We present a mathematical analysis of the transition from temporally periodic rotating waves to quasi-periodic modulated waves in rotating flows with circularly symmetric boundary conditions, applied to the flow between concentric, rotating cylinders (Taylor-Couette flow). Quasi-periodic flow (modulated wavy vortex flow) is described by two incommensurate, fundamental, temporal frequencies in an arbitrary frame, but the flow is periodic in the appropriate rotating frame. The azimuthal wavelength of the modulation may be different to that of the underlying rotating wave; hence the flow state is described by two azimuthal wavenumbers as well. One frequency and one wavenumber are determined by the wave state, but no simple physical properties have yet been associated with the parameters of the modulation. The current literature on modulated waves displays both conflicting mathematical representations and qualitatively different kinds of modulation. In this paper we use Floquet theory to derive the unique functional form for all modulated waves and show that the space-time symmetry properties follow directly. The flow can be written as a non-separable function of the two phases (θ −c1t, θ −c2t). We show that different branches of modulated wave solutions in Taylor-Couette flow are distinguished not by symmetry but by the ranges of the numerical values of c1, c2, and the spectral amplitudes of the solution. The azimuthal wavenumber associated with the modulation has a unique physical definition but is not directly expressed in the spatial symmetry of the modulated flow. Because modulated waves should occur generically in systems with rotational symmetry, this analysis has application beyond Taylor-Couette flow.

In part 1 (Coughlin & Marcus 1992a) we described the mathematics of temporally quasi-periodic fluid flows, with application to the flow between concentric cylinders (Taylor-Couette flow). In this paper we present numerical simulations of Taylor-Couette flow, motivated by laboratory experiments which show that periodic rotating waves become unstable to quasi-periodic modulated waves as the Reynolds number is increased. We find that several branches of quasi-periodic solutions exist, and not all of them occur as direct bifurcations from rotating waves. We compute solutions to the Navier-Stokes equations for both rotating waves and two branches of modulated waves; those discussed by Gorman & Swinney (1979, 1982), and those discovered by Zhang & Swinney (1985). A simple physical process is associated with both types of modulation. In the rotating frame the quasi-periodic disturbance field (defined as the total flow minus the time-averaged, steady-state piece) appears as a set of compact, coherent vortices confined almost entirely to the outflow jet region. They are adveeted (with some distortion) in the azimuthal direction along the outflow between adjacent Taylor vortices, with the frequency of modulation directly related to the mean drift speed. The azimuthal wavelength of the disturbance field determines the second wavenumber. We argue that the quasi-periodic flow arises as an instability of the essentially axisymmetric vortex outflow jet. Experimentally, modulated waves appear to bifurcate directly to low-dimensional chaos. Neither the mathematical nor the physical mechanisms of this translation have been well understood. Our numerical work suggests that the observed transition to chaos results from the existence of physically distinct Floquet modes arising from instability of the outflow jet.

We derive the conditions for the stability of strips or filaments of vorticity on the surface of a sphere. We find that the spherical results are surprisingly different from the planar ones, owing to the nature of the spherical geometr. Strips of vorticity on the surface of a sphere show a greater tendency to roll-up into vortices than do strips on a planar surface.

The results are obtained by performing a linear stability analysis of the simplest, piecewise-constant vorticity configuration, namely a zonal band of uniform vorticity located in equilibrium between two latitudes. The presence of polar vortices is also considered, this having the effect of introducing adverse shear, a known stabilizing mechanism for planar flows. In several representative examples, the fully developed stages of the instabilities are illustrated by direct numerical simulation.

The implication for planetary atmospheres is that barotropic flows on the sphere have a more pronounced tendency to produce small, long-lived vortices, especially in equatorial and mid-latitude regions, than was previously anticipated from the theoretical results for planar flows. Essentially, the curvature of the sphere's surface weakens the interaction between different parts of the flow, enabling these parts to behave in relative isolation.

