Two-dimensional finite-amplitude Kelvin–Helmholtz waves are tested for stability against three-dimensional infinitesimal perturbations. Since the nonlinear waves are time-dependent, the stability analysis is based upon the assumption that they evolve on a timescale which is long compared with that of any instability which they might support. The stability problem is thereby reduced to standard eigenvalue form, and solutions that do not satisfy the timescale constraint are rejected. If the Reynolds number of the initial parallel flow is sufficiently high the two-dimensional wave is found to be unstable and the fastest-growing modes are three-dimensional disturbances that possess longitudinal symmetry. These modes are convective in nature and focused in the statically unstable regions that form during the overturning of the stratified fluid in the core of the nonlinear vortex. The nature of the instability in the high-Reynolds-number regime suggests that it is intimately related to the observed onset of turbulence in these waves. The transition Reynolds number above which the secondary instability exists depends strongly on the initial conditions from which the primary wave evolves.

Detailed mean-flow and turbulence measurements have been made in a low-speed turbulent boundary layer in zero pressure gradient with an isolated, artificially generated vortex imbedded in it. The vortex was generated by a half-delta wing on the floor of the wind-tunnel settling chamber, so that the vortex entering the working section had the same circulation as that originally generated, while axial-component velocity variations were very much reduced, relative to the local mean velocity, from values just behind the generator. The measurements show that the circulation around the vortex imbedded in the boundary layer is almost conserved, being reduced only by the spanwise-component surface shear stress. Therefore the region of flow affected by the vortex continues to grow downstream, its cross-sectional dimensions being roughly proportional to the local boundary-layer thickness. The behaviour of the various components of eddy viscosity, deduced from measured Reynolds stresses, and of the various triple products, suggests that the simple empirical correlations for these quantities used in present-day turbulence models are not likely to yield flow predictions which are accurate in detail.

We consider the small-Rossby-number flow of a fluid past an obstacle in a coordinate frame in which the rotation rate varies linearly in the direction normal to the flow in a manner that models the variation of the Coriolis force for midlatitude planetary motions. The eastward flow is characterized by strong upstream divergence of the streamlines like that noted by Davies & Boyer (1982), and a similarly severe streamline convergence in the lee of the obstacle. Such a structure occurs for small values of the β-parameter that measures the importance of the lateral angular-velocity variation. In this parameter range, Rossby waves occur, but are confined to a narrow region in the lee of the object. The presence of these waves modifies the edge velocity ‘seen’ by the Stewartson quarter layer in such a way as to delay the onset of separation beyond what one might expect based on the analysis of Walker & Stewartson (1974) for a flow without beta-effect.

Acoustic emission by vortex–edge interaction is investigated both theoretically and experimentally. The theory of vortex sound enables us to represent the far-field pressure in terms of the vortex motion near the half-plane edge. It is found that the pressure p depends on the product of an angular factor representing directionality and a time factor representing wave profile. The pressure formula leads to the scaling law $p \propto U^{\frac{5}{2}} L^{-2}$ for the sound emitted by a vortex ring of velocity U, L being the nearest distance of the vortex path to the edge. The sound intensity is proportional to U5 and shows cardioid directionality pattern.

The vortex ring used in the experiment had radius about 4.7 mm and velocity ranging from 29 to 61 m/s. The above scaling law of the pressure and the cardioid directionality of the intensity were reproduced in the experiment with reasonable accuracy. Especially notable is the agreement between the predicted and observed wave profiles. The theoretical profile is determined by the $\frac{3}{2}\,{\rm th}$ time derivative of the volume flux (through the vortex ring) of a hypothetical potential flow around the edge.

The newly developed simulation method known as Stokesian dynamics is used to investigate the rheological behaviour of concentrated suspensions. Both the detailed microstructure (e.g. pair-distribution function) and the macroscopic properties are determined for a suspension of identical rigid spherical particles in a simple shear flow. The suspended particles interact through both hydrodynamic and non-hydrodynamic forces. For suspensions with purely hydrodynamic forces, the increase in the suspension viscosity with volume fraction ϕ is shown to be caused by particle clustering. The cluster formation results from the lubrication forces, and the simulations of a monolayer of spheres show a scaling law for the cluster size: lc ∼ [1 − (ϕ/ϕm)½]−1, where ϕm is the maximum volume fraction that can shear homogeneously. The simulation results suggest that the suspension viscosity becomes infinite at the percolation-like threshold ϕm owing to the formation of an infinite cluster. The predicted simulation viscosities are in very good agreement with experiment. A suspension with short-range repulsive interparticle forces is also studied, and is seen to have a non-Newtonian rheology. Normal-stress differences arise owing to the anisotropic local structure created by the interparticle forces. The repulsive forces also reduce particle clustering, and as a result the suspension is shear-thickening.

