Nonlinear stability of a model swept-wing boundary layer, subject to crossflow instability, is investigated by numerically solving the governing partial differential equations. The three-dimensional boundary layer is unstable to both stationary and travelling crossflow disturbances. Nonlinear calculations have been carried out for stationary vortices and the computed wall vorticity pattern results in streamwise streaks which resemble quite well the surface oil-flow visualizations in swept-wing experiments. Other features of the stationary vortex development (half-mushroom structure, inflected velocity profiles, vortex doubling, etc.) are also captured in these calculations. Nonlinear interaction of the stationary and travelling waves is also studied. When initial amplitude of the stationary vortex is larger than the travelling mode, the stationary vortex dominates most of the downstream development. When the two modes have the same initial amplitude, the travelling mode dominates the downstream development owing to its higher growth rate. It is also found that, prior to laminar/turbulent transition, the three-dimensional boundary layer is subject to a high-frequency secondary instability, which is in agreement with the experiments of Poll (1985) and Kohama, Saric & Hoos (1991). The frequency of this secondary instability, which resides on top of the stationary crossflow vortex, is an order of magnitude higher than the frequency of the most-amplified travelling crossflow mode.

The effects of viscoelasticity on the hydrodynamic stability of plane free shear flow are investigated through a linear stability analysis. Three different rheological models have been examined: the Oldroyd-B, corotational Jeffreys, and Giesekus models. We are especially interested in possible effects of viscoelasticity on the inviscid modes associated with inflexional velocity profiles. In the inviscid limit, it is found that for viscoelasticity to affect the instability of a flow described by the Oldroyd-B model, the Weissenberg number, We, has to go to infinity in such a way that its ratio to the Reynolds number, G ∞ We/Re, is finite. In this special limit we derive a modified Rayleigh equation, the solution of which shows that viscoelasticity reduces the instability of the flow but does not suppress it. The classical Orr–Sommerfeld analysis has been extended to both the Giesekus and corotational Jeffreys models. The latter model showed little variation from the Newtonian case over a wide range of Re, while the former one may have a stabilizing effect depending on the product ςWe where ς is the mobility factor appearing in the Giesekus model. We discuss the mechanisms responsible for reducing the instability of the flow and present some qualitative comparisons with experimental results reported by Hibberd et al. (1982), Scharf (1985 a, b) and Riediger (1989).

In this study of the temporal stability of Jeffery–Hamel flow, the critical Reynolds number based on the volume flux, Rc, and that based on the axial velocity, Rec, are computed. It is found that both critical Reynolds numbers decrease very rapidly when the half-angle of the channel, α, increases, such that the quantity αRc remains very nearly constant and αRecis a nearly linear function of α. For a short channel there can be more than one value of the critical Reynolds number. A fully nonlinear analysis, for Re close to the critical value, indicates that the loss of stability is supercritical. The resulting asymmetric oscillatory solutions show staggered arrays of vortices positioned along the channel.

The flow field induced by a vertical plate accelerating into a stationary fluid of finite depth with a free surface and a gravitational restoring force is investigated. This is a model problem for some technologically important design issues such as the bow splash of a ship moving at forward speed. Experimentally it is found that a thin jet forms on the plate and rises rapidly upwards. We investigate this jet in the small-time approximation and find an analytical solution for the flow field in which the jet emerges out of a thin region where the horizontal momentum of the main flow is converted by inertial effects into a rising jet.

The long-time evolution of monopolar and dipolar vortices influenced by the largescale gradient of the ambient potential vorticity (the β-effect) is studied by direct numerical solutions of the equivalent barotropic quasi-geostrophic equation. Translation and reorganization of vortical structures are shown to depend strongly on their intensity. Transport of trapped fluid by vortical structures is illustrated by calculating particle trajectories and by considering closed isolines of potential vorticity and the streamfunction in a co-moving reference frame.

The initial behaviour of strong monopoles is found to be well described by a recent approximate theory for the evolution of azimuthal mode one, even for times longer than the linear Rossby wave period. In the long-time limit, strong monopoles transport particles mainly westward, although the meridional displacement is several times larger than the initial vortex size. The appearance of an annulus with opposite radial gradient of the potential vorticity around the vortex core is demonstrated. This annulus forms owing to the meridional vortex drift on the β-plane and results in reorganization of a strong monopolar vortex into a rotating tripole. A critical value of the vortex intensity is found, below which the tripolar structure does not appear even in the case of an initially shielded vortex. Weak monopolar vortices are able to trap particles and provide some west-meridional fluid transport, even in the case when they decay like a linear Rossby wave packet.

