The estimation of the growth of propagating instability waves in laminar boundary layers is considered when the Reynolds number is sufficiently large for the mean flow to deviate only slightly from a truly parallel flow. An approximate solution for the linear perturbation is sought in the form of a scaled solution of the related locally parallel flow problem. The amplitude scaling is chosen so as to satisfy the full linearized perturbation equations as closely as possible by making the mean-square deviation of the remainder a minimum. By re-arranging the terms in the equations so that some of the small correction terms arising from the non-parallel mean flow are contained in the ordinary differential equation (ODE) defining the quasi-parallel flow solution, a useful simplification is obtained for the scaling function. Then a modified Orr–Sommerfeld equation defines the base solution and the differential expression for the scaling that can be integrated forms a simple conservation relation.

Experiments have established that an axisymmetric sharp trailing edge of a gas bubble in a viscoelastic fluid can develop into a three-dimensional ‘knifelike’ shape under certain conditions (high capillary number, large bubble size). A numerical study is conducted to discover the physics of this phenomenon. The axisymmetric deformation of a bubble rising buoyantly in a viscoelastic fluid is simulated by solving the axisymmetric flow equations coupled with the constitutive equations of the finitely extensible nonlinear elastic Chilcott–Rallison (FENE-CR) model. The three-dimensional temporal linear stability analysis of this axisymmetric base state is carried out. The dominant eigenvalue which is indicative of the growth rate of the perturbations is computed. The only unstable eigenmode has azimuthal wavenumber m equal to 2. The corresponding eigenfunction shows that indeed a sharp axisymmetric tail develops a knife-edge form. A further investigation of the energy budget of the disturbances for m = 2 is performed to determine the production and dissipation terms affecting the growth of this instability. It is shown that the normal gradient of the base-state pressure along the free surface plays an important role in the evolution of the instability.

This study of the boundary layer of steady, incompressible, plane, crossed-fields m.h.d. flow at large Reynolds number Re and magnetic Reynolds number Rm begins with a review of Hartmann's case, where a boundary layer occurs whose thickness is proportional to (Re Rm)−½. Following this clue, it is shown that in general the boundary layer is a ‘local Hartmann boundary layer’. Its profiles are always exponential and it is determined completely by local quantities. The skin friction and the total electric current in the layer are proportional to the square root of the magnetic Prandtl number, i.e. to (Rm/Re)½. Thus the exterior-flow problem, the solution of which precedes a boundary-layer solution, generally involves a current sheet at the fluid-solid interface.

This inviscid-flow problem becomes tractable if (Rm/Re)½ is small enough to permit a linearized solution. The flow field about a flat plate at zero incidence is calculated in this approximation. It is pointed out that the thin-cylinder solutions of Sears & Resler (1959), which pertain to Rm/Re = 0, can immediately be extended to small, non-zero values of this parameter by linear combination with this flat-plate solution.

The directional spectrum of wind-driven surface waves has been measured under conditions of limited fetch, in order to check the predictions of the Phillips–Miles theory of wave generation (Miles 1960). The expression obtained for the directional spectrum in this theory involves the spectrum of the atmospheric pressure fluctuations, but it is possible to obtain theoretical estimates of the major features of the directional spectrum without knowledge of the pressures. Specifically, it is possible to predict the frequency at which the power spectrum should peak, and, for the higher frequencies, the range of azimuth over which high spectral values should be observed; for the lower frequencies the theory indicates a bimodal distribution in azimuth (Phillips's resonance waves), and gives the angle of travel relative to the wind as a function of frequency.

The results of the measurements are in fairly good agreement with the theoretical predictions for the higher frequencies. The asymmetry of the fetch results in the prediction that the waves will travel at an angle to the wind which varies with frequency, and this was observed. The range of azimuth over which the spectral density is high is also close to the theoretical prediction. For the low frequencies the bimodal distribution was not observed: the waves were found to have a single predominant direction of travel at each frequency. However, this direction conformed closely to that of one of the two wave trains predicted by Phillips, and its variation with frequency was also that given by the theory. There is reason to suppose that the peculiarities of the experimental site may be responsible for the absence of the second wave train, especially as it would be difficult to account for the observed effects on any basis other than that of Phillips's theory.

