This paper describes an experimental and numerical study of an intrusion propagating along the interface of a two-layer fluid in a channel. We find that when the density of the intrusion is the depth-weighted mean of the layer densities the interface ahead of the intrusion is undisturbed, but for other densities the interface ahead of the intrusion is displaced vertically. We find that this vertical displacement, which takes the form of an upstream-propagating long wave, depends on the properties of the intrusion and not only on the relative depths of the two layers. For the case when the interface is undisturbed the intrusion propagation speed is a minimum. We develop an energy argument that describes the observed variation of the intrusion speed from this minimum speed as a function of the intrusion and layer densities and the ratio of the layer depths. We also show that if, and only if, the layer depths are equal, the speed of the intrusion is independent of the density of the intrusion.

The lattice-Boltzmann method (LBM) is used to carry out a direct numerical simulation (DNS) of grid-generated turbulence with the view to improve comparison between experimental and numerical results on approximate isotropic turbulence. The grid is made up of four by four floating flat square elements in an aligned arrangement. The Reynolds number based on the Taylor microscale is about 40 at a distance of 70 times the separation between the elements downstream of the grid; this value is comparable to that of many experiments.

While the results compare relatively well with existing experimental data on grid turbulence (grid made up of bars), they highlight the importance of the mesh resolution of the simulation and computational domain size in the decay of turbulence. For example, while a power-law decay could be identified, at least over a short distance, its decay exponent proves to be difficult to determine with good accuracy. This points out the need for simulations (and perhaps experiments too) where all scales are properly solved before conclusions can be drawn.

The interplay between inertia and elasticity is examined in this study for the transient axisymmetric flow of a thin film. The fluid is assumed to emerge from an annulus, as it is driven by axial pressure gradient and/or gravity. The substrate is assumed to be stationary and of arbitrary shape. The boundary-layer equations are generalized for a viscoelastic film obeying the Oldroyd-B constitutive model. These equations are solved by expanding the flow field in terms of orthonormal shape functions in the radial direction and using the Galerkin projection, combined with a time-stepping implicit scheme, and integration along the flow direction. It is found that the viscosity ratio and fluid elasticity can have a significant effect on steady state as well as transient behaviour. It is also found that low-inertia and/or highly elastic fluids tend to accumulate near the annulus, exhibiting a standing wave that grows with time. This behaviour clearly illustrates the difficulty associated with coating viscoelastic high-viscosity fluids. A criterion for film rupture is also established, which is based on the steepening of flow and stress gradients. The topography of the substrate has a drastic effect on the flow as well.

Thermal convection driven by centrifugal buoyancy in a rapidly rotating narrow annular channel is studied in the case of rigid cylindrical walls. The pattern of high-wavenumber thermal Rossby waves becomes increasingly modulated and finally chaotic as the Rayleigh number is increased. Retrograde as well as prograde mean zonal flows are found. The prograde mean flow dominates in fluids with large Prandtl number, while retrograde mean flows are more typical for small Prandtl numbers. Although the basic thermal Rossby waves travel in the prograde direction when the height of the annulus decreases with distance from the axis, their long-wavelength modulations often propagate in the retrograde direction.

For the Saffman–Taylor instability, the inertia of the fluid may become important for high finger speeds. We investigate the effects of inertia on the width of the viscous fingers experimentally. We find that, due to inertia, the finger width can increase with increasing speed, contrary to what happens at small Reynolds number Re. We find that inertial effects need to be considered above a critical Weber number We. In this case it can be shown that the finger width is governed by a balance between viscous forces and inertia. This allows us to define a modified control parameter $1/B'$, which takes the corrections due to inertia into account; on rescaling the experimental data with $1/B'$, they all collapse onto the universal curve for the classical Saffman–Taylor instability. Subsequently, we try to rationalize our observations. Numerical simulations, taking into account a modification of Darcy's law to include inertia, are found to only qualitatively reproduce the experimental findings, pointing to the importance of three-dimensional effects.

We have used magnetic resonance imaging to study the structure of and convection within a solidifying mushy layer formed from an aqueous sucrose solution cooled from above. We focus on the situation in which dissolution channels, known as chimneys, are created by the action of buoyancy-driven convection. We have obtained high-resolution images of the microstructure formed by individual ice platelets and coarser-grained images that average over the platelets to show the geometry of the dissolution channels. We observe that the chimneys are branched and occur only in the lower part of the mushy layer. By acquiring low-resolution images rapidly, we have made detailed measurements of the thickness of the mushy layer, its porosity distribution and the number and total area of the chimneys. The mushy layer is seen to grow in a self-similar manner until internal convection begins, whereafter the solid fraction increases in the lower part of the layer.

