The motion of a two-dimensional packet of capillary–gravity waves on water of finite depth is studied. The evolution of a packet is described by two partial differential equations: the nonlinear Schrödinger equation with a forcing term and a linear equation, which is of either elliptic or hyperbolic type depending on whether the group velocity of the capillary–gravity wave is less than or greater than the velocity of long gravity waves. These equations are used to examine the stability of the Stokes capillary–gravity wave train. The analysis reveals the existence of a resonant interaction between a capillary–gravity wave and a long gravity wave. The interaction requires that the liquid depth be small in comparison with the wavelength of the (long) gravity waves and the evolution equations describing the dynamics of this interaction are derived.

An experimental investigation of the flow around surface-mounted cubes in both uniform, irrotational and sheared, turbulent flows is described. The shear flow was a simulated atmospheric boundary layer with a height ten times the body dimension. Measurements of body surface pressures and mean and fluctuating velocities within the wake are presented. In the latter case a pulsed-wire anemometer was used extensively since the turbulent intensities were much too high for effective use of more standard instrumentation. The clear effects of upstream turbulence and shear on the wake flow are described, comparisons with the somewhat sparse measurements of previous workers are made and the relevance of recent theoretical attempts to describe the flow, as opposed to numerical calculation techniques to predict it, is briefly discussed.

The results of the linear theory for the flow of a supersonic relaxing gas past a slender body of revolution are analysed in regions where its predictions of wavelet position begin to break down. In this way new variable systems can be found which make it possible to discuss the correct nonlinear wave behaviour far from the body. The situation depends upon three especially important parameters, namely the thickness ratio ε of the body, the ratio δ of relaxing-mode energy to thermal energy and the ratio λ of a relaxation length to a typical body length. After establishing general results from the linear theory, the conical body is treated in some detail. This makes it possible to demote λ as an important parameter, although its restoration does prove useful at one point in the analysis, and results are derived for shock-wave behaviour when ord 1 [ges ] δ > ord ε4, δ = ord ε4 and δ < ord ε4. In the first range of δ fully dispersed waves are essential, although they are fully established only at great distances from the cone; in the second range of δ partly dispersed waves seem to be the most likely to appear, and in the third range relaxation effects are second-order modifications of a basically frozen-flow field. Practical situations may well fall into the first of these categories.

A theory for the evolution of the wind drift current and of the Langmuir circulations in infinitely deep water of constant density is presented. The model improves and extends a recent quasi-steady theory of Craik & Leibovich which asserts that the Langmuir circulations arise from a nonlinear interaction between surface waves and the frictional wind drift current. In turn, the development of the wind drift should be strongly influenced by Langmuir circulations, when they are present, and the two current systems are therefore treated here as a single inseparable system driven by a prescribed wind stress and surface wave field. Mixing by the vertical motions in the Langmuir circulations is shown to yield solutions for the wind drift, obtained both analytically and numerically, which are consistent with experiments and with field observations. The model yields a streaky flow pattern with a mean motion much like a turbulent wall layer, although the model is deterministic. In particular, it is found that a ‘viscous sublayer’ joins surface water to a logarithmic ‘inertial sublayer’ below. The scaling rules that emerge from the theory allow the surface speed of the wind drift to reach nearly full development in a matter of minutes.

Numerical solutions are presented for laminar incompressible fluid flow past a rotating thin elliptic cylinder either in a medium at rest at infinity or in a parallel stream. The transient period from the abrupt start of the body to some later time (at which the flow may be steady or periodic) is studied by means of streamlines and equi-vorticity lines and by means of drag, lift and moment coefficients. For purely rotating cylinders oscillatory behaviour from a certain Reynolds number on is observed and explained. Rotating bodies in a parallel stream are studied for two cases: (i) when the vortex developing at the retreating edge of the thin ellipse is in front of the edge and (ii) when it is behind the edge.

