A similarity solution is found which describes the flow impinging on a flat wall at an arbitrary angle of incidence. The technique is similar to a method used by Jeffery (1915) and discussed more recently by Peregrine (1981).

We have performed direct numerical simulations of turbulent flows in a square duct considering a range of Reynolds numbers spanning from a marginal state up to fully developed turbulent states at low Reynolds numbers. The main motivation stems from the relatively poor knowledge about the basic physical mechanisms that are responsible for one of the most outstanding features of this class of turbulent flows: Prandtl's secondary motion of the second kind. In particular, the focus is upon the role of flow structures in its generation and characterization when increasing the Reynolds number. We present a two-fold scenario. On the one hand, buffer layer structures determine the distribution of mean streamwise vorticity. On the other hand, the shape and the quantitative character of the mean secondary flow, defined through the mean cross-stream function, are influenced by motions taking place at larger scales. It is shown that high velocity streaks are preferentially located in the corner region (e.g. less than 50 wall units apart from a sidewall), flanked by low velocity ones. These locations are determined by the positioning of quasi-streamwise vortices with a preferential sign of rotation in agreement with the above described velocity streaks' positions. This preferential arrangement of the classical buffer layer structures determines the pattern of the mean streamwise vorticity that approaches the corners with increasing Reynolds number. On the other hand, the centre of the mean secondary flow, defined as the position of the extrema of the mean cross-stream function (computed using the mean streamwise vorticity), remains at a constant location departing from the mean streamwise vorticity field for larger Reynolds numbers, i.e. it scales in outer units. This paper also presents a detailed validation of the numerical technique including a comparison of the numerical results with data obtained from a companion experiment.

This paper presents the first ‘exact’ solutions to the creeping-flow equations for the transverse motion of a sphere of arbitrary size and position between two plane parallel walls. Previous solutions to this classical Stokes flow problem (Ho & Leal 1974) were limited to a sphere whose diameter is small compared with the distance of the closest approach to either boundary. The accuracy and convergence of the present method of solution are tested by detailed comparison with the exact bipolar co-ordinate solutions of Brenner (1961) for the drag on a sphere translating perpendicular to a single plane wall. The converged series collocation solutions obtained in the presence of two walls show that for the best case where the sphere is equidistant from each boundary the drag on the sphere predicted by Ho & Leal (1974), using a first-order reflexion theory, is 40 per cent below the true value when the walls are spaced two sphere diameters apart and is one order-of-magnitude lower at a spacing of 1.1 diameters.

The structure and spectrum of inertial waves of an incompressible viscous fluid inside a spherical shell are investigated numerically. These modes appear to be strongly featured by a web of rays which reflect on the boundaries. Kinetic energy and dissipation are indeed concentrated on thin conical sheets, the meridional cross-section of which forms the web of rays. The thickness of the rays is in general independent of the Ekman number E but a few cases show a scaling with E1/4 and statistical properties of eigenvalues indicate that high-wavenumber modes have rays of width O(E1/3). Such scalings are typical of Stewartson shear layers. It is also shown that the web of rays depends on the Ekman number and shows bifurcations as this number is decreased.

This behaviour also implies that eigenvalues do not evolve smoothly with viscosity. We infer that only the statistical distribution of eigenvalues may follow some simple rules in the asymptotic limit of zero viscosity.

For large classes of vorticities we prove that a steady periodic gravity water wave with a monotonic profile between crests and troughs must be symmetric. The analysis uses sharp maximum principles for elliptic partial differential equations.

A linearized stability analysis of the flow of water in a channel with a loose bed and straight banks is described. It is assumed that the wavelength of the perturbations, which develop into meanders or braids, is longer than the width of the channel. It is therefore long compared with the ripples or dunes which cover the bed of such a channel and whose wavelength is shorter than the width of the channel. The latter need be allowed for only as roughness elements creating resistance. The variation of resistance to flow and rate of transport of bed material with velocity are discussed briefly and taken into account. Instability is interpreted as leading to a meandering or braided channel and it is shown that all practicable channels are unstable. Wavelengths calculated for channels expecte to meander are compatible with those given by Inglis's empirical rule and wavelengths calculated for channels which become braided are approximately the same as those observed.

