The time-dependent form of the classic, two-dimensional stagnation-point solution of the Navier-Stokes equations is considered. If the viscosity is zero, a class of solutions of the initial-value problem can be found in closed form using Lagrangian coordinates. These solutions exhibit singular behaviour in finite time, because of the infinite domain and unbounded initial vorticity. Thus, the blow-up found by Stuart in three dimensions using the stagnation-point form, also occurs in two. The singularity vanishes under a discrete, finite-dimensional ‘point vortex’ approximation, but is recovered as the number of vortices tends to infinity. We find that a small positive viscosity does not arrest the breakdown, but does strongly alter its form. Similar results are summarized for certain Boussinesq stratified flows.

Experiments on two-layer β-plane flows are described. The regime diagrams for both easterly and westerly forcing indicate complex scenarios by which baroclinically unstable flows can become chaotic as the forcing is increased. The transition sequence can involve as many as three different vacillation mechanisms, but also exhibits the periodic window phenomena prevalent in many model dynamical systems. The fractal dimension of the chaos at low rotational Froude number F is measurable and is somewhat less than 3. The dimension increases as F is raised. A six-wave low-order model, while successfully predicting some of the observed vacillations, gives a relatively poor description of the chaos.

At high supersonic convective Mach numbers the familiar Kelvin-Helmholtz instability of a thin unconfined two-dimensional shear layer becomes neutrally stable. In this paper, it is shown that when the same shear layer is put inside a rectangular channel the coupling between the motion of the shear layer and the acoustic modes of the channel produces new two-dimensional instability waves. The instability mechanism of these waves is examined. Extensive numerical computation of the properties of these new instability waves has been carried out. Based on these results two classes of these waves are identified. Some of the important characteristic features of these waves are reported in this paper. In addition to the unstable waves, a thorough analysis of the normal modes of a supersonic shear layer inside a rectangular channel reveals that there are basically two other families of neutral acoustic waves. Examples of some of the prominent characteristics of these neutral acoustic waves are also provided in this paper. The new instability waves are the dominant instabilities of a confined mixing layer at high supersonic convective Mach number. As such they are very relevant to the supersonic mixing and combustion processes inside a ramjet engine combustion chamber.

The behaviour of passive-scalar fields resulting from mean scalar gradients in each of three orthogonal directions in homogeneous turbulent shear flow has been studied using direct numerical simulations of the unsteady incompressible Navier-Stokes equations with 128 × 128 × 128 grid points. It is found that, for all orientations of the mean scalar gradient, the sum of the pressure-scalar gradient and velocity gradient-scalar gradient terms in the turbulent scalar flux balance equation are approximately aligned with the scalar flux vector itself. In addition, the time derivative of the scalar flux is also approximately aligned with the flux vector for the developed fields (corresponding to roughly constant correlation coefficients). These alignments lead directly to a gradient transport model with a tensor turbulent diffusivity. The simulation results are used to fit a dimensionless model coefficient as a function of the turbulence Reynolds and Péclet numbers. The model is tested against two different passive-scalar fields in fully developed turbulent channel flow (also generated by direct numerical simulation) and is found to predict the scalar flux quite well throughout the entire channel.

Surface-renewal motions in the interfacial region below a gas-liquid interface were experimentally investigated in relation to bursting motions in the wall region. To estimate the frequency of the appearance of surface-renewal eddies, mass-transport experiments with methylene-blue solution, together with velocity measurements, were done in an open-channel flow. The instantaneous concentration of methylene-blue tracer emitted from a point source positioned in the buffer layer was measured at the free surface downstream from the source by an optical probe. Instantaneous streamwise velocity was measured using a laser-Doppler velocimeter at a position in the buffer region. Frequencies of both surface-renewal and bursting events were computed from these concentration and velocity signals using a conditional-averaging method. In order to clarify whether the surface-renewal eddies actually dominate mass transfer across the gas-liquid interface, gas-absorption experiments were added. Carbon dioxide was absorbed into the water flow across the calm free surface and its mass-transfer coefficient on the liquid side was measured under the same flow conditions as used in the above mass-transport experiments. The results show that the surface-renewal motions originate in the bursting motions which vigorously occur in the buffer region. That is, the decelerated fluid which is strongly lifted towards the outer layer by bursting almost always arrives at the free surface and renews the free surface. The frequency of the surface renewal, as well as the bursting frequency, is uniquely determined by the wall variables or the outer-flow variables and the Reynolds number. Mass transfer across the gas-liquid interface is dominated by the large-scale surface-renewal eddies, and the mass-transfer coefficient on the liquid side is proportional to the square-root of the surface-renewal frequency.

