It is suggested that current conceptions about unsteady rear-stagnation-point flow do not fully describe the physics, since they show discrepancies from recent numerical results. The previously neglected exponentially small rotational perturbation velocity above the boundary-layer proves to have a dominating influence on the final boundary-layer development. An asymptotic analysis reveals possible difficulties for common computational schemes for viscous flows. Failure of the usual asymptotic matching rule in the analysis is in accordance with Fraenkel's warning on logarithmic expansions.

A linear theory is developed in the time domain for vertical motions of an axisymmetric cylinder floating in the free surface. The velocity potential is obtained numerically from a discretized boundary-integral-equation on the body surface, using a Galerkin method. The solution proceeds in time steps, but the coefficient matrix is identical at each step and can be inverted at the outset.

Free-surface effects are absent in the limits of zero and infinite time. The added mass is determined in both cases for a broad range of cylinder depths. For a semi-infinite cylinder the added mass is obtained by extrapolation.

An impulse-response function is used to describe the free-surface effects in the time domain. An oscillatory error observed for small cylinder depths is related to the irregular frequencies of the solution in the frequency domain. Fourier transforms of the impulse-response function are compared with direct computations of the damping and added-mass coefficients in the frequency domain. The impulse-response function is also used to compute the free motion of an unrestrained cylinder, following an initial displacement or acceleration.

A series of melting experiments with a moving horizontal cylindrical heat source at constant surface heat flux have been performed. The heat source was designed in such a way that it could descend under its weight while melting the phase-change material (n-octadecane) surrounding it. The heat-source velocity was measured and the motion and shape of the solid–liquid interface were determined photographically. The effects of the surface heat flux, the density and initial position of the heat source, and the initial subcooling of the solid were investigated and are discussed. Conduction was found to be the dominant heat-transfer mechanism around the lower stagnation point and controlled the terminal velocity of the source. The fluid motion in the melt pool above the heat source was mainly induced by the descent of the source, while natural convection played only a relatively minor role in the motion.

A theoretical study is made of the global nonlinear growth or decay, in space and time, of an unsteady non-neutral disturbance/wavepacket when a time-dependent nonlinear viscous critical layer is present. The basic flow considered is a steady quasiparallel channel flow, boundary layer or liquid-layer flow at high Reynolds number. The unsteadiness with regard to the critical layer shows itself less in the internal dynamics than in the relatively slow movement of the layer across the flow, the temporal and spatial rate of movement discussed being sufficient to affect the nonlinear viscous balance in the layer. This greatly reduces the mean-flow distortions produced. The disturbance amplitude, in contrast, responds nonlinearly on faster time- and space-scales, both inside and outside the critical layer. These slower- and faster-scale properties inside the critical layer and outside, i.e. globally, are coupled together in general. The work addresses first the structure and nonlinear evolution equations for the growing or decaying free disturbance and the critical layer. But preliminary analysis in special cases suggests, among other things, the significant result that previous nonlinear studies based on quasineutral assumptions give unstable subcritical threshold amplitudes, above which increasingly fast disturbance growth takes place globally.

The momentum integral of a baroclinic jet of fluid in a rotating frame determines the relative size of two jets which are produced when the jet is split by a wall. Owing to lateral variation of velocity and depth of the jet, the percentage of fluid which goes to the right or left differs from that of the non-rotating jet, which is generally assumed to have no shear. For a northern hemisphere jet of zero or constant potential vorticity, much more fluid flows to the right than to the left; for a jet normal to the wall more than 65% goes to the right and less than 35% to the left.

Equilibrium shapes of two-dimensional rotating configurations of uniform vortices are numerically calculated for two to eight corotating vortices. Additionally, a perturbation series is developed which approximately describes the vortex shapes. The equilibrium configurations are subjected to a linear stability analysis. This analysis both confirms existing results regarding point vortices and shows that finite vortices may destabilize via a new form of instability derived from boundary deformations. Finally, we examine the energetics of the equilibrium configurations. We introduce a new energy quantity called ‘excess energy’, which is particularly useful in understanding the constraints on the evolution of unstable near-equilibrium configurations. This theory offers a first glance at nonlinear stability. As an example, the theory explains some features of the merger of two vortices.

