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.

The onset of convection in the form of inertial waves in a rotating fluid sphere is studied through a perturbation analysis in an extension of earlier work by Zhang (1994). Explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber are derived and new results for the case of a nearly thermally insulating boundary are obtained.

The Karlsruhe Dynamo experiment is aimed at showing that an array of columnar helical vortices in liquid sodium, confined in a cylindrical container, can generate a magnetic field by self-excitation. In three test series it has been demonstrated that magnetic self-excitation occurs and a permanent magnetic saturation field develops which oscillates about a well-defined mean value for fixed flow rates. Dynamo action is observed as an imperfect bifurcation from a seed magnetic field of the environment. Two quasi-dipolar magnetic fields of opposite direction have been realized. A transition between these two states can be enforced through imposition of a sufficiently strong external magnetic perturbation on the existent dynamo field. These perturbations were induced with the aid of two Helmholtz coils. A time series analysis of the magnetic field fluctuations shows several characteristic dynamic features, which are in agreement with theoretical predictions from turbulence models available in the literature.

We depict and analyse the successive steps of atomization of a liquid jet when a fast gas stream blows parallel to its surface. Experiments performed with various liquids in a fast air flow show that the liquid destabilization proceeds from a two-stage mechanism: a shear instability first forms waves on the liquid. The transient acceleration experienced by the liquid suggests that a Rayleigh–Taylor type of instability is triggered at the wave crests, producing liquid ligaments which further stretch in the air stream and break into droplets. The primary wavelength $\lambda\,{\sim}\,\delta (\rho_{1}/\rho_{2})^{1/2}$ is set by the vorticity thickness $\delta$, in the fast air stream and the liquid/gas density ratio $\rho_{1}/\rho_{2}$. The transverse corrugations of the crests have a size $\lperp\,{\sim}\,\delta {\We_{\delta}}^{-1/3} (\rho_{1}/\rho_{2})^{1/3}$, where $\We_{\delta}\,{=}\,\rho_{2}u_{2}^{2}\delta/\sigma$ is the Weber number constructed on the gas velocity $u_{2}$ and liquid surface tension $\sigma$. The ligament dynamics gives rise, after break-up, to a well-defined droplet size distribution whose mean is given by $\lperp$. This distribution bears an exponential tail characteristic of the broad size statistics in airblast sprays.

As the size of object diminishes, the effect of adhesional forces grows stronger for micro-manipulation and so. The capillary force generated by a liquid bridge can be greater than both the capillary force generated by another bridge and the adhesional force because the capillary force can be controlled by regulation of the liquid volume. We propose a micro-manipulation method based on the regulation of liquid bridge volume. A numerical investigation to estimate the capillary force from a given liquid volume is also presented, and four phases of capillary force curves are obtained from it. If an object is supported by two liquid bridges, we can predict which bridge collapse by a stability analysis for the wide range of liquid volume using the force curve.

The motion of membrane-bound objects is important in many aspects of biology and surface chemistry. Here we derive some general relations for objects moving in a surface film overlying a fluid of depth H. A solution to the problem of the drag can be obtained from a two-dimensional system of integral equations. Here we focus on the problem of an ideal needle moving edge-on (in the direction of its tip) or broadside-on (perpendicular to the direction of the tip). It is shown that in comparison to the drag on a circular disk a new scaling regime of the drag on a needle arises when the ratio between surface shear viscosity and subphase viscosity $\eta_s/\eta$ is smaller than the length of the needle.

A numerical model for a three-dimensional heat and fluid flow through a bank of infinitely long cylinders in yaw has been proposed to investigate complex flow and heat transfer characteristics associated with manmade structures such as extended fins and plate fins in heat transfer equipment. By exploiting the periodicity of the structure, only one structural unit has been taken as a calculation domain. An economical quasi-three-dimensional calculation procedure has been proposed to replace exhaustive full three-dimensional numerical manipulations. It has been shown that, under macroscopically uniform flow, the three-dimensional governing equations reduce to quasi-three-dimensional forms, in which all derivatives associated with the axis of the cylinder can be either eliminated or replaced by other determinable expressions. Thus, only two-dimensional storage is required for the dependent variables in question. Extensive numerical calculations were carried out for various sets of the porosity, degree of anisotropy, Reynolds number and macroscopic flow direction in a three-dimensional space. The numerical results thus obtained for periodically fully developed flow and temperature fields were integrated over a structural unit to determine the permeability tensor, Forchheimer tensor and directional interfacial heat transfer coefficient, to elucidate the effects of yaw angle on these macroscopic flow and heat transfer characteristics. Upon examining these numerical data, a useful set of explicit expressions has been established for the permeability tensor, Forchheimer tensor and directional interfacial heat transfer coefficient to characterize flow and heat transfer through a bank of cylinders in yaw.