Cavitation has been observed in the trailing vortex system of an elliptic planform hydrofoil. A complex dependence on Reynolds number and gas content is noted at inception. Some of the observations can be related to tension effects associated with the lack of sufficiently large-sized nuclei. Inception measurements are compared with estimates of pressure in the vortex obtained from LDV measurements of velocity within the vortex. It is concluded that a complete correlation is not possible without knowledge of the fluctuating levels of pressure in tip-vortex flows. When cavitation is fully developed, the observed tip-vortex trajectory shows a surprising lack of dependence on any of the physical parameters varied, such as angle of attack, Reynolds number, cavitation number, and dissolved gas content.

A buoyancy-driven system can be unstable due to two different mechanisms—one mechanical and the other involving buoyancy forces. The mechanical instability is of the type normally studied in connexion with the Orr-Sommerfeld equation. The buoyancy-driven instability is rather different and is related to the ‘Coriolis’-driven instability of rotating fluids. In this paper, the stability of a buoyancy-driven system, recently called a ‘buoyancy layer’, is examined for the whole range of Prandtl numbers, s. The buoyancy-driven instability becomes increasingly important as the Prandtl number is increased and so particular interest is attached to the limit in which the Prandtl number tends to infinity. In this limit, the system is neutrally stable to first order, but second-order effects render the flow unstable at a Reynolds number of order σ-½. Consequences of the results for the stability of convection in a vertical slot are examined.

The paper describes an investigation of a subjectively distinguishable element of high speed jet noise known as ‘crackle’. ‘Crackle’ cannot be characterized by the normal spectral description of noise. It is shown to be due to intense spasmodic short-duration compressive elements of the wave form. These elements have low energy spread over a wide frequency range. The crackling of a large jet engine is caused by groups of sharp compressions in association with gradual expansions. The groups occur at random and persist for some 10−1s, each group containing about 10 compressions, typically of strength 5 × 10−3 atmos at a distance of 50 m. The skewness of the amplitude probability distribution of the recorded sound quantifies crackle, though the recording process probably changes the skewness level. Skewness values in excess of unity have been measured; noises with skewness less than 0·3 seem to be crackle free. Crackle is uninfluenced by the jet scale, but varies strongly with jet velocity and angular position. The jet temperature does not affect crackle, neither does combustion. Supersonic jets crackle strongly whether or not they are ideally expanded through convergent-divergent nozzles. Crackle is formed (we think) because of local shock formation due to nonlinear wave steepening at the source and not from long-term nonlinear propagation. Such long-term effects are important in flight, where they are additive. Some jet noise suppressors inhibit crackle.

A derivation is given of the Eulerian equations of motion directly from the Lagrangian formulation of Hamilton's principle. The circulation round a circuit of material particles of uniform entropy appears as a constant of the motion associated with the indistinguishability of fluid elements with equal density, entropy and velocity. A discussion is given of the Lin constraint, and it is pointed out that, for a barotropic fluid, the variational principle recently suggested by Seliger & Whitham does not permit velocity fields in which the vortex lines are knotted.

We present the results of a combined experimental and theoretical investigation of the family of free-surface flows generated by obliquely colliding laminar jets. We present a parameter study of the flow, and describe the rich variety of forms observed. When the jet Reynolds number is sufficiently high, the jet collision generates a thin fluid sheet that evolves under the combined influence of surface tension and fluid inertia. The resulting flow may take the form of a fluid chain: a succession of mutually orthogonal links, each composed of a thin oval film bound by relatively thick fluid rims. The dependence of the form of the fluid chains on the governing parameters is examined experimentally. An accompanying theoretical model describing the form of a fluid sheet bound by stable rims is found to yield good agreement with the observed chain shapes. In another parameter regime, the fluid chain structure becomes unstable, giving rise to a striking new flow structure resembling fluid fishbones. The fishbones are demonstrated to be the result of a Rayleigh–Plateau instability of the sheet's bounding rims being amplified by the centripetal force associated with the flow along the curved rims.

Turbulent boundary layers along a convex surface of varying curvature were investigated in a specially designed boundary-layer tunnel. A fairly complete set of turbulence measurements was obtained.

The effect of curvature is striking. For example, along a convex wall the Reynolds stress is decreased near the wall and vanishes about midway between the wall and the edge of a boundary layer where there exists a velocity profile gradient created upstream of the curved wall.

Singular perturbation techniques are used to calculate the migration velocity of a spherical particle sedimenting, at low Reynolds numbers, in a stagnant viscous fluid bounded by one or two infinite vertical plane walls. The method is then used to study the migration of a pair of spherical particles sedimenting either in unbounded fluid or in fluid bounded by a plane vertical wall. The migration phenomenon is studied experimentally by recording the trajectory of a spherical particle settling through a viscous fluid bounded by parallel vertical plane walls. Duct- to particle-diameter ratios in the range of 27 to 48 were used with the Reynolds number based on the particle radius being between 0·03 and 0·136.