The transition from steady axisymmetric Taylor vortices to time-dependent wavy vortices is examined. The critical Taylor number and frequency at the transition point are determined in the infinite-cylinder approximation for a wide range of parameters. The results are compared with long-aspect-ratio experiments. The variation with axial wavelength is examined, and is found to be important when the radius ratio η < 0.75. A new spatially subharmonic mode is found to be the most unstable mode in some parameter regimes. This mode is identified with the jet mode recently discovered experimentally by Lorenzen, Pfister & Mullin and by Cole.

Spectral analysis and flow visualization are presented for various velocity ratios and Reynolds numbers of a jet issuing perpendicularly from a developing pipe flow into a crossflow. The results are complete with conditional averages of various turbulent quantities for one jet-to-cross-flow velocity ratio R of 0.5. A unique conditional-sampling technique separated the contributions from the turbulent jet flow, the irrotational jet flow, the turbulent crossflow and the irrotational crossflow by using two conditioning functions simultaneously. The intermittency factor profiles indicate that irrotational cross-flow intrudes into the pipe but does not contribute to the average turbulent quantities, while the jet-pipe irrotational flow contributes significantly to them in the region above the exit where the interaction between the boundary-layer eddies and those of the pipe starts to take place. Further downstream, the contributions of the oncoming boundary-layer eddies to the statistical averages reduce significantly. The downstream development depends mainly on the average relative eddy sizes of the interacting turbulent fields.

Higher-order nonlinear corrections to the Stokes pendulum problem are calculated in perturbation schemes for small values of Reynolds numbers R(2) = umd0/2ν. Here the controlling lengthscale ½d0 is the displacement amplitude of the undisturbed periodic motion, um is the velocity amplitude and ν is the kinematic viscosity. Solutions for two general types of periodic motion are found; namely orbital motion as under deep water waves and oscillatory motion as under shallow water or acoustic waves. These solutions are found by matched asymptotic expansions using the fundamental irrotational oscillation to drive a thin Stokes a.c. boundary layer over the surface of the sphere. From the boundary layer several secondary motions are excited which die away in the neighbouring fluid. Among these are an orthogonal system of steady, rotational Eulerian streaming currents, and two outwardly radiating non-dispersive waves, one having the frequency of the fundamental but with a phase shift, the other appearing at the second harmonic.

With these solutions the forces and torques on a fixed sphere were computed. One of the orthogonal components of the rotational streaming field was found to produce a rotary lift force which opposed virtual-mass forces and diminished the resultant force component in quadrature to the fundamental oscillation. The other streaming component contributed damping terms which, unlike leading-order Stokes drag, vary nonlinearly with the displacement amplitude. Steady and second-harmonic torques were found to act on the sphere about the horizontal axis transverse to the fundamental oscillation.

The nonlinear growth of periodic disturbances on a finite vortex layer is examined. Under the assumption of constant vorticity, the evolution of the layer may be analysed by following the contour of the vortex region. A numerical procedure is introduced which leads to higher-order accuracy than previous methods with negligible increase in computational effort. The response of the vortex layer is studied as a function of layer thickness and the amplitude and form of the initial disturbance. For small initial disturbances, all unstable layers form a large rotating vortex core of nearly elliptical shape. The growth rate of the disturbances is strongly affected by the layer thickness; however, the final amplitude of the disturbance is relatively insensitive to the thickness and reaches a maximum value of approximately 20% of the wavelength. In the fully developed layers, the amplitude shows a small oscillation owing to the rotation of the vortex core. For finite-amplitude initial disturbances, the evolution of the layer is a function of the initial amplitude. For thin layers with thickness less than 3% of the wavelength, three different patterns were observed in the vortex-core region: a compact elliptic core, an elongated S-shaped core and a bifurcation into two orbiting cores. For thicker layers, stationary elliptic cores may develop if the thickness exceeds 15% of the wavelength. The spacing and eccentricity of these cores is in good agreement with previously discovered steady-state solutions. The growth rate of interfacial area (or length of the vortex contour) is calculated and is found to approach a constant value in well-developed vortex layers.

