In order to investigate the stability of infinitely long fully developed salt fingers Stern (1975) has proposed a model in which the basic configuration is independent of the vertical and is sinusoidal in the horizontal direction, with constant background gradients of temperature and salinity. The present study deals with a model of finite vertical extent where τ, the ratio of the diffusivities of salt and heat, is small, and where the constant background salt gradient is replaced by a salt difference between the reservoirs above and below a salt-finger region of finite depth. Steady-state solutions in two and three dimensions are obtained for the zero-order (τ = 0) state in which rising (sinking) fingers have the salinity of the lower (upper) reservoir. For two-dimensional fingers the horizontal scale corresponding to maximum buoyancy flux turns out to be 1.7 times the buoyancy-layer scale associated with the background stable temperature gradient. Heat, salt and buoyancy fluxes are calculated. A boundary-layer analysis is given for the (salt) diffusive correction to the zero-order solution. The same set of calculations is carried out for salt fingers in a Hele-Shaw cell. An assessment of Schmitt's (1979a) model of a finger zone of finite depth shows that the parametric restrictions required by the model cannot be satisfied when Stern's idealization is used for the final state. The present model appears to be preferable for constructing a Schmitt-like theory for τ [Lt ] 1.

An earlier laboratory experiment of Ivey & Corcos (1982) on boundary mixing in a stratified fluid was extended by including the effects of rotation. An axisymmetric version of the original laboratory experiments was constructed and a series of experiments conducted on a rotating table with the tank filled with salt-stratified solution. The combination of vertical mixing at the boundary and horizontal mixing by mesoscale eddies lead to the weakening of the interior density gradient. While the mechanisms were thus complex, the experiments demonstrated that the boundary-mixing process may be parameterized by a relatively simple formulation dependent upon the turbulence properties at the boundary and the tank dimensions.

A coherent and stable baroclinic eddy in a rotating fluid was produced on a sloping bottom by releasing a dome of salt water into the ambient fresh water. A strong cyclonic vortex is produced above the heavy dome. The entire eddy system moves ‘north-westward’ (with the up-slope direction designated ‘north’) as a ‘Taylor column’. The eddy system displays long lifetimes, but it is shown that a theory of isolated systems cannot account for the experimental observations. Instead, it is demonstrated that the vortex flow above the lens is along the lines of constant depth, producing a net pressure force on the lens, which approximately balances the buoyancy force. When Ekman friction is also included, it accounts for the northward motion of the dome.

The second-moment closure applied by Gibson & Launder (1978) to buoyant turbulent flows is here employed without modification to compute the effects of Coriolis forces on fully-developed flow in a rotating channel. The augmentation of turbulent transport on the pressure surface of the channel and its damping on the suction surface seem to be well captured by the computations, provided the flow near the suction surface remains turbulent. The rather striking alteration in shape of the mean velocity profile that occurs as the Rossby number is increased from 0.06 to 0.2 is shown to be explicable in terms of the modification to the intensity of the turbulent velocity fluctuations normal to the plate; for the larger value of Rossby number these fluctuations become larger than those in the flow direction causing what at low spin rates is a source of shear stress to become a sink.

Boundary-layer transition over a stationary disk in rotating flow is studied experimentally. Circular waves are observed in the boundary layer occurring on an end disk of a cylindrical cavity during impulsive spin-down to rest. The transient flow evolves into a quasi-steady regime that exhibits the properties of the Bödewadt flow. The circular waves develop in that flow. The critical Reynolds number Re = r(Ω/v)½ is determined from frequency and wavelength measurements to be about 25. The corresponding dimensionless wavenumber 2πr/ΛRe is about 0.6 and the frequency 2πf/ ΩiRe about 0.2.

The three-dimensional nonlinear oscillations of an isolated, inviscid drop with surface tension are studied by a multiple timescale analysis and pre-averaging applied to the variational principle for the appropriate Lagrangian. Amplitude equations are derived which describe the generic cubic resonance caused by the spatial degeneracy of the eigenfrequencies of the linear normal modes. This resonant coupling leads to the instability of the finite amplitude axisymmetric oscillations to small non-axisymmetric perturbations, as is demonstrated here for the three-and four-lobed normal modes. Solutions to the interaction equations that describe finite amplitude, non-axisymmetric travelling-wave solutions are also obtained and their stability is investigated. A non-generic cubic resonance between the two-lobed and four-lobed oscillatory modes leads to quasi-periodic motions.

