of the wave frequency, wave amplitude and aspect ratio of the particles. Model predictions have been compared to wave attenuation data in frazil ice layers grown in wave tanks.

The motion of a torque-free slender axisymmetric fibre in simple shear flow is examined theoretically for small but finite ${\textit{Re}}$, where ${\textit{Re}}$ is the Reynolds number based on the fibre length, and is a measure of the inertial forces in the fluid. In the limit of zero inertia, an axisymmetric particle in simple shear is known to rotate indefinitely in any of an infinite single-parameter family of periodic orbits, originally found by Jeffery (1922) – a degenerate situation wherein the particular choice of orbit is dictated by the initial orientation of the particle. We use a generalization of the well-known reciprocal theorem for Stokes flow to derive the orbit equations, to $O({\textit{Re}})$, for the slender fibre. The structure of the equations bears some resemblance to those previously derived by Leal (1975) for a neutrally buoyant fibre in a viscoelastic (second-order) fluid. It is thereby shown that fluid inertia, for small ${\textit{Re}}$, leads to a slow $O({\textit{Re}})$ drift of the rotating fibre toward the shearing plane, thereby eliminating the aforementioned degeneracy. For Reynolds numbers above a critical value, ${\textit{Re}}_c\,{=}\, ({15}/{4 \pi})(\ln \kappa/\beta \kappa)\sin^{-2}\theta$, the fibre ceases to rotate, however, instead drifting monotonically towards the shearing plane. The limiting stationary orientation in the flow–gradient plane makes an angle $\phi_f$ with the flow direction, where $\phi_f \,{=}\, 4\pi {\textit{Re}}/(15 \ln \kappa) + \{16 \pi {\textit{Re}}^2/(225(\ln \kappa)^2-1/(\beta^2\kappa^2) \}^{1/2}$ is an increasing function of ${\textit{Re}}$. Here, $\kappa$ is the fibre aspect ratio, $\theta$ is the angle made by the fibre with the vorticity axis, and $\beta$ is an $O(1)$ coefficient related to the Jeffery period of the rotating fibre.

High-speed incompressible flow past a thin airfoil in a uniform stream is considered. When the angle of attack for a solid airfoil exceeds a certain critical value, the boundary layer in the leading-edge region separates in a process known to lead to dynamic stall. Here suction near the leading edge is studied as a means of controlling separation and thereby inhibiting dynamic stall. First, steady boundary-layer solutions are obtained to determine the nature of suction distributions required to suppress separation on an airfoil at an angle of attack beyond the critical value (for a solid wall). Unsteady boundary-layer solutions are then obtained, using a combination of Eulerian and Lagrangian techniques, for an airfoil at an angle of attack exceeding the critical value; the effects of various parameters associated with the finite-length suction slot, its location and the suction strength are considered. Major modifications of the Lagrangian numerical method are required to account for suction at the wall. It is determined that substantial delays in separation can be achieved even when the suction is weak, provided that the suction is initiated at an early stage.