In developed laminar flow in stationary curved ducts there is, in addition to the two-vortex secondary flow structure first analysed by Dean (1927), another solution branch with a four-vortex secondary flow. These two branches have recently been shown to be joined by a third branch, but stability characteristics and the possible presence of additional branches have yet to be described. In this paper orthogonal collocation is used in conjunction with continuation techniques to characterize the bifurcation structure for symmetric flows. The two- and four-vortex solutions are stable to symmetric disturbances, while the recently reported branch joining them is unstable. A more systematic exploration of the parameter space than has hitherto been reported is performed by examining the morphogenesis of the bifurcation structure within the general framework of properties described by Benjamin (1978). The starting point is the ‘perfect’ problem of flow in an infinite curved slit, which bifurcates to give rise to a cellular structure. Addition of ‘stickiness’ at the cell boundaries turns each pair of cells into a curved duct of rectangular cross-section which, by a geometry change, leads to the curved circular tube. For the perfect problem a large number of solution branches are present, but the addition of stickiness turns most of them into isolae which vanish before the no-slip limit is reached. The solution branches that remain include, in addition to the three described previously, another solution family not connected to the other one. This family comprises two branches, both four-vortex in character and unstable to symmetric disturbances.

The process by which two surfaces of the same liquid establish contact, as when two drops collide or raindrops fall on water, is studied. The mathematical formulation is based on the assumption of an incompressible, inviscid fluid with surface tension. A model problem with a simplified geometry is solved numerically by means of a boundary-integral method. The results imply that a number of toroidal bubbles form and remain entrapped between the contacting surfaces. Experimental evidence for this process, which is important for boiling nucleation and the formation of condensation nuclei for rain drops, is found in the literature.

A photochromic tracer method has been used to record pulsatile flow velocity profiles simultaneously at three axial locations along a flow channel. Two major advantages of this multiple-trace method are that it enables velocity data to be acquired in an efficient non-invasive manner and that it provides a detailed description of the spatial relationship of the flow field. The latter is found to be particularly useful in the investigation of transitional type flows; for example, in describing coherent flow structures. Studies of the flow patterns in tubes with mild to moderate degrees of vessel constriction were performed using a 2.9 Hz sinusoidal flow superimposed on a steady flow (frequency parameter of 7.5; mean and modulation Reynolds numbers of 575 and 360, respectively). With mild constrictions (< 50% area reduction), isolated regions of vortical and helical structures were observed primarily during the deceleration phase of the flow cycle and in the vicinity of the reattachment point. As expected, these effects were accentuated when the constriction was asymmetric. For moderate constrictions (50%–80%), transition to turbulence was triggered just before peak flow through the breakdown of waves and streamwise vortices that were shed in the high-shear layer. During this vortex generation phase of the flow cycle, the wall shear stress fluctuated quite intensely, especially in the vicinity of the reattachment point, and its instantaneous value increased by at least a factor of eight. Such detailed descriptions of the transition to turbulence and of the spatial and temporal variation of the wall shear stress, particularly near the reattachment point, have not been previously reported for pulsatile flow through constricted tubes. The observed wall shear stress variations support a proposal by Mao & Hanratty (1986) of an interaction of the imposed flow oscillation with the turbulent fluctuations within the viscous sublayer.

