An electrically conducting Boussinesq fluid is confined between two rigid horizontal planes. The fluid is permeated by a strong uniform horizontal magnetic field and the entire system rotates rapidly about a vertical axis. By heating the fluid from below and cooling it from above the system becomes unstable to small perturbations when the adverse temperature gradient becomes sufficiently large. Attention is restricted to small values of the Ekman number E and the ratio q of the thermal and magnetic diffusivities (see (1.2) and (1.3) below). In this parameter range marginal convection is steady and its character depends on the relative sizes of the Coriolis and Lorentz forces as measured by the parameter λ (see (1.1) below). When λ [ges ] 2/3½, motion consists of a single roll, whose axis is perpendicular to the applied magnetic field. On the other hand, when λ < 2/3½, two distinct rolls are possible: the axis of each roll lies oblique but with equal angle to the applied magnetic field. Only the latter case is discussed here.

Once the Rayleigh number R exceeds its critical value Rc only one of the two sets of single rolls remains stable, while its squared amplitude increases linearly with R – Rc. For certain values of the parameters λ and τ (see (1.6) below) a second critical value may be isolated at which the system becomes unstable to unidirectional geostrophic flow perturbations aligned with the applied magnetic field. The instability sets in as either a steady or oscillatory shear flow dependent on the values taken by λ and τ. In both cases, after the secondary instability sets in, the roll amplitude remains largely insensitive to further increase in the Rayleigh number with the consequence that the geostrophic flow is stabilized. The amplitude of the shear, on the other hand, increases with R, adjusting its magnitude to ensure stability of the convection rolls.

A rivulet is a narrow stream of liquid located on a solid surface and sharing a curved interface with the surrounding gas. Capillary instabilities are investigated by a linearized stability theory. The formulation is for small, static rivulets whose contact (common or three-phase) lines (i) are fixed, (ii) move but have fixed contact angles or (iii) move but have contact angles smooth functions of contact-line speeds. The linearized stability equations are converted to a disturbance kinetic-energy balance showing that the disturbance response exactly satisfies a damped linear harmonic-oscillator equation. The ‘damping coefficient’ contains the bulk viscous dissipation, the effect of slip along the solid and all dynamic effects that arise in contact-line condition (iii). The ‘spring constant’, whose sign determines stability or instability in the system, incorporates the interfacial area changes and is identical in cases (ii) and (iii). Thus, for small disturbances changes in contact angle with contact-line speed constitute a purely dissipative process. All the above results are independent of slip model at the liquid–solid interface as long as a certain integral inequality holds. Finally, sufficient conditions for stability are obtained in all cases (i), (ii) and (iii).

Crow & Champagne (1971), Bechert & Pfizenmaier (1975) and Moore (1977) have observed that the growth, mixing and noise production of jet flows are sensitive to harmonic forcing. This paper describes an attempt to model numerically certain features of these flows. The model flow is restricted to be axisymmetric and is consequently unrepresentative of the detailed structure in a real jet. Nonetheless, it is found that reasonable qualitative agreement exists between the results of the model and experiments as far as the large eddies are concerned. This suggests that a substantial part of the large-scale structure in a jet is essentially axisymmetric. Harmonic excitation is also applied to the model jet and the changes in frequency and amplitude of the excitation cause distinct changes in the wavelengths of the jet eddies. This resulting large-eddy behaviour is consistent with many features of the nonlinear behaviour observed experimentally in forced jets.

Measurements of mean-pressure distributions and boundary-layer development on rough-walled circular cylinders in a uniform stream are described. Five sizes of distributed sandpaper roughness have been tested over the Reynolds-number range 7 × 104 to 5·5 × 105. The results are examined together with those of previous investigators, and the observed roughness effects are discussed in the light of boundary-layer theory. It is found that there is a significant influence of surface roughness on the mean-pressure distribution even at very large Reynolds numbers. This observation is supported by an extension of the Stratford–Townsend theory of turbulent boundary-layer separation to the case of circular cylinders with distributed roughness. The pressure rise to separation is shown to be closely related, as expected, to the characteristics of the boundary layer, smaller pressure rises being associated with thicker boundary layers with greater momentum deficits. Larger roughness gives rise to a thicker and more retarded boundary layer which separates earlier and with a smaller pressure recovery.

