The flow through and around a finite row of parallel slender bodies in close proximity moving in a viscous incompressible fluid is studied. The motion occurs under creeping flow ($\hbox{\it Re}\,{\ll}\,1$) conditions. This row is a model of a comb-wing configuration found in insects of the Thrips family and being developed for use for flying vehicles of mm size, operating in the creeping flow regime. We show here that such wings utilize viscous effects to carry along enough fluid to approximate continuous surfaces. The comb is described as a row of rod-like ellipsoids of slenderness ratio smaller than 0.01 at distances apart of order 10 times the minor axis and the flow field is computed by distributing singularities along the major axes of the ellipsoids. Results for the drag on the individual rods, as well as for the full row are presented. It is shown that above a certain number of rods, dependent on the geometric parameters of the comb, the row acts very much like a continuous surface, with over 95% of the flow moving around, and not through the comb. This allows a potential saving of tens of percents in wing weight. Parametric results for number of rods, rod density (ratio of inter-rod distance to rod length) and slenderness ratio are presented demonstrating the dependence of the flow field on the configuration. It is found that 50–80 rods are required to approach the asymptotic limit of large number of rods, for various combinations of rod parameters with inter-rod distances of order of the cross-section diameter.

In this paper we propose a model to relate Eulerian spatial and temporal velocity autocorrelations in homogeneous, isotropic and stationary turbulence. We model the decorrelation as the eddies of various scales becoming decorrelated. This enables us to connect the spatial and temporal separations required for a certain decorrelation through the ‘eddy scale’. Given either the spatial or the temporal velocity correlation, we obtain the ‘eddy scale’ and the rate at which the decorrelation proceeds. This leads to a spatial separation from the temporal correlation and a temporal separation from the spatial correlation, at any given value of the correlation relating the two correlations. We test the model using experimental data from a stationary axisymmetric turbulent flow with homogeneity along the axis.

New analytical techniques for studying the motion of a point vortex in fluid domains bounded by straight walls having an arbitrary number of gaps are presented. By exploiting explicit formulae for the Kirchhoff–Routh path function in multiply connected circular domains, combined with a novel construction of conformal mappings from such circular domains to multiply connected slit domains, the governing Hamiltonians for the motion of a point vortex in a number of physically interesting fluid regions involving walls with gaps are derived. The vortex trajectories in several illustrative cases are computed. These examples include finding the vortex paths around a chain of islands sitting off an infinite coastline, around islands in an unbounded ocean and around a sequence of islands situated between two headlands.

We report on experimental observations of a gravity-wave instability forced by a highly turbulent free-surface Taylor–Couette flow. Bistability and hysteresis are observed at the bifurcation from a turbulent base state, with an axisymmetric mean flow, to a turbulent gravity-wave state, with an azimuthal $m\,{=}\,1$ pattern related to the mean flow and free surface. We show that the critical Reynolds number at which the wave state appears is not sharply defined as it depends on turbulent fluctuations.

Stokes flow of a viscous fluid in a cylindrical container driven by time-periodic forcing, either at the boundary or through oscillation of the cylinder about an axis parallel to its generators, is considered. The behaviour is governed by a dimensionless frequency parameter $\eta$ and by the geometry of the cylinder cross-section. Various cross-sections (square, rhombus, and sector of a circle) are first treated by either finite-difference or analytic techniques and typical transitions of streamline topology during flow reversals are identified. Attention is then focused on the asymptotic behaviour near any sharp corner on the boundary. For small $\eta$, a regular perturbation expansion reveals the manner in which local flow reversal proceeds during each half-cycle of the flow. The behaviour depends on the corner angle, and different regimes are identified for both types of forcing. For example, for an oscillating square domain, eddies grow symmetrically from each corner and participate in the subsequent flow reversal in the interior. For large $\eta$, the corner eddies merge into Stokes-type boundary layers which drive the interior flow-reversal process. In general, the local corner analysis provides a key to an understanding of the global flow evolution.

Previous parts studied nonlinear resonant sloshing in a prismatic tank under the academic assumption of a square base. However, a representative industrial tank may have an ‘almost’ rather than a ‘precisely’ square base. The present paper generalizes the multimodal technique of Part 1 to examine this complication. The main focus is on resonant sloshing due to lateral harmonic excitations, but several known theoretical results about free-standing waves have also been re-derived. The analysis shows that, although disturbances of the square base do not lead to new types of steady-state wave regimes, the frequency domains, where these wave regimes are stable, change considerably even with small perturbations of the breadth/width ratio around 1. The modifications of the effective frequency domains are compared with the results of two experimental series.

