A modified version of Bateman's variational formulation of the incompressible Navier-Stokes equations and boundary conditions (see Dryden, Murnaghan & Bateman 1956) is introduced. This is employed to examine a particular nonlinear problem of hydrodynamic stability which was treated previously, using a ‘direct’ approach, by Craik (1971). This problem concerns the resonant interaction at second order of a triad of wave modes in a parallel shear flow.

The present method is conceptually attractive; it also has the major advantage over the ‘direct’ method of a substantial reduction in algebraic complexity, which allows results to be derived far more readily. Also, some further improvements are made upon Craik's previous analysis. Such a variational approach may often be simpler than present conventional methods of tackling nonlinear viscous-flow problems. The present paper shows how other problems of nonlinear stability and wave interactions may be tackled in this way.

The decay of two-dimensional, homogeneous, isotropic, incompressible turbulence is investigated both by means of numerical simulation (in spectral as well as in grid-point form), and theoretically by use of the direct-interaction approximation and the test-field model. The calculations cover the range of Reynolds numbers 50 ≤ RL ≤ 100. Comparison of spectral methods with finite-difference methods shows that one of the former with a given resolution is equivalent in accuracy to one of the latter with twice the resolution. The numerical simulations at the larger Reynolds numbers suggest that earlier reported simulations cannot be used in testing inertial-range theories. However, the large-scale features of the flow field appear to be remarkably independent of Reynolds number.

The direct-interaction approximation is in satisfactory agreement with simulations in the energy-containing range, but grossly underestimates enstrophy transfer at high wavenumbers. The latter failing is traced to an inability to distinguish between convection and intrinsic distortion of small parcels of fluid. The test-field model on the other hand appears to be in excellent agreement with simulations at all wavenumbers, and for all Reynolds numbers investigated.

Wind, blowing over a water surface, induces a thin layer of high vorticity in which the wind stress is supported by molecular viscosity; the magnitude of the surface drift, the velocity difference across the layer, being of the order of 3% of the wind speed. When long waves move across the surface, there is a nonlinear augmentation of the surface drift near the long-wave crests, so that short waves, superimposed on the longer ones, experience an augmented drift in these regions. This is shown to reduce the maximum amplitude that the short waves can attain when they are at the point of incipient breaking.

Theoretical estimates of the reduction are compared with measurements in wind-wave tanks by the authors and by Mitsuyasu (1966) in which long mechanically generated waves are superimposed on short wind-generated waves. The reductions measured in the energy density of the short waves by increasing the slope of the longer ones at constant wind speed are generally consistent with the predictions of the theory in a variety of cases.

Using the framework of Lighthill's acoustic analogy, a theory has been developed for combustion-generated noise. The theory is used to derive the scaling laws for combustion-generated noise in open turbulent flames. It is shown that the scaling laws for both power and frequency are in good agreement with experiment.

A technique by which the effect of the random phase fluctuations or Doppler ambiguity can be removed from measurements of turbulence intensities by the laser-Doppler velocimeter is proposed. Because the ambiguity is often much broader band than the turbulence, it is proposed that a sequence of low-pass filters at frequencies above those of the turbulence be used in the measurement of turbulence intensities. The true turbulence intensity is given by the extrapolation of these measurements to zero frequency.

Internal waves of the fundamental mode propagating into a shoaling region have been studied experimentally in a continuously stratified fluid. The waves divide into three classes depending upon the ratio of the bottom slope γ to the wave-characteristic slope c. For γ/c < 1, the amplitude and wavenumber changes of the waves over the slope are in reasonable accord with a simple inviscid linear theory, prior to wave breakdown near the intersection of the slope and surface. Considerable mixing occurs in this corner region. When γ/c = 1, a striking instability of the bottom boundary layer is observed and the waves are heavily damped. When γ/c > 1, the waves are inhomogeneous and have complex spatial dependence.

Two attempts were made to develop a three-dimensional laminar boundary layer in the flow over a flat plate in a curved duct, establishing a negligible streamwise pressure gradient and, at the same time, an appreciable crosswise pressure gradient.

A first series of measurements was undertaken keeping the free-stream velocity at about 30 ft/s; the boundary layer was expected to be laminar, but appears to have been transitional. As was to be expected, the cross-flow in the boundary layer decreased gradually as the flow became progressively more turbulent.

