Steady liquid flow in a variable-area rectangular duct rotating rapidly about an axis perpendicular to its centre-line is treated. This problem is significant because the idea of driving a liquid through a rotating system has been largely overlooked by rotating-fluid dynamicists and because it closely resembles the flow inside the impellers of centrifugal pumps and hydraulic turbines. For a prototype formed by joining a semi-infinite constant-area duct and a semi-infinite duct with straight diverging walls, the flow in the diverging duct is carried entirely by large, O(E1/3) velocities in a boundary layer of thickness O(E1/3) adjacent to one of the (side) walls parallel to the axis of rotation, where E is the (small) Ekman number. With a vertical axis of rotation this high-velocity boundary layer is adjacent to the side wall on the right when facing in the flow direction. For a diverging or converging duct placed between two semi-infinite constant-area ducts, large, O(E1/3) velocities occur in side-wall boundary layers on both sides of the variable-area duct and on the left and right sides of the upstream and downstream constant-area ducts respectively. The existence of high-velocity side layers in rapidly rotating rectangular ducts should be relatively easy to prove experimentally and actual measurements of their velocity profiles would provide a good test of the present theory.

We consider the deformation of a thin thread of viscous liquid (viscida) as its ends are slowly moved together. Equations are deduced which are capable of describing the motion of the thread when the displacement of the axis from a straight line is either on the scale of the thread thickness (problem 1) or on the much larger scale of the thread length (problem 2). In the former case it is shown analytically that an arbitrary initial displacement evolves in such a way that, as the appropriately scaled time τ becomes large, the first mode of the disturbance emerges in a dominant role with an amplitude that is proportional to τ½ and independent of the initial amplitude. This provides the initial condition for problem 2, for which a numerical description is obtained.

In addition, we analyse the situation when the ends of the viscida are slowly pulled apart. In this case the high frequency end of the spectrum dominates as an arbitrary disturbance decays.

Two-dimensional eddies in a homogeneous fluid at large Reynolds number, if closely packed, are known to evolve towards larger scales. In the presence of a restoring force, the geophysical beta-effect, this cascade produces a field of waves without loss of energy, and the turbulent migration of the dominant scale nearly ceases at a wavenumber kβ = (β/2U)½ independent of the initial conditions other than U, the r.m.s. particle speed, and β, the northward gradient of the Coriolis frequency.

The conversion of turbulence into waves yields, in addition, more narrowly peaked wavenumber spectra and less fine-structure in the spatial maps, while smoothly distributing the energy about physical space.

The theory is discussed, using known integral constraints and similarity solutions, model equations, weak-interaction wave theory (which provides the terminus for the cascade) and other linearized instability theory. Computer experiments with both finite-difference and spectral codes are reported. The central quantity is the cascade rate, defined as \[ T = 2\int_0^{\infty} kF(k)dk/U^3\langle k\rangle , \] where F is the nonlinear transfer spectrum and 〈k〉 the mean wavenumber of the energy spectrum. (In unforced inviscid flow T is simply U−1d〈k〉−1/dt, or the rate at which the dominant scale expands in time t.) T is shown to have a mean value of 3·0 × 10−2 for pure two-dimensional turbulence, but this decreases by a factor of five at the transition to wave motion. We infer from weak-interaction theory even smaller values for k [Lt ] kβ.

After passing through a state of propagating waves, the homogeneous cascade tends towards a flow of alternating zonal jets which, we suggest, are almost perfectly steady. When the energy is intermittent in space, however, model equations show that the cascade is halted simply by the spreading of energy about space, and then the end state of a zonal flow is probably not achieved.

The geophysical application is that the cascade of pure turbulence to large scales is defeated by wave propagation, helping to explain why the energy-containing eddies in the ocean and atmosphere, though significantly nonlinear, fail to reach the size of their respective domains, and are much smaller. For typical ocean flows, $k_{\beta}^{-1} = 70\,{\rm km} $, while for the atmosphere, $k_{\beta}^{-1} = 1000\,{\rm km}$. In addition the cascade generates, by itself, zonal flow (or more generally, flow along geostrophic contours).