We present a numerical study of the structure and stability of laminar isothermal flows formed by two counterflowing jets of an incompressible Newtonian fluid. We demonstrate that symmetric counterflowing jets with identical mass flow rates exhibit multiple steady states and, in certain cases, time-dependent (periodic) steady states. Two geometric configurations were studied based on the inlet jet shapes: planar and axisymmetric. Stagnation flows formed by planar counterflowing jets exhibit both steady-state multiplicity and time-dependent behaviour, while axisymmetric jets exhibit only a steady-state multiplicity. A linearized bifurcation and stability analysis based on the continuity and Navier–Stokes equations revealed transitions between a single (symmetric) steady state and multiple steady states or periodic steady states. The dimensionless quantities forming the parameter space of this system are the inlet Reynolds number (R$e$) and a geometric aspect ratio ($\alpha$), equal to the jet inlet characteristic length (used for calculating R$e$) divided by the jet separation. The boundaries separating different flow regimes have been identified in the (R$e$, $\alpha$) parameter space. The resulting flow maps are useful for the design and operation of counterflow jet reactors.

For rotationally constrained convection, the Taylor–Proudman theorem enforces an organization of nonlinear flows into tall columnar or compact plume structures. While coherent structures in convection under moderate rotation are exclusively cyclonic, recent experiments for rapid rotation have revealed a transition to equal populations of cyclonic and anticyclonic structures. Direct numerical simulation (DNS) of this regime is expensive, however, and existing simulations have yet to reveal anticyclonic vortical structures. In this paper, we use a reduced system of equations for rotationally constrained convection valid in the asymptotic limit of thin columnar structures and rapid rotation to perform numerical simulation of Rayleigh–Bénard convection in an infinite layer rotating uniformly about the vertical axis. Visualization indicates the existence of cyclonic and anticyclonic vortical populations for all parameters examined. Moreover, it is found that the flow evolves through three distinct regimes with increasing Rayleigh number (Ra). For small, but supercritical Ra, the flow is dominated by a cellular system of hot and cold columns spanning the fluid layer. As Ra increases, the number density of these columns decreases, the up-and down-drafts within them strengthen and the columns develop opposite-signed ‘sleeves’ in all fields. The resulting columns are highly efficient at transporting heat across the fluid layer. In the final regime, lateral mixing plays a dominant role in the interior and the columnar structure is destroyed. However, thermal plumes are still injected and rejected from the thermal boundary layers. We identify the latter two regimes with the vortex-grid and geostrophic turbulence regimes, respectively. Within these regimes, we investigate convective heat transport (measured by the Nusselt number), mean temperature profiles, and root-mean-square profiles of the temperature, vertical velocity and vertical vorticity anomalies. For all Prandtl numbers investigated, the mean temperature saturates in a non-isothermal profile as Ra increases owing to intense lateral mixing.

It has recently been shown that the behaviour of a gas bubble in a uniaxial straining flow can be used as a simplified model to describe some important aspects of the more complex, turbulent bubble break-up problem, provided that the Reynolds and the Weber numbers are sufficiently large. In the present investigation, we extend that work and, using a level-set numerical scheme, we analyse the influence of the bubble Reynolds number on break-up time, $t_b$, for supercritical Weber numbers, $\hbox{\it We}\,{>}\,\hbox{\it We}_c$, where $\hbox{\it We}_c$ is the critical Weber number. It is observed that the viscosity introduces corrections of ${{O}}(1/\hbox{\it Re})$ in the break-up time obtained in the limit $\hbox{\it Re}\to \infty$. In addition, the action of other possible mechanisms of break-up at subcritical Weber numbers, $\hbox{\it We}\,{<}\,\hbox{\it We}_c$, is also explored.

We employ the full FENE-P model of the hydrodynamics of a dilute polymer solution to derive a theoretical approach to drag reduction in wall-bounded turbulence. We recapture the results of a recent simplified theory which derived the universal maximum drag reduction (MDR) asymptote, and complement that theory with a discussion of the cross-over from the MDR to the Newtonian plug when the drag reduction saturates. The FENE-P model gives rise to a rather complex theory due to the interaction of the velocity field with the polymeric conformation tensor, making analytic estimates quite taxing. To overcome this we develop the theory in a computer-assisted manner, checking at each point the analytic estimates by direct numerical simulations (DNS) of viscoelastic turbulence in a channel.

We introduce a unified variational framework in which the classical balance models for nearly geostrophic shallow water as well as several new models can be derived. Our approach is based on consistently truncating an asymptotic expansion of a near-identity transformation of the rotating shallow-water Lagrangian. Model reduction is achieved by imposing either degeneracy (for models in a semi-geostrophic scaling) or incompressibility (for models in a quasi-geostrophic scaling) with respect to the new coordinates.