Results are presented of a simplified description of the seasonal and spring–neap tidal dependence of the axial dispersion coefficients in the Severn Estuary. The coefficients are derived from salt budget calculations, which are based on 29 sets of observations of the axial salinity distributions in the estuary during 1971–1976. Regression analyses of the salinity distributions determine simple linear and logarithmic relationships for the dispersion coefficients in terms of tidal range and the total rate of input of fresh water to the estuary, the appropriate averaging periods for the freshwater inputs being computed as part of the analyses.

The results show that the coefficients generally increase with increasing run-off, and, away from the mouth, depend upon the tidal range, showing a small decrease with increasing tidal range in the seaward part of the estuary, and a large increase with increasing tidal range towards the head. The yearly averaged coefficients lie in the order-of-magnitude range 102–103 m2s−1, with the larger values occurring near the head.

The computed dispersion coefficients are applied to calculations of the seasonal dependence of residence times, the results being expressed as functions of axial distance along the estuary. It is shown that the residence time of the whole estuary varies from roughly 100 days during winter conditions to 200 days during summer conditions, and that the residence time landward of a particular section decreases rapidly as the section taken approaches the head of the estuary.

Flow induced by a body moving near a plane wall is analysed on the assumption that the normal distance from the wall of every point of the body is small compared to the body length. The flow is irrotational except for the vortex sheet representing the wake. The gap-flow problem in the case of unsteady motion is reduced to a nonlinear first-order ordinary differential equation in the time variable. In the special case of steady flow, some known results are recovered and generalized. As an illustration of the unsteady theory, the problem is solved of a flat plate falling toward the ground under its own weight, while moving forward at uniform speed.

It is shown that all parallel inviscid shear flows of constant density are unstable to a wide class of initial infinitesimal three-dimensional disturbances in the sense that, according to linear theory, the kinetic energy of the disturbance will grow at least as fast as linearly in time. This can occur even when the disturbance velocities are bounded, because the streamwise length of the disturbed region grows linearly with time. This finding may have implications for the observed tendency of turbulent shear flows to develop a longitudinal streaky structure.

An alternative to Hunt's (1973) extension of classical rapid distortion theory is used to calculate the turbulence downstream of a rapid contraction. This problem was originally studied by Batchelor & Proudman (1954) and Ribner & Tucker (1953), but their analyses were restricted to flows in which the characteristic turbulence scales were small compared to the spatial scales of the mean flow (usually the characteristic dimension of the apparatus). We now consider the case where the turbulence scale can have the same magnitude as the mean-flow spatial scale. Relatively simple formulae are obtained by calculating the turbulence only in the downstream region where the mean flow is no longer affected by the potential field of the contraction.

The results are then further simplified by assuming that the contraction is large and expanding in inverse powers of the contraction ratio. The calculations show that effects of finite turbulence scale can be quite significant. We also obtain some important new results for small-scale turbulence by expanding the solutions in inverse powers of the turbulence spatial scale.

A series of LDA measurements and visual observations of confined turbulent vortex flow are described. The experiments were performed with water as the fluid medium in a vortex tube of length-to-diameter ratio L/D = 3.8 for a range of exit diameters De between De/D = 1 and 0.18. The experiments reveal a remarkable change in the vortex structure as De is reduced: from a thick core with an axial-velocity defect in the centre, and even reversed flow, to a thin annular jet-like core with a peak axial velocity more than an order of magnitude greater than the average value and again a central velocity deficit. The corresponding swirl profiles are not remarkable and are well-represented under all conditions by the solution of Burgers (1948), albeit with a velocity maximum which is strongly dependent upon De.