The evolution of initial f-plane dipoles on the β-plane is strongly dependent on the initial direction of propagation. Strong dipoles adjust to steadily propagating modon solutions either accelerating (westward case), decelerating (eastward case) or oscillating with a decaying amplitude (meridional case), thereby carrying trapped particles predominantly eastward. A steady state is not reached if the dipole intensity is below a critical value which depends on the initial direction of propagation. Weak dipoles either decay and shrink owing to Rossby wave radiation (westward case), gradually separate and split (eastward case), or disintegrate (meridional case) without longdistance fluid transport. Thus, on the β-plane monopoles provide mainly westward transport of trapped fluid, whereas dipoles provide mainly eastward transport. Only strong monopoles are found to provide significant meridional fluid transport.

The secondary flow of a viscous fluid, caused by the Coriolis force, through a straight pipe rotating about an axis perpendicular to the pipe axis is analogous to that of a fluid, caused by the centrifugal force, through a stationary curved pipe. The quantitative analogy between these two fully developed laminar flows will be demonstrated through similarity arguments, computational studies and the use of experimental data. Similarity considerations result in two analogous governing parameters for each flow, which include a new one for the rotating flow. When one of these analogous pairs of parameters of the two flows is large, it will be demonstrated that there are strong similarities between the two flows regarding friction factors, heat transfer rates, flow patterns and flow properties for the same values of the other pair of parameters.

This study of flow just beyond a breakaway-separation point presents a description of planar nonlinear unsteady effects over a fairly wide parameter range, for a subsonic or supersonic boundary layer at large Reynolds numbers. The inviscid model thus produced essentially contains a vortex sheet near the smooth solid surface, with local inner–outer interaction. The governing equations couple the eddy velocity and pressure with the thicknesses of the detached boundary layer and the eddy. The computational method presented here uses a new adaptive gridding technique intended to capture accurately the spiky solution behaviour that develops and to compare with theory. Analysis and computations point to a breakup in the solution, suggesting an explanation for the start of transition and possible turbulent reattachment as found experimentally. The influence of the detached boundary-layer thickness proves crucial. The type of finite-time breakup encountered is studied analytically and the criterion for its occurrence is highlighted. This is guided by a characteristic analysis for a special case. The finite-time breakup is similar in spirit to, although different in detail from, a nonlinear breakup proposed earlier by one of the authors for general unsteady interactive boundary layers and it suggests a wide application of that nonlinear breakup theory and its criterion. Comparisons between computations and theory are found to be supportive.

Direct numerical simulations (DNS) and experiments are carried out to study fully developed turbulent pipe flow at Reynolds number Rec ≈ 7000 based on centreline velocity and pipe diameter. The agreement between numerical and experimental results is excellent for the lower-order statistics (mean flow and turbulence intensities) and reasonably good for the higher-order statistics (skewness and flatness factors). To investigate the differences between fully developed turbulent flow in an axisymmetric pipe and a plane channel geometry, the present DNS results are compared to those obtained from a channel flow simulation. Beside the mean flow properties and turbulence statistics up to fourth order, the energy budgets of the Reynolds-stress components are computed and compared. The present results show that the mean velocity profile in the pipe fails to conform to the accepted law of the wall, in contrast to the channel flow. This confirms earlier observations reported in the literature. The statistics on fluctuating velocities, including the energy budgets of the Reynolds stresses, appear to be less affected by the axisymmetric pipe geometry. Only the skewness factor of the normal-to-the-wall velocity fluctuations differs in the pipe flow compared to the channel flow. The energy budgets illustrate that the normal-to-the-wall velocity fluctuations in the pipe are altered owing to a different ‘impingement’ or ‘splatting’ mechanism close to the curved wall.