The paper presents a multiscale analysis of the intermediate region of the two-dimensional convectively unstable wake past a bluff body. A recent asymptotic expansion solution was used as basic flow (Tordella & Belan, Phys. Fluids, vol. 15, 2003, 1897). This solution was obtained by matching an inner to an outer flow, both of which are Navier–Stokes solutions. By introducing a spatio-temporal multiscaling into the instability problem, an inhomogeneous Orr–Sommerfeld equation and an associated modulation equation are obtained. The streamwise variation of the instability characteristics can then be deduced from the wave modulation, by considering the system to be perturbed by waves with a complex wavenumber that corresponds of the dominant saddle point of the local dispersion relation, taken at different positions downstream of the wake, and at different Reynolds numbers. The corrections of no parallelism are remarkable in the intermediate wake. When the disturbance is related to an early intermediate station, the corrections lead to absolute instability in the upstream portion of the intermediate wake, where, in addition, the spatial growth rate decreases. When the disturbance is related to a section in the far field, conditions of minimal temporal stability are reached about 20 body scales downstream. In the far field the temporal damping increases with the Reynolds number.

The mild-slope equation is an effective approximation for treating the combined effects of refraction and diffraction of infinitesimal water waves, for it reduces the spatial dimension of the linear boundary-value problem from three to two. We extend this approximation to nonlinear waves up to the second order in wave steepness, in order to simplify the inherently three-dimensional task. Assuming that the geometrical complexity is restricted to a finite, though large, horizontal domain, the hybrid-element method designed earlier for linearized problems is modified for the two-dimensional elliptic boundary-value problems at the second order. In the special case of a semi-circular peninsula (or a vertical cylinder on a cliff) in a sea of constant depth, the solution is analytical. Effects of the angle of incidence are examined for the free-surface height along the cylinder. For a cylinder standing on a shoal of radially varying depth, numerical results are discussed.

This work extends the estimator developed in Part 1 of this study to the problem of estimating a turbulent channel flow at $Re_{\tau}\,{=}\,100$ based on a history of noisy measurements on the wall. The key advancement enabling this work is the development and implementation of an efficient technique to extract, from direct numerical simulations, the relevant statistics of an appropriately defined ‘external forcing’ term on the Navier–Stokes equation linearized about the mean turbulent flow profile. This forcing term is designed to account for the unmodelled (nonlinear) terms during the computation of the (linear) Kalman filter feedback gains in Fourier space. Upon inverse transform of the resulting feedback gains computed on an array of wavenumber pairs to physical space, we obtain, as in Part 1, effective and well-resolved feedback convolution kernels for the estimation problem. It is demonstrated that, by applying the feedback so determined, satisfactory correlation between the actual and estimated flow is obtained in the near-wall region. As anticipated, extended Kalman filters (with the nonlinearity of the actual system reintroduced into the estimator model after the feedback gains are determined) outperform standard (linear) Kalman filters on the full system.

A spatial instability of parametrically excited stratified mixing layer flows is considered together with the related temporal instability problem. A relatively simple iteration procedure yielding solutions of both temporal and spatial problems is proposed. Using this procedure a parametric analysis of the temporal and spatial Kelvin–Helmholtz and Holmboe instabilities is performed and characteristic features of the instabilities are compared. Both inviscid and viscous models are considered. The parametric dependence on the mixing layer thickness and on the Richardson and Reynolds numbers is studied. It is shown that in the framework of this study the Gaster transformation is valid for the Kelvin–Helmholtz instability, but cannot be applied to the Holmboe one. The neutral stability curves are calculated for the viscous flow case. It is found that the transition between Kelvin–Helmholtz and Holmboe instabilities is continuous in the spatial case and in the temporal case occurs via the codimension-two bifurcation at which a complex pair of the leading eigenvalues merges into a multiple real eigenvalue. It is also found that for the same governing parameters the spatial upstream and downstream Holmboe waves have different amplification rates and different absolute phase velocities, with larger difference observed at larger Richardson numbers. It is shown that at large Richardson and small Reynolds numbers the primary temporal and spatial instabilities set in as a three-dimensional oblique Holmboe wave.