In all cases the particle is observed to migrate away from the walls until it reaches an equilibrium position at the axis of the duct. The experimentally determined migration velocities agree well with those predicted by the present theory.

The boundary-layer flow over a semi-infinite vertical flat plate, heated to a constant temperature in a uniform free stream, is discussed in the two cases when the buoyancy forces aid and oppose the development of the boundary layer. In the former case, two series solutions are obtained, one of which is valid near the leading edge and the other is valid asymptotically. An accurate numerical method is used to describe the flow in the region where the series are not valid. In the latter case, a series, valid near the leading edge is obtained and it is extended by a numerical method to the point where the boundary layer is shown to separate.

A method is presented for treating problems of the propagation and ultimate decay of the shocks produced by explosions and by bodies in supersonic flight. The theory is restricted to weak shocks, but is of quite general application within that limitation. In the author's earlier work on this subject (Whitham 1952), only problems having directional symmetry were considered; thus, steady supersonic flow past an axisymmetrical body was a typical example. The present paper extends the method to problems lacking such symmetry. The main step required in the extension is described in the introduction and the general theory is completed in §2; the remainder of the paper is devoted to applications of the theory in specific cases.

First, in §3, the problem of the outward propagation of spherical shocks is reconsidered since it provides the simplest illustration of the ideas developed in §2. Then, in §4, the theory is applied to a model of an unsymmetrical explosion. In §5, a brief outline is given of the theory developed by Rao (1956) for the application to a supersonic projectile moving with varying speed and direction. Examples of steady supersonic flow past unsymmetrical bodies are discussed in §6 and 7. The first is the flow past a flat plate delta wing at small incidence to the stream, with leading edges swept inside the Mach cone; the results agree with those previously found by Lighthill (1949) in his work on shocks in cone field problems, and this provides a valuable check on the theory. The second application in steady supersonic flow is to the problem of a thin wing having a finite curved leading edge. It is found that in any given direction the shock from the leading edge ultimately decays exactly as for the bow shock on a body of revolution; the equivalent body of revolution for any direction is determined in terms of the thickness distribution of the wing and varies with the direction chosen. Finally in §8, the wave drag on the wing is calculated from the rate of dissipation of energy by the shocks. The drag is found to be the mean of the drags on the equivalent bodies of revolution for the different directions.

The effect of an insoluble surfactant on the transient deformation and asymptotic shape of a spherical drop that is subjected to a linear shear or extensional flow at vanishing Reynolds number is studied using a numerical method. The viscosity of the drop is equal to that of the ambient fluid, and the interfacial tension is assumed to depend linearly on the local surfactant concentration. The drop deformation is affected by non-uniformities in the surface tension due to the surfactant molecules convection–diffusion. The numerical procedure combines the boundary-integral method for solving the equations of Stokes flow, and a finite-difference method for solving the unsteady convection–diffusion equation for the surfactant concentration over the evolving interface. The parametric investigations address the effect of the ratio of the vorticity to the rate of strain of the incident flow, the Péclet number expressing the ability of the surfactant to diffuse, the elasticity number expressing the sensitivity of the surface tension to variations in surfactant concentration, and the capillary number expressing the strength of the incident flow. At small and moderate capillary numbers, the effect of a surfactant in a non-axisymmetric flow is found to be similar to that in axisymmetric straining flow studied by previous authors. The accumulation of surfactant molecules at the tips of an elongated drop decreases the surface tension locally and promotes the deformation, whereas the dilution of the surfactant over the main body of the drop increases the surface tension and restrains the deformation. At large capillary numbers, the dilution of the surfactant and the rotational motion associated with the vorticity of the incident flow work synergistically to increase the critical capillary number beyond which the drop exhibits continuous elongation. The numerical results establish the regions of validity of the small-deformation theory developed by previous authors, and illustrate the influence of the surfactant on the flow kinematics and on the rheological properties of a dilute suspension. Surfactants have a stronger effect on the rheology of a suspension than on the deformation of the individual drops.

