This paper presents the effects of variable-area ducts and axial temperature gradients on the nonlinear distortion of a travelling wave. Quasi-one-dimensional continuity, momentum and energy equations for isentropic flow are solved using a wave front expansion technique. Anevolution equation for the slope of the wave front is obtained. This is a nonlinear ordinary differential equation that can be integrated to obtain a solution in closed form for the slope of the wave front. The solution may admit a singularity for compression wave fronts. The analysis considers pure compression and pure expansion travelling waves. A general criterion is developed for the steepening of a compression wave front into a shock. A general formula is obtained for the location of shock formation. The effects of area variation and axial temperature gradient, and their combined effect on the nonlinear distortion of travelling waves are studied. A number of examples highlighting these effects are presented in the paper.

Motivated by physiological flows in capillaries, venules and the pleural space, the pressure-driven flow of a Newtonian fluid in a two-dimensional wavy-walled channel is investigated theoretically. The sinusoidal wavy shape is due to the configuration of underlying cells, their nuclei and intercellular junctions or clefts. The walls are lined with a thin poroelastic layer that models the glycocalyx coating of the cell surface. The upper and lower wavy walls are offset axially by the phase angle $\Phi $, where $\Phi\,{ =}\, 0$ ($\upi$) yields an antisymmetric (symmetric) channel. Biphasic theory is employed for the poroelastic layer and the flow is solved by a lubrication approximation using a small parameter, $\delta\,{\ll}\,1$, where $\delta$ is the channel width/wavelength ratio. The velocity fields in the core and layer are determined as perturbation expansions in $\delta^2$ and finite-Reynolds-number effects occur at $O(\delta^2)$ assuming $\delta^2\hbox{\it Re}\,{\ll}\,1$. When the hydraulic resistivity, $\alpha$, the ratio of the channel width to the Darcy permeability, is sufficiently large and $\Phi$ is near enough to $\upi$, the flow develops a trapped recirculation eddy within the glycocalyx layer near the widest part of the channel. This can be of significance to transport through the cellular boundary, since that location corresponds to intercellular clefts through which important fluid and solute exchange occurs. Increasing $|\Phi\,{-}\,\upi |$ diminishes the recirculation region. Increasing the Reynolds number moves the recirculation slightly upstream. Both layer velocity and wall shear stresses decrease as $\alpha$ increases and support the appearance of flow recirculation. Further, the wavy geometry allows a portion of the flow to enter and exit the layer, which provides a mechanism for convective transport between these two regions that otherwise have only diffusive interactions. The relevant Péclet number is $\hbox{\it Pe}\,{=}\,V^*_nb/D$ where $D$ is molecular diffusivity and $V^*_n$ is the normal velocity to the glycocalyx layer. For large molecules, $\hbox{\it Pe}\,{=}\,O(10^2)$ or higher, so the convective transport is important. The solid displacement, dictated by the layer flow field, increases as $\alpha$ increases.

We consider flow in partially saturated heterogeneous porous media with uncertain hydraulic parameters. By treating the saturated conductivity of the medium as a random field, we derive a set of deterministic equations for the statistics (ensemble mean and variance) of fluid pressure. This is done for three constitutive models that describe the nonlinear dependence of relative conductivity on pressure. We use the Kirchhoff transform to map Richards equation into a linear PDE and explore alternative closures for the resulting moment equations. Regardless of the type of nonlinearity, closure by perturbation is more accurate than closure based on the non-perturbative Gaussian mapping. We also demonstrate that predictability of unsaturated flow in heterogeneous porous media is enhanced by choosing either the Brooks–Corey or van Genuchten constitutive model over the Gardner model.