It is shown that, in two-dimensional non-divergent flow in a bounded region, roughly 70% of triad interactions exchange more energy with longer wavelengths than with shorter wavelengths whilst roughly 40% exchange more enstrophy with longer wavelengths than with shorter wavelengths.

The velocity correlation function of a Brownian particle in a viscous compressible fluid is studied in the limit of very small compressibility. The main effect of compressibility is an initial rapid change of the particle mass from a real inertial mass to a virtual mass.

The incompressible starting flow past a downstream-facing right-angled step was investigated at Reynolds numbers R (based on step height) less than 500 by means of flow-visualization techniques. The distance between the step and the point of reattachment on the downstream wall was found to increase linearly with time at intermediate stages of the flow development. The recirculation region formed behind the step was composed of three vortex domains at earlier stages of its development when R [gap ] 140. It was observed for R [gap ] 200 that once the starting vortices had been shed downstream a steady recirculation region was established. The evolution of filaments of tracer in the flow down the step was found to vary considerably with R. The filaments were observed to develop faster in the flow down the step than in that past a symmetric model without a downstream wall when R [gap ] 140.

A method is presented for performing detailed computations of thin incompressible separation bubbles on smooth surfaces. The analysis consists of finite-difference solutions to the time-dependent boundary-layer or Navier-Stokes equations for the flow in the immediate vicinity of the bubble. The method employs the McDonald-Fish turbulence model, to predict the development of the time-mean flow field, as influenced by the free-stream turbulence level. It also employs a viscous-inviscid interaction model, which accounts for the elliptic interaction between the shear layer and inviscid free stream. The numerical method is based on an alternating-direction implicit scheme for the vorticity equation. It employs transformations, to allow the free-stream boundary to change in time with the shape of the computed shear layer, and to ensure an adequate resolution of the sublayer region. Numerical solutions are presented for transitional bubbles on an NACA 663-018 airfoil at zero angle of incidence with chordal Reynolds numbers of 2·0 × 106 and 1·7 × 106. These have a qualitative behaviour similar to that observed in numerous experiments; they are also in reasonable quantitative agreement with available experimental data. Little difference is found between steady solutions of the boundary-layer and Navier-Stokes equations for these flow conditions. Numerical studies based on mesh refinement suggest that the well-known singularity at separation, which is present in conventional solutions of the steady boundary-layer equations when the free-stream velocity is specified, is effectively removed when viscous-inviscid interaction is allowed to influence the imposed velocity distribution.

A laboratory experiment is presented which examines the role of density jumps in the reflexion and breaking of internal gravity waves. It is found that the measured phase shift of the reflected wave and the measured amplitude of the density jump are in good agreement with linear theory. Local overturning occurs when wave amplitude becomes large, and there appears to be a critical amplitude above which overturning will occur and below which it will not. The overturning seems to be due to local gravitational instability, caused by the horizontal advection of density. Overturning changes the basic flow field in the region of interaction; and it results in smaller-scale motions.

The nonlinear response of the ‘sliced-cylinder’ laboratory model for the wind-driven ocean circulation is re-examined here in part 1 for the case of strong steady forcing. Introduced by Pedlosky & Greenspan (1967), the model consists of a rapidly rotating right cylinder with a planar sloping bottom. The homogeneous contained fluid is driven by the slow rotation of the flat upper lid relative to the rest of the basin. Except in thin Ekman and Stewartson boundary layers on the solid surfaces of the basin, the horizontal flow in the interior and western boundary layer is constrained by the rapid rotation of the basin to be independent of depth. The model thus effectively simulates geophysical flows through the physical analogy between topographic vortex stretching in the laboratory model and the creation of relative vorticity in planetary flows by the β effect.

