We present the results of numerical computations of the refraction of a plane shock wave at a CO2/CH4 gas interface. The numerical method was an operator split version of a second-order Godunov method, with adaptive grid refinement. We solved the unsteady, two-dimensional, compressible, Euler equations numerically, assuming perfect gas equations of state, and compared our results with the experiments of Abd-El-Fattah & Henderson. Good agreement was usually obtained, especially when the contamination of the CH4 by the CO2 was taken into account. Remaining discrepancies were ascribed to the uncertainties in measuring certain wave angles, due to sharp curvature, poor definition, or short length of the waves at large angles of incidence. All the main features of the regular and irregular refractions were resolved numerically for shock strengths that were weak, intermediate, or strong. These include free precursor shock waves in the intermediate and strong cases, evanescent (smeared out) compressions in the weak case, and the appearance of an extra expansion wave in the bound precursor refraction (BPR). The structure of a BPR was elucidated for the first time.

It is shown experimentally that a two-dimensional momentumless wake is strongly dependent on the jet injection configuration of the model. Namely, the decay rate of mean velocity overshoot ranged from x−0.92 to x−2.0 for three different configurations, while the spreading rate ranged from x0.3 to x0.46 for those same configurations. The magnitude of axial turbulence intensity was also found to depend on model configuration. On the other hand, the rate of decay of axial turbulence intensity was the same (x−0.81) for all three models. In all cases the mean shear and Reynolds stress decayed rapidly, leaving nearly isotropic turbulence beyond 30 or 40 model diameters.

Appropriate length- and velocity scales are identified which normalize the mean velocity profiles into self-similar form. The shape of the normalized profile, however, was different for each configuration, indicating again that the initial conditions are felt very far downstream.

The linear, weakly nonlinear and strongly nonlinear evolution of unstable waves in a geostrophic shear layer is examined. In all cases, the growth of initially small-amplitude waves in the periodic domain causes the shear layer to break up into a series of eddies or pools. Pooling tends to be associated with waves having a significant varicose structure. Although the linear instability sets the scale for the pooling, the wave growth and evolution at moderate and large amplitudes is due entirely to nonlinear dynamics. Weakly nonlinear theory provides a catastrophic time ts at which the wave amplitude is predicted to become infinite. This time gives a reasonable estimate of the time observed for pools to detach in numerical experiments with marginally unstable and rapidly growing waves.

Exploratory experiments have been performed on circular cylinders executing forced rotary oscillations in a steady uniform flow. Flow visualization and wake profile measurements at moderate Reynolds numbers have shown that a considerable amount of control can be exerted over the structure of the wake by such means. In particular, a large increase, or decrease, in the resulting displacement thickness, estimated cylinder drag, and associated mixing with the free stream can be achieved, depending on the frequency and amplitude of oscillation.

This paper focuses on the résponse of bubbles to sudden changes in the ambient pressure. Trains of bubbles with radii varying between 45 and 200 μm were exposed to various pressure steps, and their response was monitored by pulsed laser holography. The experiments were performed in a specially constructed chamber, allowing generation of pressure steps ranging from 0.1 to 20 times the initial value. Air, CO2, helium and hydrogen bubbles were selected, providing a range of mass, thermal diffusivities, solubilities in water and isentropic constants. The changes in the bubbles’ diameters were determined by reconstructing the holograms, magnifying the images and measuring the sizes of individual bubbles. Most of the experiments were performed with pressure changes at the rate of 20 KPa/ms, and most data were recorded in less than 20 ms. The results confirm that, within the present range of test conditions, the bubbles can respond instantaneously to changes in the ambient pressure. The experiments also demonstrate that the response of the bubbles at the present timescales can be assumed to be isothermal (polytropic constant of 1.0), irrespective of the bubble content or size. Repeated measurements with different pressure waveforms, but with the same final pressure resulted in identical results, demonstrating that bubbles can be used as pressure sensors. Variations in timescales up to a few hundred milliseconds still resulted in the same response, confirming the isothermal assumption. The dissolved gas content had a noticeable effect on the behaviour of the CO2 bubbles, the most soluble of the gases tested, and had no detectable effect on the behaviour of air bubbles. The paper also includes a detailed error evaluation of the present experiments and an estimate of the expected error when the bubbles are utilized as pressure sensors.