Symmetric and antisymmetric periodic disturbances introduced directly into the boundary layer on a circular cylinder at low Reynolds number are shown by experiment to be capable of modifying the vortex formation process and changing the vortex shedding frequency. Spectral measurements have shown that the antisymmetric vortex shedding mode is strongly coupled to the symmetric first harmonic mode. When symmetric excitation is applied, three different shapes of the mean velocity profiles can be identified as the forcing amplitude is increased. At low forcing amplitudes nonlinear interaction between the forcing field and the natural wake oscillator produces sum and difference modes. Symmetric forcing with intermediate-amplitude disturbances suppresses the natural shedding frequency, and the dominant vortex shedding energy appears as a sinuous mode at half the excitation frequency. At high symmetric forcing amplitudes a threshold is reached, above which the large-scale vortices do not form. The symmetries of the combination modes follow two simple rules based on the symmetries of the interacting modes. The symmetry rules provide an explanation for the fundamental difference in wake structure that occurs between symmetric forcing and antisymmetric forcing.

A nonlinear computational investigation of thermal convection due to heating from below in a porous layer underlying a fluid layer has been carried out. The motion of the fluid in the porous layer is governed by Darcy's equation with the Brinkman terms for viscous effects and the Forchheimer term for inertial effects included. The motion in the fluid layer is governed by the Navier-Stokes equation. The flow is assumed to be two-dimensional and periodic in the horizontal direction, with a wavelength equal to the critical value at onset as predicted by the linear stability theory. The numerical scheme used is a combined Galerkin and finite-difference method, and appropriate boundary conditions are applied at the interface. Results have been obtained for depth ratios $\hat{d}=0, 0.1, 0.2, 0.5 $ and 1.0, where $\hat{d}$ is the ratio of the thickness of the fluid layer to that of the porous layer. For $\hat{d}=0.1 $, up to Rm (Rayleigh number of the porous layer) equal to 20 times the critical $R_{{\rm m}_{\rm e}} $, the convection is dominated by the porous layer, similar to the situation at onset, even though the Rayleigh number for the fluid layer is well into the supercritical regime. The Nusselt number for $\hat{d}$ less than the critical value (0.13 in the present case) increases sharply with Rm, whereas at larger $\hat{d}$, the increase is very moderate. Heat transfer rates predicted by the numerical scheme for $\hat{d} = 0.1$ and 0.2 show good agreement with the experimental results of Chen & Chen (1989).

A method for numerically simulating the hydroelastic behaviour of a passive compliant wall of finite dimensions is presented. Using unsteady potential flow, the perturbation pressures which arise from wall disturbances of arbitrary form are calculated through a specially developed boundary-element method. These pressures may then be coupled to a suitable solution procedure for the wall mechanics to produce an interactive model for the wall/flow system. The method is used to study the two-dimensional disturbances which may occur on a Kramer-type compliant wall of finite length. Finite-difference methods are used to yield wall solutions driven by the fluid pressure after some perturbation from the equilibrium position. Thus, histories of surface deflection and wall energy are obtained. Such a modelling of the physics of the system requires no presupposition of disturbance form.

A thorough investigation of divergence instability is carried out. Most of the results presented in this paper concern the response of the compliant wall while (and after) a point pressure pulse, carried in the applied flow, travels over the compliant panel. Above a critical flow speed and once sufficient time has passed, the compliant wall is shown to adopt the particular profile of an unstable mode. After this divergence mode has been established, instability is realized as a slowly travelling downstream wave. These features are in agreement with the findings of experimental studies. The role of wall damping is clarified: damping serves only to reduce the growth rate of the instability, leaving its onset flow speed unchanged. The present predictions provide an improvement upon some of the unrealistic aspects of predictions yielded by travelling-wave and standing-wave treatments of divergence instability.

