This paper deals with the nature of the entrainment interface of a two-layer fluid subjected to interfacial velocity shear. The shear flow was generated by driving the mixed layer over the dense layer by a disk pump such that there is no stress at the top of the mixed layer. During the entrainment process a sharp, thin-density interfacial layer developed; its thickness δ was found to increase linearly with the mixed-layer depth h, independent of the Richardson number Riu. The shear layer thickness δs was found to be much larger than δ and the ratio δs/h is also found to be constant, irrespective of Riu. At the entrainment interface, the estimated buoyancy flux and the dissipation of turbulent kinetic-energy appear to be of the same order. This result supports an entrainment law of the form E ∼ Riu−1, where E is the entrainment coefficient. The interfacial layer showed sporadic large-amplitude wave oscillations whose amplitudes scaled well with the estimated size of the undulations caused by the impingement of large eddies (of size h) on the density interface. The density-interfacial layer was found to be ‘topped’ by a layer of partially mixed fluid which had not yet incroporated into the well-mixed region.

The first- and second-order wave-diffraction problems are solved for a submerged cylinder of arbitrary shape by the integral-equation method based on Green's theorem. The integral equation is solved by an element method using cubic splines. The first- and second-order force components, transmission and reflection coefficients are presented for two fifferent contours. The results for the circular contour are compared with experimental results.

This paper proposes a combined multipole-series representation and integral-equation method for solving the low-Reynolds-number hydrodynamic interaction of a finite sphere at the entrance of a circular orifice. This method combines the flexibility of the intergral-equation method in treating complicated geometries and the accuracy and computational efficiency of the multipole-series-representation technique. For the axisymmetric case, the hydrodynamic force has been solved for the difficult case where the sphere intersects the plane of the orifice opening, which could not be treated by previous methods. For the three-dimensional case, the first numerical solutions have been obtained for the spatial variation of the twelve force and torque correction factors describing the translation or rotation of the sphere in a quiescent fluid at a pore entrance or the Sampson flow past a fixed sphere. Restricted by excessive computation time, accurate three-dimensional solutions are presented only for a sphere which is one-half the orifice diameter. However, based on an analysis of the behaviour of the force and torque correction factors for this case, approximate interpolation formulas utilizing the results on or near the orifice axis and in the far field are proposed for other diameter ratios, thus greatly extending the usefulness of the present solution.

The turbulent flow over a progressive water wave is studied using an eddy viscosity model. The governing equations are treated asymptotically for the case ε [Lt ] 1, where ε is the square root of a characteristic drag coefficient. A calculation of the phase shift between the wave-induced pressure perturbation and the surface elevation shows that the phase shift is induced by a term in the gradient of the Reynolds stress. Growth rates are determined, and are shown to agree well with observations for the most rapidly amplifying waves. However, the present model and previous turbulence calculations are found to provide significantly lower growth rates than those measured by Snyder et al. (1981) for waves with phase velocities comparable to the wind speed.

The rapid expansion of a turbulent boundary layer in supersonic flow is studied analytically and experimentally. Emphasis is placed on the effect of bulk dilatation on turbulent fluctuations. The hypotheses made in the analysis are similar to those in the rapid distortion theory and are used to simplify second-order closures. By assuming that the fluctuating velocity is solenoidal an extension of classical subsonic models is proposed. A new variable is defined, which takes into account the mean density variations, and behaves like the Reynolds stress tensor in subsonic flows with weak inhomogeneities and a weak dissipation rate. The results of the analysis are compared with turbulence measurements performed in a supersonic boundary layer subjected to an expansion fan. The proposed approximations describe correctly the evolution of turbulence intensities: bulk dilatation contributes predominantly to the Reynolds stress evolution. The boundary layer is ‘relaminarized’ by the expansion. Downstream of the latter, the layer returns to equilibrium. Measurements show that the turbulence decays slowly in the outer layer and increases rapidly in the inner layer.