In this paper we take up the question of the spectral stability of travelling water waves from a new point of view, namely that the spectral data of the water-wave operator linearized about fully nonlinear Stokes waves is analytic as a function of a height parameter. This observation was recently made rigorous by the author using a boundary perturbation approach which is amenable to approximation by a stable high-order numerical method. Using this algorithm, we investigate, for both super- and sub-harmonic disturbances, the evolution of the spectrum, in particular the ‘first collision’ of eigenvalues and the ‘smallest singularity’ in the perturbation expansion. The former is studied in response to MacKay & Saffman's (1986) work on the water-wave problem which demonstrated that instability can only arise after the collision of two eigenvalues of opposite Krein signature. However, we present results which show, quite explicitly, that eigenvalue collision (even of opposite Krein signature) is insufficient to conclude instability. With this in mind, we have identified a new criterion for the loss of spectral stability, namely the appearance of a singularity in the expansion of the spectral data (as a function of the height parameter mentioned above). We give some heuristic reasons why this should be so, and then provide complete numerical spectral stability results for four representative depths, two above (h = ∞, 2) and two below (h = 1, 1/2) Benjamin's (1967) critical value, hc ≈ 1.363, above which the Benjamin–Feir instability emerges. We find that the strongest (two-dimensional) instability appears to be among the long waves, but we notice that there is a sharp difference between ‘shallow-water’ and ‘deep-water’ waves in that first eigenvalue collision and smallest expansion singularity are synonymous for shallow water, while this is not so in deep water where ‘windows of stability’ beyond the first eigenvalue collision exist.

Integral relations are derived for steady, incompressible recirculating motions with small viscous forces. The circuit time of a fluid particle on a closed streamline in steady, inviscid flow is shown to be the same for all the closed streamlines on a surface of constant total head.

The discontinuities of velocity and velocity gradient that occur in the motion of inviscid fluid filling a closed, rotating cylinder set in a rotating support with the two rotation axes slightly misaligned are then investigated.

The instability of viscous capillary jets subject to disturbances consisting of two superposed wavenumbers, and for large disturbance amplitudes is investigated. Disturbances composed of the superposition of a fundamental disturbance (first harmonic) with either its second or third harmonic are used. The influence of the wavenumber of the fundamental disturbance on the jet breakup is studied for a disturbance composed of a first harmonic with an initial non-dimensional amplitude of ε1 = 0.01 and a second harmonic with an initial non-dimensional amplitude of ε2 = 0.05. The influence of the initial amplitudes of the first and second harmonics on the jet breakup is studied for two non-dimensional wavenumbers of the fundamental (first harmonic): k = 0.45 and k = 0.7; the second harmonic is unstable in the former and stable in the latter case. The effect of an added third harmonic is studied only for k = 0.45 but for a wide range of initial amplitudes. All cases are studied for an in-phase and a 180° out-of-phase superposition of the two waves. The nonlinear interaction between the two waves results in the formation of a variety of drop sizes and shapes. The breakup times can be controlled within a wide range using this technique.

In part 2 (Brown & Stewartson 1982) of this paper we set out the linearized solution of the critical layer that is expected to hold some time after the forced internal gravity wave of small amplitude e is initiated at an infinite distance above the shear layer. This differed from that presented in part 1 (Brown & Stewartson 1980) in that in part 2 we exploited to a much greater extent the fact that the Richardson number J was large and obtained the solution in a form consisting of explicit functions rather than infinite integrals. Also it was demonstrated that at times t = O(1) the wave did not penetrate beyond a certain level in the critical layer, and that critical-layer noise only was created above this line and transmitted below it. In this paper we examine the development of this solution on a longer time scale $\tau (\propto \epsilon^{\frac{2}{3}}t)$ and show how the reflection and transmission coefficients which are exponentially small, \[ O\{\exp(-(J-\frac{1}{4})^{\frac{1}{2}}\pi)\}, \] when τ = 0 increase with time. As in part 1 we obtain a reflection coefficient for the first harmonic that is O(τ3), and because of the simpler formulation of the linearized solution are able to obtain a reflected second harmonic. These harmonics appear as complementary functions that are induced by singularities in the particular integrals of the equations. It is shown that the interaction between the initial-noise term in the lower part of the critical layer and an induced-noise term at the sixth stage of the expansion will eventually lead to a transmitted wave. This appears at the ninth stage of the expansion and its transmission coefficient is O(τ12) though it is not explicitly calculated here.

We present a theory for the decay of the relative motion of a homogenous fluid with a free surface in a rotating cylindrical tank with a flat bottom, induced by an abrupt change in the angular velocity. We then describe a set of laboratory experiments designed to test the predictions of the theory.

At low rates of rotation the dynamics of the adjustment is well understood and measurements have verified the established theoretical results that the motion decays exponentially, with a timescale proportional to the rotation period divided by the square root of the Ekman number, and that the relative vorticity remains independent of radius. At higher rotation rates, however, the curvature and motion of the free surface complicate the dynamics, and have hindered the development of a more general theory.