The dispersion relation is found for edge waves in a rotating stratified fluid over a constant sloping bottom. The dispersion relation is then extended to the case of arbitrary gentle bottom bathymetry. Superinertial trapped modes do not exist in the rigid-lid Boussinesq case. The effect of some of the approximations that have been made in this problem is discussed.

For large classes of vorticities we prove that a steady periodic gravity water wave with a monotonic profile between crests and troughs must be symmetric. The analysis uses sharp maximum principles for elliptic partial differential equations.

The gravitational settling of small dense particles (with fall speed $v_T$, response time $\tau_p$) past an isolated spherical vortex (radius $a$, speed $U$) or a random distribution of spherical vortices translating vertical upwards, is examined. As particles sediment past a vortex, they are permanently displaced vertically and laterally a distance $X$ and $Y$, respectively. The bulk settling properties of the particles are expressed in terms of the weighted moments of displacement, denoted by $D_p$, $M_{\hbox{\scriptsize{\it xx}}}$, $M_{\hbox{\scriptsize{\it yy}}}$ and corresponding to the integral of $X$, $X^2/2$, $Y^2/2$, respectively over the particle sheet. When the particle Stokes number ${\it St}$$=U\tau_p/a \rightarrow 0$, the particles are inertialess. Particles starting outside the vortex are excluded from a spherical shadow region when $v_T/U\,{>}\,3/2$. When $v_T/U\,{<}\,3/2$, particles passing close to the particle stagnation points (in the frame moving with the vortex) are held up for a long time relative to particles far from the vortex, but are not displaced laterally. In an unbounded flow, the particle drift volume, $D_p$, is calculated using a geometrical argument, $M_{\hbox{\scriptsize{\it xx}}}=25U^2\upi a^4/8(v_T+U)^2$, and $M_{\hbox{\scriptsize{\it yy}}}=0$. As $v_T/U\,{\rightarrow}\; 0$, the results of Darwin (1953) are recovered. Results for finite values of ${\it St}$ are calculated numerically. The effect of inertia is shown to substantially increase the particle residence time near the vortex because particles overshoot the particle stagnation point, and there is a shadow region within and behind the vortex. $D_p$, $M_{\hbox{\scriptsize{\it xx}}}$, and $M_{\hbox{\scriptsize{\it yy}}}$ all substantially increase with the particle Stokes number. These results are applied to calculate the bulk settling velocity and the dispersivity of particles sedimenting through a random distribution of vortices translating vertically in a bounded flow. This is done by combining information of the particle displacements with a statistical model of their encounter with a vortex. Inertialess particles (${\it St}$$=0$) do not experience the upwards flow within the vortex and the fractional increase in fall speed is proportional to the volume of the shadow region. As ${\it St}$ increases, particles overshoot the particle stagnation point, increasing their residence time and so decreasing the bulk settling fall speed. Particle inertia significantly increases the vertical dispersivity of dense particles compared to fluid particles, but for high $v_T$, particles disperse vertically more slowly than fluid particles.

In flows of very viscous fluids, it is often justifiable to neglect inertia and solve the resulting creeping-flow or Stokes equations. For drops hanging beneath a fixed wall and extending under gravity from an initial rest state, an inevitable consequence of neglect of inertia and surface tension is that the drop formally becomes infinite in length at a finite crisis time, at which time the acceleration of the drop, which has been assumed small relative to gravity $g$, formally also becomes infinite. This is a physical impossibility, and the acceleration must in fact approach the (finite) free-fall value $g$. However, we verify here, by a full Navier–Stokes computation and also with a slender-drop approximation, that the crisis time is a good estimate of the time at which the bulk of the drop goes into free fall. We also show that the drop shape at the crisis time is a good approximation to the final shape of the freely falling drop, prior to smoothing by surface tension. Additionally, we verify that the drop has an initial acceleration of $g$, which quickly decreases as viscous forces in the drop become dominant during the early stages of fall.

Frost heave is the process by which the freezing of water-saturated soil causes the deformation and upward thrust of the ground surface. We describe the fundamental interactions between phase change and fluid flow in partially frozen, saturated porous media (soils) that are responsible for frost heave. Water remains only partially frozen in a porous medium at temperatures below $0\,^\circ$C owing both to the depression of the freezing temperature at curved phase boundaries and to interfacial premelting caused by long-range intermolecular forces. We show that while the former contributes to the geometry of fluid pathways, it is solely the latter effect that generates the forces necessary for frost heave. We develop a simple model describing the formation and evolution of the ice lenses (layers of ice devoid of soil particles) that drive heave, based on integral force balances. We determine conditions under which either (i) a single ice lens propagates with no leading frozen fringe, or (ii) a single, propagating ice lens is separated from unfrozen soil by a partially frozen fringe, or (iii) multiple ice lenses form.

