Using a coarse grained (16 × 33 × 8) numerical simulation, a lower bound on the Lyapunov dimension, Dλ, of the attractor underlying turbulent, periodic Poiseuille flow at a pressure-gradient Reynolds number of 3200 has been calculated to be approximately 352. These results were obtained on a spatial domain with streamwise and spanwise periods of 1.6π, and correspond to a wall-unit Reynolds number of 80. Comparison of Lyapunov exponent spectra from this and a higher-resolution (16 × 33 × 16) simulation on the same domain shows these spectra to have a universal shape when properly scaled. Using these scaling properties, and a partial exponent spectrum from a still higher-resolution (32 × 33 × 32) simulation, we argue that the actual dimension of the attractor underlying motion on the given computational domain is approximately 780. The medium resolution calculation establishes this dimension as a strong lower bound on this computational domain, while the partial exponent spectrum calculated at highest resolution provides some evidence that the attractor dimension in fully resolved turbulence is unlikely to be substantially larger. These calculations suggest that this periodic turbulent shear flow is deterministic chaos, and that a strange attractor does underly solutions to the Navier–Stokes equations in such flows. However, the magnitude of the dimension measured invalidates any notion that the global dynamics of such turbulence can be attributed to the interaction of a few degrees of freedom. Dynamical systems theory has provided the first measurement of the complexity of fully developed turbulence; the answer has been found to be dauntingly high.

The large-amplitude rectilinear ‘slow-drift’ oscillation of a floating body constrained by a weak restoring force in random waves is considered. The free-surface flow is approximated by a perturbation series expansion for a small slow-drift velocity and wave steepness. A model slow-drift equation of motion is derived, the time-dependent slow-drift excitation force and wave damping coefficient are defined and the complete series of free-surface problems governing their magnitude are formulated. The free-surface problem governing the wave-drift damping coefficient in monochromatic waves is studied and an explicit solution is obtained for a vertical circular cylinder of infinite draught. This solution is extended for arrays of vertical circular cylinders by employing an exact interaction theory. The wave-drift damping coefficient is evaluated for configurations of interest in practice and an expression is derived for the steady drifting velocity of an unconstrained body in regular waves.

The ability of a large-eddy simulation to represent the large-scale motions in the interior of a turbulent flow is well established. However, concerns remain for the behaviour close to rigid surfaces where, with the exception of low-Reynolds-number flows, the large-eddy description must be matched to some description of the flow in which all except the larger-scale ‘inactive’ motions are averaged. The performance of large-eddy simulations in this near-surface region is investigated and it is pointed out that in previous simulations the mean velocity profile in the matching region has not had a logarithmic form. A number of new simulations are conducted with the Smagorinsky (1963) subgrid model. These also show departures from the logarithmic profile and suggest that it may not be possible to eliminate the error by adjustments of the subgrid lengthscale. An obvious defect of the Smagorinsky model is its failure to represent stochastic subgrid stress variations. It is shown that inclusion of these variations leads to a marked improvement in the near-wall flow simulation. The constant of proportionality between the magnitude of the fluctuations in stress and the Smagorinsky stresses has been empirically determined to give an accurate logarithmic flow profile. This value provides an energy backscatter rate slightly larger than the dissipation rate and equal to idealized theoretical predictions (Chasnov 1991).

Thermohaline instabilities produced by horizontal gradients of temperature and salinity in a saturated homogeneous isotropic infinite porous medium are studied using linear stability analysis. In the basic state horizontal gradients of temperature and salinity are taken to be mutually compensating, so that the basic-state fluid density does not vary horizontally. It is found that under these conditions the fluid is always unstable. In a porous medium, assuming the solid matrix to be impervious to dissolved salts, the effective advection rates of heat and dissolved salts are different. Because of this difference any disturbance involving a horizontal component of displacement creates net horizontal density gradients, and thus destabilizes the predominantly hydrostatic force balance. When the vertical Rayleigh number is positive, the typical velocity field consists of almost vertical layers of fluid sliding past each other in opposite directions (salt fingers). When the vertical Rayleigh number is negative the fluid layers are almost horizontal, similar to the interleaving observed in Newtonian fluids. Resulting perturbation fluxes of heat and salt always tend to reduce the basic-state concentration gradients, and typically also the gravitational potential energy of the fluid. We also make some tentative estimates regarding properties of these instabilities at the fully developed state. It seems that thermohaline fine structures, similar to oceanic observations, are also possible in porous media.