An experimental study has been made of the growth and collapse of a bubble near a composite surface consisting of two viscoelastic materials. The migratory characteristics of the bubble are examined by means of streak photography. The bubble migration depends not only on the properties of the composite surface but also on bubble size and distance from the surface. Both the surface stiffness and surface inertia are considered to be effective parameters contributing to the bubble migration: the former seeming to be a particularly important factor. The state of neutral bubble collapse, with no migration towards or away from a surface, can be made to occur with an appropriate combination of the surface stiffness and inertia.

Viscous gravity currents resulting from the introduction of fluid between an upper layer of fluid of lesser density and a lower layer of greater density are analysed. The nonlinear equations governing the spread and shape of the intrusion are formulated for the cases of intrusion at low Reynolds number between deep ambient layers and of flow over a shallow layer of viscous fluid with a rigid lower boundary. Similarity solutions of these equations are obtained in both two-dimensional and axisymmetric geometries, under the assumption that the volume of intruding fluid increases with time like tα. The theoretical predictions are shown to be in reasonable agreement with experimental observations of the spreading of glucose syrups and of viscous hydrocarbons between fluid layers of differing densities. Scaling arguments are used to derive many new results for the rates of spread of intrusions in a wide variety of further situations. A compendium of spreading relations, including some previously isolated results, is derived within a coherent framework and tabulated.

The equations of the steady, adiabatic, one-dimensional flow of an equilibrium mixture of a perfect gas and incompressible particles, in constant-area ducts with friction, are derived taking into account the effects of gravity and of the finite volume of the particles. As is the case for a pure gas, the mixture is shown to be subject to the phenomenon of choking, and the possibility of an adiabatic heating of the mixture in a subsonic expansion is also theoretically predicted for certain flow inlet conditions. The model may be used to approximately describe the conditions existing in portions of volcanic conduits during the Plinian phases of explosive eruptions. Some results of the numerical integration of the equations for a typical application of this type are briefly discussed, thus showing the potential of the model for carrying out rapid analyses of the influence of the main geometrical and flow parameters describing the problem. A non-volcanological application is also analysed to illustrate the possibility of the adiabatic heating of the mixture.

In this work, we study the interference of two scalars diffusing in homogeneous isotropic turbulence. We use the method described by Kaplan & Dinar (1988b) to calculate the cross-correlation coefficient ρ between the concentration fluctuations of two sources. The dependence of ρ on the source separation, shapes and sizes, and its time evolution is calculated for different points in space. Results for the case of two line sources are compared with data from wind tunnel experiments (Warhaft 1984), and seen to be in good agreement. At the centreline ρ is shown to increase as overlap of the two plumes increases. ρ may be either negative or positive depending on the separation between the two sources.

In this paper the Dean (1928) equations are extended to the case of a helical pipe flow, and it is shown that they depend not only on the Dean number K but also on a new parameter λ/[Rscr ] where λ is the ratio of the torsion τ to the curvature κ of the pipe axis and [Rscr ] the Reynolds number referred in the usual way to the pipe radius a and to the equivalent maximum speed in a straight pipe under the same axial pressure gradient. The fact that the torsion has no first-order effect on the flow is confirmed, but it is shown that this is peculiar to a circular cross-section. In the case of an elliptical cross-section there is a first-order effect of the torsion on the secondary flow, and in the limit λ/[Rscr ] → ∞ (twisted pipes, provided only with torsion), the first-order ‘displacement’ effect of the walls on the secondary flow, analysed in detail by Choi (1988), is recovered.

Different systems of coordinates and different orders of approximations have recently been adopted in the study of the flow in a helical pipe. Thus comparisons between the equations and the results presented in different reports are in some cases difficult and uneasy. In this paper the extended Dean equations for a helical pipe flow recently derived by Kao (1987) are converted to a simpler form by introducing an appropriate modified stream function, and their equivalence with the present set of equations is recovered. Finally, the first-order equivalence of this set of equations with the equations obtained by Murata et al. (1981) is discussed.