We report some experiments undertaken to investigate the origin of ordered and chaotic laminar vortex streets behind circular cylinders at low Reynolds numbers. We made simultaneous measurements of near wake longitudinal velocity and cylinder lateral vibration amplitude spectra for cylinder Reynolds numbers in the range from 40 to 160. For a non-vibrating cylinder the velocity energy spectra contained only a single peak, at the Strouhal frequency. When the cylinder was observed to vibrate in response to forcing by the vortex wake, additional dominant spectral peaks appeared in the resulting ‘ordered’ velocity spectra. Cylinder vibrations too small to be noticed with the naked eye or from audible Aeolian tones produced a coupled wake-cylinder response with dramatic effects in hot-wire and cylinder vibration detector signals. The velocity spectra associated with these coupled motions had dominant peaks at the Strouhal frequency fs, at a frequency fc proportional to the fundamental cylinder vibration frequency, and at sum and difference combinations of multiples of fs and fc. In windows of chaos the velocity spectra were broadened by switching between different competing coupling modes. The velocity spectra were very sensitive to the nature of the boundary conditions at the ends of the cylinder. Our measurements strongly suggest that the very similar regions of ‘order’ and ‘chaos’ observed by Sreenivasan and interpreted by him as transition through quasi-periodic states in the sense of the Ruelle, Takens, and Newhouse theory were also due to aeroelastic coupling of the vortex wake with cylinder vibration modes.

A solution has been obtained for steady propagation of a two-dimensional fluid fracture driven by buoyancy in an elastic medium. The problem is formulated in terms of an integro-differential equation governing the elastic deformation, coupled with the differential equation of lubrication theory for viscous flow in the crack. The numerical treatment of this system is carried out in terms of an eigenfunction expansion of the cavity shape, in which the coefficients are found by use of a nonlinear constrained optimization technique. When suitably non-dimensionalized, the solution appears to be unique. It exhibits a semi-infinite crack of constant width following the propagating fracture. For each value of the stress intensity factor of the medium, the width and propagation speed are determined. The results are applied to the problem of the vertical ascent of magma through the earth's mantle and crust. Values obtained for the crack width and ascent velocity are in accord with observations. This mechanism can explain the high ascent velocities required to quench diamonds during a Kimberlite eruption. The mechanism can also explain how basaltic eruptions can carry large mantle rocks (xenoliths) to the surface.

The way in which light, scattered by a thin layer of fluid, may be used to obtain quantitative information about the temperature field in the fluid is investigated. Expressions for the phase shift imposed by the fluid, and the intensity of the scattered light are derived in terms of the Fourier representation of the temperature field, under the assumption of small variations in the refractive index. The method is applied to the particular case of chaotic convection, with a view to studying the connection between strange attractors and turbulence. Two simple mathematical models of chaotic convection are studied; particular emphasis is attached to the statistical properties of the flow and of the scattered light field, which are calculated numerically.

Linear three-dimensional instabilities of nonlinear two-dimensional uniform gravitycapillary waves are studied using numerical methods. The eigenvalue system for the stability problem is generated using a Galerkin method and differs in detail from techniques used to study the stability of pure gravity waves (McLean 1982) and pure capillary waves (Chen & Saffman 1985). It is found that instabilities develop in the neighbourhood of the linear (triad, quartet and quintet) resonance curves. Further, both sum and difference triad ressonances are unstable for sufficiently steep waves in consequence of which Hasselmann's (1967) theorem is restricted to weakly nonlinear waves. The appearance of a superharmonic two-dimensional instability and bifurcation to three-dimensional waves are noted.

This paper first describes an apparatus for measuring the Nusselt number N versus the Rayleigh number R of convecting normal liquid 4He layers. The most important feature of the apparatus is its ability to provide layers of different heights d, and hence different aspect ratios [Gcy ]. The horizontal cross-section of each layer is circular, and [Gcy ] is defined by [Gcy ] = D/2d where D is the diameter of the layer. We report results for 2.4 [les ] [Gcy ] [les ] 16 and for Prandtl numbers Pr spanning 0.5 [lsim ] Pr [lsim ] 0.9 These results are presented in terms of the slope N1 = RcdN/dR evaluated just above the onset of convection at Rc. We find that N1 is only a slowly increasing function of [Gcy ] in the range 6 [lsim ] [Gcy ] [lsim ] 16, and that it has a value there which is quite close to 0.72. This value of N1 is in good agreement with variational calcuations by Ahlers et al. (1981) pertinent to parallel convection rolls in cylindrical geometry. Particularly for [Gcy ] [lsim ] 6, we find additional small-scale structure in N1 associated with changes in the number of convection rolls with changing [Gcy ]. An additional test of the linearzied hydrodynamics is given by measurements of Rc. We find good agreement between theory and our data for Rc.