Both the theoretical predictions and the experiments show that at high rotation rates the decay of the relative vorticity is independent of radius and exponential in time, but with a decay timescale, τe, that increases linearly with the rotational Froude number F, i.e. $\tau_{\rm e} = 1+\frac{1}{16} F$. An analysis of the vorticity dynamics during spin-up indicates that, near the centre of the tank, this simple behaviour is the result of vigorous competition between the rate of vortex line stretching by Ekman-layer pumping and surface deformation. Near the boundary, these mechanisms cooperate, but are partially offset by the stretching produced by the secondary radial circulation.

Systematic measurements of the surface drift current, the wind profile over the water surface and the wave spectra have been made for (i) pure wind-waves, (ii) a coexisting system of wind-waves and swell propagating against the wind, and (iii) a coexisting system of wind-waves and swell propagating in the direction of the wind. The surface drift current is gradually intensified by the swell propagating against the wind when the swell steepness increases. The maximum increase of the surface drift velocity caused by the opposing swell is about 46 % of the surface drift velocity for pure wind-waves at the same wind speed. Such a phenomenon was not observed when the swell was propagating in the direction of the wind.

The cubic Schrödinger equation, which governs the complex amplitude of a cross-wave that is subharmonically excited by a symmetric wavemaker in a deep wave tank (Jones 1984), is solved for prescribed boundary conditions at the wavemaker and a null condition at infinity. This solution (which is a limiting case of a one-parameter family of cnoidal waves) typically describes a trapped wave that decays exponentially in a semi-infinite tank, although it may be fitted to a tank of finite length for a particular combination of parameters. It is closely related to the solution for trapped waves excited through vertical oscillation of a long channel (Wu, Keolian & Rudnick 1984).

The present investigation deals with the development stage (x/d = 20–100) of the porous-body wake, generated by a mesh strip. Pattern-recognition analysis was applied to the velocity signals from hot-wire measurements in the vertical centreplane and various horizontal planes in the flow field. Spanwise coherent structures (with ωz) and lateral coherent structures (ωy) have been identified. An examination of the fine-scale turbulence is used to show that the two kinds of structures are indeed connected. In the near region of the wake, the lateral structures are ‘ribs’ attached to the spanwise vortices. In the intermediate region, the interaction of the two structures changes the original spanwise vortices into three-dimensional structures. Beyond this zone, in the far region, the’ legs’ of the original spanwise structure merge with the’ ribs’, and it can be inferred that the coherent structures within the flow are essentially randomly distributed hairpin-like vortices.

A stably stratified interface, with strong turbulence below and quiescent air above, is studied in a wind tunnel with the aim of simulating the conditions at the inversion cap at the top of the atmospheric boundary layer. The interfacial layer was generated by means of a composite grid, with small mesh size above and a large one below (Veeravalli & Warhaft 1989). A temperature step generated in the plenum of the wind tunnel, was located at the centre of the layer. There is no shear and thus turbulence interactions, usually masked by turbulent production in traditional mixing layers, are highlighted. Close to the grid where the velocity fluctuations are strong, buoyancy effects are insignificant, but as the turbulence decays they become dominant. The bulk Richardson number, N2B/(〈u2〉2/L2u), where NB is the Brunt—Väisälä frequency across the layer, and 〈u2〉2 and Lu are the velocity variance and integral lengthscale, respectively, of the turbulence on the lower side of the layer, varied from approximately zero close to the grid to 80 far downstream. The stratification inhibited the turbulent penetration into the layer, reducing the high skewness and kurtosis of the velocity field for the neutral case, to Gaussian values. The layer, which initially thickened with downstream distance, thinned when buoyancy became pronounced, owing to the collapse of the heat flux. Significant regions of countergradient heat flux, and reversals in sign of the triple moment transport terms were observed in the upper part of the layer. An analysis of the value of the heat flux conditioned on the temperature fluctuations, showed that the large temperature fluctuations associated with weak turbulence became affected by stratification first. Cospectral analysis shows that these fluctuations are associated with large scales. We also show that although the joint normal approximation between velocity and temperature fluctuations is sound for a passive scalar field, it becomes less good with the onset of stratification, failing completely when the stratification is strong.

The hydrodynamic stability of the developing laminar flow in the entrance region of a parallel-plate channel is investigated using the theory of small disturbances. The stability of the fully developed flow is also re-examined. A wide range of analytical (i.e. asymptotic) and numerical methods are employed in the stability investigation. Among the asymptotic methods, each of three viscous solutions (singular, regular and composite) is used along with the inviscid solution to provide critical Reynolds numbers and complete neutral stability curves. Two numerical methods, finite differences and stepwise integration, are applied to calculate critical Reynolds numbers. The basic flow in the development region is treated from two stand-points: as a channel velocity profile and as a boundary-layer velocity profile. Extensive comparisons among the various methods and flow models disclose their various strengths and ranges of applicability. As a general result, it is found that the critical Reynolds number decreases monotonically with increasing distance from the channel entrance, approaching the fully developed value as a limit.