A linear analysis of rotating stratified quasi-geostrophic flow past a circular cylinder on a β-plane is performed for moderate stratification, i.e. for σS = O(E½), covering effectively the range $E^{\frac{2}{3}} \ll \sigma \ll 1$, and for strong stratification such that σS = O(1). E [Lt ] 1 is the Ekman number and σS is the product of the Prandtl number and the inverse rotational Froude number. The parameter β measures the importance of the production of relative vorticity by meridional motion. The analysis pivots about a range of β which constrains the interior motion to follow geostrophic contours. For moderate stratification β = O(E1/4), covering effectively the range E½ [Lt ] β [Lt ] E⅛, while for strong stratification E [Lt ] β [Lt ] 1. The dominance of β in the interior is responsible for creating a narrow intense boundary layer along the eastern sector of the cylinder and an extensive blocked flow region surrounded by intense free shear layers west of the cylinder. These narrow regions which channel horizontally O(1) mass flux communicate through corner-like regions centred about the extreme meridional locations of the cylinder. The effect of stratification is to shear the flow vertically and to induce counter-flows laterally. When the stratification is strong the z-dependence is parametric. Nonlinear effects can be ignored when the Rossby number, ε, satisfies the constraints $\epsilon \ll E^{\frac{11}{12}}$ for moderate stratification and $\epsilon \ll \beta^{\frac{1}{7}} E^{\frac{6}{7}}$ for strong stratification. When expressed in terms of the Reynolds number, Re = ε/E, smallness of nonlinear effects can be assured also for high-Reynolds-number flows.

The oscillations are driven by the sinusoidal motion of a piston at one end of the tube. Near half the fundamental frequency the first overtone, driven by nonlinear effects, becomes resonant. For small boundary-layer friction the amplitude of this resonant part is comparable with the non-resonant acoustic solution and shocks are formed. Theoretical results are compared with pressure signals measured at the closed end of the tube. The viscous effects are large for air at atmospheric pressure and the nonlinear effects remain small. Experiments with xenon, sulphurhexafluoride (SF6) and Freon RC-318 (C4F8) were therefore conducted and shocks formed as predicted. The comparison of the nonlinear theory by Keller (1975) with the experiments shows very good agreement.

Ambient fluid of a submerged water jet was continuously tagged with fluorescent dye at a point outside the turbulent region (at 33 jet nozzle diameters from the jet exit). This made it possible to follow the tagged entrained fluid to 73 jet diameters downstream of the exit, a distance unattainable by other methods. The dispersion of the tagged fluid in a plane containing the jet axis and the tagging source was observed and recorded using photography and simple digital image-processing techniques. Most of the entrainment activity appeared to be the result of engulfment by the large-scale structures over an axial distance of ± 1.7B from the source where B is the half-peak velocity radius. The entrained fluid crossed the jet centreline within a downstream distance of Δx = 1.5B.

Downstream of the entrainment region, the spread rate of the tagged entrained fluid was close to that of the turbulent jet fluid. However, the peak mean concentration of the tagged entrained fluid was located near the r/x = 0.1 line closest to the tagging source and shifted very slowly towards the jet centreline. A self-preserving distribution of the mean concentration appears to have been approached after a distance of 6B downstream from the tagging source but further verification is needed owing to experimental uncertainties.

A small fraction of the tagged entrained fluid was found on the side of the jet remote from the tagging source. On rare occurrences, tagged entrained fluid was observed at the interface most remote from the source.

An approach utilizing multiple scales and matched asymptotic expansions is developed for the description of small perturbations at large distances from a thin airfoil oscillating harmonically in a uniform supersonic flow. The problem of determining the unsteady perturbation potential is formulated in general, and an analytical solution is derived for an airfoil with parabolic or flat surfaces. The results describe the flow ahead of the region influenced by the trailing edge. The variation in the pressure jump across an attached leading-edge shock wave is also obtained.

Various reasons have been given as to why the trajectories and circulations of vortices generated at sharp edges do not follow classical similarity-theory predictions for at least an initial short time. Amongst these are the effect of the particular flow geometry (e.g. duct with wedge, nozzle) distant from the salient edge (for rectilinear vortices); axisymmetry (for ring vortices); end effects (for rectilinear vortices); viscous diffusion; finite thickness of the detaching shear layer, as well as secondary vorticity caused by the interaction of the primary vortex with the edge at which it was generated. A further process that may be active is that of viscous entrainment. Experiments, in which essentially straight-line vortices were generated, indicate that of the seven possibilities mentioned, the first five do not play a significant part. All models consist of a basic flow onto which the modelled vortex is superposed. Thus either the basic flow or the vortex model are at fault. The basic flow onto which the vortex is superposed may well not be a pure edge flow, but one that is already taking on the character of an entraining jet flow. On the other hand, the vortex model fails to incorporate secondary vorticity which, particularly when rolled up, might be expected to be dynamically important.