A rivulet is a narrow stream of liquid flowing down a solid surface. When its flow rate exceeds a certain limit, it tends to meander, exhibiting instability of the interface. Here we report a perturbation analysis of this meandering rivulet. We find that the combined effects of the tangential velocity difference across the interface and the dynamic wetting are responsible for the instability. The Weber number represents the ratio of the destabilizing force, inertia, to the stabilizing force, surface tension. As the Weber number increases, both the wavenumber of maximum instability and the cutoff wavenumber increase. The effects of the capillarity are such that the wavenumber of maximum instability asymptotically increases as the sensitivity of the dynamic contact angles to the contact line speed increases. However, the cutoff wavenumber remains constant despite wetting parameter changes when the Weber number is constant.

From observations of swimming of the shell-less pteropod mollusc Clione antarctica we compare swimming velocities achieved by the organism using ciliated surfaces alone with velocities achieved by the same organism using a pair of flapping wings. Flapping dominates locomotion above a swimming Reynolds number ${\it Re}$ in the range 5–20. We test the hypothesis that ${\it Re} \approx$5–20 marks the onset of ‘flapping flight’ in these organisms. We consider the proposition that forward, reciprocal flapping flight is impossible for locomoting organisms whose motion is fully determined by a body length $L$ and a frequency $\omega$ below some finite critical value of the Reynolds number ${\it Re}_\omega = \omega L^2/\nu$. For a self-similar family of body shapes, the critical Reynolds number should depend only upon the geometry of the body and the cyclic movement used to locomote. We give evidence of such a critical Reynolds number in our data, and study the bifurcation in several simplified theoretical models. We argue further that this bifurcation marks the departure of natural locomotion from the low Reynolds number or Stokesian realm and its entry into the high Reynolds number or Eulerian realm. This occurs because the equilibrium swimming or flying speed $U_f$ obtained at the instability is determined by the mechanics of a viscous fluid at a value of ${\it Re}_f=U_f L/\nu$ that is not small.

Asymptotic approximations to the velocity and pressure fields generated by steadily towing a body through a uniformly stratified fluid in an arbitrary direction are calculated. This is done explicitly for point sources, both oscillatory and non-oscillatory in strength, and for a spherical source of non-oscillatory strength. The flow fields are divided into regions by causality and radiation conditions which have contributions containing up to three significant internal waves in the non-oscillatory cases, and up to six in the oscillatory cases. A method for describing the caustics is given. The drag due to the internal wave field induced by towing a body through the fluid is calculated for a spherical source, of non-oscillatory strength, being towed at a general tow angle and Froude number. Dependence of the drag coefficient on the tow angle and the Froude number is shown in particular the tow angle can significantly affect the magnitude of the drag coefficient for a given Froude number.

We derive an improved rigorous upper bound for the long-time-averaged vertical buoyancy flux for stably stratified Couette flow; i.e. the flow of a Boussinesq fluid (with reference density $\rho_0$, kinematic viscosity $\nu$, and thermal diffusivity $\kappa$) confined between two parallel horizontal plates separated by a distance $d$, which are driven at a constant relative velocity $\uDelta U$, and are maintained at a constant (statically stable) temperature difference leading to a constant density difference $\uDelta \rho$. We construct the bound by means of a numerical solution to the ‘background method’ variation problem as formulated by Constantin and Doering using a one-dimensional uni-directional background. The upper bound so constructed is the best possible bound with the imposed constraints for streamwise independent mean flows that are statistically steady, and is calculated up to asymptotically large Reynolds numbers. We find that the associated (dimensional) upper bound ${\cal B}^*_{\hbox{\scriptsize\it max}}$ on the long-time-averaged and volume averaged buoyancy flux ${\cal B}^*:=\lim_{t \rightarrow \infty} (1/t) \int^t_0 \langle \rho u_3 \rangle g/\rho_{0} {\rm d}\skew2\tilde{t}$ (where $u_3$ is the vertical velocity, $g$ is the acceleration due to gravity, and angled brackets denote volume averaging) does not depend on either the bulk Richardson number $J=g\uDelta \rho d/(\rho_0 \uDelta U^2)$ of the flow, or the Prandtl number $\sigma=\nu/\kappa$ of the fluid. We show that ${\cal B}^*_{\hbox{\scriptsize\it max}}$ has the same inertial characteristic scaling as the (dimensional) mechanical energy dissipation rate ${\cal E}^*_B$, and ${\cal B}^*_{\hbox{\scriptsize\it max}}=0.001267\uDelta U^3/d$ as $\hbox{\it Re}\rightarrow\infty$. The associated flow structure exhibits velocity boundary layers embedded within density boundary layers, with local gradient Richardson numbers $\hbox{\it Ri} =O(\sigma/\hbox{\it Re})\ll 1$ in the vicinity of the horizontal plates. There is a correspondence between the predicted flow structure and the flow structure at a lower Reynolds number associated with the upper bound on the mechanical energy dissipation rate ${\cal E}^*_{\hbox{\scriptsize\it max}}$ in an unstratified fluid. We establish that, for the flow that maximizes the buoyancy flux, the flux Richardson number $\hbox{\it Ri}_f \rightarrow 1/3$ as $\hbox{\it Re} \rightarrow \infty$, independently to leading order of both $\hbox{\it Re}$ and $J$. There is a generic partition of the energy input by the shear into the fluid into three equal parts: viscous dissipation of kinetic energy by the mean flow; viscous dissipation of kinetic energy by perturbation velocities; and vertical buoyancy flux.