Turbulent boundary layers separating from a blunt trailing edge give rise to organized vortical structures in the downstream wake. The perturbation of this inherent flow-instability at f0 by controlled oscillations of the edge at fe produces corresponding, organized components of unsteady surface pressure along the edge. For edge excitation near the ‘natural’ vortex shedding frequency f0, the phase between the local pressure fluctuations and the edge displacement shows large changes for small changes in excitation frequency. Moreover, in this range of excitation, there is quenching (or attenuation) of the surface pressure component at f0 and resonant peaking of the component at fe. These phenomena are related to the change in sign of the energy transfer between the fluid and the body. Integration of the instantaneous pressure distributions along the surfaces of the edge leads to the instantaneous lift at fe and f0 acting upon the oscillating trailing edge. The location of the lift varies as the cotangent of the dimensionless time during an oscillation cycle. When the edge is excited near, or at, the natural vortex shedding frequency, there is a resonant peak in the amplitude of oscillation of the lift location at fe; that at f0 is invariant. Moreover, the mean location of the lift at fe undergoes abrupt changes in this region of excitation. Flow visualization allows determination of the phasing of the organized vortical structures shed from the trailing edge relative to the edge displacement. Modulation of the flow structure at the frequencies f0 and fe, as well as interaction of small-scale vortices at high excitation frequencies, was observed. These aspects of the near-wake structure are related to the instantaneous pressure field.

Basaltic volcanoes erupt in several different regimes which have not been explained. At Kilauea (Hawaii), eruption can take the form of either fire fountaining, where gas-rich jets propel lava clots to great heights in the atmosphere, or quiet effusive outflow of vesicular lava. Another regime is commonly observed at Stromboli, where large gas slugs burst intermittently at the vent. In an attempt to provide a unifying framework for these regimes, we investigate phenomena induced by degassing in a reservoir which empties into a small conduit. Laboratory experiments are done in a cylindrical tank topped by a thin vertical tube. Working liquids are silicone oils and glycerol solutions to investigate a range of viscosity and surface tension. Gas bubbles are generated at the tank bottom with known bubble diameter and total gas flux. The bubbles rise through the tank and accumulate in a foam layer at the roof. Depending on the behaviour of this foam layer, three different regimes can be distinguished: (i) steady horizontal flow of the foam leading to bubbly flow in the conduit; (ii) alternating regimes of foam build-up and collapse leading to the eruption of a single, large gas pocket; (iii) flow of the foam partially coalesced into larger gas pockets leading to intermittent slug flow in the conduit. These regimes have natural counterparts in basaltic volcanoes.

A simple theory is proposed to explain regimes (i) and (ii). The bubbles in contact with the roof deform under the action of buoyancy forces, developing flat contact areas whose size increases as a function of foam thickness. Maximum deformation corresponds to a critical thickness hc = 2σ/ερlgR, where σ is the coefficient of surface tension, ρl the liquid density, g the acceleration due to gravity, R the bubble radius and ε the gas volume fraction in the foam. The foam thickness is determined by a balance between the input of bubbles from below and the output into the conduit, and is proportional to (μlQ/ε2 ρlg)¼, where μl is the liquid viscosity and Q the gas flux. A necessary and sufficient condition for collapse is that it exceeds the critical value hc. In a liquid of given physical properties, this occurs when the gas flux exceeds a critical value which depends on viscosity, surface tension and bubble size. Experimental determinations of the critical gas flux and of the time between two events of foam collapse are in agreement with this simple theory.