In this paper a numerical model for the viscous flow past a cylinder in a rotating frame is discussed when both the Rossby number Ro and Ekman number E are small. The results of this model are analysed and compared to an inviscid study by Page (1987) applicable in the limit E → 0 with Ro = O(E½). The detailed structure of the separated flow is also examined and compared to the proposals for the higher-order flow in E¼ layers in Page (1987) which were based, in part, on the theory of Smith (1979, 1985) for the non-rotating flow past bluff bodies. Some discrepancies between this theory and the numerical results are noted.

The effect of Mach number on the evolution of instabilities in the compressible mixing layer is investigated. The full time-dependent compressible Navier–Stokes equations are solved numerically for a temporally evolving mixing layer using a mixed spectral and high-order finite difference method. The convective Mach number Mc (the ratio of the velocity difference to the sum of the free-stream sound speeds) is used as the compressibility parameter. Simulations with random initial conditions confirm the prediction of linear stability theory that at high Mach numbers (Mc > 0.6) oblique waves grow more rapidly than two-dimensional waves. Simulations are then presented of the nonlinear temporal evolution of the most rapidly amplified linear instability waves. A change in the developed large-scale structure is observed as the Mach number is increased, with vortical regions oriented in a more oblique manner at the higher Mach numbers. At convective Mach numbers above unity the two-dimensional instability is found to have little effect on the flow development, which is dominated by the oblique instability waves. The nonlinear structure which develops from a pair of equal and opposite oblique instability waves is found to resemble a pair of inclined A-vortices which are staggered in the streamwise direction. A fully nonlinear computation with a random initial condition shows the development of large-scale structure similar to the simulations with forcing. It is concluded that there are strong compressibility effects on the structure of the mixing layer and that highly three-dimensional structures develop from the primary inflexional instability of the flow at high Mach numbers.

We report the results of a comprehensive comparative study of the inviscid spatial stability of a parallel compressible mixing layer using various models for the mean flow. The models are (i) the hyperbolic tangent profile for the mean speed and the Crocco relation for the mean temperature, with the Chapman viscosity–temperature relation and a Prandtl number of one; (ii) the Lock profile for the mean speed and the Crocco relation for the mean temperature, with the Chapman viscosity-temperature relation and a Prandtl number of one; and (iii) the similarity solution for the coupled velocity and temperature equations using the Sutherland viscosity–temperature relation and arbitrary but constant Prandtl number. The purpose of this study was to determine the sensitivity of the stability characteristics of the compressible mixing layer to the assumed thermodynamic properties of the fluid. It is shown that the qualitative features of the stability characteristics are quite similar for all models but that there are quantitative differences resulting from the difference in the thermodynamic models. In particular, we show that the stability characteristics are sensitive to the value of the Prandtl number and to a particular value of the temperature ratio across the mixing layer.

Toroidal bubbles with circulation are studied numerically and by means of a physically motivated model equation. Two series of computations are performed by a boundary-integral method. One set shows the starting motion of an initially spherical bubble as a gravitationally driven liquid jet penetrates through the bubble from below causing a toroidal geometry to develop. The jet becomes broader as surface tension increases and fails to penetrate if surface tension is too large. The dimensionless circulation that develops is not very dependent on the surface tension. The second series of computations starts from a toroidal geometry, with circulation determined from the earlier series, and follows the motion of the rising and spreading vortex ring. Some modifications to the boundary-integral formulation were devised to handle the multiply connected geometry. The computations uncovered some unexpected rapid oscillations of the ring radius. These oscillations and the spreading of the ring are explained by the model equation which provides a more general description of vortex ring bubbles than previously available.