Direct numerical simulations (DNS) are conducted of a model hydrocarbon–nitrogen mixing layer under supercritical conditions. The temporally developing mixing layer configuration is studied using heptane and nitrogen supercritical fluid streams at a pressure of 60 atm as a model system related to practical hydrocarbon-fuel/air systems. An entirely self-consistent cubic Peng–Robinson equation of state is used to describe all thermodynamic mixture variables, including the pressure, internal energy, enthalpy, heat capacity, and speed of sound along with additional terms associated with the generalized heat and mass transport vectors. The Peng–Robinson formulation is based on pure-species reference states accurate to better than 1% relative error through comparisons with highly accurate state equations over the range of variables used in this study (600 [les ] T [les ] 1100 K, 40 [les ] p [les ] 80 atm) and is augmented by an accurate curve fit to the internal energy so as not to require iterative solutions. The DNS results of two-dimensional and three-dimensional layers elucidate the unique thermodynamic and mixing features associated with supercritical conditions. Departures from the perfect gas and ideal mixture conditions are quantified by the compression factor and by the mass diffusion factor, both of which show reductions from the unity value. It is found that the qualitative aspects of the mixing layer may be different according to the specification of the thermal diffusion factors whose value is generally unknown, and the reason for this difference is identified by examining the second-order statistics: the constant Bearman–Kirkwood (BK) thermal diffusion factor excites fluctuations that the constant Irwing–Kirkwood (IK) one does not, and thus enhances overall mixing. Combined with the effect of the mass diffusion factor, constant positive large BK thermal diffusion factors retard diffusional mixing, whereas constant moderate IK factors tend to promote diffusional mixing. Constant positive BK thermal diffusion factors also tend to maintain density gradients, with resulting greater shear and vorticity. These conclusions about IK and BK thermal diffusion factors are species-pair dependent, and therefore are not necessarily universal. Increasing the temperature of the lower stream to approach that of the higher stream results in increased layer growth as measured by the momentum thickness. The three-dimensional mixing layer exhibits slow formation of turbulent small scales, and transition to turbulence does not occur even for a relatively long non-dimensional time when compared to a previous, atmospheric conditions study. The primary reason for this delay is the initial density stratification of the flow, while the formation of strong density gradient regions both in the braid and between-the-braid planes may constitute a secondary reason for the hindering of transition through damping of emerging turbulent eddies.

This paper deals with an experimental investigation of a turbulent low-speed jet of air spreading out radially over a flat smooth plate: a flow which has been discussed by Glauert (1956) in his theory of the wall jet.

The aim of the experiments has been to determine the mean velocity distribution and rate of growth of the jet. It is found, within the experimental range and accuracy, that the velocity profiles are similar and that the rate of change of velocity and width of the jet can be expressed by simple power-laws.

We present simulation results of vortex-induced vibrations of an infinitely long flexible cylinder at Reynolds number Re = 1000, corresponding to a ‘young’ turbulent wake (i.e. exhibiting a small inertial subrange). The simulations are based on a new class of spectral methods suitable for unstructured and hybrid grids. To obtain different responses of the coupled flow–structure system we vary the structure's bending stiffness to model the behaviour of a vibrating inflexible (rigid) cylinder, a cable, and a beam. We have found that unlike the laminar flow previously studied, the amplitude of the cross-flow oscillation is about one diameter for the cable and the beam, close to experimental measurements, but is lower for the rigid cylinder. We have also found that for the latter case the flow response corresponds to parallel shedding, but for the beam and cable with free endpoints a mixed response consisting of oblique and parallel shedding is obtained, caused by the modulated travelling wave motion of the structure. This mixed shedding pattern which alternates periodically along the span can be directly related to periodic spatial variation of the lift force. In the case of structures with pinned endpoints a standing wave response is obtained for the cylinder; lace-like flow structures are observed similar to the ones seen in the laminar regime. Examination of the frequency spectra in the near wake shows that at Re = 1000 all cases follow a −5/3 law in the inertial range, which extends about half a decade in wavenumber. However, these spectra are different in all three cases both in low and high frequencies, with the exception of the beam and cable, for which the high-frequency portion is identical despite the differences in the displacement time history and the large-scale features of the corresponding flow.

A two-dimensional model for the flapping of an elastic flag under axial flow is described. The vortical wake is accounted for by the shedding of discrete point vortices with unsteady intensity, enforcing the regularity condition at the flag's trailing edge. The stability of the flat state of rest as well as the characteristics of the flapping modes in the periodic regime are compared successfully to existing linear stability and experimental results. An analysis of the flapping regime shows the co-existence of direct kinematic waves, travelling along the flag in the same direction as the imposed flow, and reverse dynamic waves, travelling along the flag upstream from the trailing edge.