It has been suggested that in a rapidly rotating fluid sphere, convection would be in the form of slowly drifting columnar rolls with small azimuthal scale (Roberts 1968; Busse 1970). The results in this paper show that there are two alternative convection modes which are preferred at small Prandtl numbers. The two new convection modes are, at leading order, essentially those inertial oscillation modes of the Poincaré equation with the simplest structure along the axis of rotation and equatorial symmetry: one propagates in the eastward direction and the other propagates in the westward direction; both are trapped in the equatorial region. Buoyancy forces appear at next order to drive the oscillation against the weak effects of viscous damping. On the basis of the perturbation of solutions of the Poincaré equation, and taking into account the effects of the Ekman boundary layer, complete analytical convection solutions are obtained for the first time in rotating spherical fluid systems. The condition of an inner sphere exerts an insignificant influence on equatorially trapped convection. Full numerical analysis of the problem demonstrates a quantitative agreement between the analytical and numerical analyses.

We consider steady two-dimensional fluid flow and heat transfer near contact lines in single-phase and two-phase systems. Both single- and double-wedge geometries admit separable solutions in plane polar coordinates for both thermal and flow fields. We consider the class of functions which have bounded temperatures and velocities at the corner. When free surfaces are present, we seek local solutions, those that satisfy all local boundary conditions, and partial local solutions, those that satisfy all but the normal-stress boundary condition. Our aim in this work is to describe local fluid and heat flow in problems where these fields are coupled by determining for which wedge angles solutions exist, identifying singularities in the heat flux and stress which are present at contact lines, and determining the dependence of these singularities on the wedge angles. For thermal fields in two phases we identify two modes of heat transfer that are analogous to the two modes identified by Proudman & Asadullah (1988) for two-fluid flow. For non-isothermal flow, locally, convection does not play a role but coupling through thermocapillary effects on non-isothermal free surfaces can arise. We find that under non-isothermal conditions a planar free surface must leave a planar rigid boundary at an angle of π, the same angle found by Michael (1958) for an isothermal rigid/free wedge, in order to satisfy all local boundary conditions. Finally, we find that situations arise where no coupled solutions of the form sought can be found; we discuss means by which alternative solutions can be obtained.

The first highly accurate solutions for the resistance tensor of an oblate or prolate spheroid moving near a planar wall obtained by Hsu & Ganatos (1989) are used to compute the translational and angular velocities and trajectories of a neutrally buoyant spheroid in shear flow and the gravitational settling motion of a non-neutrally buoyant spheroid adjacent to an inclined plane. The neutrally buoyant spheroid in shear flow undergoes a periodical motion toward and away from the wall as it continually tumbles forward. For some orientation angles it is found that the wall actually enhances the angular velocity of the particle. For certain inclinations a spheroid settling under gravity near an inclined plane reaches an equilibrium position, after which it translates parallel to the wall without rotation.

A new roll-type instability has been discovered experimentally. When fluid between two closely spaced, parallel plates is oscillated about an axis midway between the plates, it exhibits an instability that takes the form of longitudinal rolls aligned perpendicular to the axis of rotation. The basic-state oscillatory shear flow, before the onset of rolls, may be viewed as driven by the $\dot{\bm \Omega}\times \hat{\bm r}$ term of the Navier–Stokes equation in the oscillatory reference frame. A regime diagram is presented in a parameter space defined by the maximum amplitude of angular oscillation, α, and the non-dimensional frequency, Φ = ωd2/ν. The equilibrium wavelength of the rolls scales with d, the gap spacing between the plates, and it increases as Φ increases. Supercritical to a weak-roll onset, an abrupt transition to stronger roll amplitude occurs. Photographs of the cell after an impulsive start show the roll development and initial increase in roll wavelength. A variety of phenomena are observed, including wavelength selection via defect creation and elimination, front propagation, secondary wave instabilities, and the transition to turbulence. We also present solutions of the Navier–Stokes equation for the basic-state shear flow in a near-axis approximation. We develop a simple resonance model which shows some promise in understanding the low-α, high-Φ behaviour of strong rolls. A theoretical analysis of this instability is presented by Hall (1994).

The instability of a viscous fluid inside a rectangular tank oscillating about an axis parallel to the largest face of the tank is investigated in the linear regime. The flow is shown to be unstable to both longitudinal roll and standing wave instabilities. The particular cases of low and high oscillation frequencies are discussed in detail. The relationship between the roll instability and convective or centrifugal instabilities in unsteady boundary layers is discussed. The eigenvalue problems associated with the roll and standing wave instabilities are solved using Floquet theory and a combination of numerical and asymptotic methods. The results obtained are compared to the recent experimental investigation of Bolton & Maurer (1994) which indeed provided the stimulus for the present investigation.