The present theory provides an asymptotic-expansion method for inviscid compressible flows with shock in arbitrary two-dimensional slender nozzles. The flow in front of the shock is assumed to be potential, whereas the flow behind the shock is considered to be rotational owing to the presence of the shock. A parameter that measures the slenderness of the nozzle is used as the expansion quantity. It is found that, except for the region immediately behind the shock, the same coordinate scale can be used for the flows both in front of and further downstream behind the shock. The flows for the regions thus obtained show that all the streamlines are approximately affinely similar to the nozzle wall, and the leading term of the transverse pressure gradient is determined by the local wall shape. For the flow region immediately behind the shock, however, the transverse pressure gradient just behind the shock is determined by the shock conditions rather than by the local wall shape, and a solution is found for that region which transforms the transverse pressure gradient from that determined by the shock conditions to that determined by the local wall shape. The well-known flow singularity at the intersection of the wall and the shock is involved in the solution. Meanwhile, a critical shock location at which the flow has no singularity is derived. A numerical example shows also that the inviscid flow may separate from the wall, owing to the different entropy increase across the shock for different streamlines. The predicted separation point, however, is only of qualitative value, since our theory does not account for reverse flows.

The properties of steady cellular flows between a rotating cylinder and a stationary outer one of square cross-section have been investigated experimentally. Results for the primary-flow selection process, which involves four-cell and six-cell states, are intricate but bear a reassuring qualitative resemblance to those obtained previously for the standard Taylor–Couette model. In addition, the existence of anomalous modes disconnected from the primary flow has been demonstrated and their dependence upon the length of the flow domain studied. The phenomena are found to be in accord with the abstract theoretical framework that has been successfully used to interpret previous experimental observations on steady-flow problems with multiple solutions.

Experimental techniques, including flying-hot-wire anemometry, have been used to determine the pressure and velocity characteristics of a flow designed to simulate the trailing-edge region of an airfoil at high angle of attack. Emphasis is placed on the region of recirculating flow and on the downstream wake. It is shown that the effect of this recirculation is large even though the details of the flow within it may be unimportant. Normal stresses and cross-stream pressure gradients are important immediately upstream and downstream of the recirculating flow and are associated with strong streamline curvature. The relative importance of the terms in the transport equations for mean momentum and turbulence energy are quantified and the implications for procedures which solve potential-flow and boundary-layer equations and for alternative calculation methods are discussed.

A finite-difference solution method for nonlinear wave generation in the near field of ships of arbitrary three-dimensional configuration is developed. Momentum equations of finite-difference form in a fixed rectangular cell system are solved by a time-marching scheme. The exact inviscid free-surface condition is approximately satisfied at the actual location of the free surface, and the free-slip body boundary condition is implemented by use of approximation of the body configuration and a special pressure computation in body boundary cells. The degree of accuracy is raised by employing a variable-mesh system in the vertical direction. Computed results are presented for three hull forms: a mathematical and two practical hull forms. Agreement with experiment seems to be fairly good. In particular, the computed wave profiles and contour maps of bow waves show excellent resemblance to the measured ones, having some typical characteristics of nonlinear ship waves.

We consider the two-phase flow of a suspension in a rotating cylinder with inclined endplates for which inertial and viscous effects are small. It is shown that, when the Coriolis force is dominant, flow in the core is essentially unaffected by geometry. If a fluid particle can make a complete circuit about the rotation axis, the sedimentation velocity cannot be augmented by geometrical effects as it can in gravitational settling. However, with the insertion of a complete meridional barrier to block movement around the centre, separation becomes more sensitive to the shape of the container walls. In this case, behaviour similar to that in a gravitational field is possible once again.