Considered below is the plane, high-Reynolds-number (Re), flow of an incompressible fluid through a channel suffering a severe non-symmetric constriction, ‘severe’ meaning ‘of typical dimensions comparable with the channel width a*’. The (mainly inviscid) flow description is governed by free streamline theory, to be consistent with the viscous incompressible separation from the constriction surface and with a relatively slow eddy motion beyond. Separation also occurs far ahead of the constriction, at a distance $O(Re^{\frac{1}{7}} a^{*})$ upstream. Attention is given primarily to the flow features for a particular class of slowly varying severe constrictions, from which however the features for the probably more useful classes of slender (i.e. of large length but O(a*) height) and moderately severe (i.e. of O(a*) length but small height) constriction follow, as do those for curved or cornered channel flows. In all these cases a long-scale flow response is induced upstream and downstream and in many cases remarkably simple universal formulae for the separation and reattachment positions result. The corresponding O(a*) corrections to the upstream separation distance above are also derived.

We consider a head-on collision between two solitary waves on the surface of an inviscid homogeneous fluid. A perturbation method which in principle can generate an asymptotic series of all orders, is used to calculate the effects of the collision. We find that the waves emerging from (i.e. long after) the collision preserve their original identities to the third order of accuracy we have calculated. However a collision does leave imprints on the colliding waves with phase shifts and shedding of secondary waves. Each secondary wave group trails behind its primary, a solitary wave. The amplitude of the wave group diminishes in time because of dispersion. We have also calculated the maximum run-up amplitude of two colliding waves. The result checks with existing experiments.

In many polymer processing applications, the generation of heat by viscous losses in the flowing molten polymer is highly significant. The heating reduces the viscosity of the melt sharply and the flow patterns are different from the isothermal case. In this paper, a finite element scheme based on the Galerkin method is developed and is used to explore the effects of thermally induced property changes in extrusion.

In the program we solve simultaneously for the flow and temperature fields at each iteration. To check the program for accuracy and correctness, some simple problems were first attempted. A solution for viscous heating in Poiseuille flow was used to check the variable-viscosity part of the program. The crucial convection (‘radiation’) boundary condition was checked using the solution for cooling of a moving rod. Finally, the swelling of extruded jets with self-heating was investigated. A new phenomenon, thermal extrudate swell, was thereby discovered. We have found extrudate expansions up to 70% of the die diameter in a Newtonian fluid with thermal properties similar to those of low density polyethylene. It is clear that this phenomenon will affect many experimental interpretations of extrudate swelling.

A three-dimensional ‘turbulent spot’ has been induced in the axisymmetric free mixing layer of a 12.7 cm diameter air jet by a spark generated at the nozzle boundary layer upstream of the exit. The spot coherent-structure signature, buried in the large-amplitude random fluctuating signal, has been educed at three downstream stations within the apparent self-preserving region of the mixing layer (i.e. x/D = 1.5, 3.0 and 4.5) at the jet exit speed of 20 ms−1. The eduction has been performed through digital phase averaging of the spot signature from 200 realizations. In order to reduce the effect of the turbulence-induced jitter on the phase average, individual filtered signal arrays were optimally time-aligned through an iterative process of cross-correlation of each realization with the ensemble average. Further signal enhancement was achieved through rejection of realizations requiring excessive time shifts for alignment. The number of iterations required and the fraction of realizations rejected progressively increase with the downstream distance and the radial position.

The mixing-layer spot is a large-scale elongated structure spanning the entire width of the layer but does not appear to exhibit a self-similar shape. The dynamics of the mixing-layer spot and its eduction are more complicated than those of the boundary-layer spot. The spot initially moves downstream essentially at a uniform speed across the mixing layer, but further downstream it accelerates on the high-speed side and decelerates on the low-speed side. This paper discusses the data acquisition and processing techniques and the results based on the streamwise velocity signals. Phase average distributions of vorticity, pseudo-streamlines, coherent and background Reynolds stresses and further dynamics of the spot are presented in part 2 (Hussain, Kleis & Sokolov 1980).

