Flow visualization has been used quantitatively to determine the flow relative to a piston and a free surface started from rest. The discharge of water from a cylindrical reservoir was investigated. Flow with a free surface started from rest was found to have a critical Reynolds number (based on tube diameter and surface speed) of about 450 above which a ring vortex was produced just below the surface.

Measurements at Reynolds numbers of 525 and 1200 were compared with computations made by the methods described in Part 1. The computed drift of tracer particles agreed well with observed values. The largest discrepancies occurred in the radial component of the drift in the early stages of the motion and amounted to 2½% of the tube diameter.

We study theoretically and experimentally how a thin layer of liquid flows along a flexible beam. The flow is modelled using lubrication theory and the substrate is modelled as an elastica which deforms according to the Euler–Bernoulli equation. A constant flux of liquid is supplied at one end of the beam, which is clamped horizontally, while the other end of the beam is free. As the liquid film spreads, its weight causes the beam deflection to increase, which in turn enhances the spreading rate of the liquid. This feedback mechanism causes the front position ${\it\sigma}(t)$ and the deflection angle at the front ${\it\phi}(t)$ to go through a number of different power-law behaviours. For early times, the liquid spreads like a horizontal gravity current, with ${\it\sigma}(t)\propto t^{4/5}$ and ${\it\phi}(t)\propto t^{13/5}$. For intermediate times, the deflection of the beam leads to rapid acceleration of the liquid layer, with ${\it\sigma}(t)\propto t^{4}$ and ${\it\phi}(t)\propto t^{9}$. Finally, when the beam has sagged to become almost vertical, the liquid film flows downward with ${\it\sigma}(t)\propto t$ and ${\it\phi}(t)\sim {\rm\pi}/2$. We demonstrate good agreement between these theoretical predictions and experimental results.

An oblique shock impinging on a shear layer that separates two uniform supersonic streams, of Mach numbers $M_{1}$ and $M_{2}$, at an incident angle $\unicode[STIX]{x1D70E}_{i}$ can produce regular and irregular interactions with the interface. The region of existence of regular shock refractions with stable flow structures is delineated in the parametric space $(M_{1},M_{2},\unicode[STIX]{x1D70E}_{i})$ considering oblique-shock impingement on a supersonic vortex sheet of infinitesimal thickness. It is found that under supercritical conditions, the oblique shock fails to deflect both streams consistently and to provide balanced flow properties downstream. In this circumstance, the flow renders irregular configurations which, in the absence of characteristic length scales, exhibit self-similar pseudosteady behaviours. These cases involve shocks moving upstream at constant speed and increasing their intensity to comply with equilibrium requirements. Differences in the variation of propagation speed among the flows yield pseudosteady configurations that grow linearly with time. Supercritical conditions are described theoretically and reproduced numerically using highly resolved inviscid simulation.

An almost rigid rotation of a viscous fluid is produced by dividing the containing cylinder into two sections and rotating them at slightly different speeds. The fluid velocity can be separated into two parts, a swirl about the axis and a streaming motion in the axial planes. When the difference in the speeds of rotation of the two sections is small, the equations of motion can be linearized. The solution is found for large Reynolds numbers and provides an illustration of the way in which the conditions of geostrophic flow (no velocity variation in the axial direction and an inability to insist on undistrubed flow at infinity) are approached as the Reynolds number tends to infinity.

This paper describes the boundary-layer structure of flow through a porous medium in a two-dimensional rectangular cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated. In the limit of large Darcy–Rayleigh number, the solution involves a horizontal boundary layer near the upper surface where the main thermal gradients occur. For a monotonic temperature distribution at the upper surface, these drive fluid to the colder end of the cavity where it descends within a narrow vertical boundary layer before returning to the horizontal layer. The horizontal and vertical layers form an interactive system which is solved by a combination of asymptotic analysis and numerical computation. A complete solution is obtained for the case of a quadratic temperature distribution at the upper surface. The solution of the interactive boundary-layer system determines the almost constant temperature in the core region below the horizontal and vertical layers, which contains relatively weak variations in both the thermal and velocity fields.

Large-amplitude nonlinear oscillations of an axially symmetric water drop of volume 7.33 cm3, initial aspect ratio 3.4, with surfactant Triton X-100 of 1.4×10−4 g ml−1 (1 CMC), in microgravity are compared with predictions of the boundary-integral method. The small shear viscosity of the bulk phase, as well as the surface dilatational viscosity and surface shear viscosity are considered. When a very specific set of material properties is assumed, numerical simulations of the drop oscillations are in good agreement with the experimental results of drop oscillations measured in space during the second United States Microgravity Laboratory, USML-2. The obtained surface viscosities are in rough agreement with literature values.