This paper revisits the classical problem of convectively driven one-dimensional (parallel) flow along an infinite vertical plate. We consider flows induced by an impulsive (step) change in plate temperature, a sudden application of a plate heat flux, and arbitrary temporal variations in plate temperature or plate heat flux. Provision is made for pressure work and vertical temperature advection in the thermodynamic energy equation, processes that are generally neglected in previous one-dimensional studies of this problem. The pressure work term by itself provides a relatively minor refinement of the Boussinesq model, but can be conveniently combined with the vertical temperature advection term to form a single term for potential temperature advection. Vertical motion of air in a statically stable environment (stable potential temperature stratification) is associated with a simple negative feedback mechanism: warm air rises, expands and cools relative to the environment, whereas cool air subsides, compresses and warms relative to the environment. Exact solutions of the viscous equations of motion are obtained by the method of Laplace transforms for the case where the Prandtl number is unity. Pressure work and vertical temperature advection are found to have a significant impact on the structure of the solutions at later times.

Secondary currents are a characteristic feature of flow in open-channel bends. Besides the classical helical motion (centre-region cell), a weaker and smaller counter-rotating circulation cell (outer-bank cell) is often observed near the outer bank, which is believed to play an important role in bank erosion processes. The mechanisms underlying the circulation cells, especially the outer-bank cell, are still poorly understood, and their numerical simulation still poses problems, not least due to lack of detailed experimental data. The research reported herein provides detailed experimental data on both circulation cells in an open-channel bend such as found in nature. Furthermore, the underlying dynamics are investigated by simultaneously analysing the vorticity equation and the kinetic energy transfer between the mean flow and the turbulence. This shows that turbulence plays a minor role in the generation of the centre-region cell, which is mainly due to the centrifugal force. By accounting for the feedback between the downstream velocity profile and the centre-region cell, a strongly simplified vorticity balance is shown to yield accurate predictions of the velocities in the centre region. For strong curvatures, however, a fully three-dimensional flow description is required. Due to the non-monotonic velocity profiles, the centrifugal force favours the outer-bank cell. Moreover, terms related to the anisotropy of the cross-stream turbulence, induced by boundary proximity, are of the same order of magnitude and mainly enhance the outer-bank cell. Both mechanisms strengthen each other. The occurrence of the outer-bank cell is shown to be not just due to flow instability, like in the case of curved laminar flow, but also to kinetic energy input from turbulence.

The exact steady two-dimensional solutions of the Euler equations due to Stuart (1967) are generalized to the surface of a sphere. The solutions are parametrized by three parameters $N$,$\theta_0$ and $\Omega_{\hbox{\scriptsize\it max}}$; the integer $N > 1$ denotes the number of smooth vorticity extrema, with extremal vorticity $\Omega_{\hbox{\scriptsize\it max}}$, that are equally spaced in longitudinal angle around a latitude circle at latitudinal spherical polar angle $\theta_0$. The solutions have two equal point-vortex singularities at the north and south spherical poles. Like Stuart's, the solutions are exact and explicit. The solutions are expected to be useful in geophysical and astrophysical applications where curvature effects are important.

The period $T$ of the Helmholtz mode in a circular well that is bounded above by a free surface and below by a semi-infinite reservoir is determined in terms of elliptic integrals. It is shown that $T$ decreases monotonically with increasing amplitude $A$ and is within 1% (10%) of the linear limit $T_0$ for $A{/}h_0\,{<}\,0.4(1.0)$, where $h_0$ is the ambient depth.