The response of a long compliant panel after a single-point pressure-pulse initiation, applied at its midpoint, is simulated. At flow speeds higher than a critical value, parts of the formerly (at subcritical flow speeds) upstream-travelling wave system change to travel downstream and show amplitude growth. The development of this ‘upstream-incoming’ wave illustrates how divergence instability can occur at locations upstream of the point of initial excitation. Faster flexural waves transmit energy upstream, thereafter these disturbances can evolve into slow downstreamtravelling divergence waves. The spread of the instability to locations both downstream and upstream of the point of initial excitation indicates that divergence is an absolute instability. This behaviour and the effects of wall damping clarified by the present work strongly suggest that divergence is a Class C instability.

Analysis of secondary instabilities of natural convection in a shallow cavity heated from a side has been carried out. For mercury with Prandtl number equal to 0.027 analysis of the primary instabilities by linear theory shows that an instability sets in as transverse cells at Grashof number equal to 9157.6. Instability resulting in oscillatory longitudinal rolls is also possible, their critical Grashof number being equal to 10608.4. The secondary instabilities of the equilibrium states of transverse cells for mercury have been determined. The results show roughly that stable transverse cells with wavelength shorter than the critical become unstable by subharmonic resonance, but the instability for longer cells sets in by a combination resonance. The instability as longitudinal oscillatory rolls reappears at larger values of Grashof number, although slightly delayed by the presence of the transverse cells.

The collisional dynamics of equal-sized water and normal-alkane droplets, in the 150 μm radius range, have been experimentally studied for situations involving 0(1) droplet Weber numbers and head-on to grazing impact parameters. Results show that in the parametric range investigated the behaviour of hydrocarbon droplets is significantly more complex than that of water droplets. For head-on collisions, while permanent coalescence always results for water droplets, the outcome is quite nonmonotonic for the hydrocarbon droplets in that, with increasing droplet Weber number, the collision can result in permanent coalescence, bouncing, permanent coalescence again, and coalescence followed by separation with or without production of satellite droplets. Similar complexities exist for off-centre collisions. Phenomenological explanations are offered for these observations based on the material properties of the fluids, the relative influences of the normal and shearing aspects of the collision, and the nature and extent of energy dissipation due to droplet deformation during collision.

Aspect ratio effects on the vortex shedding flow from a circular cylinder have been studied by using moveable end plates. Experiments were carried out to measure fluctuating forces, shedding frequency and spanwise correlation whilst varying end plate separation and Reynolds number. The aspect ratio (0.25–12) was found to have a most striking effect on the fluctuating lift. Within a certain range of Reynolds number an increase of the sectional fluctuating lift was obtained for reduced aspect ratio, and showed a maximum for an aspect ratio of 1, where the fluctuating lift could be almost twice the value for very large aspect ratios. This increase of the lift amplitude was found to be accompanied by enhanced spanwise correlation of the flow. The measurements were carried out over the Reynolds number range 8 × 103 < Re < 1.4 × 105. The strong increase in fluctuating lift with small aspect ratio did not occur at the lower and upper boundaries of this range. In the lower Reynolds number range (Re < 2 × 104) the trend could be reversed, i.e. the fluctuating lift decreased with decreasing aspect ratio. Also, with small aspect ratio, a shedding breakdown was found in the upper Reynolds number range (Re = 1.3 × 105). The main three-dimensional feature observed was a spanwise variation in the phase of vortex shedding, accompanied by amplitude modulation in the lift signal. However, the level of three-dimensionality can be reduced by using a small aspect ratio. Three-dimensional vortex shedding features are discussed and comparison of the results with those from both two-dimensional numerical simulations and other experiments using large aspect ratios are presented.

Axisymmetric viscous flow around ellipsoids of circular section is studied in detail by the method of matched asymptotic expansion and a deterministic hybrid vortex method. The main feature of the hybrid vortex method consists in solving the viscous vorticity equation by interlacing a finite-difference method for diffusion and a vortex-in-cell method for convection and stretching. Numerical results are presented for an ellipsoid of axis ratio 2:1 and the limiting case of a sphere at various Reynolds numbers between 100 and 3000. Special consideration is given to the evaluation of the drag coefficient which is computed with three different approaches, denoted by S, P and V respectively. Particular emphasis is placed on the numerical implications and physical significance of each of these different approaches. Comparisons between numerical and analytical results at small times show very close agreement in each case. Separation angles, wake lengths and stationary drag coefficients for the sphere are also found to be in good agreement with previous results by a finite-difference method and with the standard drag curve.