An investigation of turbulent wakes was conducted and phase-averaged velocity vector fields are presented, as well as phase-averaged and global Reynolds normal and shear stresses. The topology of the phase-averaged velocity fields is discussed in terms of critical point theory. Here in Part 1, the vortex formation process in the cavity region of several nominally two-dimensional bluff bodies is investigated and described using phase-averaged streamlines where the measurements were made in a nominal plane of symmetry. It was found that the flows encountered were always three-dimensional and that the mean-flow patterns in the cavity region were quite different from those expected using classical two-dimensional assumptions.

An investigation of a selection of high-Reynolds-number bluff-body flows was conducted. Here in Part 2 phase-averaged velocity-field results will be presented for several far-wake flows generated by nominally two-dimensional and three-dimensional bodies. In these far-wake flows the shed vortices have approached a nearly constant convection velocity. Some mean velocity and phase-averaged and global Reynoldsstress measurements are also presented. The turbulent wake of a lift-producing three-dimensional body has been examined. Also included are the phase-averaged wake patterns behind a flapping flag and a windmill. The topological structure of these patterns is discussed and a preliminary classification of wake patterns is presented.

The modified Zakharov equation is used to study the coupled evolution of class I and class II instabilities of surface gravity waves on infinitely deep water. In contrast to single class (I or II) evolution, the coupled behaviour is non-periodic. Except for the very steep waves a dominance of class I modes over those of class II is observed. Energy calculations show that the Hamiltonian of the wave field considered is nearly constant. Thus the Zakharov and the modified Zakharov equations represent consistent approximations of the original water-wave problem.

The instabilities of convection columns (also called thermal Rossby waves) in a cylindrical annulus rotating about its axis and heated from the outside are investigated as a function of the Prandtl number P and the Coriolis parameter η*. When this latter parameter is sufficiently large, it is found that the primary solution observed at the onset of convection becomes unstable when the Rayleigh number exceeds its critical value by a relatively small amount. Transitions occur to columnar convection which is non-symmetric with respect to the mid-plane of the small-gap annular layer. Further transitions introduce convection flows that vacillate in time or tend to split the row of columns into an inner and an outer row of separately propagating waves. Of special interest is the regime of non-symmetric convection, which exhibits decreasing Nusselt number with increasing Rayleigh number, and the indication of a period doubling sequence associated with vacillating convection.

We consider the oscillatory motion of a solid plate into and out of a bath of liquid. Assuming that the displacement amplitude of the plate motion is small and that the capillary number is small, the problem reduces to solving an interfacial boundaryvalue problem for the response of the contact line. The characteristic contact angle versus contact-line speed relationship includes contact-angle hysteresis which is assumed small and comparable to the amplitude of the plate motion. Sinusoidal and square-wave plate motions are considered. We find that the contact line moves with the plate if the contact line is fixed, but has relative motion otherwise. It would then advance part of the time, recede part of the time, and remain stationary in the transition periods. Further, we find that both contact-angle hysteresis and steepening of the contact angle with increasing contact-line speed are dissipative effects.

Steady and oscillatory convection in a rectangular box heated from below are studied by means of a numerical solution of the three-dimensional, time-dependent Boussinesq equations. The effect of the rigid sidewalls of the box on the spatial structure and the dynamical behaviour of the flow is analysed. Both conducting and adiabatic sidewalls are considered. Calculated streamlines illustrate the three-dimensional structure of the steady flow with Prandtl numbers 0.71 and 7. The onset and the frequency of the oscillatory instability are calculated and compared with available experimental and theoretical data. With increasing Rayleigh number a subharmonic bifurcation and the onset of a quasi-periodic flow can be observed. A comparison of the different time-dependent solutions shows some interesting relations between the spatial structure and the dynamical behaviour of the confined flow.

