A ship, towing a heavier-than-water cable with a neutrally buoyant slender cylinder attached to the downstream end of the cable, is considered. The neutrally buoyant element contains a sonar array. Linear changes in the ship's velocity cause perturbations of both the cable and cylinder. The form of transverse vibrations of the neutrally buoyant cylinder was determined in Part 1. The propagation of disturbances along the cable is investigated in this paper. In particular, the effectiveness of the cable at isolating the sonar array from forcing due to unsteady ship motion is examined.

The cable and cylinder are found to be stable under constant towing conditions. It is therefore appropriate to investigate their response to forcing. Meanderings in the ship's track produce transverse displacements of both the cable and the cylinder. These transverse oscillations entirely decouple from any in-plane motion. The propagation of disturbances of frequency ω along the cable depends strongly on the value of the non-dimensional frequency ωlC/U, where lC is the cable length and U is the towing speed, and only weakly on the other cable parameters. The cable acts as an effective low-pass filter to transverse oscillations, the amplitude of disturbances with non-dimensional frequency greater than 10 being reduced by at least 90% as they propagate along the cable.

Unsteadiness in the ship's speed can result in in-plane deflections of the cable, and vertical oscillations of the cylinder containing the sonar array. In contrast to the transverse oscillations a significant proportion of the in-plane disturbances at the ship travels to the sonar array at all values of the frequency. Low- and high-frequency analytical forms are derived to explain why this occurs. Perturbations in the ship's position are most effectively transformed into vertical oscillations of the array at a frequency of 2.8U/lC. The effect of cable properties on transmission along the cable is investigated. The transmission again depends on the value of the non-dimensional frequency ωlC/U. Parameter changes, which increase the cable critical angle, increase the proportion of the disturbance at the towing point that is transformed into vertical array motion, for a fixed value of ωlC/U. This is explained by reference to the low- and high-frequency analytical solutions.

In this paper a turbulent plane channel flow modified by a distributed transpiration at the wall, with zero net mass flux, is studied through direct numerical simulation (DNS) using the incompressible Navier–Stokes equations. The transpiration is steady, uniform in the spanwise direction, and varies sinusoidally along the streamwise coordinate. The transpiration wavelength is found to dramatically affect the turbulent flow, and in particular the frictional drag. Long wavelengths produce large drag increases even with relatively small transpiration intensities, thus providing an efficient means for improved turbulent mixing. Shorter wavelengths, on the other hand, yield an unexpected decrease of turbulent friction. These opposite effects are separated by a threshold of transpiration wavelength, shown to scale in viscous units, related to a longitudinal length scale typical of the near-wall turbulence cycle. Transpiration is shown to affect the flow via two distinct mechanisms: steady streaming and direct interaction with turbulence. They modify the turbulent friction in two opposite ways, with streaming being equivalent to an additional pressure gradient needed to drive the same flow rate (drag increase) and direct interaction causing reduced turbulent activity owing to the injection of fluctuationless fluid. The latter effect overwhelms the former at small wavelengths, and results in a (small) net drag reduction. The possibility of observing large-scale streamwise-oriented vortical structures as a consequence of a centrifugal instability mechanism is also discussed. Our results do not demonstrate the presence of such vortices, and the same conclusion can be arrived at through a stability analysis of the mean velocity profile, even though it is possible that a higher value of the Reynolds number is needed to observe the vortices.

The symmetric near wake of a flat plate with a circular trailing edge, exhibiting pronounced shedding of wake vortices, is investigated via a direct numerical simulation. The separating plate boundary layers are turbulent and statistically identical. The present study focuses on the vortical structures, the distribution of phase-averaged velocity statistics and vorticity fluctuations in the near wake. Braid and vortex core regions are investigated via contours of instantaneous vorticity and pressure with an emphasis on the three-dimensionality of near-wake structures. Rib vortices are observed in the braids. The vortex cores are populated with both ribs that wrap themselves around the cores and spanwise-elongated regions of spanwise vorticity near the centres of the cores. Production terms in the transport equations for phase-averaged fluctuating vorticity show that stretching of rib vortices via the phase-averaged strain rate produces significantly less turbulent vorticity than turbulent stretching in the braids and cores.

We investigate the effect of intergranular cohesive forces on the properties of self-diffusion in dense granular flows. The study is based on a series of simulated plane shear flows at different inertial and cohesion numbers, in which transverse diffusivities are measured. Results evidence an increase in diffusivity by up to two orders of magnitude when introducing cohesion. This strong effect is analysed using the Green–Kubo framework, expressing the diffusivity in terms of instantaneous grain velocity fluctuations and their time correlation. This analysis shows that cohesion, by forming enduring clusters in the flow, enhances the velocity fluctuations and their time persistence, which both contribute to enhancing grain mixing and self-diffusion.