This paper presents experimental results on the wake of a propelled bluff body towed at a constant horizontal speed in a linearly stratified fluid. Three regimes of the wake have been found, depending on the angle of attack and on the ratio of drag force to propeller thrust. Most of the experiments were obtained in a first regime where a strong momentum flux is created in the wake, which can be oriented backward or forward depending on the ratio of drag force to thrust of the propeller. The velocity amplitude, wake width and Strouhal number of the wake can be predicted by defining a momentum thickness based on the drag coefficient of the bluff body and the thrust of the propeller. A second regime is obtained for a narrow band of towing velocities, with a relative width of 4%, in which the momentum flux is found to vanish. The wake is characterized by the velocity fluctuations; the scaling exponents of the velocity, vorticity and width of the wake are measured. A third regime is obtained for wakes with a small angle of attack, with a null momentum flux. The mean profile of the wake is found to be asymmetric and its amplitude and wake width are measured. Finally, the relevance of these results to the case of a real self-propelled bluff body is discussed. The presence of weak internal waves or of weak fluctuations of background velocity would lead to a wake in the regime with momentum flux, and would allow prediction of the amplitude, width and Strouhal number of the wake.

Time-dependent flows of a Newtonian fluid through periodic arrays of spheres were simulated using the lattice-Boltzmann scheme. By applying a constant body force per unit mass to the fluid, a steady background fluid flow through the array of stationary spheres was first established. Subsequently, a small-amplitude perturbation to the body force, which varied periodically in time, was added and the long-time behaviour of the unsteady flow fields and the forces on the particles were determined. From the simulations, the pressure and friction (shear) forces acting on the particles were determined for a range of conditions. Results on simple cubic lattices are presented. Computations spanned a range of particle volume fractions ($0.1\,{<}\,\phi\,{<}\,0.4$), background flow Reynolds numbers ($0.25 \,{\le}\, \hbox{\it Re}_p\,{\le}\, 60$, where $\hbox{\it Re}_p\,{=}\,2 a u_f / \nu$) and oscillatory flow Reynolds numbers ($0.9 \,{\leq}\, {Re_\omega} \,{\leq}\, 420$ with ${Re_\omega}\,{=}\,2a^2 \omega / \nu$). Here $u_f$ is the superficial velocity of the fluid through the bed, $a$ is the particle radius, $\nu$ is the kinematic viscosity of the fluid, and $\omega$ is the oscillation frequency.

In the limit of ${Re_\omega} \,{\to}\, 0$ the quasi-steady-state drag force was obtained. At low $\hbox{\it Re}_p$ this force approached the steady-state drag force, while its increase with $\hbox{\it Re}_p$ was stronger than the steady-state drag force, similar to that for isolated spheres given by Mei et al. (J. Fluid Mech., vol. 233, 1991, p. 613).

The unsteady force was decomposed into pressure and friction components. The phase angles of these components in the limit ${Re_\omega} \,{\to}\, \infty$ indicate that the virtual mass force contributes to the unsteady pressure force while the history force contributes to the friction force. The remainder of the unsteady friction and pressure forces is attributed to unsteady drag force.

The apparent virtual mass coefficient was found to vary from ${\sim} 0.5$ at high ${Re_\omega}$, which is the well-known limit for isolated spheres in inviscid flows, to ${\sim} 1.0$ at low ${Re_\omega}$. This change is clearly a consequence of viscous effects. The ${Re_\omega}$ at which the transition between these limits occurs increases with $\phi$. The history force exhibits a strong decay towards lower values of ${Re_\omega}$ in accordance with the results of Mei et al. (1991) for isolated spheres; however, the ${Re_\omega}$ value at which this decay sets in increases appreciably with $\phi$. This $\phi$-dependence is associated with the limited separation between the particles available for the Stokes boundary layer.

It was found that the unsteady drag coefficient $\beta'$ varies with ${Re_\omega}$. At low $Re_p$, the drag coefficient initially decreases with increasing ${Re_\omega}$, passes through a minimum and then increases strongly. With increasing ${Re_\omega}$ the relative contribution of pressure and friction forces to the unsteady drag force changes.