The separating flow induced on a ramp under a supersonic main stream is discussed, for high Reynolds numbers, according to the interactive-boundary-layer approach. There are two principal motivations for the study, apart from the recent general upsurge of interest in compressible boundary-layer separation and stall. The first is the need for more progress to be made in the numerical determination of strongly reversed flows than has proved possible in computations hitherto. This is tackled by means of a new computational scheme based in effect on a global Newton iteration procedure but coupled with linearized shooting, second-order accurate windward differencing and linear multi-sweeping. The specific case addressed is the triple-deck version for steady laminar two-dimensional motion, although the present scheme, like the theory described below, also has broader application, for example to subsonic, hypersonic and/or unsteady interactive flows. The second motivation is to compare closely with the recent theoretical prediction (Smith 1988a) of a local breakdown or stall occurring in any interactive boundary-layer solution at a finite value of the controlling parameter, a say, within the reversed-flow region; the breakdown produces a large adverse pressure gradient and minimum negative surface shear locally. The first quantitative comparisons are made between the theory and computational results, derived in this work at values of a, here the scaled ramp angle, greater than those obtainable before, but with a fixed outer boundary. The agreement, while not complete, seems to prove fairly affirmative overall and tends to support the suggestion (in Smith 1988a) that, contrary to most earlier expectations, in general there is a finite upper limit on the extent to which the interacting boundary-layer approach can be taken on its own. A similar conclusion holds for unsteady interactive boundary layers concerning a finite-time breakdown (Smith 19886) and boundary-layer transition, and in the present context the local nonlinear breakdown provides an explanation for the severe computational difficulties encountered previously as well as for a form of airfoil stall.

Surface waves in a channel can be produced by the horizontal motion of a plane wavemaker at one end of the channel. The amplitude and the frequency of the waves depend on both surface tension and gravity, as well as on the condition imposed at the contact line between the free surface and the wavemaker. Some of the previous work on the generation of capillary–gravity waves has been based on the unjustified assumption that the slope of the free surface at the contact line can be prescribed. A more acceptable condition is one that relates the slope to the motion of the contact line relative to the wavemaker; in this way the dynamic properties of the contact angle can be incorporated. The waves generated by a plane wavemaker in fluid of infinite depth and in fluid of a depth equal to that of the wavemaker are determined. An important reason for including surface tension is that in its absence the transient motion initiated by an impulsive start is singular; when surface tension is included this singularity is removed.

Far from a shoreline, the spreading of a contaminant plume in a shallow-water flow can be predicted easily and accurately by a ray-tracing method. Unfortunately, the concentration predictions become singular at a beach, where the ray paths have a caustic. In this paper a uniform approximation is derived which remains valid at a beach. It is shown how the singular ray solutions corresponding to rays incident to and transmitted from the beach can be combined to construct the uniform approximation.

The development of a small three-dimensional disturbance in plane Poiseuille flow is considered. Its kinetic energy is expressed in terms of the velocity and vorticity components normal to the wall. The normal vorticity develops according to the mechanism of vortex stretching and is described by an inhomogeneous equation, where the spanwise variation of the normal velocity acts as forcing. To study specifically the effect of the forcing, the initial normal vorticity is set to zero and the energy density in the wavenumber plane, induced by the normal velocity, is determined. In particular, the response from individual (and damped) Orr–Sommerfeld modes is calculated, on the basis of a formal solution to the initial-value problem. The relevant timescale for the development of the perturbation is identified as a viscous one. Even so, the induced energy density can greatly exceed that associated with the initial normal velocity, before decay sets in. Initial conditions corresponding to the least-damped Orr–Sommerfeld mode induce the largest energy density and a maximum is obtained for structures infinitely elongated in the streamwise direction. In this limit, the asymptotic solution is derived and it shows that the spanwise wavenumbers at which the largest amplification occurs are 2.60 and 1.98, for symmetric and antisymmetric normal vorticity, respectively. The asymptotic analysis also shows that the propagation speed for induced symmetric vorticity is confined to a narrower range than that for antisymmetric vorticity. From a consideration of the neglected nonlinear terms it is found that the normal velocity component cannot be nonlinearly affected by the normal vorticity growth for structures with no streamwise dependence.

