We consider fully developed steady laminar flow through a pipe that is rotating slowly about a line perpendicular to its own axis. The solution is expanded for low Reynolds numbers in powers of a single combined similarity parameter and the series extended to 34 terms by computer. Analysis shows that convergence is limited by a square-root singularity on the negative real axis of the similarity parameter. An Euler transformation and extraction of the leading, secondary and tertiary singularities at infinity render the series accurate for all values of the similarity parameter. The major conclusion of this investigation is that the friction ratio in a slowly rotating pipe grows asymptotically as the ⅛ power of the similarity parameter and not as the ¼ power as previously deduced from boundary-layer analysis. This discrepancy between the present computer-extended method and boundary-layer analysis has also occurred in the similar problem of flow through a loosely coiled pipe (Van Dyke 1978).

The large-scale structures that occur in a forced turbulent mixing layer at moderately high Reynolds numbers have been modelled by linear inviscid stability theory incorporating first-order corrections for slow spatial variations of the mean flow. The perturbation stream function for a spatially growing time-periodic travelling wave has been numerically evaluated for the measured linearly diverging mean flow. In an accompanying experiment periodic oscillations were imposed on the turbulent mixing layer by the motion of a small flap at the trailing edge of the splitter plate that separated the two uniform streams of different velocity. The results of the numerical computations are compared with experimental measurements.

When the comparison between experimental data and the computational model was made on a purely local basis, agreement in both the amplitude and phase distribution across the mixing layer was excellent. Comparisons on a global scale revealed, not unexpectedly, less good accuracy in predicting the overall amplification.

A systematic study is made of the effect of latitude on the linear, normal mode stability characteristics of the laminar barotropic Ekman layer. The outcome depends upon the direction of the geostrophic flow (in the case of flows modelling the atmospheric Ekman layer) or, alternatively, upon the direction of the applied stress (in the case of flows modelling the oceanic Ekman layer). The minimum critical Reynolds number Rc is a function of latitude. For the atmospheric Ekman layer Rc = 30.8 for all latitudes less than 26.2° and increases monotonically with latitude to 54.2. At a latitude of 45° N, Rc is 33.9 and arises for a geostrophic wind directed towards a compass heading 252° (clockwise from north), corresponding to rolls with axes pointing due west and having wavenumber k (with unit of length taken to be the Ekman layer depth) of 0.594. The minimum Rc for the oceanic boundary layer is 11.6 for latitudes less than 81.1°, and increases with latitude to 11.8. At 45° N latitude, the critical condition arises for a surface-current compass heading of 345.2°, roll axis of 351° and a wavenumber k = 0.33. The results for Rc are all symmetric about the equator, with roll axes and associated basic flow directions rotated by 180°. As the Reynolds number R increases, the effects of the perturbation Coriolis acceleration on the instability diminish, as has been previously shown, and the error caused by neglect of the horizontal component of angular velocity therefore decreases. The high Reynolds number limit is systematically explored. It is shown that the lower branch of the neutral curve is not inviscid as R → ∞; rather kR → constant. The upper branch is inviscid in the limit R → ∞, and corresponds to a regular or singular neutral mode depending on whether the angle ε between the outer geostrophic flow and the roll axis is greater or less than 15.93°. ‘Inflectional’ modes, thought to be relevant by some investigators, do not exist for ε < 15.93°. Lastly, the most unstable inviscid mode corresponding to zero phase speed, a condition to which certain well-known experiments are sensitive, occurs at ε = 11.8° with wavenumber k = 0.6. This is in good agreement with published experimental data.

Experimental results on free convection in a vertical annulus filled with a saturated porous medium are reported for height-to-gap ratios of 1.46, 1 and 0.545, and radius ratio of 5.338. In these experiments, the inner and outer walls are maintained at constant temperatures. The use of several fluid–solid combinations indicates a divergence in the Nusselt-number–Rayleigh-number relation, as also reported by previous investigators for horizontal layers and vertical cavities. The reason for this divergence is the use of the stagnant thermal conductivity of the fluid-filled solid matrix. A simple model is presented to obtain an effective thermal conductivity as a function of the convective state, and thereby eliminate the aforementioned divergence. A reasonable agreement between experimentally and theoretically determined Nusselt numbers is then achieved for the present and previous experimental results. It is thus concluded that a unique relationship exists between the Nusselt and Rayleigh numbers unless Darcy's law is inapplicable. The factors that influence the breakdown of Darcian behaviour are characterized and their effects on heat-transfer rates are explained. It is observed that, once the relation between the Nusselt and Rayleigh numbers branches out from that obtained via the mathematical formulation based on Darcy's law, its slope approaches that for a fluid-filled enclosure of the same geometry when the Rayleigh number is large enough. An iterative scheme is also presented for estimation of effective thermal conductivity of a saturated porous medium by using the existing results for overall heat transfer.