An analysis is presented of the mechanics of red blood cells flowing in very narrow tubes. Mammalian red cells are highly flexible, but their deformations satisfy two significant constraints. They must deform at constant volume, because the contents of the cell are incompressible, and also at nearly constant surface area, because the red cell membrane strongly resists dilation. Consequently, there exists a minimal tube diameter below which passage of intact cells is not possible. A cell in a tube with this diameter has its critical shape: a cylinder with hemispherical ends. Here, flow of red cells in tubes with near-minimal diameters is analysed using lubrication theory. When the tube diameter is slightly larger than the minimal value, the cell shape is close to its shape in the critical case. However, the rear end of the cell becomes flattened and then concave with a relatively small further increase in the diameter. The changes in cell shape and the resulting rheological parameters are analysed using matched asymptotic expansions for the high-velocity limit and using numerical solutions. Predictions of rheological parameters are also obtained using the assumption that the cell is effectively rigid with its critical shape, yielding very similar results. A rapid decrease in the apparent viscosity of red cell suspensions with increasing tube diameter is predicted over the range of diameters considered. The red cell velocity is found to exceed the mean bulk velocity by an amount that increases with increasing tube diameter.

Steady, one-dimensional flow of a compressible fluid through a collapsible tube is analysed. A general model is employed, incorporating axial variations in the parameters of the conducting system, such as the tube unstressed cross-section area and wall stiffness, the external pressure and energy exchange with the environment. The flow variables are described in differential form as functions of the conduit system parameters. A coupled set of equations for the dependent flow variables is summarized in a table of influence coefficients, which provides a clear and simple description of the effects produced by the system parameters. Examples of the effects of fluid compressibility in the respiratory system are presented for forced expiration manoeuvres. The effects are found to be generally small, but are most accentuated when breathing heavy gases and when the airways are pathologically stiffened.

By use of a virial equation introduced in a recent paper (Benjamin 1987), the main results of a second-order perturbation theory developed by Longuet-Higgins (1989a) are recovered in comparatively simple fashion. Asymmetric capillary vibrations of a gas bubble in an infinite incompressible liquid are confirmed to generate an increase in the volume of the bubble, a lowering of the mean pressure of the gas and a monopole component in the motion of the liquid. It is shown that the second effect remains when the bubble is incompressible.

An experiment has been conducted to study the occurrence, configuration and dynamics of large-scale coherent vortical motions in the fully developed region of a turbulent axisymmetric jet. The key idea is to use vorticity signals from a spatial grid to detect and sample large-scale vortical structures and then use the (smoothed) vorticity peaks of spatial vorticity patterns to align and ensemble average successive realizations to determine structure configuration and dynamics. Measurements were made in an air jet at ReD = 69000 by employing a radial rake of seven × -wires to obtain the azimuthal vorticity map. Two additional conditioning probes were placed ± 90° away from the rake to determine the three-dimensional phase and hence the structure configuration. Structures with axisymmetric, helical and double helical configurations have been educed. Among them, the helical structures are far more dominant than the others, and the jet dynamics are thus discussed in terms of these helical structures. Helical structures move radially outward as they advect downstream. This radial movement, in conjunction with simultaneous local ejection of turbulent fluid and subsequent entrainment of the ejected fluid with ambient fluid, appears to be a major means of jet spreading. The shear strain rate is strong on the downstream side of the structure, causing intense small-scale turbulence production and mixing there.

The complete stress tensor has been measured using a computer simulation of an assemblage of rough, inelastic spheres in an imposed simple shear flow. Only five components of the stress tensor were found to be significantly different from zero. These represent the disperssive normal stresses τxx, τyy and τzz and the in-the-shear-plane shear stresses τxy and τyx; furthermore, the two off-diagonal stresses, τxy and τyx, were found to be equal so that the resultant stress tensor is symmetric. Two modes of microscopic momentum transport produce the final macroscopic stress tensor: the streaming or kinetic mode by which particles carry the momentum of their motion as they move through the bulk material, and the collisional mode by which momentum is transported by interparticle collisions. The contribution of each to the final result is examined separately. The friction coefficient, the ratio of shear to normal force, is shown to decrease at dense packings for both the collisional and streaming modes. Also observed were normal stress differences, both in and out of the shear plane, reflecting anisotropies in the granular temperature.