Propagating surface evolution in isotropic turbulence is studied using velocity fields generated by direct numerical simulations. The statistics of tangential strain rate, fluid velocity, characteristic curvature and area-following propagating surface elements are investigated. The one-time statistics of strain rate and fluid velocity pass monotonically from Lagrangian value at low propagation speeds to Eulerian values at high speeds. The strain-rate statistics start deviating significantly from the Lagrangian values only for propagating velocities greater than the Kolmogorov velocity scale vη, whereas, with fluid-velocity statistics the deviation occurs only for velocities greater than the turbulence intensity u′. The average strain rate experienced by a propagating surface decreases from a positive value to near zero with increasing propagation velocity. The autocorrelation function and frequency spectrum of fluid velocity and strain rate scale as expected in the limits of small and large propagation velocities. It is also found that for the range of propagation velocities considered, an initially plane surface element in turbulence develops a cusp in finite time with probability nearly one. The evolution of curvature is studied using the concept of hitting time. Initially plane propagating surfaces end up being almost cylindrical in shape. Highly curved surface elements are associated with negative strain rates and small surface areas.

An effective and practical method of producing strong constant-shear flow with a low turbulence level in the laboratory has been developed. A pair of closely spaced shaped gauzes with the upstream gauze of non-uniform porosity and the downstream gauze of uniform porosity was used for the generation of the flow. The method presented can be generalized to specify the gauze shape and porosity distribution necessary to generate other velocity distributions desired in a duct flow.

It is well known that a system of N point vortices arranged in a circular row, so that the vortices are at the vertices of a regular polygon, is stable if N < 7, neutrally stable if N = 7 and unstable if N > 7 (Havelock 1931). The effect on this result of taking account of the finite size of the vortices is considered analytically. The vortices are considered to be uniform with small but finite core. Approximate equations for the shape and motion of a vortex subjected to an external velocity field are given and used to evaluate the shape and angular velocity of rotation of the system and to study its stability to plane infinitesimal disturbances. It is found that the system is stable if N < 7 and unstable if N ≥ 7. These asymptotic results for small core area are in general consistent with Dritschel (1985) where the motion and stability of up to N = 8 finite vortices is evaluated numerically; the steady configuration and the stability results for these values of N are in agreement except in a region of parameter space where a high degree of accuracy is required in the numerical calculation to resolve the growth rate of small disturbances. The case of a linear array of finite vortices is obtained as a special limiting case of the system. The growth rate of plane infinitesimal disturbances for this case is given.

A finite-element method has been developed to model inertial waves in a frustum of a cone, since analytical methods have proved inadequate. The governing Poincaré problem is posed as a variational principle and approximate eigensolutions are computed. The numerical results are used to complete the interpretation of the experimental results of Beardsley (1970) for a frustum of a cone. The significant role played by characteristic surfaces partly explains the enigmatic nature of the ill-posed boundary-value problem that describes inertial wave resonances.

The asymptotic structure of the three-dimensional turbulent boundary layer near a plane of symmetry is considered in the limit of large Reynolds number. A selfconsistent two-layer structure is shown to exist wherein the streamwise velocity is brought to rest through an outer defect layer and an inner wall layer in a manner similar to that in two-dimensional boundary layers. The cross-stream velocity distribution is more complex and two terms in the asymptotic expansion are required to yield a complete profile which is shown to exhibit a logarithmic region. The flow in the inner wall layer is demonstrated to be collateral to leading order; pressure gradient effects are formally of higher order but can cause the velocity profile to skew substantially near the wall at the large but finite Reynolds numbers encountered in practice. The governing set of ordinary differential equations describing a self-similar flow is derived. The calculated numerical solutions of these equations are matched asymptotically to an inner wall-layer solution and the results show trends that are consistent with experimental observations.