The incompressible three-dimensional Navier–Stokes equations are solved numerically for a fluid-filled cylindrical cannister that is spinning and nutating. The motion of the cannister is characteristic of that experienced by spin-stabilized artillery projectiles. Equations for the internal fluid motion are derived in a non-inertial aeroballistic coordinate system. Steady-state numerical solutions are obtained by an iterative finite-difference procedure. Flow fields and liquid induced moments have been calculated for viscosities in the range of 0.9 × 104−1 × 109 cSt. The nature of the three-dimensional fluid motion inside the cylinder is discussed, and the moments generated by the fluid are explained. The calculated moments generally agree with experimental measurements.

For steady airflows at moderate Dean number (100 [les ] κ [les ] 500) axial- and transversevelocity profiles were delineated by anemometry in the entry region of curved circular pipes, having curvature ratios R/a of 4.66 or 8. Uniform inlet velocity profiles were found to evolve downstream over two inviscid length scales. Axial profiles develop on the lengthscale a, the pipe radius. The upstream influence extends at least 6a into a bend. Within this region axial development is largely inviscid and apparently independent of the radius of curvature R. Secondary currents develop on the scale (aR)½. Their initial growth is consistent with a model of vorticity transport based on streamline curvature, but instead of progressing monotonically to an asymptotic value, this swirl first overshoots, then subsides approximately 2(aR)½ downstream from the inlet. Hawthorne's model of streamline twist appears to account qualitatively for this dampening.

It is shown that reduction of broadband (even total) far-field jet noise can be achieved via controlled excitation of a jet at a frequency in the range 0.01 < Stθ < 0.02, where Stθ is the Strouhal number based on the exit momentum thickness of the shear layer. Hot-wire measurements in the noise-producing region of the jet reveal that the noise suppression is a direct consequence of turbulence suppression, produced by early saturation and breakdown of maximally growing instability modes.

Steady small-amplitude thermal convection in a fluid-saturated. infinitely extended porous layer is investigated theoretically in the wavenumber range 1/√2−1. It was shown that the point of multiple bifurcation Ra0 = (3/√2 + 2)π2, α0 = 2−0.25 leads to secondary bifurcation when the wavenumber decreases.

As a result a new branch of a stable, complicated, three-dimensional flow in the square cell was discovered for α close to α0. This branch joins two adjacent branches of three-dimensional flows emanating from the trivial solution and causes their stability transition at the branching points.

This paper describes a sampling technique which is non-periodic, but also non-random, and which allows a pulsed wire anemometer to be used to obtain time-domain information in separated flows. Data free from aliasing can be obtained at frequencies considerably higher than that associated with the maximum possible (periodic) sampling rate achievable with a pulsed-wire probe. Autocorrelation and spectral data obtained in three quite distinct types of flow are presented, as a demonstration of the technique. Some of the data serve to emphasize the special character of separated flows. It is shown that in some circumstances it is possible to obtain spectral data that are adequate for estimating the turbulent-energy dissipation rate (provided that an inertial subrange exists). The measurement of all the terms necessary to examine the balance of turbulent energy in separated flows – or, at least, those terms which can be obtained by standard hot-wire anemometry in low-intensity flows – is therefore a distinct possibility.

A method is presented for following the self-induced motion of vortex sheets. In this method, we use a piecewise analytic representation of the sheet consisting of circular arcs with trigonometric polynomials for the circulation. The procedure is used to study the evolution of the motion in two special cases: a circular vortex sheet with sinusoidal circulation distribution and an infinite plane vortex sheet subject to periodic disturbances. The first problem was studied by Baker (1980) as a test of the method of Fink & Soh (1978), while the second has been studied by a number of authors, notably Meiron, Baker & Orszag (1982). In each case, we follow the motion of the sheet up to the appearance of a singularity at a finite time. The singularity takes the form of an exponential spiral with the simultaneous development of singularities in the curvature and in the circulation distribution. In the final stages of the calculations, up to 155 marker points are used to specify the position of the sheet. If it were possible to execute a stable calculation with equally spaced point vortices, approximately 106 points would be required to achieve the same resolution. Problems with instabilities have been reduced, but not entirely eliminated, and prevent a rigorous verification of the results obtained.