In studying head-sea diffraction by a slender body, Haren & Mei (1981) found, at a special frequency, a singularity inherent in the cross-section (inner) mathematical problem. In this note this singularity is interpreted, and it is shown that the head-sea problem is similar to the problem of an acoustic duct being excited at one end by a pressure field. As a consequence, the head sea is not always refracted away from the body, since the solution remains oscillatory for frequencies above that special frequency.

The motions of vortices around single cylinders and around pairs of cylinders in relative sinusoidal flow are investigated in this paper. Using simultaneous flow visualization and force measurements, the vortex motions are related to the fluid-induced lift and in-line forces. For the single cylinder, several repeatable patterns of vortex shedding are identified within particular ranges of flow amplitude. The process of pairing of vortices from a previous half cycle with those in a present half cycle is fundamental to all the patterns. Visualization is shown to be more effective in a reference frame which is fixed with respect to the undisturbed fluid rather than with respect to the cylinders. For this reason, the examples of vortex motions are taken from a rig in which vertical cylinders are oscillated in a tank of fluid. By oscillating a pair of cylinders over a range of gaps, orientations and amplitudes, it is found that the vortex-shedding patterns identified for a single cylinder can synchronize either in phase or in antiphase between the two cylinders. Such observations help to explain how lift and in-line forces are influenced by cylinder proximity and in some cases these forces are significantly magnified. Force coefficients are evaluated for both the single cylinder and the pair of cylinders.

The steady planar symmetric motion of an incompressible fluid, past a symmetric bluff body fixed in an otherwise uniform stream, is considered for large Reynolds numbers Re. A laminar-flow structure is proposed which consists primarily of (a) the large-scale flow and (b) the smaller, body-scale, flow. Here (a) involves a pair of massive, effectively inviscid, recirculating eddies set up behind the body and bounded by viscous shear layers. Each eddy has small constant vorticity and its length and width both increase linearly with Re, so that the large-scale potential flow outside the eddies is significantly disturbed from the oncoming stream. This reduces the effective free stream acting on (b). The latter has the Kirchhoff property of a parabolic growth in the eddy width downstream; but its eddy vorticity is non-uniform and substantial, contrary to the Kirchhoff and Prandtl–Batchelor models, and secondary separation is possible. The non-uniform vorticity is provoked by the thick return jet, which is forced back along the centreline in (a) from downstream. Buffer zones, e.g. of length ∝ Re½, are required to join (b) fully to (a). The resulting drag coefficient cD is believed to be O(1) generally, and is controlled, along with the eddy length and vorticity, by a combination of the viscous–inviscid flow problems posed in both (a) and (b). A special case of small cD is also covered. The structure seems self-consistent so far, and tends to compare reasonably well with recent numerical solutions of the Navier–Stokes equations at increased Re.

An Appendix describes the inviscid parts of (a) for relatively thin eddies.

The unsteady two-dimensional heat-transfer problem of melting around a horizontal heated cylinder is studied numerically. The cylinder is heated isothermally. A physical model is introduced which accounts for the effects of density change upon melting and subcooling effect, as well as natural convection. Most previous work has assumed that the density number (ratio of solid density to liquid density) is unity. In practice, all solid materials exhibit some density change upon melting. If the density number is greater than one, this induces a blowing effect at the phase-change boundary. If the density number is less than one, a suction effect is produced. This study indicates that the density-change effect on heat transfer during melting is minor. Subcooling results when the solid is at a temperature below the melting temperature. When the melting process begins, some of the available thermal energy must be used as sensible heat, to raise the solid's temperature to the melting point. As a result, less thermal energy is available for melting. Subcooling effects are found to have a substantial effect on the heat-transfer process. The effects of natural convection have been clearly documented, and indicate that natural convection must be included in any realistic model of the melting process. Detailed predictions of the effects of density change and subcooling on the melting process are given. Information on the temperature and flow fields for representative values of Stefan, Rayleigh, Prandtl, subcooling and density number is given. Further results from the numerical solutions include information on local and average heat-transfer rates and sensibleheat gain as well as melt volume as a function of time. Comparisons are made with earlier numerical and analytical results.