As the forcing is increased, the flow in both the sliced-cylinder laboratory and numerical models first exhibits downstream intensification in the western boundary layer. At greater forcing, separation of the western boundary current occurs with the development of stationary topographic Rossby waves in the western boundary-layer transition regions. The observed flow ultimately becomes unstable when a critical Ekman-layer Reynolds number is exceeded. We first review and compare the experimental and numerical descriptions of this low-frequency instability, then present a simple theoretical model which successfully explains this observed instability in terms of the local breakdown of the finite-amplitude topographic Rossby waves embedded in the western boundary current transition region. The inviscid stability analysis of Lorenz (1972) is extended to include viscous effects, with the consequence that dissipative processes in the sliced-cylinder problem (i.e. lateral and bottom friction) are shown to inhibit the onset of the instability until the topographic Rossby wave slope exceeds a finite critical value.

It is argued on the basis of exact solutions for uniform vortices in straining fields that vortices of finite cross-section in a row will disintegrate if the spacing is too small. The results are applied to the organized vortex structures observed in turbulent mixing layers. An explanation is provided for the disappearance of these structures as they move downstream and it is deduced that the ratio of average spacing to width should be about 3·5, the width being defined by the maximum slope of the mean velocity. It is shown in an appendix that walls have negligible effect.

The linear stability of a slowly varying flow, the flow in a diverging straight-walled channel, is studied using a modification of the ‘WKB’ or ‘ray’ method. It is shown that ‘quasi-parallel’ theory, the usual method for handling such flows, gives the formally correct lowest-order growth rate; however, this growth rate can be substantially in error if its magnitude is comparable to that of the rate of change of the basic state. The method used clearly demonstrates the dependence of the growth rate, wavenumber, neutral curves, etc., on the cross-stream variable and on the flow quantity under consideration. When applied to the divergent channel, the method yields a much wider ‘unstable’ region and a much lower ‘critical’ Reynolds number (depending on the flow quantity used) than those predicted by quasi-parallel theory. The determination of the downstream development of waves of constant frequency shows that waves of all frequencies eventually decay.

An experimental and theoretical study is made of a suddenly separating and reattaching two-dimensional turbulent boundary layer on a flat surface. A separation bubble is formed on the floor of a wide parallel-sided wind-tunnel duct with the pressure field causing the bubble formation produced by fixing the shape of the flexible roof of the duct. Boundary layers on the roof are controlled and remain attached. It is found that a very satisfactory model for the flow is an inviscid one.

The boundary layer on the floor of the duct is represented by a region of constant vorticity with slip at the boundary, and it is assumed that the separation process is dominated by the interaction between this ‘vortical’ region and the irrotational field between the vortical region and the roof (of prescribed shape). The interface between the rotational and irrotational regions is a free boundary and may be located when all necessary boundary conditions are given. These conditions include two characteristic parameters for the adverse-pressure-gradient turbulent boundary layer which is developing upstream of the region of interest.

The problem is solved by an electrical analog method. The theoretical size and shape of the bubble and positions of separation and reattachment are in agreement with observations. The advantage of the model over most previous attempts to predict separation is that the governing equations are elliptic rather than parabolic or hyperbolic and therefore the interaction between the boundary-layer flow and the irrotational free stream is included in the calculations.

The equilibrium spectra of two-dimensional numerical model flows are studied from the viewpoint of microcanonical ensemble averages. The method leads to accurate numerical verification of the ergodic, or mixing, hypothesis in the case of systems constrained to a finite number of degrees of freedom.

Measurements are presented of the mean and fluctuating pressure field acting on a two-dimensional square cylinder in uniform and turbulent flows. The addition of turbulence to the flow is shown to raise the base pressure and reduce the drag of the body. It is suggested that this is attributable to the manner in which the increased turbulence intensity thickens the shear layers, which causes them to be deflected by the downstream corners of the body and results in the downstream movement of the vortex formation region. The strength of the vortex shedding is shown to be reduced as the intensity of the incident turbulence is increased.