A new model is presented to describe flow in segments of collapsible tube mounted between two rigid tubes and surrounded by a pressurized container. The new features of the model are the inclusion of (a) longitudinal wall tension and (b) energy loss in the separated flow downstream of the time-dependent constriction in a collapsing tube, in a manner which is consistent with the one-dimensional equations of motion. As well as accurately simulating steady-state collapse, the model predicts self-excited oscillations whose amplitude is large enough to be observable only if the flow in the collapsible tube becomes supercritical somewhere (fluid speed exceeding long-wave propagation speed). The dynamics of the oscillations is dominated by longitudinal movement of the point of flow separation, in response to the adverse pressure gradient associated with waves propagating backwards and forwards between the (moving) narrowest point of the constriction and the tube outlet.

Measurements are presented for low-Reynolds-number turbulent boundary layers developing in a zero pressure gradient on the sidewall of a duct. The effect of rotation on these layers is examined. The mean-velocity profiles affected by rotation are described in terms of a common universal sublayer and modified logarithmic and wake regions.

The turbulence quantities follow an inner and outer scaling independent of rotation. The effect appears to be similar to that, of increased or decreased layer development. Streamwise-energy spectra indicate that, for a given non-dimensional wall distance, it is the low-wavenumber spectral components alone that are affected by rotation.

Large spatially periodic spanwise variations of skin friction are observed in the destabilized layers. Mean-velocity vectors in the cross-stream plane clearly show an array of vortex-like structures which correlate strongly with the skin-friction pattern. Interesting properties of these mean-flow structures are shown and their effect on Reynolds stresses is revealed. Near the duct centreline, where we have measured detailed profiles, the variations are small and there is a reasonable momentum balance.

Large-scale secondary circulations are also observed but the strength of the pattern is weak and it appears to be confined to the top and bottom regions of the duct. The evidence suggests that it has minimally affected the flow near the duct centreline where detailed profiles were measured.

A two-dimensional model of flow and bed topography in sinuous channels with erodible boundaries is developed and applied in order to investigate the mechanism of meander initiation. By reexamining the problem recently tackled by Ikeda, Parker & Sawai (1981), a previously undiscovered ‘resonance’ phenomenon is detected which occurs when the values of the relevant parameters fall within a neighbourhood of certain critical values. It is suggested that the above resonance controls the bend growth, and it is shown that it is connected in some sense with bar instability. In fact, by performing a linear stability analysis of flow in straight erodible channels, resonant flow in sinuous channels is shown to occur when curvature ‘forces’ a ‘natural’ solution represented by approximately steady perturbations of the alternate bar type. A comparison with experimental observations appears to support the idea that resonance is associated with meander formation.

In this paper we generalize the von Kármán solution for flow above a single rotating disk, to include non-axisymmetric solutions. These solutions contain an arbitrary parameter; for zero value of the parameter the asymmetric flow degenerates into the classical von Kármán solution. Thus the classical solution is never isolated when considered within the scope of the full Navier–Stokes equations; there are asymmetric solutions in every neighbourhood of the von Kármán solution. Calculations are reported here for s = 0, 0.02 and 0.06, where s represents the ratio of angular velocity of the fluid at infinity to the angular velocity of the disk. A subset of the solutions obtained here corresponds to flow induced by the rotation of a disk when the latter is placed in a fluid that is moving with a constant uniform velocity.

The most detailed theoretical description of the steady flow at high Reynolds number about a circular cylinder is given by Smith (1979). It appears to give a satisfactory description of much of the flow field. Numerical solutions for this flow computed by Fornberg (1985) are in conflict with some aspects of Smith's model.

An acknowledged weak point in Smith's argument is the lack of a detailed flow for the interior of the eddies, particularly with respect to their rear closure. This paper discusses this aspect of the flow with particular emphasis on the vorticity distribution and with the aid of a solution of the Navier–Stokes equations which is valid at the rear of the eddies. As a result an interpretation of Fornberg's results is possible. The self-induced velocity of translation of the vorticity distribution is considered and discussed with reference to the flow as Re → ∞.