The multiplicity, stability and bifurcations of low-Prandtl-number steady natural convection in a two-dimensional rectangular cavity with partially and symmetrically heated vertical walls are studied numerically. The problem represents a simple model of a set-up in which the height of the heating element is less than the height of the molten zone. The calculations are carried out by the global spectral Galerkin method. Linear stability analysis with respect to two-dimensional perturbations, a weakly nonlinear approximation of slightly supercritical states and the arclength path-continuation technique are implemented. The symmetry-breaking and Hopf bifurcations of the flow are studied for aspect ratio (height/length) varying from 1 to 6. It is found that, with increasing Grashof number, the flow undergoes a series of turning-point bifurcations. Folding of the solution branches leads to a multiplicity of steady (and, possibly, oscillatory) states that sometimes reaches more than a dozen distinct steady solutions. The stability of each branch is studied separately. Stability and bifurcation diagrams, patterns of steady and oscillatory flows, and patterns of the most dangerous perturbations are reported. Separated stable steady-state branches are found at certain values of the governing parameters. The appearance of the complicated multiplicity is explained by the development of the stably and unstably stratified regions, where the damping and the Rayleigh–Bénard instability mechanisms compete with the primary buoyancy force localized near the heated parts of the vertical boundaries. The study is carried out for a low-Prandtl-number fluid with $\hbox{\it Pr}\,{=}\,0.021$. It is shown that the observed phenomena also occur at larger Prandtl numbers, which is illustrated for $\hbox{\it Pr}\,{=}\,10$. Similar three-dimensional instabilities that occur in a cylinder with a partially heated sidewall are discussed.

Maximum drag reduction (MDR) in a turbulent channel flow by polymer additives is studied using direct numerical simulation. An Oldroyd-B model is adopted to express the polymer stress because MDR is closely related to the elasticity of the polymer solution. The Reynolds number considered is 4000, based on the bulk velocity and the channel height, and the amount of MDR from the present study is 44%, which is in good agreement with Virk's asymptote at this Reynolds number. For ‘large drag reduction’, the variations of turbulence statistics such as the mean streamwise velocity and r.m.s. velocity fluctuations are quite different from those of ‘small drag reduction’. For example, for small drag reduction, the r.m.s. streamwise velocity fluctuations decrease in the sublayer but increase in the buffer and log layers with increasing Weissenberg number, but they decrease in the whole channel for large drag reduction. As the flow approaches the MDR limit, the significant decrease in the production of turbulent kinetic energy is compensated by the increase in energy transfer from the polymer elastic energy to the turbulent kinetic energy. This is why turbulence inside the channel does not disappear but survives in the MDR state.

The interaction of a pair of spherical particles of different densities and/or sizes with microscopic surface roughness sedimenting due to gravity in a viscous fluid is analysed by theory and experiment. The surface topography is modelled as a combination of small uniformly distributed bumps of uniform height and larger bumps that are more sparsely distributed. The existence of these surface asperities allows the spheres to physically contact each other, so that both hydrodynamic and solid-contact forces are important. When the angle between the line of centres and vertical is small, the spheres may rotate as a rigid body because they are not able to roll up and over a large bump. As this angle increases, however, the heavy sphere rolls and slips past the lighter sphere, and the separation between the nominal surfaces of the spheres varies between the heights of the small and large asperities. When considering many encounters, there is a distribution of nominal separations at each angle due to the distribution of initial conditions and surface topography. The average nominal separation generally increases with increasing angle between the line of centres and vertical because the normal component of gravity, which drives the spheres close together after an encounter with a large bump lifts them apart, decreases as this angle increases.

We compare two-dimensional vorticity and passive scalar cascades seen as a gradient enhancement process. Our criteria are based on conditional averages of the first and second Lagrangian derivatives of vorticity and passive scalar gradients in relation to the local flow geometry. In order to interpret these criteria, transient properties are derived for random vorticity and scalar fields, showing that the second-order Lagrangian derivatives of vorticity and passive scalar gradients may behave differently. Cascades obtained in numerical simulations of decaying and forced incompressible turbulence are analysed. First-order analysis reveals that the direct cascade in elliptic domains is more efficient than previously suspected. While several first-order diagnostics collapse to a single curve for vorticity and passive scalars, second-order diagnostics consistently show that the vorticity gradient exhibits faster temporal fluctuations than the passive scalar gradient, a property which we anticipate qualitatively in the study of random fields.