Thermally induced flow instabilities in uniformly heated boiling channels have been studied analytically. The classical homogeneous equilibrium model was used. This distributed model was transformed into an integrodifferential equation for inlet velocity. A linear analysis showed interesting features (i.e. islands of instability) of the marginal stability boundary which appear when the effects of gravity and friction were systematically considered. A quasilinear Hopf-bifurcation analysis, valid near the marginal-stability boundaries, gives the amplitude and frequency of limit-cycle oscillations that can appear on the unstable side of the boundary. The analysis also shows cases where a finite-amplitude perturbation can cause a divergent instability on the stable side of the linear-stability boundary.

The axisymmetric flow inside a rotating cavity with radial outflow or inflow of fluid is discussed. The basic theoretical model of Hide (1968) is extended, using the integralmomentum techniques of von Kármán (1921), to include laminar and turbulent flows; both linear and nonlinear equations are considered. The size of the source region is estimated using a ‘free disk’ model for the outflow case and a free vortex for the inflow case. In both cases, the estimates are in good agreement with available experimental data. Theoretical values of the tangential component of the velocity outside the Ekman layers on the disks, obtained from solutions of the laminar and turbulent integral equations, are compared with experimental values. The experiments were conducted in a number of rotating-cavity rigs, with a radial outflow or inflow of air, and laser-Doppler anemometry was used to measure the velocity in the ‘interior core’ between the Ekman layers. The measurements provide good support for the theoretical models over a wide range of flow rates, rotational speeds and radial locations. Although only isothermal flow is considered in this paper, the methods can be readily extended to non-isothermal flow and heat transfer.

Previous experiments on natural convection in a differentially heated porous layer with large lateral dimensions gave evidence for different configurations of flow. Depending on the values of the Rayleigh number, the inclination and the longitudinal extension of the layer, the three main structures observed correspond to a two-dimensional unicellular flow, polyhedral convective cells and longitudinal coils. In this paper there is a definition of the conditions necessary for these types of flow to exist using a linear stability theory and it is shown that the experimentally observed structures can be theoretically predicted by a three-dimensional numerical model based upon Galerkin's spectral method. Finally, the results of this model are used to show the influence of initial conditions on the setting up of the stationary flow.

The stability of a capillary jet of an ideal liquid with a linear variation of axial velocity is investigated. Because of the time dependence in the basic extensional flow the evolution of surface perturbations in the jet is an initial-value problem instead of an eigenvalue one (as in the case of non-stretching jets). The amplification of any given peturbation is found to depend on the elative effects of surface tension and intertia terms associated with the extensional flow as well as on the initial wavenumber and the specific time when the perturbation is introduced in the flow field. The simulation of a shaped-charge jet by the present model is discussed. The esults obtained are found to give a good description of the essential features of the breakup phenomenon of such jets.

The dispersion of fluid particles in a turbulent flow is described by transition-probability density functions. A renormalized expansion is made to establish the evolution equations of these functions. The resulting equations are nonlinear integrodifferential equations written in terms of Eulerian velocity-correlation functions. For the dispersion of a single particle, the equation at the zeroth order is the same as the one obtained by Roberts (1961). For the relative dispersion of a pair of particles, the equation is more convenient for applications than those of other theories. With this equation, Richardson's $\frac{4}{3}$-power law for relative diffusion is recovered analytically based upon the Kolmogoroff spectrum and numerically based upon the von Kármán spectrum and a smoothed experimental spectrum.

This paper has three main objectives. First, it aims to show that basic general conservation principles for viscous flow can be formulated in terms of diffusion and convection. Secondly, it aims to show that three scalar conservation principles suffice to provide a method for characterizing swirling axisymmetric flows in terms of axial and boundary production of the conserved quantities. Thirdly, it aims to exemplify these two objectives by giving a complete specification of the axial causes for swirl-free conically similar flow in otherwise free space.

This series of papers, overall, is concerned with the analysis and characterization of swirling conically similar flows in terms of the singularities that generate the conserved quantities. In conically similar flows there is no natural lengthscale, and the sole parameters governing the flow are provided by the strengths of the singularities that cause the flow. These are required to have the same dimensions as a power of the kinematic viscosity v. The axisymmetric flow generated by uniform production of swirl angular momentum per unit mass along a half-axis at a constant rate provides a simple example.

In conically similar flow the three conservation principles for axisymmetric flow provide a sixth-order non-autonomous system of two ordinary differential equations governing the flow. Here, in Part 1, these equations are derived for the general case of swirling flow, and are shown to reduce to a fourth-order system when swirl is absent. The two scalar conservation principles describing swirl-free flow are used to classify the basic axial causes for this system.