Convection of two-dimensional rolls in an infinite horizontal layer of fluid-saturated porous medium heated from below is studied numerically. Several important finite-amplitude states are isolated, and their bifurcation properties are shown. Effects of the temperature-dependent viscosity are included. The stability of these states is investigated with respect to the class of disturbances that have a ½π phase shift relative to the basic state. In particular, the oscillatory mechanism and the mean-flow generating mechanism through the variable viscosity are discussed.

A study of two-layer quasi-geostrophic vortex flow is performed to determine the effect of a current difference between the layers on a vortex initially extending through both layers. In particular, the conditions under which the vortex can resist being torn by the current difference are examined. The vortex evolution is determined using versions of the contour dynamics and discrete vortex methods which are modified for two-layer quasi-geostrophic flows. The vortex response is found to depend upon the way in which the current difference between the layers is maintained. In the first set of flows studied, the current difference is generated by a (stronger) third vortex in the upper layer located at a large distance from the (weaker) vortex under study. Flows of this type are important for understanding the interactions of vortices of different sizes in geophysical turbulence. A set of flows is also considered in which an ambient geostrophic current difference is produced by a non-uniform background potential vorticity field. In this case, an additional (secondary) flow field about the vortex patch in each layer is generated by redistribution of the ambient potential vorticity field.

It is found that a vortex that initially extends through both layers will undergo a periodic motion, in which the two parts of the initial vortex in the different layers (called the ‘upper’ and ‘lower’ vortices) oscillate about each other, provided that the current difference between the layers is less than a critical value. When the current difference exceeds this critical value, the upper and lower vortices separate permanently and the initial vortex is said to ‘tear’. The effects of various dimensionless parameters that characterize the flow are considered, including the ratio of core radius to internal Rossby radius, the ratio of layer depths and the ratio of the strengths of the upper and lower vortices. These parameters affect both the critical current difference for tearing and the deformation of the vortex cores by their interaction. It is found that for small values of inverse internal Rossby deformation radius, calculations with circular non-deformable vortices (convected at their centrepoints) give results in good agreement with the contour dynamics simulations, since the vortex deformation is small. The results of the study will be useful in determining the conditions under which large geophysical vortex structures, such as cyclones and ocean rings, can extend to large heights (depths) even though the mean winds (currents) in the ambient flow change significantly along the vortex length.

Mean flow and turbulence measurements have been made in three-dimensional turbulent boundary layers in curved ducts, simulating adverse pressure gradients on two ‘infinite’ swept curved wing surfaces with concave and convex curvature respectively. The ratio of the initial boundary-layer thickness to the surface radius of curvature in both cases is approximately 0.01, the value used in the earlier two-dimensional turbulent boundary-layer studies on the effects of concave and convex curvature by Hoffmann, Muck & Bradshaw (1985) and Muck, Hoffmann & Bradshaw (1985) respectively. The pressure-driven crossflow has nearly the same streamwise distribution as in the ‘infinite’ swept flat-surface experiment of Bradshaw & Pontikos (1985), which used a similar duct. The results of the present study show that the coupled effects of mean flow three-dimensionality and prolonged mild surface curvature of either sign have rather a weak influence on the turbulence structure, unlike the significant influence of the above extra strain rates when applied individually. In the concave case, the effect of the crossflow appears to oppose the destabilizing effect of curvature in addition to suppressing spanwise wavy inhomogeneities In contrast, the weak combined influence of convex curvature and crossflow, both of which, separately, tend to attenuate turbulence, implies that the interaction between the two effects is grossly nonlinear. Implications of the present results for turbulence modelling are briefly discussed.

Application of Landau's ideas to the theory of weakly nonlinear instabilities shows that the amplitude of the unstable modes behaves as the square root of the reduced control parameter ε, its critical value being ε = 0. When applied to cellular structures the theory has been improved by taking into account the slow spatial variations of the amplitude and phase of the unstable modes. Until now the case of thermo-convective instabilities in high vertical channels has not been studied using this approach. In high vertical structures the nonlinear terms disappear in the limit of an infinite height, and the supercritical behaviour requires a specific treatment. It differs from the standard analysis valid for the case of fluid layers of infinite horizontal extent, where the nonlinearities and the finite-size effects are disconnected. In the limit of high aspect ratios (height [Gt ] horizontal extent) we have derived an amplitude equation for convective systems where the nonlinear terms contain derivatives at the lowest order. As a consequence the amplitude equation cannot be put into a variational form and the stability of the stationary solutions cannot be deduced from an ordering in decreasing values of a Lyapunov functional.