The low-Reynolds-number wake dynamics and stability of the flow past toroids placed normal to the flow direction are studied numerically. This bluff body has the attractive feature of behaving like the sphere at small aspect ratios, and locally like the straight circular cylinder at large aspect ratios. Importantly, the geometry of the ring is described by a single parameter, the aspect ratio ($\hbox{\it Ar}$), defined as a ratio of the torus diameter to the cross-sectional diameter of the ring. A rich diversity of wake topologies and flow transitions can therefore be investigated by varying the aspect ratio. Studying this geometry allows our understanding to be developed as to why the wake transitions leading to turbulence for the sphere and circular cylinder differ so greatly. Strouhal–Reynolds-number profiles are determined for a range of ring aspect ratios, as are critical Reynolds numbers for the onset of flow separation, unsteady flow and asymmetry. Results are compared with experimental findings from the literature. Calculated Strouhal–Reynolds-number profiles show that ring wakes shed at frequencies progressively closer to that of the straight circular cylinder wake as aspect ratio is increased from $\hbox{\it Ar}\,{ =}\, 3$. For $\hbox{\it Ar} \,{>}\, 8$, the initial asymmetric transition is structurally analogous to the mode A transition for the circular cylinder, with a discontinuity present in the Strouhal–Reynolds-number profile. The present numerical study reveals a shedding-frequency decrease with decreasing aspect ratio for ring wakes, and an increase in the critical Reynolds numbers for flow separation and the unsteady flow transition. A Floquet stability analysis has revealed the existence of three modes of asymmetric vortex shedding in the wake of larger rings. Two of these modes are analogous to mode A and mode B of the circular cylinder wake, and the third mode, mode C, is analogous to the intermediate wavelength mode found in the wake of square section cylinders and circular cylinder wakes perturbed by a tripwire. Furthermore, three distinct asymmetric transition modes have been identified in the wake of small aspect ratio bluff rings. Fully developed asymmetric simulations have verified the unsteady transition for rings that exhibit a steady asymmetric wake.

Computations are performed to examine the instabilities of baroclinically sheared Eady basic flows with respect to banded normal-mode perturbations in three-dimensional space in the presence of eddy diffusivity with two (free-slip and non-slip) types of boundary conditions. The non-dimensional model system contains four external parameters: the Richardson number, the Ekman number, the Prandtl number and the ratio between inertial and buoyancy frequencies. The solutions are controlled mainly by the first three parameters. Growth rate patterns are computed for unstable modes as functions of the horizontal wavelength, $l$, and tilt angle $\alpha $ of the band orientation with respect to the basic shear (measured negative clockwise from the basic-shear direction). It is found that the main growth rate pattern (for non-propagating modes with respect to the middle-level basic flow) has only one maximum unless the Ekman number is sufficiently small. The growth rate pattern obtained with the free-slip boundary conditions has a slightly larger global maximum and is more symmetric with respect to the symmetric axis in the ($l$, $\alpha$) space than that obtained with the non-slip boundary conditions. When the Richardson number is increased from 0.25 to 1.0, the maximum growth rate decreases and the associated instability changes gradually from a nearly symmetric type to a nearly baroclinic type as manifested by the continuous increase of $l$ (from mesoscale to synoptic scale) and continuous change of $\alpha $ (from nearly zero to nearly ${-}90^{\circ})$. When the Ekman number is sufficiently small, the main growth rate pattern can have two local maxima if the Richardson number is within a subrange $0.8 < {\hbox{\it Ri}} < 1.0$. One of the local maxima is near the symmetric axis and the other is near the baroclinic axis in the wavenumber space. When the Richardson number increases through a transitional value in the subrange, the global maximum growth rate decreases continuously but the maximum point jumps from one local maximum to the other and the associated instability switches from a nearly symmetric type to a nearly Eady baroclinic type. The subrange depends on the smallness of the Ekman number and it diminishes as the Ekman number increases to 0.0025 (for the non-slip case). The computed growth rates and ($l$, $\alpha $) are compared with the nearly inviscid results of Miller & Antar and the inviscid results of Stone.