Part 2 analyses these basic exact one-parameter swirl-free families of solutions, and Part 3 extends the analysis to the remaining one-parameter family of swirling flows associated with uniform swirl angular-momentum production on a half-axis. Each of the families is characterized by a single independent cause, and two of them provide new non-trivial solutions of the Navier–Stokes equations. The effects of nonlinear coupling of these basic one-parameter causes and of conically similar distributions over conical boundaries will be examined in later papers.

It was shown in Part 1 of this series that in swirl-free flow there are three different types of axial causes of steady conically similar viscous flow. The three corresponding swirl-free one-parameter families of exact solutions to the Navier–Stokes equations are presented and analysed here in terms of the basic conservation principles for volume and ring circulation. The simplest is the irrotational flow generated by a uniform distribution of volume sources along a half-axis. A second, independent, one-parameter family of solutions is provided by Landau's (1943) solution, where the second moment of ring circulation about the axis is produced at the origin at a finite constant rate. Fresh insight into the nature of this flow is gained by separating and comparing the roles of the diffusion and convection terms in the flux vector for ring circulation. A similar analysis is applied to the remaining independent one-parameter family caused by an antisymmetric (about the origin) conically similar axial distribution of the singularity in Landau's solution. This simple new family of exact solutions is characterized by opposed jets neighbouring the axis of symmetry. When the axial jets are directed inwards, they always erupt into an emergent axisymmetric jet normal to the axis of symmetry. Solutions fail to exist, however, for sufficiently strong axial jets directed outwards.

In Part 1 of this series conservation principles for ring circulation and kinematic swirl angular momentum were developed for general axisymmetric incompressible viscous flow. These principles were then used to classify the four independent axial causes of swirl-free conically similar viscous flow. Part 2 provided a detailed analysis of the one-parameter swirl-free flows that are generated by each one of the axial singularities acting alone. The present paper extends, to swirling flow, the description of the axial singularities that drive axisymmetric viscous flow. In the special case of conically similar viscous flow, two independent half-line sources of swirl angular momentum suffice to complete the set of axial singularities that can generate such swirling flows. The individual strengths of the six independent axial causes provide a complete characterization of all conically similar viscous flows that can be generated in this way. This Part 3 completes the task of analysing in detail the independent one-parameter flows generated by axial causes by studying the flow caused by uniform production of kinematic swirl angular momentum on a half-axis. This flow demonstrates how swirl may induce an axial half-plane flow. For large swirl circulation strengths, swirl angular momentum diffuses and convects so as to fill slightly more than half the space with an almost constant density of swirl angular momentum. A well-developed internal boundary layer, in the form of an outward radial jet, then separates this region from one in which the flow is almost irrotational. The jet entrains two impinging convection fields. The angular location of the jet is determined by relating the axial component of moment of whirl produced at the origin to the strength of the swirling circulation singularity on the axis.

The problem of slow draining of incompressible fluid from a rotating cylindrical vessel is considered. Transient effects, internal viscosity, boundary friction and finite cylindrical boundaries are included. In addition a tornado model formulated by Turner is solved and compared with some experiments on rotating turbulence of Hopfinger, Browand & Gagne.

Two classic problems in unsteady boundary layers, the Stewartson and the Lam & Crocco problems, are formulated with a unified new semi-similar transformation using velocity and static enthalpy as dependent variables. By this formulation, the resulting governing equations – singular parabolic in nature – for these two physically different problems are shown to closely resemble one another in all essential aspects. For both cases, the domain of the streamwise independent variable is mapped onto [0, 1] for all t. The existence of the Reynolds analogy and the exact energy integral are given; their relations are shown to be different from those in the steady boundary layers. Uniformly valid solutions are shown to be obtainable, accurately, by a standard relaxation method commonly applied to the solution of elliptical partial differential equations. Characteristics of the transition from non-similar solutions to downstream similar solutions are discussed.

We consider the flow driven by a thermally induced surface stress in a fluid held in a shallow two-dimensional slot, and show that, for low Maragoni number, the extent of surface stagnation due to the presence of a non-diffusing surfactant depends on a single parameter E, the elasticity number defined in Part 1 of this analysis (Homsy & Meiburg 1984). For situations in which the adsorbed species are insufficient to result in a fully covered surface, we find that the interface is either clean and subject to constant stress, or contaminated and no-slip. There is a region in which one type of surface is replaced by the other. The Wiener–Hopf technique is used to obtain an analytic expression for the stream function in the vicinity of the leading edge of the stagnant surface in the limit of creeping flow. This result shows that the flow dies off under the stagnant surface at a distance of the order of the depth of the fluid, in a series of vortices of exponentially decreasing magnitude.