The numerical method of Longuet-Higgins & Cokelet (1976), for waves on deep water, is extended to account for a horizontal bottom contour, and used to investigate breaking waves in water of finite depth. It is demonstrated that a variety of overturning motions may be generated, ranging from the projection of a small-scale jet at the wave crest (of the type that might initiate a spilling breaker) to large-scale plunging breakers involving a significant portion of the wave. Although there seems to be a continuous transition between these wave types, a remarkable similarity is noticed in the overturning regions of many of the waves.

Three high-resolution computations are also discussed. The results are presented in the form of interrelated space-, velocity- and acceleration-plane plots which enable the time evolution of individual fluid particles to be followed. These computations should be found useful for the testing of analytical theories, and may also be applied, for example, to studies of slamming forces on shipping and coastal structures.

A theoretical study is made of the structure of a steady planar deflagration downstream of a specific origin location from which a compressible reactive gas flow emanates. The chemistry is modelled by a high-activation-energy Arrhenius reactionrate law without the introduction of an ignition temperature. Chemically derived heat addition is significant relative to the initial thermal energy of the flow. Perturbation methods, based on the limit of high activation energy, are used to construct solutions for sub- and supersonic values of the Mach number [ ] at the origin. With the exception of a thin layer adjacent to the origin in which very small changes occur, the structure of the deflagration is determined by a fundamental balance of convection, reaction and compressibility effects. Transport processes have an insignificant effect on the energetics of the flow. The upstream portion of the deflagration is dominated by an ignition event reminiscent of the induction period of an adiabatic thermal explosion. Subsequently in the neighbourhood of a well-defined ignition delay (or explosion) location a very rapid reaction takes place with order-unity changes in all the dependent variables. Compressibility effects are shown to be the source of basic limitations on the maximum temperature rise permitted in a flow with a particular value of [ ]. Chapman–Jouguet deflagrations are found to appear when the chemical heat addition is maximized for a given [ ]. Subsonic combustion is shown to exist for fairly general initial conditions at the origin. In contrast, a purely supersonic reaction is found to be possible only for specifically defined values of the initial strain rate and temperature gradient which would be difficult to control in the experimental environment.

The influence of periodic excitation on a plane turbulent one-stream shear layer with turbulent separation was investigated. For the qualitative study flow visualization was employed. Quantitative data were obtained with hot-wire anemometry and spectrum analysis. It was found that sinusoidal perturbations with frequencies of order f0 [lsim ] u0/100θ0 (depending on excitation strength), introduced at the trailing edge are always amplified. Maximum amplification factors are observed for the lowest perturbation levels. The frequency and amplitude of excitation determine the downstream location of the amplification maximum in the flow. At sufficient amplitude two-dimensional vortices are formed which subsequently decay without pairing. The development of the periodic r.m.s. values along x follows a universal curve for all frequencies and amplitudes when properly normalized.

At high excitation amplitudes the flow development depends strongly on the geometrical conditions of the excitation arrangement at the trailing edge. Thus regular vortex pairing as well as suppression of pairing can be achieved.

The excited shear layer has considerably stronger, yet nonlinear, spread than the neutral. The region of vortex formation, irrespective of whether it includes pairing or not, is associated with a step-like increase in width, while after the position of maximum vortex energy, i.e. in the region of decay, the spread is reduced to values below the neutral. There the overall lateral fluctuation energy is increased, while the longitudinal may be decreased as compared with the neutral flow.

Experimental and analytical results are presented for the self-excited oscillations that occur in a partially filled centrifuge when centrifugal forces interact with shallow-water waves. Periodic and aperiodic modulations of the basic whirl phenomena are both observed and calculated. The surface waves are found to be hydraulic jumps, undular bores or solitary waves.

An experiment has been performed to determine the effect of yaw upon transition in the boundary layer formed on the windward face of a long cylinder. The china-clay-evaporation and surface-oil-flow techniques have been used to study the development of the fixed-wavelength stationary disturbances which are characteristic of cross-flow instability. It has been found that the boundary layer is also susceptible to time-dependent disturbances which grow to very large amplitudes prior to the onset of transition. These disturbances have been studied with a hot-wire anemometer. The conditions necessary for the onset and completion of transition have been determined by the use of surface Pitot tubes. Data from the experiment have been compared with the simple criteria for instability and transition which were proposed by Owen & Randall over thirty years ago. In general it has been found that these criteria are inadequate, and, where possible, improvements have been proposed. The raw data are presented in sufficient detail for them to be used to test, or calibrate, future theoretical models of the transition process in three-dimensional boundary-layer flows.