The spectral energy transfer of turbulent velocity fields has been examined over a wide range of Reynolds numbers by experimental and empirical methods. Measurements in a high Reynolds number grid flow were used to calculate the energy transfer by the direct Fourier-transform method of Yeh & Van Atta. Measurements in a free jet were used to calculate energy transfer for a still higher Reynolds number. An empirical energy spectrum was used in conjunction with a local self-preservation approximation to estimate the energy transfer at Reynolds numbers beyond presently achievable experimental conditions.

Second-order spectra of the grid measurements are in excellent agreement with local isotropy down to low wavenumbers. For the first time, one-dimensional third-order spectra were used to test for local isotropy, and modest agreement with the theoretical conditions was observed over the range of wavenumbers which appear isotropic according to second-order criteria. Three-dimensional forms of the measured spectra were calculated, and the directly measured energy transfer was compared with the indirectly measured transfer using a local self-preservation model for energy decay. The good agreement between the direct and indirect measurements of energy transfer provides additional support for both the assumption of local isotropy and the assumption of self-preservation in high Reynolds number grid turbulence.

An empirical spectrum was constructed from analytical spectral forms of von Kármán and Pao and used to extrapolate energy transfer measurements at lower Reynolds number to Rλ = 105 with the assumption of local self preservation. The transfer spectrum at this Reynolds number has no wavenumber region of zero net spectral transfer despite three decades of $k^{-\frac{5}{3}}$. behaviour in the empirical energy spectrum. A criterion for the inertial subrange suggested by Lumley applied to the empirical transfer spectrum is in good agreement with the $k^{-\frac{5}{3}}$ range of the empirical energy spectrum.

The linearized stability of plane Couette flow is investigated here, without using the Orr–Sommerfeld equation. Rather, an unusual symmetry of the problem is exploited to obtain a complete set of modes for perturbations of the unbounded (no walls) flow. An explicit Green's function is constructed from these modes. The unbounded flow is shown to be rigorously stable. The bounded case (with walls) is investigated by using a ‘method of images’ with the unbounded Green's function; the stability problem in this form reduces to an algebraic characteristic equation (not a differential-equation eigenvalue problem), involving transcendental functions defined by integral representations.

Nonlinear oblique interactions between two slightly dispersive gravity waves (in particular, solitary waves) of dimensionless amplitudes α1 and α2 (relative to depth) and relative inclination 2ϕ (between wave normals) are classified as weak if sin2ϕ α1,2 or strong if ϕ2 = O(α1,2). Weak interactions permit superposition of the individual solutions of the Korteweg-de Vries equation in first approximation; the interaction term, which is O(α1α2), then is determined from these basic solutions.

Strong interactions are intrinsically nonlinear. It is shown that these interactions are phase-conserving (the sum of the phases of the incoming waves is equal to the sum of the phases of the outgoing waves) if |α2-α1 > (2ϕ)2 but not if |α2-α1| (2ϕ)2 (e.g. the reflexion problem, for which the interacting waves are images and α2 = α1). It also is shown that the interactions are singular, in the sense that regular incoming waves with sech2 profiles yield singular outgoing waves with - csch2 profiles, if \[ \psi_{-}< |\psi| < \psi_{+},\quad{\rm where}\quad\psi_{\pm}={\textstyle\frac{1}{2}}\left|(3\alpha_2)^{\frac{1}{2}}\pm(3\alpha_1)^{\frac{1}{2}}\right|. \]

Regular interactions appear to be impossible within this singular regime, and its end points, |ϕ| = ϕ±, are associated with resonant interactions.

Uniformly valid asymptotic solutions for linearized disturbances valid in a finite region containing the critical point u = c are developed using the method of multiple scales for a given isoenergetic basic flow. While uniformity of approximation will certainly improve the quantitative accuracy of a typical computation of stability characteristics of the flow, a major feature of such solutions is that they afford a more accurate description of the behaviour of the linearized disturbances in the neighbourhood of the critical point u = c.

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.