The paper presents a linear analytical model to predict the frequency characteristics of the discrete oscillations of the jet-edge feedback cycle. The jet is idealized as having top-hat profile with vortex-sheet shear layers, and the nozzle from which it issues is represented by a parallel plate duct. At a stand-off distance h, a flat plate is inserted along the centreline of the jet, and a sinuous instability wave with real frequency w is assumed to be created in the vicinity of the nozzle and to propagate towards the splitter plate. Its interaction with the splitter plate produces an irrotational feedback field which, near the nozzle exit, is a periodic transverse flow producing singularities at the nozzle lips. Vortex shedding is assumed to occur, alleviating the singularities and allowing a trailing-edge Kutta condition to be satisfied; this Kutta condition is claimed to be the phase-locking criterion. The shed vorticity develops into a sinuous spatial instability, and the cycle of events is repeated periodically.

Problems corresponding to the various physical processes described are analysed, for in viscid flow with vortex-sheet shear layers and aligned flat-plate boundaries, and solved in an appropriate asymptotic sense by Wiener-Hopf methods. Calculation of the phase changes occurring in the constituents of the cycle gives an equation for the frequency w in the Nth ‘stage’ as a function of jet width 2b, jet velocity U0, standoff distance h, and stage label N: \[ \omega b/U_0 = (b/h)^{\frac{3}{2}}[4\pi(N-{\textstyle\frac{3}{8}})]^{\frac{3}{2}}. \] The variations with b, U0, h and N are in excellent agreement with edge-tone experiments; the principal disagreement lies in the overall numerical factor $(4\pi)^{\frac{3}{2}}$ and explanations are given for this. Possible effects associated with the inclusion of displacement thickness fluctuations in the splitter-plate boundary layers, and the enforcement of a leading-edge Kutta condition, are also considered and shown not to affect the frequencies of the operating stages.

The transition between hexagons and rolls in convective patterns has been studied. The transition thresholds and changes in the Nusselt number are discussed theoretically in terms of calculations made by Busse (1967a) and with amplitude equations. Experiments have been done in a shallow layer of pure water under non-Boussinesq conditions using complementary techniques: shadowgraph (qualitative), optical (based on the deflections of a laser beam) and calorimetric. The experimental values of the critical Rayleigh number Rc and the critical wavenumber kc are in agreement with theoretical ones. However, theory and experiments show some discrepancies in the slopes of the non-dimensional convective heat flow curve and in the thresholds of the hysteretic hexagons-rolls transition. These discrepancies are discussed in terms of lateral effects and of the presence of defects in the pattern.

Besonant interactions in flows over rigid and flexible walls are studied. Attention is concentrated on symmetric, three-dimensional wave triads as proposed by Craik (1971). Location of resonant triads and evaluation of interaction coefficients are performed numerically, for a wide range of Reynolds numbers and wavenumbers, considering the temporal stability problem. Good agreement is found with previous work. It is shown that triads comprising various combinations of Tollmien-Schlichting and/or wall modes are possible, and have some interesting features; also, the possibility of interaction with Squire modes must not be overlooked.

The Kraichnan-Phillips theorem asserts that, for incompressible, homogeneous, turbulent flow over a plane, rigid wall, the wall-pressure wavenumber-frequency spectrum P(k, ω) → 0 as the planar wavenumber k → 0 provided the frequency ω ≠ 0. A proof of this theorem is given by use of a general formula that expresses the normal force on an arbitrary rigid body in terms of volume and surface integrals involving the vorticity. Implications for the theory of flow-induced surface vibrations are briefly discussed.

When the fluid inside a completely filled cylinder is set in motion by the rotation of one endwall, steady and unsteady axisymmetric vortex breakdown is possible. Nonlinear dynamical systems theory is used to describe the changing kinematics of the flow as the speed of the rotating endwall is increased. Two distinct modes of oscillation have been found in the unsteady regime and the chaotic advection caused by the oscillations has been investigated. The results of this study are used to describe the filling and emptying processes of the vortex breakdown bubbles observed in flow visualization experiments.