The ability of a drop to stick to a solid surface is investigated when the surrounding fluid is in motion. The specific problem analysed consists of a small drop on a planar surface immersed in a second immiscible fluid which is flowing parallel to the solid surface at a constant rate of strain. An expression is obtained, in terms of experimentally measurable quantities, for the value of the rate of strain beyond which the drop cannot maintain contact with a fixed position on the solid. The most limiting restrictions assumed in the analysis are that both the advancing contact angle and the contact angle hysteresis must be small.

Coherent structures of turbulent open-channel flow in the wall region of a channel bed were examined quantitatively using experimental data obtained by flow visualization. Successive pictures of flow patterns in two horizontal cross-sections at different levels near the channel bed were taken, and then were digitized and analysed by a computer.

This method of flow visualization and picture processing enabled us to calculate the distributions of the three components of the velocity vectors. The distributions of velocities, streamlines, two-dimensional divergence and three components of vorticity could be calculated and are displayed as graphical output. In our numerical analyses, the idea of a two-dimensional correlation coefficient is introduced, through which the degree of similarity of turbulence structures can be better estimated than with the usual one-dimensional coefficient. Use of the data was based on the premise that the essential element in a turbulence structure is vortex motion.

We propose a conceptual model of turbulence structure in which the elementary unit of coherent structure in the buffer layer is presumed to be a horseshoe vortex and in which the characteristics of the multiple structure of turbulence are shown with respect to the scale, arrangement and generating process of horseshoe vortices and longitudinal vortices. Our model clearly explains the generating mechanism and mutual relations of low-speed regions, high-speed regions, ejections, sweeps and localized free-shear layers.

The average settling velocity in homogeneous turbulence of a small rigid spherical particle, subject to a Stokes drag force, is shown to depend on the particle inertia and the free-fall terminal velocity in still fluid. With no inertia the particle settles on average at the same rate as in still fluid, assuming there is no mean flow. Particle inertia produces a bias in each trajectory towards regions of high strain rate or low vorticity, which affects the mean settling velocity. Results from a Gaussian random velocity field show that this produces an increased settling velocity.

Equations for the steady flow of an incompressible, inviscid fluid through a collapsible tube under longitudinal tension are derived by treating the tube longitudinally as a membrane, and taking the collapsibility of the tube into account in an approximate way by replacing in the equation for an axisymmetric membrane a term representing the resistance of the tube to area change by the tube law for collapsible tubes. The flow is assumed to be uniform in a cross-section. A nonlinear differential equation is obtained for the shape of the tube for given values of total pressure p0, flow rate q, longitudinal tension τ and tube law P = P(ρ); where ρ = (A/πR2)½ is the equivalent radius of the tube (A = area of a cross-section, R = radius of the unloaded, then circular tube). The equation can be integrated and analysed in the phase plane. Equilibrium points correspond to uniform flow through cylindrical tubes; saddle points correspond to subcritical flow (S < 1), centrepoints to supercritical (S > 1) and a higher-order point to critical flow (S = 1). Here S is the speed index, the ratio of the flow speed to the speed of long waves. Near centrepoints there are solutions, that represent area-periodic tubes. For a finite tube, held open at the ends, the steady flow is formulated as a two-point boundary-value problem. On the basis of numerical calculations, and a bifurcation analysis using the method of Lyapunov–Schmidt, the existence and multiplicity of the solutions of this problem are discussed and the process of flow limitation studied. For negative total pressures two collapsed solutions are found that disappear at the flow-limitation value of the flow rate. For positive total pressures a distinction is made between subcritical, critical and supercritical total pressures. In all these cases there is a multiplicity, proportional to the ratio of the tube length to [Lscr ]1(0), the wavelength of the collapsed periodic solution for vanishing flow rate, and having maximum radius ρ = 1. For subcritical total pressures increase of the flow rate leads to a gradual loss of all solutions in higher-order flow limitations until final flow limitation occurs by the mergence of two collapsed solutions. For supercritical total pressures increase of the flow rate also leads to a gradual loss of all solutions in higher-order flow limitations in a process which now also depends upon the ratio of the tube length to the wavelength L of periodic solutions with vanishing amplitude and ρ ≡ 1.