A new form is given for the general solution to the thin-shock-layer equations for the flow over a nearly plane delta wing. Using this, the solution described conjecturally by Hayes & Probstein for symmetrical flow with attached shock waves over a plane delta wing is realized numerically. The flow construction devised for this purpose is applied also to yawed flows. The solutions obtained are found to agree moderately well with the results of numerical calculations from the full equations, but contain a number of anomalous features characteristic of the thin-shock-layer approximation.

Turbulent entrainment at the density interface of a stable two-layer stratified fluid is studied in the laboratory, a constant surface stress being applied at the free surface. Conservation of mass requires that the overall Richardson number Ri = Dgδρ/ρu*2 is constant in each experiment, where D is the depth of the mixed layer, gδρ/ρ the buoyancy difference and u* the friction velocity. If the entrainment rate E = ue/u* is a function only of Ri, it is therefore constant in each experiment and can be measured with a greater accuracy than has previously been attained. The functional dependence of ue/u* on Ri is established over the range 30 < Ri < 1000; it is found not to follow any simple power law. The entrainment rates are considerably higher than those measured by Kato & Phillips (1969), for which the fluid below the mixed layer was linearly stratified. Such a condition allows internal gravity waves to be radiated downwards and the reduction in entrainment rate is consistent with that found by Linden (1975).

We derive the Rayleigh number RHS for thermal convection in a Hele-Shaw cell with gap width d and full width (gap plus walls) Y. For the state of marginal stability, the system of equations is found to be formally identical to that describing flow through a uniform porous medium, if d3/12Y is identified as the Hele-Shaw permeability. Thus Lapwood's (1948) thermal-instability analysis should apply, and the critical Rayleigh number should be 4π2 when the cell has impermeable isothermal boundaries.

Baker's (1966) pH-indicator method for visualizing fluid flow has been adapted for use in a Hele-Shaw cell. In addition to revealing the convection pattern clearly, this technique proves to be an especially sensitive detector of incipient flow, and a highly accurate means of verifying the onset of convection. Our experiments confirm that the critical Hele-Shaw Rayleigh number is 40 ± 2, thereby validating our theoretically derived expression for the Rayleigh number. We also measure the vertical flow velocity wm and find that wm∝ (R2HS−402)½ closely fits our data for 40 < RHS < 140.

Measurements of a separating two-dimensional incompressible boundary layer with an airfoil-type pressure distribution are reported. Unique mean and fluctuation velocity measurements and the distribution of the fraction of the time γp during which the flow moves downstream were obtained in the separated region using a directionally sensitive laser anemometer. Linearized hot-film anemometer measurements of mean velocities, turbulent shearing stress and intensities, eddy speeds, spectra and dissipation were made for γp > 0·8. The wall shearing stress, bursting frequencies, wall speed and spanwise structure were obtained using flush-surface hot-film sensors. The turbulent/non-turbulent interfacial intermittency γ and the frequency of passage of turbulent bulges were determined using smoke as a turbulence marker and the laser anemometer system for illumination and signal detection.

Upstream of separation the velocity profile correlations of Perry & Schofield (1973) are supported within the uncertainty of the data. Normal-stress effects are very important, influencing $-\overline{uv}/\overline{q^2} $ and the dissipation length correlations, and directly providing sizable terms in the momentum and turbulence energy equations. The criteria of Sandborn for turbulent separation and fully developed separation are found to hold. Downstream of separation there is apparent similarity of $\overline{u^2}$, U and γp throughout the shear flow. The passive low velocity backflow near the wall apparently just serves to satisfy continuity requirements after the energetic outer-region flow has deflected away from the wall upon separation.

The wall bursting frequency nA scales on outer velocity and length scales, with U∞/δnA ≈ 10, or about twice the value observed for zero-pressure-gradient flows. The non-dimensional spanwise spacing of wall eddies is given approximately by the relation λzUM/v ≈ 100 upstream of separation, where $U_M = (- \overline{uv}_{\max})^{\frac{1}{2}}$. The speed of wall eddies is found to be about 14Uτ.