The rapid shearing of a mixture of cohesionless glass spheres and air or water was studied in an annular, parallel-plate shear cell designed after Savage (1978). Two types of flow were observed. In the first type of flow the entire mass of the granular material was mobilized. At high shear rates the shear and normal stresses were found to be quadratically dependent upon the mean shear rate (at constant volume concentration), in general agreement with the observations of Bagnold (1954) and Savage & Sayed (1984), and the ‘kinetic’ theory of Jenkins & Savage (1983). The stresses were found to be weakly dependent on the volume concentration up to approximately 0.5, and strongly dependent above this concentration. For flows in which water was the interstitial fluid, the ratio of the shear stress to the normal stress was slightly higher (than in air), and the stresses at lower shear rates were found to be more nearly linearly related to the shear rate. It is suggested that these effects are contributed to by the viscous dampening of grain motions by the water. The second type of flow was distinguished by the existence of an internal boundary above which the granular material deformed rapidly, but below which the granular material remained rigidly locked in place. The thickness of the shearing layer was measured to be between 5 and 15 grain diameters. The stress ratio at the bottom of the shearing layer was found to be nearly constant, suggesting the internal boundary is a consequence of the immersed weight of the shearing grains, and may be described by a Coulomb yield criterion. A scaled concentration is proposed to compare similar data obtained using different-sized materials or different apparatus. An intercomparison of the two types of flow studied, along with a comparison between the present experiments and those of Bagnold (1954) and Savage & Sayed (1984), suggests that the nature of the boundaries can have a significant effect upon the dynamics of the entire flow.

We consider the flow of two immiscible fluids lying between concentric cylinders when the outer cylinder is fixed and the inner one rotates. The interface is assumed to be concentric with the cylinders, and gravitational effects are neglected. We present a numerical study of the effect of different viscosities, different densities and surface tension on the linear stability of the Couette flow. Our results indicate that, with surface tension, a thin layer of the less-viscous fluid next to either cylinder is linearly stable and that it is possible to have stability with the less dense fluid lying outside. The stable configuration with the less-viscous fluid next to the inner cylinder is more stable than the one with the less-viscous fluid next to the outer cylinder. The onset of Taylor instability for one-fluid flow may be delayed by the addition of a thin layer of less-viscous fluid on the inner wall and promoted by a thin layer of more-viscous fluid on the inner wall.

In existing experiments it is known that the slow evolution of nonlinear deep-water waves exhibits certain asymmetric features. For example, an initially symmetric wave packet of sufficiently large wave slope will first lean forward and then split into new groups in an asymmetrical manner, and, in a long wavetrain, unstable sideband disturbances can grow unequally to cause an apparent downshift of carrier-wave frequency. These features lie beyond the realm of applicability of the celebrated cubic Schrödinger equation (CSE), but can be, and to some extent have been, predicted by weakly nonlinear theories that are not limited to slowly modulated waves (i.e. waves with a narrow spectral band). Alternatively, one may employ the fourth-order equations of Dysthe (1979), which are limited to narrow-banded waves but can nevertheless be solved more easily by a pseudospectral numerical method. Here we report the numerical simulation of three cases with a view to comparing with certain recent experiments and to complement the numerical results obtained by others from the more general equations.

In the two-dimensional sedimentation process beneath an inclined wall, the mixture of the participate and liquid phases is separated from the wall by a boundary layer of the clear liquid. This paper contains a simple mathematical model giving waves on the interface between the clear liquid and the mixture. These waves are caused by a discontinuity in the gradient of the tangential velocity of the clear liquid, across the interface. In the limiting case of small concentration of the particulate phase in the mixture, the model gives a dispersive wave running upward along the interface in the direction of the flow in the boundary layer. The effect of finite concentration is to introduce a damping.

Research on the abstract properties of the Navier–Stokes equations in three dimensions has cast a new light on the time-asymptotic approximate solutions of those equations. Here heuristic arguments, based on the rigorous results of that research, are used to show the intimate relationship between the sufficient number of degrees of freedom describing fluid flow and the bound on the fractal dimension of the Navier–Stokes attractor. In particular it is demonstrated how the conventional estimate of the number of degrees of freedom, based on purely physical and dimensional arguments, can be obtained from the properties of the Navier–Stokes equation. Also the Reynolds-number dependence of the sufficient number of degrees of freedom and of the dimension of the attractor in function space is elucidated.