We study the statistics of a passive scalar $T({\bm{x},t)$ governed by the advection–diffusion equation with variations in the scalar produced by a steady source. Two important statistical properties of the scalar are the variance, $\sigma^2\equiv \langle T^2 \rangle$, and the entropy production, $\chi \equiv \kappa \langle|\bm{\nabla} T|^2\rangle$. Here $\langle\rangle$ denotes a space–time average and $\kappa$ is the molecular diffusivity of $T$. Using variational methods we show that the system must lie above a parabola in the $(\chi, \sigma^2)$-plane. The location of the bounding parabola depends on the structure of the velocity and the source. To test the bound, we consider a large-scale source and three two-dimensional model velocities: a uniform steady flow; a statistically homogeneous and isotropic flow characterized by an effective diffusivity; a time-periodic model of oscillating convection cells with chaotic Lagrangian trajectories. Analytic solution of the first example shows that the bound is sharp and realizable. Numerical simulation of the other examples shows that the statistics of $T({\bm{x},t)$ lie close to the parabolic frontier in the $(\chi, \sigma^2)$-plane. Moreover in the homogenization limit, in which the largest scale in the velocity field is much less than the scale of the source, the results of the simulation limit to the bounding parabola.

The analysis of the linear stability of a planar Chapman–Jouguet detonation wave is reformulated for an arbitrary caloric (incomplete) equation of state in an attempt to better represent the stability properties of detonations in condensed-phase explosives. Calculations are performed on a ‘stiffened-gas’ equation of state which allows us to prescribe a finite detonation Mach number while simultaneously allowing for a detonation shock pressure that is substantially larger than the ambient pressure. We show that the effect of increasing the ambient sound speed in the material, for a given detonation speed, has a stabilizing effect on the detonation. We also show that the presence of the slow reaction stage, a feature of detonations in certain types of energetic materials, where the detonation structure is characterized by a fast reaction stage behind the detonation shock followed by a slow reaction stage, tends to have a destabilizing effect.

This paper considers the capillary drainage of a thin annular film on the inside or outside of a circular cylinder of radius $a$. A film of uniform thickness and axial length greater than $\pi a$ suffers a Rayleigh instability and evolves to form an axisymmetric structure in which the film thickness varies with axial distance. The fluid is gathered into collars, having axial length $2\pi a$, and shorter lobes; the pressure within each collar or lobe is spatially uniform and adjacent collars and lobes are separated by thin necks. We examine numerically the evolution of this structure and demonstrate that, for sufficiently short cylinders, lobes drain into collars as described by Hammond (J. Fluid Mech. vol. 137, 1983, p. 363). For longer cylinder lengths we find that, in spite of the energetic advantage, neighbouring collars do not drain into one another, and that the neck region between adjacent collars is governed by a similarity solution of the thin-film equation having axial length that varies as $t^{-1/2}$ after time $t$, and film thickness that varies as $t^{-1}$, which is different from that found by Jones & Wilson (J. Fluid Mech. vol. 87, 1978, p. 263).

We also find a new phenomenon: a collar can spontaneously and episodically translate back and forth along the cylinder, on each occasion consuming the lobe ahead and leaving a smaller daughter lobe behind. This motion takes place on several different timescales: the relatively rapid translation is governed by Landau–Levich equations; the collision with a neighbouring collar is governed by the similarity equation for the neck regions ahead and behind; and the delay between one episode of translation and the next is governed via the Landau–Levich equation by a slow peeling process. Asymptotic results for each of the processes of translation, collision and peeling are obtained and are compared with a full numerical solution. Each episode of translation reduces the thickness of the daughter lobe by a factor 0.115, and successive translations back and forth give rise to a lobe thickness that decays on average and on very long timescales like $t^{-1/2}$.

A thin film of fluid trapped beneath a two-dimensional drop sedimenting towards a rigid horizontal plane is described by the same evolution equation, and analogous lobe and collar dynamics are found (Lister, Morrison & Rallison, J. Fluid Mech. vol. 552, 2006, p. 345).

We consider the drainage of fluid trapped beneath a two-dimensional drop that sediments towards a horizontal plane. The governing equation is closely related to that for capillary drainage (without gravity) of an annular film discussed in a companion paper (Lister et al., J. Fluid Mech. vol. 552, 2006, p. 311). When drainage starts, dynamical structures rapidly appear that are usually called dimples in the context of sedimentation. Dimples are constant-pressure regions to which most of the fluid in the film is confined, which are analogous to the collars and lobes that appear in annular capillary drainage.