We present a theoretical study of a fundamentally new wave–mean or wave–vortex interaction effect able to force persistent, cumulative change in mean flows in the absence of wave breaking or other kinds of wave dissipation. It is associated with the refraction of non-dissipating waves by inhomogeneous mean (vortical) flows. The effect is studied in detail in the simplest relevant model, the two-dimensional compressible flow equations with a generic polytropic equation of state. This includes the usual shallow-water equations as a special case. The refraction of a narrow, slowly varying wavetrain of small-amplitude gravity or sound waves obliquely incident on a single weak (low Froude or Mach number) vortex is studied in detail. It is shown that, concomitant with the changes in the waves' pseudomomentum due to the refraction, there is an equal and opposite recoil force that is felt, in effect, by the vortex core. This effective force is called a ‘remote recoil’ to stress that there is no need for the vortex core and wavetrain to overlap in physical space. There is an accompanying ‘far-field recoil’ that is still more remote, as in classical vortex-impulse problems. The remote-recoil effects are studied perturbatively using the wave amplitude and vortex weakness as small parameters. The nature of the remote recoil is demonstrated in various set-ups with wavetrains of finite or infinite length. The effective recoil force ${\bm R}_V$ on the vortex core is given by an expression resembling the classical Magnus force felt by moving cylinders with circulation. In the case of wavetrains of infinite length, an explicit formula for the scattering angle $\theta_*$ of waves passing a vortex at a distance is derived correct to second order in Froude or Mach number. To this order ${\bm R}_V\,{\propto}\,\theta_*$. The formula is cross-checked against numerical integrations of the ray-tracing equations. This work is part of an ongoing study of internal-gravity-wave dynamics in the atmosphere and may be important for the development of future gravity-wave parametrization schemes in numerical models of the global atmospheric circulation. At present, all such schemes neglect remote-recoil effects caused by horizontally inhomogeneous mean flows. Taking these effects into account should make the parametrization schemes significantly more accurate.

Deformation and breakup of drops in an isotropic turbulent flow has been studied by numerical simulation. The numerical method involves a pseudospectral representation of the turbulent outer flow field coupled to three-dimensional boundary integral simulations of the local drop dynamics. A statistical analysis based on an ensemble of drop trajectories is presented; results include breakup rates, the distribution of primary daughter drops produced by breakup events, and stationary distributions for drop deformation and orientation. Depending on the local flow history, drops may break at modest length or become highly elongated and relax without breaking. Drop deformation is the dominant mechanism of drop reorientation. The volume of the primary daughter drops, produced by a given fluctuation in flow strength, scales with the volume of the corresponding critical drop size for the fluctuation. A simplified description for the evolution of the drop size distribution, based on this scaling, is presented.