At first order, we recover the classical semi-geostrophic and quasi-geostrophic equations, Salmon's $L_1$ and large-scale semi-geostrophic equations, as well as a one-parameter family of models that interpolate between the two. We identify one member of this family, different from previously known models, that promises better regularity – hence consistency with large-scale vortical motion – than all other first-order models. Moreover, we explicitly derive second-order models for all cases considered. While these second-order models involve nonlinear potential vorticity inversion and do not obviously share the good properties or their first-order counterparts, we offer an explicit survey of second-order models and point out several avenues for exploration.

Columnar vortices are known to support a family of waves initially discovered by Lord Kelvin (1880) in the case of the Rankine vortex model. This paper presents an exhaustive cartography of the eigenmodes of a more realistic vortex model, the Lamb–Oseen vortex. Some modes are Kelvin waves related to those existing in the Rankine vortex, while some others are singular damped modes with a completely different nature. Several families are identified and are successively described. For each family, the underlying physical mechanism is explained, and the effect of viscosity is detailed. In the axisymmetric case (with azimuthal wavenumber $m\,{=}\,0$), all modes are Kelvin waves and weakly affected by viscosity. For helical modes ($m\,{=}\,1$), four families are identified. The first family, denoted D, corresponds to a particular wave called the displacement wave. The modes of the second family, denoted C, are cograde waves, except in the long-wave range where they become centre modes and are strongly affected by viscosity. The modes of the third family, denoted V, are retrograde, singular modes which are always strongly damped and do not exist in the inviscid limit. The modes of the last family, denoted L, are regular, counter-rotating waves for short wavelengths, but they become singular damped modes for long wavelengths. In an intermediate range of wavelengths between these two limits, they display a particular structure, with both a wave-like profile within the vortex core and a spiral structure at its periphery. This kind of mode is called a critical layer wave, and its significance is explained from both a physical and a mathematical point of view. Double-helix modes ($m\,{=}\,2$) can similarly be classified into the C, V and L families. One additional mode, called F, plays a particular role. For short wavelenghs, this mode corresponds to a helical flattening wave, and has a clear physical significance. However, for long wavelenghts, this mode completely changes its structure, and becomes a critical layer wave. Modes with larger azimuthal wavenumbers $m$ are all found to be substantially damped.

Global modes of the thermal convection type in the Rayleigh–Bénard–Poiseuille (RBP) system are analysed for the case of non-uniform heating of the lower wall. Specifically, a single two-dimensional ‘hot spot’ or ‘temperature bump’ giving rise to a finite region of local instability is considered. For the case of the lower wall temperature varying slowly on the scale of the RBP cell height, i.e. for a gentle temperature bump, WKBJ asymptotics are used to construct an analytical approximation of the linear global mode. At the same time, an analytical selection criterion for the critical global mode is derived from the breakdown of the WKBJ expansion at a double turning point located at the top of the temperature bump. The analytical construction and the underlying assumptions are supported by comparison with direct numerical simulations for Gaussian temperature bumps of elliptical planform not necessarily aligned with the mean flow. From these comparisons, it is concluded that the proposed analytical construction indeed yields the most amplified global mode, which is characterized by an essentially transverse orientation (normal to the mean flow direction) of the convection rolls, independent of the planform of the temperature bump. The paper concludes with preliminary DNS results on the saturated global mode shape and a discussion of possible connections to the ‘steep’ fully nonlinear global modes found for one-dimensional inhomogeneities of the basic state.

Effects of wall suction/injection on the linear stability of flat Stokes layers are investigated. A semi-analytical method is used to examine the stability of time-periodic boundary flows with wall suction/injection. Results show that the onset of instability of the flat Stokes layers can be suppressed by wall suction and enhanced by wall injection.

The motion of a spherical particle suspended in an incompressible Newtonian fluid flowing longitudinally between hexagonally arranged circular cylinders has been numerically analysed by a finite-element method in the Stokes flow regime. The results are applied to study the diffusive and convective transport of spherical solutes across the vascular endothelial surface glycocalyx, based on the quasi-periodic ultrastructural model. The obtained values of diffusive permeability and reflection coefficient of the solutes show a reasonable agreement with experimental observations, and conform to the hypothesis that the endothelial surface glycocalyx forms the primary size selective structure to solutes in microvascular permeability.