The stability boundary of a density-stratified fluid contained in an inclined slot subjected to a lateral temperature gradient was determined experimentally. The initial stratification due to salt was stable and linear in the vertical direction. Experiments were conducted in a 1·0 × 11.1 × 25·7 cm slot with the inclination angle θ from the vertical varying from −75° to + 75°. A positive angle denotes heating from the lower wall while a negative angle denotes heating from the upper wall. The temperature difference across the slot was increased slowly until the onset of instability was observed by means of a shadowgraph. The critical thermal Rayleigh number was found to be non-symmetrical with respect to θ = 0°, with heating of the upper wall less stable than heating of the lower wall. This is because there is a larger vertical solute gradient in the steady-state regime prior to the onset of instabilities when the lower wall is heated. The secondary flow consisting of horizontal convecting layers was very stable θ < 0° cases because of the stabilizing temperature effect. The motion of the layers when θ > 0° was quite vigorous. At θ = +75°, the secondary flow became unstable in a rather dramatic manner not observed heretofore.

Correction of an error in the matrix elements used by Salwen & Grosch (1972) has brought the results of the matrix-eigenvalue calculation of the linear stability of Hagen–Poiseuille flow into complete agreement with the numerical integration results of Lessen, Sadler & Liu (1968) for azimuthal index n = 1. The n = 0 results were unaffected by the error and the effect of the error for n > 1 is smaller than for n = 1. The new calculations confirm the conclusion that the flow is stable to infinitesimal disturbances.

Further calculations have led to the discovery of a degeneracy at Reynolds number R = 61·452 ± 0·003 and wavenumber α = 0·9874 ± 0·0001, where the second and third eigenmodes have equal complex wave speeds. The variation of wave speed for these two modes has been studied in the vicinity of the degeneracy and shows similarities to the behaviour near the degeneracies found by Cotton and Salwen (see Cotton 1977) for rotating Hagen-Poiseuille flow. Finally, new results are given for n = 10 and 30; the n = 1 results are extended to R = 106; and new results are presented for the variation of the wave speed with αR at high Reynolds number. The high-R results confirm both Burridge & Drazin's (1969) slow-mode approximation and more recent fast-mode results of Burridge.

Energy stability theory has been formulated for two-dimensional buoyancy–thermocapillary convection in a layer with a free surface. The theory yields a critical Rayleigh number RE for which R < RE is a sufficient condition for stability of the layer. RE emerges from the variational formulation as an eigenvalue of a nonlinear system of Euler–Lagrange equations. For the case of small capillary number (large mean surface tension) explicit values are obtained for RE. The analogous linear-theory results for this case are obtained in terms of a critical Rayleigh number RL. These are compared. It is found that the existence of the deformable interface can lead to a stabilization relative to the case of a planar interface. This result is explained in physical terms. The energy theory is then generalized to include general flow problems having three-dimensional disturbances, non-Newtonian bulk fluids and general interfacial mechanics such as surface viscosity and elasticity.

The mechanics of a spark-induced coherent structure (called a ‘spot’) in the turbulent mixing layer of a 12.7 cm diameter incompressible air jet has been investigated through phase-locked measurements at three streamwise stations. Phase averages have been obtained from 200 realizations of X-wire (time-series) data after these are optimally time-aligned with respect to one another through an iterative process of maximization of cross-correlation of individual realizations with the ensemble average. Realizations that are grossly out of alignment owing to turbulence-induced distortions have been rejected; the rejection ratio increases with increasing radial position. Data include phase-average time series of background turbulence intensities, coherent and background Reynolds stresses, vorticity and intermittency at different transverse positions. Spatial distributions of these properties over the extent of the spot have been presented as contour maps. The computed pseudo-stream-functions have been compared with the phase-average streamlines inferred from the measured distributions of the velocity vector. Comparison with the phase-average intermittency contours show that the pseudo-stream-functions are reliable and, even though the integration involved produces smoothed-out stream functions, are most useful in deducing the structure dynamics and its convection velocity.