The stability of finite-amplitude double–diffusive interleaving driven by linear gradients of salinity and temperature is considered. We show that as the sinusoidal interleaving predicted by linear analysis grows to finite amplitude it is subject to instabilities centred along the lines of minimum vertical density gradient and maximum shear. These secondary instabilities could lead to the step-like density profiles observed in experiments. We show that these instabilities can occur for large Richardson numbers and hence are not driven by shear, but are driven, by double-diffusive effects.

The applicability of the method of matched asymptotic expansions to both propeller aerodynamics and acoustics is investigated. The method is applied to a propeller with blades of high aspect ratio, in a uniform axial flow. The first two terms of the inner expansion and the first three terms of the outer expansion are considered. The matching yields an expression for the spanwise distribution of the downwash velocity. A numerical application shows that the first two terms of the inner solution do not yield an acceptable approximation for the downwash velocity. However, recasting the analytical expressions into an integral equation, similar to Prandtl's lifting line equation for wings, yields results for both aerodynamic and acoustic quantities, which agree well with experimental results. The method thus constitutes a practical analysis method for conventional propellers.

In this theoretical and computational study of the flow of a liquid layer, under the influence of surface tension and gravity most notably, the nonlinear equations governing an interaction between viscous effects and the effects of surface tension, gravity and streamline curvature for the limit of large Reynolds numbers are derived. The aim is to make a comparison between the predictions of this theory and the experiments of Craik et al. on the axisymmetric hydraulic jump. Such a jump is commonly encountered in the everyday context of the initial filling of a kitchen sink, for example, and it is found in the present work that initially all the effects listed above can play a primary role in practice in the local jump phenomenon. As a first step here, the flow of the layer over a small obstacle is considered. It is seen that as surface tension becomes increasingly significant the upstream influence becomes more wave-like. Second, calculations and analysis of the nonlinear free interaction are presented and show wave-like behaviour upstream, followed downstream by a depth profile not unlike that in the typical hydraulic jump. The effects of gravity dominate those of surface tension downstream. Finally, comparisons are made with the experiments and show fair quantitative agreement, supporting the present proposition that these hydraulic jumps are caused by boundary-layer separation due to a viscous–inviscid interaction forced by downstream boundary conditions on, in this case, a fully developed, high-Froude-number liquid layer.

We present results of experiments on a turbulent grid flow and a few results on measurements in the outer region of a boundary layer over a smooth plate. The air flow measurements included three velocity components and their nine gradients. This was done by a twelve-wire hot-wire probe (3 arrays × 4 wires), produced for this purpose using specially made equipment (micromanipulators and some other auxiliary special equipment), calibration unit and calibration procedure. The probe had no common prongs and the calibration procedure was based on constructing a calibration function for each combination of three wires in each array (total 12) as a three-dimensional Chebishev polynomial of fourth order. A variety of checks were made in order to estimate the reliability of the results.

Among the results the most prominent are the experimental confirmation of the strong tendency for alignment between vorticity and the intermediate eigenvector of the rate-of-strain tensor, the positiveness of the total enstrophy-generating term ωiωjsij (sij = ½(∂ui/∂xj+∂uj/∂xi), ωi = εijk∂uj/∂xk) even for rather short records and the tendency for alignment in the strict sense between vorticity and the vortex stretching vector Wi = ωjsij. An emphasis is put on the necessity to measure invariant quantities, i.e. independent of the choice of the system of reference (e.g. sijsij and ωiωjsij) as the most appropriate to describe physical processes. From the methodological point of view the main result is that the multi-hot-wire technique can be successfully used for measurements of all the nine velocity derivatives in turbulent flows, at least at moderate Reynolds numbers.

The capillary instability and break-up of an annular liquid layer coating the inner surface of a cylindrical tube while surrounding another core fluid is studied. In the limiting case where the thickness of the layer is almost equal to the radius of the tube, we obtain a thread of core fluid suspended in a virtually unbounded ambient liquid. The evolution of a layer or thread with a cylindrical or unduloidal interface subject to an axisymmetric perturbation is computed using a boundary integral method. It is shown that for large and moderate layer thicknesses the instability causes the core to transform into an alternating array of primary and secondary spherical drops along the centreline of the tube. The relative volume of the drops depends primarily on the ratio of the core radius to the tube radius, a/R, and the type of initial perturbation. The evolution of thin layers with a/R > 0.82 leads to formation of an array of lobes or collars. These results are used to assess the accuracy of previous approximate analyses based on lubrication flow and minimization of interfacial area.