The motion of a small, rigid sphere in a linear shear flow is considered. Saffman's analysis is extended to other asymptotic cases in which the particle Reynolds number based on its slip velocity is comparable with or larger than the square root of the particle Reynolds number based on the velocity gradient. In all cases, both particle Reynolds numbers are assumed to be small compared to unity. It is shown that, as the Reynolds number based on particle slip velocity becomes larger than the square root of the Reynolds number based on particle shear rate, the magnitude of the inertial migration velocity rapidly decreases to very small values. The latter behaviour suggests that contributions that are higher order in the particle radius may become important in some situations of interest.

Caflisch & Luke (1985) showed that, owing to the long range of the hydrodynamie interactions, the variance of the sedimentation velocity in a random suspension with uniform probability for the positions of the particles is divergent in the sense that it grows without bound as the macroscopic linear dimension of the settling vessel is increased. It is shown here, however, that a Debye-like screening of a particle's velocity disturbance, leading to a finite variance, will occur if the pair probability reflects a net deficit of one particle in the vicinity of each particle. The three-particle interactions, which determine the structure of a dilute, monodisperse suspension of spheres, lead to a deficit of neighbouring particles. The magnitude and range of this deficit are shown to be sufficient to lead to a Debye-like screening of the velocity disturbance at a radial distance of order aϕ−1, where a is the particle radius and ϕ their volume fraction. A self-consistent approximation to the screened conditional average velocity field and pair distribution is presented. The screening leads to a variance of the particle velocity and a particle tracer diffusion coefficient that are finite and of order Us2 and Usaϕ−1, respectively, where Vs is the Stokes settling velocity of the particles in unbounded fluid.

The axisymmetric electrophoretic motion of a dielectric sphere along the axis of an orifice in a large conducting plane or of a conducting disk is considered. The radius of the orifice or the disk may be either larger or smaller than that of the sphere. The assumption of thin electrical double layers at the solid surfaces is employed. To solve the electrostatic and hydrodynamic governing equations both the electric and the flow fields are partitioned at the plane of the orifice or the disk. For each field, separate solutions are developed on both sides of the plane that satisfy the boundary conditions in each region and the unknown functions for the field at the fluid interface. The continuities of the electric current flux and the fluid stress tensor at the matching interface lead to sets of dual integral equations which are solved analytically to determine the unknown functions for the fields at the matching interface. Then, a boundary–collocation technique is used to satisfy the boundary conditions on the surface of the sphere.

The numerical solutions for the electrophoretic velocity of the colloidal sphere are presented for various values of a/b and a/d, where a is the particle radius, b is the radius of the orifice or the disk, and d is the distance of the particle centre from the plane of the wall. For the limiting case of electrophoresis of a sphere perpendicular to an infinite plane wall, our results for the boundary effects agree very well with the exact calculations using spherical bipolar coordinates. Interestingly, the electrophoretic velocity of a sphere approaching an orifice of a larger radius increases when the sphere is close to the orifice, and this velocity can be even larger than that for an identical sphere undergoing electrophoresis in an unbounded fluid. If the sphere has a radius larger than that of the orifice, or if the sphere has a smaller radius and is located sufficiently far from the orifice, its electrophoretic mobility will decrease with the increase of the spacing parameter a/d. For the electrophoretic motion of a sphere along the axis of and close to a disk of finite radius, the resistance to the particle movement can be stronger than that for an equal sphere undergoing electrophoresis normal to an infinite plane wall at the same value of a/d. As the particle approaches the disk wall, its mobility decreases steadily and vanishes at the limit a/d → 1. The boundary effects on the particle mobility and the fluid flow pattern in electrophoresis differ significantly from those of the corresponding sedimentation problem with which a comparison is made.