In a second experiment, at a lower free-stream velocity of approximately 10 ft/s, the boundary layer was laminar. Its streamwise profile resembled closely the Blasius form, but the cross-flow near the edge of the boundary layer appears to have exceeded that predicted theoretically. However, there was a substantial experimental scatter in the measurements of the yaw angle, which in laminar boundary layers is difficult to obtain accurately.

Simultaneous measurements of the longitudinal (sx) and transverse (sz) components of the turbulent fluctuating velocity gradient at the wall of a pipe are presented for ten locations in the transverse direction. Separate measurements of sx and of sz are also presented for twenty locations. The sz pattern reveals alternating positive and negative values over lengths Λ of the magnitude suggested by the spacing of the streaky wall structure observed in a number of laboratories. Of particular interest is the observation of regular eddies for which the sz pattern is accompanied by a spatial variation of sx out of phase by about 1/4Λ. These regular eddies are associated with intermittent production of Reynolds stress and with the occurrence of large negative values of sx which could be identified with wall bursts previously reported. The time response of the axial flow appears to influence the development of these regular eddies in that the development of an sx pattern lags behind that of an sz pattern. Measurements of spatial correlations of sx and sz with different time delays, as well as direct observations of the temporal variation of sx and sz patterns, support these notions.

Inertialess incompressible flow through a rapidly rotating, variable-area conduit of circular cross-section is treated. For the practical case of an expansion (or contraction) placed between two pipes, the flow is strongly asymmetrical and involves regions of weak reverse flow in the expansion and downstream pipe, while the disturbance to the fully developed pipe flows persists for large, O(E−½) distances upstream and downstream, where E is the (small) Ekman number. The flow in the two pipes depends only on the ratio of their radii and is independent of the shape and length of the expansion. The startling implication of the disturbance's persistence is that, in practice, fully developed flow will almost never be realized in rapidly rotating pipes.

We obtain the solutions, under the Oseen and Boussinesq approximations, for the flow field disturbance due to a line singularity in an otherwise uniform, horizontal, inviscid, incompressible flow of a vertically stratified fluid. The results obtained show no upstream influence for those singularities across which [xdtri ]2ψ + (N/U)2ψ is continuous. Doublet and vortex singularities are examples of these. Uniform flows past doublets and vortices are considered for a range of internal Froude numbers, including the calculation of the pressure distributions and drag for the doublet. An application of the vortex solution to flows in the β-plane is discussed.

The parametric limit process for Laplace's tidal equations (LTE) is considered, starting from the full equations of motion for a rotating, gravitationally stratified, compressible fluid. The boundary-value problem for free oscillations of angular frequency σ is not well posed if σ2 < N2 + 4ω2, where N is the Väisälä frequency and ω is the rotational speed of the Earth, and the governing partial differential equation is elliptic/hyperbolic on the polar/equatorial sides of the inertial latitudes given by ± σ = f (vertical component of 2ω) if σ < 2ω [Lt ] N. The solution of this ill-posed problem is considered for a global ocean of uniform depth, with the effects of ellipticity, the ‘traditional’ approximation and stratification measured by the small parameters m = ω2a/g, δ = h/a and s = hN2/g (g = acceleration due to gravity, h = depth of ocean, a = radius of Earth). LTE represent the joint limit m, δ, s ↓ 0 and yield bounded solutions for all latitudes. It is argued that the parametric expansion in m is regular. The joint expansion in δ and s with LTE as the basic approximation is singular at the inertial latitudes if σ < 2ω, which difficulty is traced to the failure of LTE to provide an adequate description of the characteristics in the hyperbolic domain. It is shown that an alternative formulation, in which the buoyancy force is retained in the basic equations in the joint limit s↓0, δ↓0 with N [Gt ] 2ω, yields solutions that are uniformly valid in the neighbourhoods of the inertial latitudes. The resulting representation comprises a barotropic mode, which satisfies LTE, and an infinite discrete set of baroclinic modes, each of which has Airy turning points at the inertial latitudes and is trapped between them. The barotropic and baroclinic modes are coupled by the Coriolis acceleration associated with the horizontal component of the Earth's rotation. The relative effects of this coupling are uniformly O(δ) if σ > 2ω, but it induces currents O(δ/s1/4) and vertical displacements O(δ/s3/4) between the inertial latitudes if σ < 2ω [Lt ] N. It appears that resonant amplification of the baroclinic modes forced by the barotropic modes could imply internal displacements that dominate those of the basic motion.