The response of the ‘sliced-cylinder’ laboratory model for the wind-driven ocean circulation is studied here in part 2 for the case of an oscillatory ‘wind’ stress. The model consists of a rapidly rotating right cylinder with a planar sloping bottom. This basin geometry contains no closed geostrophic contours, so that low frequency topographic Rossby wave modes possessing mean vorticity exist in the sliced-cylinder model by the physical analogy between topographic vortex stretching and the β effect for large-scale planetary flows. The interior flow in the laboratory model is driven by the time-dependent Ekman-layer suction produced by the periodic relative angular velocity of the upper lid. The frequency of the forcing is sufficiently small that the interior motion is quasi-geostrophic with the horizontal velocities being independent of depth. Simple two-dimensional analytic and numerical models are developed and compared very favourably with the laboratory results. The observed horizontal velocity field exhibits both (i) westward intensification and decreased horizontal scale when the forcing frequency is decreased, and (ii) a significant resonant magnification when the forcing frequency is tuned to the natural frequency of one of the lower inviscid topographic Rossby wave modes. The observed westward phase speed of the driven motion is accurately predicted and shows little dependence on the amplitude of the forcing. The instantaneous and mean Lagrangian fluid particle trajectories were measured in the laboratory model. The general derivation by Moore (1970) of the governing equations for the mean Lagrangian motion are extended to incorporate forcing and Ekman-layer dissipation. The results suggest that the mean Lagrangian flow should be significantly reduced near resonant frequencies, since the mean Eulerian motion is partially offset by the Stokes drift associated with the topographic Rossby wave modes. This result is consistent with the small observed amplitude of the mean Lagrangian motion. Also presented are the results for a laboratory experiment conducted using a combined steady and oscillatory ‘wind’ stress.

This paper reports the development of practical methods of viscometry to characterize non-Newtonian fluids in slow flow. It is shown that measurements of the free surface near rods rotating in STP and polyacrylamide are accurate, reproducible, and in excellent agreement with a theory of rod climbing. Results are presented that establish the theory and experiment as a viscometer for determining the values of certain (Rivlin-Ericksen) constants that arise in the theory of slow flow. The variation of these constants with temperature in our sample of STP has been explicitly and accurately determined. The experiments in STP show that there is a range of rotational speeds for which STP may be well described by the fluids of grade four. Depth-averaged equations are derived from the equations governing steady axisymmetric flow of any incompressible simple fluid. From the depth-averaged equations, we prove a theorem about the variation of the torque required to turn the rod.

This account of the Colloquium on Coherent Structures in Turbulence held at Southampton from 26-29 March 1974 presents a brief summary of two invited lectures and 42 formal presentations on turbulent shear-flow structure. A number of shorter contributions and discussions are also outlined. The present position of the study of turbulent shear-flow structure is reviewed and some new experimental techniques are discussed.

Any straight continental shelf of monotonic depth profile is shown to have as its entire complement of barotropic trapped modes (i) an infinite discrete set of ‘continental-shelf waves’, (ii) a single ‘Kelvin wave’, and (iii) an infinite discrete set of ‘edge waves’. The decomposition of energy density and fluxes into modal constituents is discussed.

The stability of a compressible non-dissipative swirling flow to adiabatic infinitesimal disturbances of arbitrary orientation is considered. The resulting sufficient condition for stability is the general form of the effective Richardson criterion for swirling flows, first obtained, for axisymmetric modes only, by Howard. In addition, upper bounds to the growth rate of unstable modes are obtained and some extensions of the semicircle theorem to azimuthal disturbances are stated.

Conditionally sampled hot-wire measurements were taken in a pipe at low Reynolds numbers (2700 > Re > 2000) corresponding to the onset of turbulence as a result of a large perturbation in the flow. This type of transition gives rise to a turbulent puff which maintains itself indefinitely at around Re = 2200. The structure of puffs was investigated in some detail and was found to be very different from the structure of fully developed turbulent pipe flow. Nevertheless, it is independent of the character of the disturbance which created it. The purpose of the study was to gain some insight into the mechanism of transition in a pipe.