The generalized channel Boussinesq (gcB) two-equation model and the forced channel Korteweg–de Vries (cKdV) one-equation model previously derived by the authors are further analysed and discussed in the present study. The gcB model describes the propagation and generation of weakly nonlinear, weakly dispersive and weakly forced long water waves in channels of arbitrary shape that may vary both in space and time, and the cKdV model is applicable to unidirectional motions of such waves, which may be sustained under forcing at resonance of the system. These two models are long-wave approximations of a hierarchy set of section-mean conservation equations of mass, momentum and energy, which are exact for inviscid fluids. Results of these models are demonstrated with four specific channel shapes, namely variable rectangular, triangular, parabolic and semicircular sections, in which case solutions are obtained in closed form. In particular, for uniform channels of equal mean water depth, different cross-sectional shapes have a leading-order effect only on the variations of a k-factor of the coefficient of the term bearing the dispersive effects in the model equations. For this case, the uniform-channel analogy theorem enunciated here shows that long waves of equal (mean) height in different uniform channels of equal mean depth but distinct k-shape factors will propagate with equal velocity and with their effective wavelengths appearing k times of that in the rectangular channel, for which k = 1. It also shows that the further channel shape departs from the rectangular, the greater the value of k. Based on this observation, the solitary and cnoidal waves in a k-shaped channel are compared with experiments on wave profiles and wave velocities. Finally, some three-dimensional features of these solitary waves are presented for a triangular channel.

We study the stability of a motionless liquid below its vapour between heated horizontal plates. The temperature of the bottom plate is held below the vaporization temperature and the top plate is hotter than the vaporization temperature. A water film is on the cold plate and a vapour film on the hot plate. We find a basic solution depending only on the variable y normal to the plates, with steady distributions of temperature, a null velocity and no phase change. The linear stability of this basic state is studied in the frame of incompressible fluid dynamics, without convection, but allowing for phase change. An ambiguity in the choice of the conditions to be required of the temperature at the phase-change boundary is identified and discussed. It is shown that the basic state of equilibrium is overstable under conditions of large temperature gradient, when the other parameters have suitable values. An analysis of the energy of the most dangerous disturbance shows that the source of the instability is associated with change of phase.

The second variation of a linear combination of energy and angular momentum is used to investigate the formal stability of circular vortices. The analysis proceeds entirely in terms of Lagrangian displacements to overcome problems that otherwise arise when one attempts to use Arnol'd's Eulerian formalism. Specific attention is paid to the simplest possible model of an isolated vortex consisting of a core of constant vorticity surrounded by a ring of oppositely signed vorticity. We prove that the linear stability regime for this vortex coincides with the formal stability regime. The fact that there are formally stable isolated vortices could imply that there are provable nonlinearly stable isolated vortices. The method can be applied to more complicated vortices consisting of many nested rings of piecewise-constant vorticity. The equivalent expressions for continuous vorticity distributions are also derived.

Solitary internal waves in a density-stratified fluid of shallow depth are considered. According to the classical weakly nonlinear long-wave theory, the propagation of each long-wave mode is governed by the Korteweg–de Vries equation to leading order, and locally confined solitary waves with a ‘sech’ profile are possible. Using a singular-perturbation procedure, it is shown that, in general, solitary waves of mode n > 1 actually develop oscillatory tails of infinite extent, consisting of lower-mode short waves. The amplitude of these tails is exponentially small with respect to an amplitude parameter, and lies beyond all orders of the usual long-wave expansion. To illustrate the theory, two special cases of stratification are discussed in detail, and the amplitude of the oscillations at the solitary-wave tails is determined explicitly. The theoretical predictions are supported by experimental observations.

Wind-tunnel experiments on the flows created by a number of slightly tapered models of circular cross-section have shown the presence of spanwise cells (regions of constant shedding frequency) at Reynolds numbers of the order of 100. The experiments have also shown a number of other interesting features of these flows: the cellular flow configuration is dependent on the base Reynolds number and independent of the tip Reynolds number, the frequency jump between adjacent cells is a function of flow speed, taper angle and kinematic viscosity, but is constant along a cone's span, and the unsteady hot-wire anemometer signal is both amplitude and phase modulated. A mathematical model is proposed based on the complex Landau—Stuart equation with a spanwise diffusive coupling term. Numerical solutions of this equation have shown many of the qualitative features observed in the experiments.