The stability of an incompressible parallel flow with a linear wall slip – where the amount of wall slip is proportional to the velocity gradient – is analysed. Numerical computations are performed for microchannel flows in the slip-flow region. The wall slip causes short-wave instability while the slip-flow model is stable for long waves. This instability disappears once the slip wall condition is replaced by the no-slip condition.

A linear stability analysis of the laminar flow in the boundary layer at the bottom of a solitary wave is made to determine the conditions leading to transition and the appearance of turbulence. The Reynolds number of the phenomenon is assumed to be large and a ‘momentary’ criterion of instability is used. The results show that the laminar regime becomes unstable during the decelerating phase, when the height of the solitary wave exceeds a threshold value which depends on the ratio between the boundary layer thickness and the local water depth. A comparison of the theoretical results with the experimental measurements of Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231) supports the analysis.

This paper deals with nearly inviscid, capillary–gravity, modulated waves parametrically excited by monochromatic horizontal vibrations in liquid containers whose width and depth are both large compared with the wavelength of the excited waves. A general linear amplitude equation is derived with appropriate boundary conditions that provides the threshold acceleration and associated spatiotemporal patterns, which compare very well with experimental measurements and visualizations. The primary instability is associated with a pair of complex Floquet multipliers that are close to (but strictly different from) −1, meaning that the instability is not strictly (2:1) subharmonic. The resulting (quasi-periodic) waves are generally oblique, not perpendicular to the vibrating endwalls. The extension of the theory to other confined systems such as vibrating containers of arbitrary shape and vibrating drops is also considered.

An angular spectrum model for predicting the transformation of Stokes waves on a mildly varying topography is developed, including refraction, diffraction, shoaling and nonlinear wave interactions. The equations governing the water-wave motion are perturbed using the method of multiple scales and Stokes expansions for the velocity potential and free-surface displacement. The first-order solution is expressed as an angular spectrum, or directional modes, of the wave field propagating on a beach with straight iso-baths whose depth is given by laterally averaged depths. The equations for the evolution of the angular spectrum due to the effects of bottom variation and cubic resonant interaction are obtained from the higher-order problems. Comparison of the present model with existing models is made for some simple cases. Numerical examples of the time-independent version of the model are presented for laboratory experiments for wave diffraction behind a breakwater gap and wave focusing over submerged shoals: an elliptic shoal on a sloping beach and a circular shoal on a flat bottom.

The unsteady, three-dimensional, incompressible, viscous flow interactions between a vortical (initially cylindrical) structure advected by a uniform free stream and a spherical particle held fixed in space is investigated numerically for a range of particle Reynolds numbers 20 ≤ Re ≤ 100. The counter-clockwise rotating vortex tube is initially located ten sphere radii upstream from the sphere centre. The finite-difference computations yield the flow properties and the temporal distributions of lift, drag, and moment coefficients of the sphere. Initially, the lift force is positive owing to the upwash on the sphere, then becomes negative owing to the downwash as the vortex tube passes the sphere. Varying the size of the vortex core (σ) shows that the r.m.s. lift coefficient is linearly proportional to the circulation of the vortex tube at small values of σ. At large values of σ, the r.m.s. lift coefficient is linearly proportional to the maximum fluctuation velocity (vmax) induced by the vortex tube but independent of σ. For intermediate values of σ, the r.m.s. lift coefficient depends on both σ and vmax (or equivalently both σ and the circulation). We observe some interesting flow phenomena in the near wake as a function of time owing to the passage of the vortex tube.

The family of shear flows comprising the jet, wake, and the mixing layer are subjected to perturbations in an inviscid incompressible fluid. By modelling the basic mean flows as parallel with piecewise linear variations for the velocities, complete and general solutions to the linearized equations of motion can be obtained in closed form as functions of all space variables and time when posed as an initial-value problem. The results show that there is a continuous spectrum as well as the discrete spectrum that is more familiar in stability theory and therefore there can be both algebraic and exponential growth of disturbances in time. These bases make it feasible to consider control of such flows. To this end, the possibility of enhancing the disturbances in the mixing layer and delaying the onset in the jet and wake is investigated. It is found that growth of perturbations can be delayed to a considerable degree for the jet and the wake but, by comparison, cannot be enhanced in the mixing layer. By using moving coordinates, a method for demonstrating the predominant early and long time behaviour of disturbances in these flows is given for continuous velocity profiles. It is shown that the early time transients are always algebraic whereas the asymptotic limit is that of an exponential normal mode. Numerical treatment of the new governing equations confirm the conclusions reached by use of the piecewise linear basic models. Although not pursued here, feedback mechanisms designed for control of the flow could be devised using the results of this work.

Depending on the flow conditions, the liquid film in annular two-phase flow in coiled tubes may be pushed towards the outer or the inner side by the centrifugal force. It is important to understand the mechanism of this ‘film inversion’ in order to develop a predictive model for the film thickness distribution. In this paper, this phenomenon is studied analytically, and a new criterion, based on the secondary flow in the thin liquid film, is proposed to predict its occurrence. The criterion shows good agreement with available experimental data. It is suggested that the analytical model can readily be extended to predict the distribution of the film thickness and film flow rate in coiled tubes.