The process of drainage is controlled by a Bond number, $B$, that measures the relative importance of gravity and surface tension for the sedimenting drop. When $B$ is sufficiently small, all the fluid ultimately drains from a single small dimple and the drop takes a static sessile equilibrium shape. The dimple-drainage process is the same as that of a lobe. When $B$ is sufficiently large, several permanent dimples are formed under the drop, and these exhibit complex dynamics of collision and interaction analogous to that of collars and lobes. No static drop shape is reached, even for long times. For critical values of $B$, fluid may be permanently trapped in one or more stationary dimples (analogous to collars), and families of equilibrium drop shapes are found that depend upon the quantity of trapped fluid.

Turbulent boundary layer skin friction in liquid flows may be reduced when bubbles are present near the surface on which the boundary layer forms. Prior experimental studies of this phenomenon reached downstream-distance-based Reynolds numbers ($Re_{x}$) of several million, but potential applications may occur at $Re_{x}$ orders of magnitude higher. This paper presents results for $Re_{x}$ as high as 210 million from skin-friction drag-reduction experiments conducted in the USA Navy's William B. Morgan Large Cavitation Channel (LCC). Here, a near-zero-pressure-gradient flat-plate turbulent boundary layer was generated on a 12.9 m long hydraulically smooth flat plate that spanned the 3 m wide test section. The test surface faced downward and air was injected at volumetric rates as high as 0.38 m$^{3}$ s$^{-1}$ through one of two flush-mounted 40 $\mu$m sintered-metal strips that nearly spanned the test model at upstream and downstream locations. Spatially and temporally averaged shear stress and bubble-image-based measurements are reported here for nominal test speeds of 6, 12 and 18 m s$^{-1}$. The mean bubble diameter was $\sim$300 $\mu$m. At the lowest test speed and highest air injection rate, buoyancy pushed the air bubbles to the plate surface where they coalesced to form a nearly continuous gas film that persisted to the end of the plate with near-100% skin-friction drag reduction. At the higher two flow speeds, the bubbles generally remained distinct and skin-friction drag reduction was observed when the bubbly mixture was closer to the plate surface than 300 wall units of the boundary-layer flow without air injection, even when the bubble diameter was more than 100 of these wall units. Skin-friction drag reduction was lost when the near-wall shear induced the bubbles to migrate from the plate surface. This bubble-migration phenomenon limited the persistence of bubble-induced skin-friction drag reduction to the first few metres downstream of the air injector in the current experiments.

Gill's semigeostrophic equations of one-layer, rotating channel hydraulics are exactly solved at the critical section, for flows with a finite uniform potential vorticity (PV). A detailed characterization of the critical flow at the sill is consequently obtained, together with a simple, explicit expression for the associated mass transport, that provides a direct extension of the well-known transport formula of zero-PV theory.

The dependence of the mass transport on the PV, for a given average flow energy, is found to be non-monotonic. In particular, the transport increases when the PV is raised from zero, attaining a maximum value that can be up to twice those predicted by the zero-PV theory. This suggests that, contrary to common belief, finite-PV effects may be important in determining the mass transport in real oceanic outflows.

Evolution equations are derived for weakly nonlinear, multi-frequency and directional surface gravity waves propagating from deep to shallow water over weakly two-dimensional bottom topography. A uniform transition from cubic resonances in deep–intermediate water (Stokes regime) to quadratic near resonances in shallow water (Boussinesq regime) is obtained by extending the ordered solution to include additional higher-order terms for the bound wave components. The model assumes a leading-order, alongshore-uniform bottom with a two-dimensional depth perturbation that is incorporated through a Taylor series expansion of the bottom boundary condition. Numerical implementations of the model and comparisons to experimental data are presented that demonstrate the model's ability to describe: (i) cubic wave–wave interactions in deep–intermediate water depth; (ii) harmonic generation over a one-dimensional submerged obstacle; (iii) harmonic generation over two-dimensional topography.

A report of experimental results on dynamo action is presented, obtained at the Karlsruhe dynamo test facility. Former observations concerning the properties of the dynamo magnetic field are corroborated and complemented. The feedback of the dynamo magnetic field on the sodium velocity in the test module is analysed based on measurements employing permanent magnet potential sensors. The evaluation of time signals recorded by these probes shows that a growing dynamo magnetic field transforms hydrodynamic turbulent velocity profiles in channels into magnetohydrodynamic slug flow profiles, indicating a balance between pressure and Lorentz forces. Cross-correlations between time signals of the magnetic induction and the local velocity reveal a significant coherency between apparently random fluctuations of these relevant dynamo quantities.