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.

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.

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.

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 flow of a viscous incompressible fluid contained between a stationary outer sphere and rotating inner sphere is studied theoretically and experimentally. Previous theoretical results concerning the basic laminar flow (part 1) are compared with experimental results. Small and large Reynolds number results are compared with Stokes-flow and boundary-layer solutions. The effect of the radius ratio of the two spheres is demonstrated. A linearized theory of stability for the laminar flow is formulated in terms of toroidal and poloidal potentials; the differential equations governing these potentials are integrated numerically. It is found that the flow is subcritically unstable and that the observed instability occurs at a Reynolds number close to the critical value of the energy stability theory. Observations of other flow transitions, at higher values of the Reynolds number, are also described. The character of the stability of the spherical annulus flow is found to be strongly dependent on the radius ratio.

In a recent paper (Griffin & Ramberg 1974) the authors studied the vortex-street wakes behind forced vibrating rigid cylinders. All experimental conditions were within the regime of wake capture or synchronization between the vibration and vortex frequencies. Both mean and fluctuating velocities in the wake together with the length of the vortex formation region were measured as functions of vibration amplitude and frequency at a Reynolds number of 144. The viscous vortex strength, age and spacing at this Reynolds number were then obtained by matching a model for the vortex street with the mean and r.m.s. velocity profiles obtained from hot-wire measurements. These results are employed here to determine the steady drag force on the vibrating cylinder by means of the von Kármán drag formulation. The drag coefficients determined in this way are in agreement with the recently published direct force measurements of Tanida, Okajima & Watanabe (1973) and Griffin, Skop & Koopmann (1973) at Reynolds numbers of 80, 500-900 and 4000. From these results a direct relation is drawn between the increased drag on resonantly vibrating structures and changes in the vortex strength, spacing and formation in their wakes.

This paper is a theoretical investigation of the stable laminar decay of a fully established channel or pipe flow following a sudden blockage such as would be caused by the rapid closure of a valve or imposition of an end wall or gate. The development of the subsequent velocity and pressure fields is examined from the instant the initial pressure wave passes until the final decay of all motion. Three time scales of hydrodynamic interest are identified and the relevant solutions are obtained. The time scales are as follows: (i) a very short time characteristic of the passage of the pressure wave during which the velocity field adjusts inviscidly to the new boundary conditions imposed by the presence of the end wall, (ii) a short diffusion time during which the displacement interaction generated by the diffusion of the primary Rayleigh layer induces a substantial secondary motion with distinct side-wall boundary layers and an inviscid core and (iii) a long diffusion time during which the boundary layers fill the entire channel or pipe and the residual motion then dies out. The secondary flow for short diffusion times is of special interest in that it is an example of an unsteady boundary layer where the external pressure gradient and inviscid outer flow are unknown and determined by the integrated time history of the combined mass flow displacement generated by the primary- and secondary-flow boundary layers. The paper closes with some preliminary comments and experimental observations on decelerating pipe flows.

Measurements have been made of the vertical velocity and simultaneous temperature fluctuation at fixed points in turbulent, statistically steady convection in water over an ice surface. Convection was confined to a horizontal layer of constant depth that lay under a stably stratified layer. Measurements of vertical velocities less than 1 mm/s were accomplished using a laser-Doppler velocimeter with a frequency-shifted reference beam. Vertical profiles of all relevant one-point moments, including joint temperature-velocity moments, up to third order were obtained. In the stable layer internal gravity waves were observed, and certain aspects of their behaviour were found to agree with existing theoretical analyses. Typical r.m.s. values of the temperature and velocity fluctuations were nearly equal to the appropriate ‘convection’ scales in the convection layer while in the conduction layer immediately above the ice surface they were more nearly equal to the ‘molecular’ scales. All of the observations were consistent with the existence of buoyant plume-like masses which dominated the structure of the flow. An analysis of the data using a conditional averaging technique to determine the mean properties of these masses showed that below the stable layer they accounted for over 80% of the mean-square fluctuations in temperature and velocity.

A Galerkin method is proposed for a class of boundary-layer flow problems. In this method, the assumed solution is composed of an auxiliary function and a series solution. The representing functions used in the series solution are orthonormal eigenfunctions, closely related to that of the boundary-layer equation, and are independent of the initial condition, as well as the boundary conditions. The reduced system of stiff, first-order, nonlinear, ordinary differential equations then has diagonal dominance for the first part of the flow region. The proposed method has been tested on two representative flows. Numerical experiments show that highly accurate results can be obtained for the entire boundary-layer flow region, if the auxiliary function satisfying the initial and boundary conditions is chosen to satisfy the first compatibility condition of the upstream flow region. In fact the computation is rather simple, and the numerical integration of the reduced initial-value problem can be carried out up to separation with a fairly large step.

Numerical solutions of the Navier-Stokes equations are presented for two-dimensional viscous flow past semicircular and semielliptical projections attached to a plane wall on which a laminar boundary layer has developed. Since the major axis is in the direction normal to the wall and is chosen to be twenty times as long as the minor axis in the present case, the flow around the semielliptical projection will approximately correspond to that around a normal flat plate. It is assumed that the height of each obstacle is so small in comparison with the local boundary-layer thickness that the approaching flow can be approximated by a uniform shear flow. Numerical solutions are obtained for the range 0·1-100 of the Reynolds number, which is defined in terms of the undisturbed approaching velocity at the top of the obstacle and its height. The geometrical shapes of the front and rear standing vortices, the drag coefficients and the pressure and shear-stress distributions are presented as functions of the Reynolds number. The computed results are discussed in connexion with the data already obtained in the other theoretical solutions and an experimental observation.