The spark-induced spot is an elongated large-scale coherent vortical structure spanning the entire thickness of the mixing layer, which moves downstream at a convection velocity of about 0.68Ue. The dynamics of the turbulent mixing layer spot, whose signature is buried in the large-amplitude background fluctuations, is much more complicated than that of the boundary-layer spot. The spot transports jet-core fluid outwards at its front and entrains ambient fluid primarily at its back; the outward-momentum transport dominates the inward transport. The Reynolds stress contribution by the spot structure is noticeably larger than that due to the background turbulence. The coherent structure vorticity is significantly modified by the structure-induced organization of the background Reynolds stress at the locations of ‘saddle points’ of the latter's distribution. The vorticity, intermittency and other turbulence measures, zone averaged over the extent of the spot, compare well with the time-average values, thus suggesting that the spark-induced ‘spot’ is probably not different from a naturally occurring large-scale coherent structure.

Results from idealized ocean models indicate that equatorially trapped baroclinic waves incident on an eastern boundary may be partially transmitted north and south along the coast as boundary-trapped internal Kelvin waves. The offshore scale of the coastal internal Kelvin waves is the internal Rossby radius of deformation δR, which decreases as the Coriolis parameter f increases. The effect of the presence of a continental slope of width Ls, along a north–south oriented coastline, on the poleward propagation of coastal trapped internal Kelvin waves is studied in a two-layer β-plane model. The waves propagate from regions near the equator where δR > Ls to mid-latitudes where δR < Ls. It is assumed that f varies slowly on the alongshore scale of the waves L, that L [Gt ] Ls, and that either the topographic slope is weak or that the upper-layer depth is small compared to the lower-layer depth. All of the coastal trapped waves present in the model are non-dispersive. For most values of f, the cross-shelf eigen-functions consist of the internal Kelvin wave and an infinite set of continental shelf waves whose vertical structure depends on δR/Ls. For δR/Ls [Gt ] 1, the shelf waves are bottom trapped while for δR/Ls [Lt ] 1 they are barotropic. The wave speeds Cn of the shelf waves vary linearly with f, whereas the wave speed c0 of the internal Kelvin wave is independent of f. As f increases through critical values fCn, where Cn approaches C0, the phase speeds and the eigenfunctions vary so that the eigenfunctions represent a different type of wave on either side of fCn. In the slowly-varying approximation, the alongshore energy flux in each eigenfunction is a constant. It follows that an internal Kelvin wave which has a wavelength short enough that the slowly-varying approximation remains valid and which propagates poleward from the equatorial region where f < fC1 will transform into a shelf wave, at values of f near fC1, and will continue propagation poleward in that form. As a result, coastal trapped baroclinic disturbances may be able to propagate efficiently from the equatorial region to mid-latitudes where they may take the form of barotropic shelf waves.

Linear stability analysis is applied to the problem of a density-stratified fluid contained in an inclined slot being subjected to a lateral temperature gradient. Stability equations are solved using the Galerkin technique with 12 terms in the truncated expansion series. Within the range of θ considered, |θ| < 75°, critical instability was found to be of the stationary type. Results of critical thermal Rayleigh numbers and wavenumbers at all inclination angles are in good agreement with the experimental results obtained earlier (Paliwal & Chen 1980). Contrary to intuition, these results show that the system is more stable when the lower wall is heated. This is shown to be the result of the increased vertical solute gradient in the steady state prior to the onset of instabilities when the heating is from below.

Miles’ (1977b) model of the Mach reflexion of a solitary wave by a vertical wall is tested by laboratory experiments. The model over-predicts the measured run-up at the wall, and no evidence of the predicted maximum was found. The measurements provide support for the predicted critical angle of incidence at which Mach reflexion is replaced by regular reflexion. It is shown that mass and energy conservation determine the length of the reflected wave in Miles’ model and that this is not consistent with momentum conservation in the neighbourhood of the end point of the reflected wave. It is suggested that the discrepancy between the measurements and the model may result from this failure of the model.