A simple model for the experimentally observed instability of the vortex ring to azimuthal bending waves of wavelength comparable with the core size is presented. Short-wave instabilities are discussed for both the vortex ring and the vortex pair. Instability for both the ring and the pair is predicted to occur whenever the self-induced rotation of waves on the filament passes through zero. Although this does not occur for the first radial bending mode of a vortex filament, it is shown to be possible for bending modes with a more complex radial structure with at least one node at some radius within the core. The previous work of Widnall & Sullivan (1973) is discussed and their experimental results are compared with the predictions of the analysis presented here.

The pressure field due to a large circular capped air bubble rising in water has been determined experimentally. The results verify the Davies & Taylor cap boundary condition and the frontal pressure field is well approximated by that due to irrotational flow around an oval body.

The pressure field extends axially as far as ten bubble half-widths below the bubble floor. Immediately below the floor the pressure is constant for about two-thirds of a bubble height. The wake is closed and contains symmetric pressure minima. For the bubbles studied, turbulence, as well as the diffusion of vorticity, probably controls the momentum distribution and energy dissipation in the wake.

The stability of small travelling-wave disturbances in the flow over a flat plate is discussed. An iterative method is used to generate an asymptotic series solution in inverse powers of the Reynolds number Rx = Ux/v to the power one half. The neutral-stability boundaries given by the first two terms of this series are obtained and compared with experimental data. It is shown that the parallel flow approximation leads to a valid solution at very large Reynolds numbers.

Grant (1973) showed that the expansion giving the profile of a steady Stokes wave near a 120° corner was more complicated than had previously been assumed. This paper gives further terms in such an expansion, and shows that generating them cannot introduce transcendental quantities beyond those noted by Grant.

This paper deals with a survey of mean flow and fluctuating quantities in a turbulent boundary layer developing on a smooth wall in a pressure domain P(x), where both dP/dx and d2P/dx2 are positive (increasingly adverse). The two-dimensional nature of the flow field was checked by momentum balance, as well as velocity traverses either side of the working section centre-line. Using the integrated form of the momentum integral equation, it was found that the skinfriction term and the summed momentum and pressure terms differed by at most 19%; but for the majority of measuring points they differed by less than 14%. The off-centre-line velocity profiles were indistinguishable from those taken on the centre-line. The flow field was also surveyed for fluctuating components $(\overline{u^2_1})^{\frac{1}{2}}, (\overline{u^2_2})^{\frac{1}{2}}, (\overline{u^2_3})^{\frac{1}{2}}$, and $\overline{u_1u_2}$, as well as for u1 spectra. Wherever possible, the results were compared with existing models of boundary-layer development. These comparisons indicated that the only all-embracing model for boundary-layer development is the law of the wall.

Rotational flow of an inviscid fluid over an irregularity in the bottom is investigated. The flow is regarded as a perturbed unidirectional flow, and the shape of the irregularity is described using Fourier transforms. The velocity profile in the unidirectional flow is determined using the eddy-viscosity concept and a finite wall slip velocity.

Two different examples of irregularities are considered: (a) an infinitely long straight irregularity which forms an arbitrary angle with the direction of the basic flow and (b) a hump in a channel with impermeable walls. The influence of rotation on the two- and three-dimensional waves which are formed downstream of these irregularities is analysed and experimentally verified. Further, it is shown that the gradient of the basic velocity profile increases the transverse movement of the fluid particles at the bottom, while at the surface this transverse movement is decreased.

Employing a linear stability analysis, Dudis (1973) has recently suggested that shear-flow instability might exist within the upper stratosphere of Venus owing to destabilization by radiative transfer. We have incorporated a more realistic formulation for radiative transfer into his stability analysis and conclude that such an instability is unlikely.

A general characteristic of rapidly rotating fluids is that accurate experimental measurements can only be made of the main (azimuthal) flow. The secondary flow is then usually deduced from theory, although this is often incomplete in the boundary regions where the secondary flow is of most interest.

In this paper we consider the case of source-sink flow between the porous walls of a rapidly rotating annular container and numerically integrate the full equations of motion in order to determine the complete structure of the secondary flow. The results are compared with the (approximate) analytic studies of Hide (1968) and Bennetts & Hocking (1973) to show the differences between the two approaches.

A defect of many previous numerical papers has been the inability to check the solution in the nonlinear case. To overcome this, new experimental measurements of the azimuthal velocity profile for a Rossby number of 0·238 have been obtained and these are compared with the numerical results.