This paper concerns the flow about a sphere placed in a weak shear flow of an inviscid fluid. The secondary velocity resulting from advection of vorticity by the irrotational component of the flow is computed on the sphere surface, and on the upstream axis. The resulting lift force on the sphere is evaluated, and the result is confirmed by an analytical far-field calculation. The displacement of the stagnation streamline, far upstream of the sphere, is calculated more accurately than in previous papers.

In displacing a viscous fluid from the gap between two closely spaced parallel plates, a thin film of the original fluid remains on the surface of each plate. Boundary conditions which connect the approximate equations in the region in front of the interface with the approximate solutions in the thin-film region are determined from local solutions of the equations in the vicinity of the interface edge. These interface conditions depend on both b/R (gap half-width/radius of curvature) and μUn/T, where μ is the viscosity of the original fluid, Un is the normal velocity of the interface edge, and T is the interfacial tension. These conditions are determined using perturbation method when μUn/T [Lt ] 1 and numerical methods when μUn/T is O(1). Though previous theories have shown qualitative agreement with experiments, it is hoped that these new boundary conditions improve the quantitative agreement.

In order to elucidate the turbulent structure below a shear-free gas-liquid interface, turbulence measurements were made in a 50 cm square by 40 cm deep tank stirred by a vertically oscillating grid well below the surface, using a split-film anemometer probe rotating in a horizontal circle. This instrument is able to measure both vertical and horizontal velocity fluctuations to within 0.4 mm of the surface, from which spatial spectra and profiles of r.m.s. velocity fluctuations and integral lengthscales can be calculated. The turbulent structure is affected by the presence of the surface within a ‘surface-influenced layer’ roughly one integral scale, or ten per cent of the distance from the surface to the centre of the grid stroke, in thickness. The shapes of the spectra and profiles within the surface-influenced layer are predicted to a good first approximation by the source theory of Hunt & Graham (1978), which treats the turbulent structure as the superposition of homogeneous turbulence with an irrotational velocity field driven by a source distribution at the surface which cancels the vertical velocity fluctuations there. The magnitudes (as opposed to the shapes) of the profiles scale according to the values that would otherwise occur in the vicinity of the surface-influenced layer were the surface not present. These magnitudes are adequately predicted by the bulk relations determined by Hopfinger & Toly (1976) and Thompson & Turner (1975), with no apparent dependence on turbulent Reynolds number. There are some minor discrepancies between the measured profiles and those of Hunt & Graham. A thin layer of reduced velocity fluctuations below what would be expected from the theory was observed near the surface. Also, anisotropy in the velocity spectra at depths within the surface-influenced layer extended well into the inertial subrange, whereas the Hunt & Graham theory predicts no anisotropy at high wavenumbers.

The linear temporal stability of incompressible semi-bounded inviscid parallel flows over passive compliant walls is studied. It is shown that some of the well-known classical results for inviscid parallel flows with rigid boundaries can, in fact, be extended in modified form to passive compliant walls. These include a result of Rayleigh (1880) which shows that the real part of the phase velocity of a non-neutral disturbance must lie within the range of the velocity distribution; the semi-circle theorem of Howard (1961) and a result of Høiland (1953) which places a bound on the temporal amplification rates of unstable disturbances. The bounds on the phase velocity and the temporal amplification rates of unstable two-dimensional disturbances provide useful guides for numerical studies.

The results are valid for a large class of passive compliant walls. This generality is achieved through a variational-Lagrangian formulation of the essential dynamics of wall motion. A general treatment of the marginal stability of thin shear flows over general passive compliant walls is given. It represents a generalization of the analysis given by Benjamin (1963) for membrane and plate surfaces. Sufficient conditions for the stability of thin shear flows over passive compliant walls are deduced. The applications of the stability criteria to simple cases of compliant wall are described to illustrate the use and the effectiveness of these criteria.