Primary instability of the three-dimensional boundary layer on a rotating disk introduces periodic modulation of the mean flow in the form of stationary crossflow vortices. Here we study the stability of this modulated mean flow with respect to secondary disturbances. These secondary disturbances are found to have quite large growth rates compared to primary disturbances. Both fundamental and subharmonic resonance cases are considered and their corresponding results indicate that the growth rate and the frequency of the secondary instability are insensitive to the exact nature of the resonance condition. The threshold primary stationary crossflow vortex amplitude for secondary instability found in this three-dimensional incompressible boundary layer is significantly larger than that for a two-dimensional boundary layer which is subjected to Tollmien–Schlichting instability. The secondary instability results in a pair of travelling counter-rotating vortices, tilted up and oriented at an angle to the primary stationary crossflow vortices. The computed velocity signals and flow visualization, evaluated based on this secondary disturbance structure, are compared with experimental results.

Compressible rapid distortion theory is used to examine pressure fluctuations and the pressure–dilatation correlation in a field of turbulence subjected to rapid homogeneous compression. It is shown how a one-dimensional compression produces large solenoidal pressure fluctuations. As the dimensionality of the compression increases, the magnitude of these fluctuations decreases — it vanishes for a spherically symmetric compression. By contrast the dilatational, or acoustic, pressure fluctuations depend mainly on the net volumetric compression, and are relatively insensitive to the dimensionality of the compression. These same comments apply to the pressure–dilatation correlation.

The pressure–dilatation correlation appears in the compressible turbulent kinetic energy equation and is significant in rapidly evolving flows; Reynolds stress closure models require that it be represented. The continuity equation provides a relation between pressure dilatation and the rate of change of pressure fluctuation variance. This relation is the basis for our RDT analysis. That analysis leads to a proposal for modelling the rapid contribution to pressure dilatation.

Density gradients modify the flow and hence the shear dispersion of one miscible fluid in another. A solution procedure is given for calculating the effects of weak buoyancy for vertical laminar parallel shear flows. A particular extrapolation to large buoyancy gives an exactly solvable nonlinear diffusion equation. For the particular case of vertical plane Poiseuille flow explicit formulae are derived for the flow, for the nonlinear shear dispersion coefficient and for the onset of instability. The exactly solvable model gives reasonably accurate results for the buoyancy-modified shear dispersion over a range from half to one-and-a-half times the non-buoyant value.

When orthogonal, plane sound waves of the same frequency, wavelength and amplitude, but with phase difference ½π, are incident upon a circular cylinder there is a time-independent streaming about the cylinder which is in the form of a potential vortex separated from the cylinder by a thin, viscous Stokes layer. As the amplitude of the waves in one of the beams decreases, an additional viscous boundary layer is involved until, when the amplitude is sufficiently small, this flow structure is destroyed as fluid erupts from the boundary layer. In the limit of a single beam it is known that this eruption results in opposing jets perpendicular to the wavefronts of the oncoming wave.

The instability of an anticyclonic solid-body rotating eddy embedded on a quiescent environment is studied, for all possible values of the parameters of the unperturbed state, i.e. the vortex's relative thickness and rotation rate. The Coriolis force is fundamental for the existence of the eddy (because the pressure force has a centrifugal direction) and therefore this analysis pertains to the study of mesoscale vortices in the ocean or the atmosphere, as well as those in other planets.

These eddies are known to be stable when the ‘second’ layer is assumed imperturbable (infinitely deep); however, here these vortices are found to be unstable in the more realistic case of an active environment layer, which may be arbitrarily thick.

Three basic types of instability are found, classified according to the dynamic structure of the growing perturbation field, in both layers: baroclinic instability (Rossby-like in both layers), Sakai instability (Poincaré-like in the vortex layer and Rossby-like in the environment), and Kelvin–Helmholtz instability (Poincaré-like in both layers). In addition, there is a hybrid instability, which goes continuously from the baroclinic to the Sakai types, as the rotation rate is increased.

The problem is constrained by the conservation of pseudoenergy and angular pseudomomentum, which are quadratic (to lowest order) in the perturbation. The requirement that both integrals of motion vanish for a growing disturbance, determines the structure of the latter in both layers. Furthermore, that constraint restricts the region, in parameter space, where each type of instability is present.

A thermal boundary layer, in which the temperature and velocity fields are coupled by buoyancy, flows along a horizontal, insulated wall. For sufficiently low local Froude number the solution terminates in a singularity with rising skin friction and falling pressure. The structure of the singularity is obtained and the results are compared with numerical solutions of the horizontal boundary-layer equations. A novel feature of the analysis is that the powers of the streamwise coordinate involved in the structure of the singularity do not appear to be simple rational numbers and are determined from the solution of a pair of ordinary differential equations which govern the flow in an inner viscous region close to the wall. Modifications of the theory are noted for cases where either the temperature or a non-zero heat transfer are specified at the wall.