The effects of viscosity on Kelvin–Helmholtz instability in a channel are studied using three different theories; a purely irrotational theory based on the dissipation method, an exact rotational theory and a hybrid irrotational–rotational theory. These new results are compared with previous results from a viscous irrotational theory. An analysis of the neutral state is conducted and its predictions are compared with experimental results related to the transition from a stratified-smooth to a stratified-wavy or slug flow. For values of the gas fraction greater than about 0.20, there is an interval of velocity differences for which the flow is unstable for an interval of wavenumbers between two cutoff wavenumbers, k− and k+. For unstable flows with a velocity difference above that interval or with gas fractions less than 0.20, k− = 0. The maximum critical relative velocity that determines the onset of instability can be found when the kinematic viscosity of the gas and liquid are the same. This critical value is surprisingly achieved when both fluids are inviscid. The neutral curves from the analyses of potential flow of viscous fluids and the hybrid method, the only theories that account for the viscosity of both fluids in this work, indicate that the critical velocity does not change with the viscosity ratio when the kinematic viscosity of the liquid is greater than a critical value. For smaller liquid viscosities, the critical relative velocity decreases.

A tiny air bubble can be entrapped at the bottom of a solid sphere that impacts onto a liquid pool. The bubble forms due to the deformation of the liquid surface by a local pressure buildup inside the surrounding gas, as also observed during the impact of a liquid drop on a solid wall. Here, we perform a perturbation analysis to quantitatively predict the initial deformations of the free surface of a liquid pool as it is approached by a solid sphere. We study the natural limits where the gas can be treated as a viscous fluid (Stokes flow) or as an inviscid fluid (potential flow). For both cases we derive the spatiotemporal evolution of the pool surface, and recover some of the recently proposed scaling laws for bubble entrapment. On inserting typical experimental values for the impact parameters, we find that the bubble volume is mainly determined by the effect of gas viscosity.

An experimental study on localized secondary instability of unsteady streamwise streaks in bypass boundary layer transition under an elevated level of free-stream turbulence has been carried out mainly using the particle image velocimetry (PIV) technique. Simultaneous orthogonal dual-plane PIV measurements were performed for a concurrent examination of the transitional flow features in both wall-normal and spanwise planes. These quantitative and simultaneous visualizations clearly show the wall-normal view of a low-speed streak undergoing sinuous/varicose motion in the spanwise plane. An oscillating shear layer in the wall-normal plane is found to be associated with the sinuous/varicose streak oscillation in the spanwise plane. Further, these measurements indicate that a localized secondary instability wavepacket can originate near the boundary layer edge. The time-resolved PIV measurements in the wall-normal plane clearly show how an instability develops on a lifted-up inclined shear layer and leads to flow breakdown. The estimated wavelength and convection velocity of such instabilities are found to compare well with those calculated from the one-dimensional linear stability analysis of the spatially averaged velocity profiles associated with the lifted-up shear layers. The time-resolved PIV measurements in the spanwise plane also facilitate quantitative visualizations of sinuous and varicose instabilities. These measurements experimentally confirm that a varicose instability at the juncture of an incoming high-speed streak and a downstream low-speed streak can eventually lead to the formation of lambda structures. The estimated convection velocity, wavelength and growth rate of these instabilities are found to be consistent with the numerical results reported in the literature. Moreover, the streak secondary instability is found to be apparent in the velocity contours, while the estimated streak amplitude is approximately 30 % of the free-stream velocity.