Resonant (phase-locked) interactions among three obliquely oriented solitary waves are studied. It is shown that such interactions are associated with the parametric end points of the singular regime for interactions between two solitary waves. The latter include regular reflexion at a rigid wall, which is impossible for ϕi < (3α)½ (ϕ = angle of incidence, α = amplitude/depth [Lt ] 1), and it is shown that the observed phenomenon of ‘Mach reflexion’ can be described as a resonant interaction in this regime. The run-up at the wall is calculated as a function of ϕi/(3α)½ and is found to have a maximum value of 4αd for ϕi = (3α)½. This same resonant interaction also describes diffraction of a solitary wave at a corner of internal angle π − ψi, −(3α)½, and suggests that a solitary wave cannot turn through an angle in excess of (3α)½ at a convex corner without separating or otherwise losing its identity.

Self-diffusion coefficients were determined experimentally for lateral dispersion of spherical and disk-like particles in linear shear flow of a slurry at very low Reynolds number. Using a concentric-cylinder Couette apparatus, recurrent observations were made of the lateral position of a particular radioactively labelled particle. The self-diffusion coefficient D was calculated by means of random-walk theory, using the ergodic hypothesis. Owing to great experimental difficulties, the calculated values of D are not of high accuracy, but are correct to within a factor of two. In the range 0 < ϕ < 0·2, D/a2ω increases from zero linearly with ϕ up to D/a2ω ≈ 0.02 (where ϕ = volumetric concentration of particles, a = particle radius, ω = mean shear rate of suspending fluid). In the range 0.2 < ϕ < 0.5, the trend of D/a2ω is not clear because of experimental scatter, but in this range D/a2ω ≈ 0·025 to within a factor of two. Within the experimental accuracy, spheres and disks have the same value of D/a2ω.

An investigation of the existence of purely thermal intermittency has been undertaken for the flow produced by a step change in temperature in a decaying homogeneous turbulent field. The results show that a thermal interface can be defined and that it is a random function of space and time, having gross characteristics similar to the turbulent interface in a free shear flow. A technique for identifying the hot and cold regions of the fluid in a systematic way was developed and this permitted conditional averages of the mean and fluctuating temperature to be formed.

A two-dimensional configuration has been investigated in which air flows through a convergent nozzle and expands abruptly into a rectangular duct of larger cross-section which terminates in a plenum chamber. Three different types of oscillation have been observed in the downstream duct. At low plenumchamber pressures an oscillation occurs towards the exit of the duct as the boundary layer of the flow becomes alternately separated and attached. At increasing plenum pressure a shock-pattern oscillation takes place in which a change from a normal shock to oblique shocks occurs during a cycle. At still greater plenum pressures a base-pressure oscillation occurs which influences the entire duct flow downstream of the abrupt change in cross-section. The amplitudes of the oscillation can be as high as 10% of the rest state, and the frequency of the base-pressure oscillations can be predicted approximately from one-dimensional gasdynamic theory.

The unsteady duct phenomena have been studied by synchronizing instantaneous pressures measured by quartz pressure transducers with interferograms obtained with a Mach–Zehnder interferometer.

The flow in a square cavity is studied by solving the full Navier–Stokes and energy equations numerically, employing finite-difference techniques. Solutions are obtained over a wide range of Reynolds numbers from 0 to 50000. The solutions show that only at very high Reynolds numbers (Re [ges ] 30000) does the flow in the cavity completely correspond to that assumed by Batchelor's model for separated flows. The flow and thermal fields at such high Reynolds numbers clearly exhibit a boundary-layer character. For the first time, it is demonstrated that the downstream secondary eddy grows and decays in a manner similar to the upstream one. The upstream and downstream secondary eddies remain completely viscous throughout the range of Reynolds numbers of their existence. It is suggested that the behaviour of the secondary eddies may be characteristic of internal separated flows.