The stability of circular Couette flow when the outer cylinder is at rest and the inner is modulated both with and without a mean shear is examined in the narrow-gap limit. Disturbances are assumed to be axisymmetric. Two criteria are used to determine conditions for stability; the first requires that the motion be strongly stable, the second only that disturbances of arbitrary initial energy decay from cycle to cycle. The behaviour of critical parameters as a function of frequency is similar for the linear and the energy analysis. The range of Reynolds numbers bounded above by certain instability and below by conditional nonlinear stability is enlarged by modulation.

In Part 1 (Woodley & Peake 1999) we described a method for predicting the occurrence of resonant states in a system comprising twin cascades in zero relative motion. We now demonstrate how that work can be extended to account for the case of more practical interest, in which the upstream cascade (rotor) is rotating in the transverse direction relative to the downstream cascade (stator). Time periodicity now forces the temporal frequency of any disturbance to be an integer multiple of the rotor passing frequency in the stator frame, and vice versa, and this leads to the requirement to sum over a discrete set of temporal modes, as well as over the spatial modes already described in Part 1. The mechanisms by which temporal and spatial modes are scattered by the blade rows is made clear by the analytical approach adopted here; the scattering of the incident pressure (and, for the stator, vorticity) fields by each row in its own frame is completed using results similar to those presented in Part 1, and the fields in the two frames then matched across the inter-row gap to provide a single matrix equation. Specimen results for the conditioning of this equation are given, and although it seems more difficult to obtain very strong excitation than it was for zero rotation, the significance of Parker resonance of the stator is again apparent.

Within the same framework we calculate the mean induction of a magnetic field and the mean diffusivity of a passive scalar, for two families of flows in which the degree of spatial decorrelation can be systematically adjusted. We investigate the dependence of these quantities both on the spatial decoherence and on the molecular diffusivity. We demonstrate that for flows with similar global properties, the mean induction is dramatically reduced as the flows become less spatially correlated; the mean diffusivity, on the other hand, shows no significant or systematic variation.

Binder & Vanden-Broeck (2005) showed there are no subcritical or critical solutions satisfying the radiation condition for steady flows past a flat plate. By using a weakly nonlinear analysis, it is shown that such flows exist for a curved plate. Fully nonlinear solutions are computed by a boundary integral equation method, and new nonlinear solutions for supercritical and generalized critical flows past a curved plate are presented.

The group vaporization of a monodisperse fuel-spray jet discharging into a hot coflowing gaseous stream is investigated for steady flow by numerical and asymptotic methods with a two-continua formulation used for the description of the gas and liquid phases. The jet is assumed to be slender and laminar, as occurs when the Reynolds number is moderately large, so that the boundary-layer form of the conservation equations can be employed in the analysis. Two dimensionless parameters are found to control the flow structure, namely the spray dilution parameter λ, defined as the mass of liquid fuel per unit mass of gas in the spray stream, and the group vaporization parameter ϵ, defined as the ratio of the characteristic time of spray evolution due to droplet vaporization to the characteristic diffusion time across the jet. It is observed that, for the small values of ϵ often encountered in applications, vaporization occurs only in a thin layer separating the spray from the outer droplet-free stream. This regime of sheath vaporization, which is controlled by heat conduction, is amenable to a simplified asymptotic description, independent of ϵ, in which the location of the vaporization layer is determined numerically as a free boundary in a parabolic problem involving matching of the separate solutions in the external streams, with appropriate jump conditions obtained from analysis of the quasi-steady vaporization front. Separate consideration of dilute and dense sprays, corresponding, respectively, to the asymptotic limits λ ≪ 1 and λ ≫ 1, enables simplified descriptions to be obtained for the different flow variables, including explicit analytic expressions for the spray penetration distance.