Governing equations for a mushy layer are analysed in the asymptotic regime Rm [Gt ] 1, where Rm is an appropriately defined Rayleigh number. A model is proposed in which there is downward flow everywhere in the mushy layer except in and near localized chimneys, which are characterized by having zero solid fraction. Upward, convective flow within the chimneys is driven by compositional buoyancy. The radius of each chimney is determined locally by thermal balances within a boundary layer that surrounds it. Simple solutions are derived to determine the structure of the mushy layer away from the immediate vicinity of chimneys in order to demonstrate the gross effects of convection upon the solidification within the layer.

A theory is presented to describe the linear baroclinic instability of coupled density fronts on a sloping continental shelf. The new baroclinic model equations used to study the instability process correspond to an ‘intermediate lengthscale’ dynamical balance. Specifically, the frontal dynamics, while geostrophic, is not quasigeostrophic because frontal height deflections are not small in comparison with the frontal scale height. The evolution of the frontal height is strongly coupled to the geostrophic pressure in the surrounding slope water through the hydrostatic balance which expresses the continuity of the dynamic pressures across the frontal interface. The deeper surrounding slope water evolves quasi-geostrophically and is coupled to the front by baroclinic vortex-tube stretching/compression associated with the perturbed density front (allowing the release of mean frontal potential energy) and the topographic vorticity gradient associated with the sloping bottom. It is shown that the baroclinic stability characteristics are principally determined by a so-called non-dimensional interaction parameter (denoted μ) which physically measures the ratio of the destabilizing baroclinic vortex-tube stretching/compression to the stabilizing topographic vorticity gradient. For a given along-front mode wavenumber it is shown that a minimum μ is required for instability. Several other general stability results are presented: necessary conditions for instability, growth rate and phase speed bounds, the existence of a high wavenumber cutoff, and a semicircle theorem for the unstable modes. The linear stability equations are solved exactly for a parabolic coupled density front and a detailed description of the spatial and temporal characteristics of the instabilities is given. For physically realistic parameter values the instabilities are manifested as amplifying topographic Rossby waves in the slope water, and on the density front the unstable perturbations take the form of amplifying anticyclones which have maximum amplitude on the offshore side.

In this paper we discuss ‘exotic’ rotationally symmetric containers that admit an entire continuum of distinct equilibrium capillary free surfaces. The paper extends earlier work to a larger class of parameters and clarifies and simplifies the governing differential equations, while expressing them in a parametric form appropriate for numerical integration. A unified presentation suitable for both zero and non-zero gravity is given. Solutions for the container shapes are depicted graphically along with members of the free-surface continuum, and comments are given concerning possible physical experiments.

In experiments to measure the surface energy of a magnetically levitated molten metal droplet by observation of its oscillation frequencies, Rayleigh's equation is usually used. This assumes that the equilibrium shape is a sphere, and the surface restoring force is due only to surface tension. This work investigates how the vibrations of a non-rotating liquid droplet are affected by the asphericity and additional restoring forces that the levitating field introduces. The calculations show that the expected single frequency of the fundamental mode is split into either three, when there is an axis of rotational symmetry, or five unequally spaced bands. Frequencies, on average, are higher than those of an unconstrained droplet; the surface tension appears to be increased over its normal value. This requires a small correction to be made in all analyses of surface energy. A frequency sum rule is derived from a simplified model of the magnetic field which allows the corresponding Rayleigh frequency to be evaluated from the observed frequencies of the fundamental and translational modes. A more detailed analysis shows a similar correction but one that is also sensitive to the position of the droplet in the field.