An instability theory is presented for the initiation of longitudinal vortices due to waves interacting with a turbulent current in an open channel of finite depth. With a simple model of eddy viscosity the dimensionless shearing rate of the basic current is much weaker than the dimensionless velocity almost everywhere except near the bed. We shall show that in addition to the internal vortex force, a mean shear stress exists on the mean sea level, owing to nonlinear wave–current interactions, despite the absence of wind. This stress adds vertical vorticity to the interior and augments the vortex force mechanism of instability significantly. Effects of current strength and wave conditions on the unstable growth are studied by numerical examples.

The motion of a heavy rigid ellipsoidal particle settling in an infinitely long circular tube filled with an incompressible Newtonian fluid has been studied numerically for three categories of problems, namely, when both fluid and particle inertia are negligible, when fluid inertia is negligible but particle inertia is present, and when both fluid and particle inertia are present. The governing equations for both the fluid and the solid particle have been solved using an arbitrary Lagrangian-Eulerian based finite-element method. Under Stokes flow conditions, an ellipsoid without inertia is observed to follow a perfectly periodic orbit in which the particle rotates and moves from side to side in the tube as it settles. The amplitude and the period of this oscillatory motion depend on the initial orientation and the aspect ratio of the ellipsoid. An ellipsoid with inertia is found to follow initially a similar oscillatory motion with increasing amplitude. Its orientation tends towards a flatter configuration, and the rate of change of its orientation is found to be a function of the particle Stokes number which characterizes the particle inertia. The ellipsoid eventually collides with the tube wall, and settles into a different periodic orbit. For cases with non-zero Reynolds numbers, an ellipsoid is seen to attain a steady-state configuration wherein it falls vertically. The location and configuration of this steady equilibrium varies with the Reynolds number.

The linear stability of flow in a vertical rectangular duct subject to homogeneous internal heating, constant-temperature no-slip walls and a driving pressure gradient is investigated numerically in the framework of the Boussinesq hypothesis. A Galerkin method based upon modified Chebyshev polynomials is used to discretize the linearized Navier–Stokes equations in their most compact form, i.e. eliminating the pressure and the streamwise velocity and making use of the intrinsic symmetry properties. A classification of the basic flow in Grashof and Reynolds space in terms of inflectional properties is proposed. It is found that the flow loses stability at all aspect ratios for a combination of finite thermal buoyancy and pressure forces with opposed signs. In the square duct, the unstable region lies inside the range where the basic velocity profile exhibits additional inflection lines. Unstable eigenfunctions are obtained for all basic symmetry modes, consisting of rapidly propagating large-scale structures located in the vicinity of the inflection lines near the centre of the duct.

A theoretical model of electromigrative, diffusive and convective transport in polymer-gel composites is presented. Bulk properties are derived from the standard electrokinetic model with an impenetrable charged sphere embedded in an electrolyte-saturated Brinkman medium. Because the microstructure can be carefully controlled, these materials are promising candidates for enhanced gel-electrophoresis, chemical sensing, drug delivery, and microfluidic pumping technologies. The methodology provides solutions for situations where perturbations from equilibrium are induced by gradients of electrostatic potential, concentration and pressure. While the volume fraction of the inclusions should be small, Maxwell's well-known theory of conduction suggests that the model may also be accurate at moderate volume fractions. In this work, the theory is used to compute ion fluxes, electrical current density, and convective flow driven by an electric field applied to an homogeneous composite. The electric-field-induced (electro-osmotic) flow is a sensitive indicator of the inclusion $\zeta$-potential and size, electrolyte concentration, and Darcy permeability of the gel, while the electrical conductivity is usually independent of the polymer gel and is relatively insensitive to characteristics of the inclusions and electrolyte.

The behaviour of turbulent flows within the single-layer quasi-geostrophic (Charney–Hasegawa–Mima) model is shown to be strongly dependent on the Rossby deformation wavenumber $\lambda$ (or free-surface elasticity). Herein, we derive a bound on the inverse energy transfer, specifically on the growth rate ${\rm d}\ell/{\rm d} t$ of the characteristic length scale $\ell$ representing the energy centroid. It is found that ${\rm d}\ell/{\rm d} t\le2\|q\|_\infty/(\ell_s\lambda^2)$, where $\|q\|_\infty$ is the supremum of the potential vorticity and $\ell_s$ represents the potential enstrophy centroid of the reservoir, both invariant. This result implies that in the potential-energy-dominated regime ($\ell\ge\ell_s\,{\gg}\,\lambda^{-1}$), the inverse energy transfer is strongly impeded, in the sense that under the usual time scale no significant transfer of energy to larger scales occurs. The physical implication is that the elasticity of the free surface impedes turbulent energy transfer in wavenumber space, effectively rendering large-scale vortices long-lived and inactive. Results from numerical simulations of forced-dissipative turbulence confirm this prediction.