A class of unsteady boundary layers that form on flat extensible surfaces of finite but increasing length in otherwise stagnant surroundings is considered. The surface length $\overline{R}$ is assumed to grow with time as tp where p > 0 and the velocity at any location $0 \le \overline{r} \le \overline{R}$ on the surface as tp-np-1r-n, where n [ges ] 0. The problem is cast into similarity variables and the governing parabolic differential equation shown to exhibit, for various combinations of n and p, regions of mixed mathematical diffusivity and reversals in the direction of convection of vorticity. Equations depicting such behaviour are usually termed singular parabolic and are here classified as follows: type-0, in which the mathematical diffusivity may be either positive or mixed but in which there are no reversals in the direction of convection of vorticity; type-1, in which the mathematical diffusivity may be either positive or mixed but in which there are reversals in the direction of convection of vorticity. Both types are shown to occur. Moreover while type-0 flows occur only when n = 1 and form with an unsteady separated stagnation point at the origin, type-1 flows occur only for 0 [les ] n < 1 and form with a steady stagnation point at the origin. Type-1 flows are further characterized by boundary layers with zero displacement thickness both at the origin and leading edge. Because singular parabolic equations require two initial conditions plus boundary conditions to ensure uniqueness, they are here treated numerically in a manner akin to elliptic boundary value problems. A successive-approximation implicit scheme was thus used and a wide range of cases solved in the parameter range n ∈ [0,1], p ∈ (0,2]. Amongst other things, it is shown that type-0 flows have lower drag than their type-1 counterparts. It is further shown that the drag on a flat rigid surface of finite length moving in its own plane at constant velocity and being continuously produced at the origin is higher than on a corresponding length of either a semi-infinite surface likewise produced or a semi-infinite plate in an aligned uniform stream; however if the surface is extensible and $n > \frac{1}{2}$ the converse is true.

We consider the spatio-temporal evolution of patterns in the marginally unstable Ekman layer driven by an applied shear stress. Both the normal and tangential components of the Earth's angular velocity are included in a tangent plane approximation of the oceanic boundary layer at latitude $\lambda$. The fluid motion in a layer of finite depth as well as one of infinite depth is considered. The linear instability in the infinite depth case is known to depend on the direction of the applied stress for $\lambda \ne 90\dg$, but this dependence is weak for the stress-driven Ekman layer. By contrast, the weakly nonlinear motion exhibits for finite and infinite depths qualitatively different dynamics for different stress directions.

The problem is treated by the method of multiple scales. In the case of finite depth, this leads to the Davey–Hocking–Stewartson equation, an amplitude equation of complex Ginzburg–Landau type coupled to a Poisson equation. In the case of infinite depth, it leads to the anisotropic complex Ginzburg–Landau equation for the amplitude of the roll motion. Motions in both finite and infinite depth basins are explored by numerical simulation, and are shown to lead to chaotic dynamics for the modulation envelope in most cases. The statistics and the nature of the patterns produced in this motion are discussed.

Low-Rossby-number flow past a circular cylinder in a rapidly rotating frame is studied when 1 < N < 2, where N is equal to E½/Ro in terms of the Ekman number E and Rossby number Ro. For this parameter range the E¼ boundary layer contains a singularity at the rear stagantion point. The asymptotic structure of this singularity is shown to consist of three distinct asymptotic regions, one of which is viscous while the others are inviscid. New accurate numerical solutions of the boundary-layer equation confirm this singularity structure. The use of Von Mises coordinates both simplifies the analysis, and enables numerical solutions to be found closer to the critical value N = 1, beneath which the flow separates upstream of the rear stagnation point.

Using stationary rakes of hot-wire anemometer probes, a series of temperature measurements have been conducted within rising buoyant thermals created by detonation of large-diameter gas-filled balloons. These atmospheric detonation simulation tests were performed at the SANDIA Laboratories Aerial Cable Test Facility, Kirtland AFB, during the late fall of 1973. All measurements were performed after pressure equilibrium (t > 1 s) but prior to completion of the toroid formation process and within the time required (t > 4 s) for the fireball to rise one diameter.

Based on an extensive processing of measured data for two detonation events, unique results describing the mean flow field and turbulent structure statistics for the early-time large-scale buoyant thermal have been determined. Final results include: mean-velocity and temperature data; temperature and temperature gradient intensities; space, auto- and ‘eddy-lifetime’ correlation scales; power-spectral densities; probability density distributions and approximate microscale data.

The present temperature data illustrate that the early-time rising fireball is characterized by a highly intermittent irregular top or cap, a constant (high-) temperature core and a thermal wake exhibiting a linear decay in mean temperature. Mean convection velocities for the fireball wake were measured to be approximately twice as large as fireball cap velocities (13.1 m/s). Maximum fireball temperatures (1100 °C) were in good agreement with peak brightness-temperature data determined from an independent IR measurements experiment. Core leading-edge mean-temperature gradients compare favourably with predictions (10 °C/cm) whereas trailing-edge gradients (0.4 °C/cm) are smaller than predicted results (3 °C/cm).