We study the dynamics of line plumes on the bottom horizontal plate in turbulent convection over six decades of Rayleigh number $(10^5<\mathit{Ra}_w<10^{11})$ and two decades of Prandtl number or Schmidt number ($0.7<\mathit{Pr}<5.3$, $\mathit{Sc}=602$). From the visualisations of these plumes in a horizontal plane close to the plate, we identify the main dynamics as (i) motion along the plumes, (ii) lateral merging of the plumes and (iii) initiation of the plumes; various other minor types of motion also occur along with these main dynamics. In quantifying the three main motions, we first find that the spatiotemporal mean velocity along the length of the plumes ($\overline{V_{sh}}$) scales as the large-scale flow velocity ($V_{LS}$), with the fraction of the length of the plumes affected by shear increasing with $\mathit{Ra}_w$ as $L_{ps}/L_p\sim \mathit{Ra}_w^{0.054} \mathit{Pr}^{-0.12}$. The mean time of initiation of the plumes $\overline{t^{*}}$, scales as the diffusive time scale near the plate, $Z_w^2/\alpha $, where $Z_w$ is the appropriate length scale near the plate, in agreement with Howard (Proc. 11th Int. Congress Applied Mechanics, Munich, 1964, pp. 1109–1115). Merging occurs in a large fraction of the area of the plate, with ${\sim }70\, \%$ of the length of the plumes undergoing merging at $\mathit{Ra}_w\approx 10^{11}$ and $\mathit{Sc}= 602$. The fraction of the length of the plumes that undergoes merging decreases with increase in $\mathit{Ra}_w$ as, $L_{pm}/L_p \sim \mathit{Ra}_w^{-0.054} \mathit{Pr}^{0.12}$; the exponents of $\mathit{Ra}_w$ and $\mathit{Pr}$ being of the same magnitude but of opposite sign as that in the relation for $L_{ps}/L_p$. Measurements of the locational means of the velocities of merging of the plumes $(V_m)$ show that $V_m$ is a constant during each merging cycle at any location. However, the values of these constant velocities depend on the location and the time of measurement, since the merging velocities are affected by the local shear, which is a function of space and time at any $\mathit{Ra}_w-\mathit{Pr}$ combination. The merging velocities at all $\mathit{Ra}_w$ and $\mathit{Pr}$ have a common lognormal distribution, but their mean and variance increased with increasing $\mathit{Ra}_w$ and decreasing $\mathit{Pr}$. Using mass and momentum balance of the region between two merging plumes, we show that the spatiotemporal mean merging velocities ($\overline{V_m}$), which are an order lower than $\overline{V_{sh}}$, scale as the entrainment velocity at the sides of the plumes, averaged over the height of the diffusive layer near the plate. This implies that $\overline{V_m}$ scales as the diffusive velocity scale near the plate $\nu /Z_w$. The Reynolds number in terms of $\overline{V_m}$ and the layer height $H(\mathit{Re}_H)$ scales as $\mathit{Ra}_w^{1/3}$, in the same way as the Nusselt number ($\mathit{Nu}$) scales approximately; therefore $\mathit{Re}_{H}\sim \mathit{Nu}$. These relations imply that $\mathit{Re}_w= \overline{V_m}Z_w/\nu $ a Reynolds number near the plate, is an invariant for a given fluid in turbulent convection.

The steady motion of a flat surfboard propelled by a solitary wave is considered. The shape of the free surface and the flow of the fluid are determined numerically by series truncation for flows without spray or splash. These flows all bifurcate from the uniform horizontal flow at the critical value of the Froude number. Various limiting cases of these special flows are described analytically. Flows past submerged hydrofoils are discussed also.

The model developed in Part 1 (Lister 1994) for the solidification of hot fluid flowing in a thin buoyancy-driven layer between cold solid but freely deformable boundaries is extended to study the case of continual release of fluid. In this model lubrication theory was applied to reduce the equations of mass and heat conservation to a kinematic-wave equation and an advection-diffusion equation, which were coupled by the rate of solidification. The equations allow the source flux to be specified, and the cases of constant input and of flux proportional to a power of time are considered here. The structure of the flow differs significantly from the case of constant-volume release considered in Part 1. The advective resupply of heat prevents the flow from solidifying completely at the source and, if the initial fluid temperature is greater than the melting temperature of the solid, will in fact lead to rapid melting near the source. A perturbation expansion is used to describe the development of thermal boundary layers at the flow margins and the initial self-similar extension of the zone of melting. As the flow propagates beyond its thermal entry length, the fluid temperature falls to the liquidus value and melting gives way to solidification. At large times nearly all of the fluid supplied solidifies against the margins of the flow but, provided the source flux decreases less rapidly than t−½, sufficient reaches the nose of the flow that the flow continues to increase in length indefinitely. Analytic solutions are given for this long-time regime showing, for example, that the length increases asymptotically like t1/2 for constant-flux input. The theoretical solutions, which are calculated by a combination of analytic and numerical methods, may be used to describe the propagation of a dyke fed by a large body of magma through the Earth's lithosphere or the flow of lava down the flanks of a volcano during an extensive period of eruption.

The coupled Kuramoto–Sivashinsky (CKS) equations for multilayer downflowing films are derived and explored. The CKS equations exhibit a wealth of dynamical behaviour, displaying travelling periodic waves, regular and chaotic-like patterns, coexistence of different attractors, and perfect and imperfect synchronization of the interfaces. New physical effects are found, such as suppression of the Rayleigh–Taylor instability for heavy-top stratified films, and new surface-tension-driven instability.

Guided by computation, we theoretically calculate the steady flow driven by the Marangoni stress due to a surfactant introduced on a fluid interface at a constant rate. Two separate extreme cases, where the surfactant dynamics is dominated by the adsorbed phase or the dissolved phase, are considered. We focus on the case where the size of the surfactant source is much smaller than the size of the fluid domain, and the resulting Marangoni stress overwhelms the viscous forces so that the flow is strongest in a boundary layer close to the interface. We derive the resulting flow in a region much larger than the surfactant source but smaller than the domain size by approximating it with a self-similar profile. The radially outward component of fluid velocity decays with the radial distance $r$ as $r^{-3/5}$ when the surfactant spreads in an adsorbed phase, and as $r^{-1}$ when it spreads in a dissolved phase. Universal flow profiles that are independent of the system parameters emerge in both the cases. Three hydrodynamic signatures are identified to distinguish between the two cases and verify the applicability of our analysis with successive stringent tests.