In this paper, the stability of core–annular flows consisting of two immiscible fluids in a cylindrical tube with circular cross-section is examined. Such flows are important in a wide range of industrial and biomedical applications. For example, in secondary oil recovery, water is pumped into the well to displace the remaining oil. It is also of relevance in the lung, where a thin liquid film coats the inner surface of the small airways of the lungs. In both cases, the flow is influenced by a surface-tension instability, which may induce the breakup of the core fluid into short plugs, reducing the efficiency of the oil recovery, or blocking the passage of air in the lung thus inducing airway closure. We consider the stability of a thin film coating the inner surface of a rigid cylindrical tube with the less viscous fluid in the core. For thick enough films, the Rayleigh instability forms a liquid bulge that can grow to eventually create a plug blocking the tube. The analysis explores the effect of an oscillatory core flow on the interfacial dynamics and particularly the nonlinear stabilization of the bulge. The oscillatory core flow exerts tangential and normal stresses on the interface between the two fluids that are simplified by uncoupling the core and film analyses in the thin-film high-frequency limit of the governing equations. Lubrication theory is used to derive a nonlinear evolution equation for the position of the air–liquid interface which includes the effects of the core flow. It is shown that the core flow can prevent plug formation of the more viscous film layer by nonlinear saturation of the capillary instability. The stabilization mechanism is similar to that of a reversing butter knife, where the core shear wipes the growing liquid bulge back on to the tube wall during the main tidal volume stroke, but allows it to grow back as the stoke and shear turn around. To be successful, the leading film thickness ahead of the bulge must be smaller than the trailing film thickness behind it, a requirement necessitating a large enough core capillary number which promotes a large core shear stress on the interface. The core capillary number is defined to be the ratio of core viscous forces to surface tension forces. When this process is tuned correctly, the two phases balance and there is no net growth of the liquid bulge over one cycle. We find that there is a critical frequency above which plug formation does not occur, and that this critical frequency increases as the tidal volume amplitude of the core flow decreases.

We report experimental results on highly supercritical thermal convection in a rapidly rotating hemispherical shell with parabolic gravity. Using silicone oil as the working fluid and an Ekman number $Ek \,{=}\, 4.7 \,{\times}\, 10^{-6}$ we reach Rayleigh numbers up to $1.2 \,{\times}\, 10^{10}$, over 600 times critical. In-situ temperature measurements show that, at these highly supercritical states where convective heat transfer becomes dominant, the time-averaged temperature in the fluid becomes nearly uniform except in a thin thermal boundary layer near the inner spherical boundary. Heat transfer measurements show that Nusselt number $\hbox{\it Nu}$ increases with Rayleigh number $\hbox{\it Ra}$ as $\hbox{\it Nu} \,{\propto}\, Ra^{0.4}$. The measured amplitudes of temperature fluctuations scale well with a model of geostrophic convective turbulence. We also examine convection in a two-layer fluid in the same geometry, using layers of water and silicone oil to produce a stable density stratification. We determine the dependence of heat transfer on the thickness ratio of the layers.

We consider the fluid flow in a spherical shell in rapid overall rotation, with additionally a differential rotation imposed on the inner sphere. The basic state consists of the axisymmetric Stewartson shear layer situated on the tangent cylinder, the cylinder parallel to the axis of rotation and just touching the inner sphere. In this work we consider the non-axisymmetric instabilities that arise when the differential rotation becomes sufficiently large. We find that the sign of the differential rotation, that is, whether the inner sphere is rotating slightly faster or slightly slower than the outer sphere, is crucial, with positive differential rotations yielding a progression to higher wavenumbers $m$ as the overall rotation rate increases, but negative differential rotations yielding $m\,{=}\,1$ over almost the entire range of rotation rates. This difference is particularly intriguing, as it has been seen before in one closely related experimental study, but not in another. A prior asymptotic analysis also suggested there should be no difference. We therefore try to understand what subtle features of the flow structures and/or geometries should cause this difference in results. We show that the geometry is the critical feature, with the height along the axis of rotation changing abruptly across the tangent cylinder. We are not able to identify why this should make such a difference, and why only for negative differential rotations. We suggest instead additional experiments and asymptotics to further clarify this point.