Consider a background state which consists of a spatially uniform chemically reactive mixture in a general state of disequilibrium. The analytical method of characteristics is used to show that a plane finite amplitude disturbance propagates through this system at the frozen sound speed and, if the degree of disequilibrium is sufficient, is amplified by the chemical reaction. Some comments are made about the time to shock-wave formation and its relation to the homogeneous explosion ignition time, and also about expansion waves, which are found to have a tendency towards fixed-strength ‘quenching waves’, their strength being proportional to the extent of the ambient disequilibrium.

To investigate reflection of a shallow-water soliton at a sloping beach, the edge-layer theory is developed to obtain a ‘reduced’ boundary condition relevant to the simplified shallow-water equation describing the weakly dispersive waves of small but finite amplitude. An edge layer is introduced to take account of the essentially two-dimensional motion that appears in the narrow region adjacent to the beach. By using the matched-asymptotic-expansion method, the edge-layer theory is formulated to cope with the shallow-water theory in the offshore region and the boundary condition at the beach. The ‘reduced’ boundary condition is derived as a result of the matching condition between the two regions. An explicit edge-layer solution is obtained on assuming a plane beach.

Recently, Leibovich & Stewartson (1983) developed a sufficient condition for the instability of columnar vortices with radial shears in both the azimuthal and axial velocities, while others (e.g. Staley & Gall 1984) have found instabilities in numerical simulations which conform exactly to expectations based on the Leibovich-Stewartson theory. The purpose of this brief note is to show that this three-dimensional stability problem is isomorphic to the classical two-dimensional inertialThe instability discussed here is sometimes referred to as ‘centrifugal instability’. stability problem when viewed in an appropriate local coordinate system. The instability is therefore clearly inertial in character, as suggested by Pedley (1969).

A vertically standing freely-rotating ellipsoidal vortex of uniform anomalous potential vorticity in a rotating stratified fluid under quasi-geostrophic conditions of small Rossby and Froude numbers steadily rotates without change of form. The vortex can have arbitrary axis lengths, but must have one axis parallel to the vertical $z$-axis along the direction of gravity. The rotation rate is proportional to the potential vorticity anomaly but otherwise depends on only two independent aspect ratios characterizing the shape of the vortex. The linear stability of this class of vortex equilibria was first determined semi-analytically more than a decade ago. It was found that vortices are unstable over a wide range of the parameter space and are stable only when strongly oblate and of nearly circular cross-section.

New results, presented here, using a complementary approach and backed by nonlinear simulations of the full quasi-geostrophic equations indicate that these ellipsoidal vortices are in fact stable over a much wider range of parameter space. In particular, a mode previously thought to be unstable over much of the parameter space is evidently stable. Moreover, we have determined that this mode is just the difference between two neighbouring equilibrium states having slightly different horizontal aspect ratios; hence, this mode must be neutrally stable. Agreement is found for all other modes. However, by an independent analysis considering only ellipsoidal (though time-varying) disturbances, we have identified one unstable mode as purely ellipsoidal, i.e. it does not change the form of the ellipsoid, only its shape. Under this instability, the vortex quasi-periodically tilts over while undergoing mild changes in shape.

The range of parameters leading to non-ellipsoidal instabilities turns out to be narrow, with instability principally occurring for highly eccentric (horizontally squashed, prolate) vortices. The long-term fate of these instabilities is examined by nonlinear contour-dynamical simulations. These reveal a wealth of complex phenomena such as the production of a sea of small-scale vortices, yet, remarkably, the dominant vortex often tends to relax to a stable rotating ellipsoid.

We investigate the kinematic dynamo properties of interacting vortex tubes. These flows are of great importance in geophysical and astrophysical fluid dynamics: for a large range of systems, turbulence is dominated by such coherent structures. We obtain a dynamically consistent $2 \frac{1}{2}$-dimensional velocity field of the form $\left(u(x,y,t),v(x,y,t),w(x,y,t)\right)$ by solving the $z$-independent Navier–Stokes equations in the presence of helical forcing. This system naturally forms vortex tubes via an inverse cascade. It has chaotic Lagrangian properties and is therefore a candidate for fast dynamo action. The kinematic dynamo properties of the flow are calculated by determining the growth rate of a small-scale seed field. The growth rate is found to have a complicated dependence on Reynolds number $\Rey$ and magnetic Reynolds number $\Rem$, but the flow continues to act as a dynamo for large $\Rey$ and $\Rem$. Moreover the dynamo is still efficient even in the limit $\Rey \,{\gg}\, \Rem$, providing $\Rem$ is large enough, because of the formation of coherent structures.

We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.