We report the results of an experimental study of magnetohydrodynamic damping of sidewall convection in a rectangular enclosure filled with gallium. In particular we investigate the suppression of convection when a steady magnetic field is applied separately in each of the three principal directions of the flow. The strongest damping of the steady flow is found for a vertical magnetic field, which is in agreement with theory. However, we observe that the application of a field transverse to the flow provides greater damping than a longitudinal one, which seems to contradict available theory. We provide a possible resolution of this apparent dichotomy in terms of the length scale of the experiment.

We explore the three-dimensional interaction of higher acoustic modes with bias-flow perforated liners in cylindrical and annular ducts. Pressure fluctuations in the vicinity of the liners excite the production and shedding of vorticity from the rims of apertures in the liners. An effective liner compliance is used which accounts for this transfer of acoustical into vortical energy. The investigation is facilitated by a Green's function solution of the Helmholtz equation in a lined section of duct, allowing calculation of the amplitudes of exiting wave modes due to incoming acoustic disturbances. A system containing an arbitrary number of concentric liners and the hollow cavities formed between them can be modelled. The results for an incident plane wave are compared with those from our previously developed one-dimensional model, with excellent agreement. We demonstrate that results for all modes that travel parallel to the liner, including higher circumferential modes in narrow annular gaps, exhibit self-similar behaviour, and that liner design rules developed for planar duct modes can be adapted accordingly. The acoustic absorption can be strongly enhanced by downstream duct reflection for wavelengths larger than twice the liner length, but is less affected at higher frequencies that allow persistent pressure minima along the liner. Across larger frequency ranges, the liner systems are shown to permit two types of resonance associated with the duct and the cavities, respectively, and a third of Helmholtz type, associated with the system as a whole. The effect of these resonances on incident modes is demonstrated, and, in particular, we explore their enhancement of acoustic absorption.

A freely floating motion-trapping structure can be defined as one or more rigid bodies floating on the surface of a fluid which extends to infinity in at least one direction, whose free motion under its natural hydrostatic restoring force is coupled to that of the surrounding fluid in such a way that no waves are radiated to infinity. The resulting local time-harmonic oscillation of the structure and the surrounding fluid is called a motion-trapped mode. Such a structure would, if displaced slightly from its equilibrium position and released, ultimately oscillate indefinitely at the trapped-mode frequency. Previous examples of motion-trapping structures have been devised using an inverse approach in which the shape of pairs of such structures is determined implicitly by sketching certain streamlines. In this paper an alternative direct approach to the construction of motion-trapping structures in the form of a pair of identical floating cylinders of rectangular cross-section in two dimensions is presented. It is also shown that a thick-walled axisymmetric heaving circular cylinder can act as a motion-trapping structure.

When very light particles are sprinkled on a resonating horizontal plate, inverse Chladni patterns are formed. Instead of going to the nodal lines of the plate, where they would form a standard Chladni pattern, the particles are dragged to the antinodes by the air currents induced by the vibration of the plate. Here we present a detailed picture of the mechanism using numerical simulations involving both the particles and the air. Surprisingly, the time-averaged Eulerian velocity, commonly used in these type of problems, does not explain the motion of the particles: it even has the opposite direction, towards the nodal lines. The key to the inverse Chladni patterning is found in the averaged velocity of a tracer particle moving along with the air: this Lagrangian velocity, averaged over a vibration cycle, is directed toward the antinodes. The Chladni plate thus provides a unique example of a system in which the Eulerian and Lagrangian velocities point in opposite directions.