This paper deals theoretically with a filmwise condensation of a vapour on the endwall of a shock tube behind a reflected shock wave. The gas dynamics, accompanied by heat and mass transfer at the vapour-liquid interface, is treated by the method of matched asymptotic expansions. The first and second approximate solutions are obtained and evaluated numerically. It is clarified that there exists a transition process on the growth of a liquid film, that is, the liquid film grows approximately in proportion to the time t in the early stages after the reflection of the shock wave, and after some time, it grows in proportion to the square root of the time. This transition process from the t-dependent growth to the t½-dependent one is mainly controlled by the intensity of condensation. In the t-dependent growth region, the growth rate of the liquid film is proportional to the condensation parameter, depending both upon an initial condition and upon thermal properties of the vapour and the liquid film, while in the t½-dependent growth region it becomes independent of the condensation parameter and is controlled only by thermal properties of the vapour, liquid film and shock-tube endwall. This result suggests that the measurement of the condensation parameter by shock tubes should be made in the t-dependent growth region immediately after the reflection of the shock wave.

It is shown how the unsteady, nonlinear critical-layer equation determines the evolution of instability waves in a weak adverse-pressure-gradient boundary layer. Numerical solutions show that the nonlinearity halts the growth of these inviscidly unstable waves. The stabilizing effect of nonlinearity, in the present case, can be described as a consequence of either the increase (toward zero) of the phase jump across the critical layer or the roll-up of the critical-layer disturbance vorticity.

The one-layer reduced gravity (or ‘shallow water’) equations in the f-plane have solutions such that the active layer is horizontally bounded by an ellipse that rotates steadily. In a frame where the height contours are stationary, fluid particles move along similar ellipses with the same revolution period. Both motions (translation along an elliptical path and precession of that orbit) are anticyclonic and their frequencies are not independent; a Rossby number (R0) based on the combination of both of them is bounded by unity. These solutions may be taken, with some optimism, as a model of ocean warm eddies; their stability is studied here for all values of R0 and of the ellipse eccentricity (these two parameters determine uniquely the properties of the solution).

Sufficient stability conditions are derived from the integrals of motion; f-plane flows that satisfy them must be either axisymmetric or parallel. For the model vortex, the circular case simply corresponds to a solid-body rotation, and is found to be stable to finite-amplitude perturbations for all values of R0. This includes R0 > ½, which implies an anticyclonic absolute vorticity.

The stability of the truly elliptical cases are studied in the normal modes sense. The height perturbation is an n-order polynomial of the horizontal coordinates; the cases for 0 ≤ n ≤ 6 are analysed, for all possible values of the Rossby number and of the eccentricity. All eddies are stable to perturbations with n ≤ 2. (A property of the shallow-water equations, probably related to the last result, is that a general finite-amplitude n-order field is an exact nonlinear solution for n ≤ 2.) Many vortices - noticeably the more eccentric ones - are unstable to perturbations with n ≥ 3; growth rates are O(R02f) where f is the Coriolis parameter.

The problem of a cylindrical falling film, descending vertically outside an infinitely long cylinder is considered. The linear stability of the fully developed flow is studied, first with a perturbation technique for small wavenumbers, and then by direct numerical computation. The numerical results are in agreement with other published values for the cylindrical jet and flat plate limits. The study shows that the cylindrical falling film is unstable for all Reynolds numbers, Weber numbers and radius ratios. Stability and amplification curves are calculated for different values of the parameters. With increasing curvature of the film the range of unstable wavenumbers and the wavenumber of the most amplified wave increase. For low curvature the wavenumber of the most amplified wave decreases with Reynolds number or Weber number, while for high curvatures it increases.

The dynamics of vorticity in two-dimensional turbulence is studied by means of semi-direct numerical simulations, in parallel with passive-scalar dynamics. It is shown that a passive scalar forced and dissipated in the same conditions as vorticity, has a quite different behaviour. The passive scalar obeys the similarity theory à la Kolmogorov, while the enstrophy spectrum is much steeper, owing to a hierarchy of strong coherent vortices. The condensation of vorticity into such vortices depends critically both on the existence of an energy invariant (intimately related to the feedback of vorticity transport on velocity, absent in passive-scalar dynamics, and neglected in the Kolmogorov theory of the enstrophy inertial range); and on the localness of flow dynamics in physical space (again not considered by the Kolmogorov theory, and not accessible to closure model simulations). When space localness is artificially destroyed, the enstrophy spectrum again obeys a k−1 law like a passive scalar. In the wavenumber range accessible to our experiments, two-dimensional turbulence can be described as a hierarchy of strong coherent vortices superimposed on a weak vorticity continuum which behaves like a passive scalar.