We consider a thin layer of a viscous fluid flowing down a uniformly heated planar wall. The heating generates a temperature distribution on the free surface which in turn induces surface tension gradients. We model this thermocapillary flow by using the Shkadov integral-boundary-layer (IBL) approximation of the Navier–Stokes/energy equations and associated free-surface boundary conditions. Our linear stability analysis of the flat-film solution is in good agreement with the Goussis & Kelly (1991) stability results from the Orr–Sommerfeld eigenvalue problem of the full Navier–Stokes/energy equations. We numerically construct nonlinear solutions of the solitary wave type for the IBL approximation and the Benney-type equation developed by Joo et al. (1991) using the usual long-wave approximation. The two approaches give similar solitary wave solutions up to an $O(1)$ Reynolds number above which the solitary wave solution branch obtained by the Joo et al. equation is unrealistic, with branch multiplicity and limit points. The IBL approximation on the other hand has no limit points and predicts the existence of solitary waves for all Reynolds numbers. Finally, in the region of small film thicknesses where the Marangoni forces dominate inertia forces, our IBL system reduces to a single equation for the film thickness that contains only one parameter. When this parameter tends to zero, both the solitary wave speed and the maximum amplitude tend to infinity.

A numerical model for two-dimensional flow around an airfoil undergoing prescribed heaving motions in a viscous flow is presented. The model is used to examine the flow characteristics and power coefficients of a symmetric airfoil heaving sinusoidally over a range of frequencies and amplitudes. Both periodic and aperiodic solutions are found. Additionally, some flows are asymmetric in that the upstroke is not a mirror image of the downstroke. For a given Strouhal number – defined as the product of dimensionless frequency and heave amplitude – the maximum efficiency occurs at an intermediate heaving frequency. This is in contrast to ideal flow models, in which efficiency increases monotonically as frequency decreases. In accordance with Wang (2000), the separation of the leading-edge vortices at low heaving frequencies leads to diminished thrust and efficiency. At high frequencies, the efficiency decreases similarly to inviscid theory. Interactions between leading- and trailing-edge vortices are categorized, and the effects of this interaction on efficiency are discussed. Additionally, the efficiency is related to the proximity of the heaving frequency to the frequency of the most spatially unstable mode of the average velocity profile of the wake; the greatest efficiency occurs when the two frequencies are nearly identical. The importance of viscous effects for low-Reynolds-number flapping flight is discussed.

As a model for the precession-driven motion in the Earth's core, the flow of incompressible fluid inside a spheroidal shell with imposed rotation and precession is investigated by direct numerical simulation. In one set of simulations, free-slip boundary conditions are used in order to isolate inertial instabilities. These occur as triad resonances involving pairs of inertial modes which have the form of columnar vortices. The simulations reproduce the phenomenon of ‘resonant collapses’ in which the excited modes periodically grow and suddenly decay into turbulence. The experiments of Malkus (1968) are simulated using a hyperviscosity. A hysteretic transition towards developed turbulence observed in one of these experiments can be interpreted as a feature of the basic laminar flow rather than the instability itself. A similar transition can be excluded for Earth's parameters.

In this paper, the stability of a uniformly driven granular layer is examined by linear stability analysis. This includes two main steps: first the base state at various values of mass holdup ($M_{t}$) and energy input ($Q_{t}$) is calculated; and, secondly, small perturbations are introduced to verify the stability of the base state by solving the linearized governing equations and corresponding boundary conditions. Results from the base-state solution show that, for a given pair of $M_{t}$ and $Q_{t}$, solid fraction tends to increase at first up the layer height and then decrease after a certain vertical position. In contrast, granular temperature decreases rapidly from the bottom plate to the top surface. The stability diagram is constructed by checking the eigenvalues at different points in the ($M_{t}$, $Q_{t}$) plane, and their dependence on the operating conditions and materials properties is also investigated. For the unstable regime, pattern formation is illustrated with the variation of solid fraction with vertical position. For the layer mode, there are no variations at different horizontal positions. In contrast, a periodic feature is found for the stationary mode in which alternating particle clusters and voids are observed in the horizontal direction. By introducing perturbations in different directions, we have produced surface patterns such as stripes, squares and hexagons. Besides the solid fraction distribution, other variables such as the profiles of velocities and granular temperatures are also examined.