Flow around a high-speed rotating circular cylinder for $Re\leqslant 500$ is investigated numerically. The Reynolds number is defined as $UD/\unicode[STIX]{x1D708}$ with $U$, $D$ and $\unicode[STIX]{x1D708}$ being the free-stream flow velocity, the diameter of the cylinder and the kinematic viscosity of the fluid, respectively. The aim of this study is to investigate the effect of a high rotation rate on the wake flow for a range of Reynolds numbers. Simulations are performed for Reynolds numbers of 100, 150, 200, 250 and 500 and a wide range of rotation rates from 1.6 to 6 with an increment of 0.2. Rotation rate is the ratio of the rotational speed of the cylinder surface to the incoming fluid velocity. A systematic study is performed to investigate the effect of rotation rate on the flow transition to different flow regimes. It is found that there is a transition from a two-dimensional vortex shedding mode to no vortex shedding mode when the rotation rate is increased beyond a critical value for Reynolds numbers between 100 and 200. Further increase in rotation rate results in a transition to three-dimensional flow which is characterized by the presence of finger-shaped (FV) vortices that elongate in the wake of the cylinder and very weak ring-shaped vortices (RV) that wrap the surface of the cylinder. The no vortex shedding mode is not observed at Reynolds numbers greater than or equal to 250 since the flow remains three-dimensional. As the rotation rate is increased further, the occurrence frequency and size of the ring-shaped vortices increases and the flow is dominated by RVs. The RVs become bigger in size and the flow becomes chaotic with increasing rotation rate. A detailed analysis of the flow structures shows that the vortices always exist in pairs and the strength of separated shear layers increases with the increase of rotation rate. A map of flow regimes on a plane of Reynolds number and rotation rate is presented.

Building upon the recent experimentally verified modelling of turbulent plumes which are subject to decreases in their source strength (Scase et al., J. Fluid Mech., vol. 563, 2006b, p. 443), we consider the complementary case where the plume's source strength is increased. We consider the effect of increasing the source strength of an established plume and we also compare time-dependent plume model predictions for the behaviour of a starting plume to those of Turner (J. Fluid Mech., vol. 13, 1962, p. 356).

Unlike the decreasing source strength problems considered previously, the relevant solution to the time-dependent plume equations is not a simple similarity solution. However, scaling laws are demonstrated which are shown to be applicable across a large number of orders of magnitude of source strength increase. It is shown that an established plume that is subjected to an increase in its source strength supports a self-similar ‘pulse’ structure propagating upwards. For a point source plume, in pure plume balance, subjected to an increase in the source buoyancy flux F0, the rise height of this pulse in terms of time t scales as t3/4 while the vertical extent of the pulse scales as t1/4. The volume of the pulse is shown to scale as t9/4. For plumes in pure plume balance that emanate from a distributed source it is shown that the same scaling laws apply far from the source, demonstrating an analogous convergence to pure plume balance as that which is well known in steady plumes. These scaling law predictions are compared to implicit large eddy simulations of the buoyancy increase problem and are shown to be in good agreement.

We also compare the predictions of the time-dependent model to a starting plume in the limit where the source buoyancy flux is discontinuously increased from zero. The conventional model for a starting plume is well approximated by a rising turbulent, entraining, buoyant vortex ring which is fed from below by a ‘steady’ plume. However, the time-dependent plume equations have been defined for top-hat profiles assuming only horizontal entrainment. Therefore, this system cannot model either the internal dynamics of the starting plume's head or the extra entrainment of ambient fluid into the head due to the turbulent boundary of the vortex ring-like cap. We show that the lack of entrainment of ambient fluid through the head of the starting plume means that the time-dependent plume equations overestimate the rise height of a starting plume with time. However, by modifying the entrainment coefficient appropriately, we see that realistic predictions consistent with experiment can be attained.

The onset of thermal convection in a finite rotating cylinder is investigated using direct numerical simulations of the Navier–Stokes equations with the Boussinesq approximation in a regime in which spatio-temporal complexity is observed directly after onset. The system is examined in the non-physical limit of zero centrifugal force as well as with an experimentally realizable centrifugal force, leading to two different paths to Küppers–Lortz-like spatio-temporal chaos. In the idealized case, neglecting centrifugal force, the onset of convection occurs directly from a conduction state, resulting in square patterns with slow roll switching, followed at higher thermal driving by straight roll patterns with faster roll switching. The case with a centrifugal force typical of laboratory experiments exhibits target patterns near the theoretically predicted onset of convection, followed by a rotating wave that emerges via a Hopf bifurcation. A subsequent Hopf bifurcation leads to ratcheting states with sixfold symmetry near the axis. With increasing thermal driving, roll switching is observed within the ratcheting lattice before Küppers–Lortz-like spatio-temporal chaos is observed with the dissolution of the lattice at a slightly stronger thermal driving. For both cases, all of these states are observed within a 2% variation in the thermal driving.

When traditional linearized theory is used to study gravity–capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude-number or low-Bond-number limits can be derived, but here, the solutions invariably predict a waveless surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. By applying techniques in exponential asymptotics to this problem, we have discovered the existence of new classes of gravity–capillary waves, from which the usual linear solutions form but a special case. In this paper, we present the initial theory for deriving these waves through a study of gravity–capillary flow over a linearized step. This will be done using two approaches: in the first, we derive the surface waves using the standard method of Fourier transforms; in the second, we derive the same result using exponential asymptotics. Ultimately, these two methods give the same result, but conceptually, they offer different insights into the study of the low-Froude-number, low-Bond-number problem.