Normalized temperature intensities at the fireball top and within the thermal wake were substantial (≈ 0.6) in contrast with quiescent fireball core levels (0.1). Moving-frame autocorrelation times (‘eddy lifetimes’, 0.17 + 0.04 s) were approximately four times larger than measured autocorrelation times. This result illustrates that the early-time fireball eddy structure is highly persistent, a finding which tends to substantiate those radar models for large-scale fireballs which postulate relatively slow ‘smearing’ of early-time electron density gradients. Wake integral scales, determined from vertical space correlation data, were approximately 5% of the fireball mean radius. Probability density distributions for the fireball thermal wake were ‘spiky’ in nature, with some evidence of bimodal behaviour, and slightly skewed to the negative side of their normalized values. Normalized spectra data, in general, follow a − &5/3; falloff behaviour for several decades in spectral power. The noted large ‘spread’ in the inertial subrange is consistent with measured turbulence Reynolds numbers (2000) and with the large-scale fully developed turbulent structure for the GEST fireballs (ReD = 3 × 107). Wake microscale data, estimated from temperature and temperature-gradient intensity data and the isotropic turbulence assumption, varied from 2 to 4 ern and were in favourable agreement with predicted microscale magnitudes.

The formation of patterns of granules on the bottom of a differentially rotating tank, partially filled with a fluid, is described. The pattern formation is initiated when the tank is spun up from an initial constant rotation rate by a sufficiently large increment to another constant higher rotation rate. The granules are observed to be set into motion, slide across the bottom of the tank and organize themselves into two distinct, superposed geometrical patterns. The first pattern constitutes a system of tightly wound equi-angular spirals and is identified as being simply a visualization of the well-known stationary Class B waves of the unstable boundary layer. This pattern is not considered further here. The second pattern, which is considered here, is shown not to be a visualization of some other known stationary wave mode. It consists of a number of large-scale spiral arms extending outward towards the wall of the tank and originating at a patch of salt in the tank's centre. The presence of Class B waves is apparently a necessary condition for the spiral arm formation. The experiments show that the radius r0 of the inner salt patch is inversely proportional to the increment Δω of the rotation rate for the larger values of Δω. The angle ε(r) between the direction of the spiral arms and the tangential direction is observed to decrease with the distance r from the centre as rb where b = −0.64 ± 0.1. The relationship between the number n of spiral arms and the associated Reynolds number Re and Rossby number Ro is found from the experimental observations and dimensional considerations to be \[n^2 = Re \, Ro. \] Various aspects related to the mechanism involved in the pattern formation are discussed. However, no conclusive model for the spiral arm formation can at present be suggested.

The flow past a circular cylinder in a rotating frame is examined when the Rossby number Ro is O(E½), where E is the Ekman number. Previous studies of the configuration have shown that, provided the ratio Ro/E½ is less than a certain critical value, the flow around the cylinder is determined by the classical potential-flow solution. However, once Ro/E½ is greater than that critical value the E1/4 layer on the surface of the cylinder, which is rather like a boundary layer in a high-Reynolds-number non-rotating fluid, can separate from the cylinder and distort the potential flow. In this study the form of the flow once separation has occurred is examined using a method analogous to the Kirchhoff free-streamline theory in a non-rotating fluid. The results are compared with published experimental and numerical data on the flow for various values of Ro/E½.

The second variation of a linear combination of energy and angular momentum is used to investigate the formal stability of circular vortices. The analysis proceeds entirely in terms of Lagrangian displacements to overcome problems that otherwise arise when one attempts to use Arnol'd's Eulerian formalism. Specific attention is paid to the simplest possible model of an isolated vortex consisting of a core of constant vorticity surrounded by a ring of oppositely signed vorticity. We prove that the linear stability regime for this vortex coincides with the formal stability regime. The fact that there are formally stable isolated vortices could imply that there are provable nonlinearly stable isolated vortices. The method can be applied to more complicated vortices consisting of many nested rings of piecewise-constant vorticity. The equivalent expressions for continuous vorticity distributions are also derived.

The influence of uniform flow acceleration on the stability and the characteristics of circular-cylinder wakes over a Reynolds-number range, 20 < R < 330, was investigated. Experiments were performed to examine the temporal evolution of the wake before, during, and after the onset of the wake instability. We have demonstrated in several ways that the wake is stabilized by flow acceleration: (i) the onset of the wake instability occurs at larger Reynolds numbers than in the steady flow case, (ii) the closed wake develops to states that would be unstable in a steady flow, and (iii) once vortex shedding does occur there is a reduction in instantaneous Strouhal number. We have also examined the temporal growth rate of the wake instability and find that it is directly proportional to the applied flow acceleration. Physical mechanisms are proposed to describe the experimental observations.