Drag reduction phenomena, in which 14% drag reduction of tap water flowing in a 16 mm-diameter pipe occurs in the laminar flow range, have been clarified. Experiments were carried out to measure the pressure drop and the velocity profile of tap water and an aqueous solution of glycerin flowing in pipes with highly water-repellent walls, by using a pressure transducer and a hot-film anemometer, respectively. The same drag reduction phenomena also occurred in degassed tap water when using a vacuum tank. The velocity profile measured in this experiment gives the slip velocity at the pipe wall, and it was shown that the shear stress is directly proportional to the slip velocity.

The friction factor formula for a pipe with fluid slip at the wall has been obtained analytically from the exact solution of the Navier–Stokes equation, and it agrees well qualitatively with the experimental data.

The main reasons for the fluid slip are that the molecular attraction between the liquid and the solid surface is reduced because the free surface energy of the solid is very low and the contact area of the liquid is decreased compared with a conventional smooth surface because the solid surface has many fine grooves. Liquid cannot flow into the fine grooves owing to surface tension. These concepts are supported by the experimental result that drag reduction does not occur in the case of surfactant solutions.

Convective instability of a ferromagnetic fluid is predicted for a fluid layer heated from below in the presence of a uniform vertical magnetic field. Convection is caused by a spatial variation in magnetization which is induced when the magnetization of the fluid is a function of temperature and a temperature gradient is established across the layer. A linearized convective instability analysis predicts the critical temperature gradient when only the magnetic mechanism is important, as well as when both the magnetic and buoyancy mechanisms are operative. The magnetic mechanism predominates over the buoyancy mechanism in fluid layers about 1 mm thick. For a fluid layer contained between two free boundaries which are constrained flat, the exact solution is derived for some parameter values and oscillatory instability cannot occur. For rigid boundaries, approximate solutions for stationary instability are derived by the Galerkin method for a wide range of parameter values. It is shown that in this case the Galerkin method yields an eigenvalue which is stationary to small changes in the trial functions, because the Galerkin method is equivalent to an adjoint variational principle.

We treat the problem of slow flow through a periodic array of spheres. Our interest is in the drag force exerted on the array, and hence the permeability of such arrays. It is shown to be convenient to formulate the problem as a set of two-dimensional integral equations for the unknown surface stress vector, thus lowering the dimension of the problem. This set is solved numerically to obtain the drag as a function of particle concentration and packing characteristics. Results are given over the full concentration range for simple cubic, body-centred cubic and face-centred cubic arrays and these agree well with previous limited experimental, asymptotic and numerical results.

An investigation is carried out to determine the conditions marking the onset of convective motion in a horizontal fluid layer in which a negative temperature gradient occurs somewhere within the layer. In such cases, fluid of greater density is situated above fluid of lesser density. Consideration is given to a variety of thermal and hydrodynamic boundary conditions at the surfaces which bound the fluid layer. The thermal conditions include fixed temperature and fixed heat flux at the lower bounding surface, and a general convective-radiative exchange at the upper surface which includes fixed temperature and fixed heat flux as special cases. The hydrodynamic boundary conditions include both rigid and free upper surfaces with a rigid lower bounding surface. It is found that the Rayleigh number marking the onset of motion is greatest for the boundary condition of fixed temperature and decreases monotonically as the condition of fixed heat flux is approached. Non-linear temperature distributions in the fluid layer may result from internal heat generation. With increasing departures from the linear temperature profile, it is found that the fluid layer becomes more prone to instability, that is, the critical Rayleigh number decreases.

This paper describes an investigation of the turbulent forced plumes generated by steady release of mass, momentum and buoyancy from a source situated in an extensive region of uniform or stably stratified fluid. The treatment, which is an extension of earlier work on buoyant plumes, also brings out the relationship between the jet and the Plume as special cases of forced plumes.

The analysis shows that the behaviour of a forced plume from a source of finite size which delivers buoyancy, mass and momentum can in a uniform environment be related to that from a virtual point source of buoyancy and momentum only, and a treatment is given for the latter type of forced plume. When the environment is weakly stratified it is inferred that forced plumes can be related to point definitely; but when the environment is stably stratified, increasing the release of mass and momentum from a given source of buoyancy has the effect at first of reducing the total height of the plume, and only for very large flux of momentum does the height increase again, without limit. A description is given for the behaviour of vertical jets in a stably stratified environment, and for forced plumes of fluid with negative buoyancy.

We consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate. The motion is represented as a stepwise Monte Carlo process on a finite two-dimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. The displacement process exhibits considerable fingering and trapping of displaced phase at all length scales, leading to high residual retention of the displaced phase. Many features of our results are well described by percolation-theory concepts. In particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic-size distribution of the lattice elements.

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.

Careful reassessment of new and pre-existing data shows that recorded scatter in the hot-wire-measured near-wall peak in viscous-scaled streamwise turbulence intensity is due in large part to the simultaneous competing effects of the Reynolds number and viscous-scaled wire length l+. An empirical expression is given to account for these effects. These competing factors can explain much of the disparity in existing literature, in particular explaining how previous studies have incorrectly concluded that the inner-scaled near-wall peak is independent of the Reynolds number. We also investigate the appearance of the so-called outer peak in the broadband streamwise intensity, found by some researchers to occur within the log region of high-Reynolds-number boundary layers. We show that the ‘outer peak’ is consistent with the attenuation of small scales due to large l+. For turbulent boundary layers, in the absence of spatial resolution problems, there is no outer peak up to the Reynolds numbers investigated here (Reτ = 18830). Beyond these Reynolds numbers – and for internal geometries – the existence of such peaks remains open to debate. Fully mapped energy spectra, obtained with a range of l+, are used to demonstrate this phenomenon. We also establish the basis for a ‘maximum flow frequency’, a minimum time scale that the full experimental system must be capable of resolving, in order to ensure that the energetic scales are not attenuated. It is shown that where this criterion is not met (in this instance due to insufficient anemometer/probe response), an outer peak can be reproduced in the streamwise intensity even in the absence of spatial resolution problems. It is also shown that attenuation due to wire length can erode the region of the streamwise energy spectra in which we would normally expect to see kx−1 scaling. In doing so, we are able to rationalize much of the disparity in pre-existing literature over the kx−1 region of self-similarity. Not surprisingly, the attenuated spectra also indicate that Kolmogorov-scaled spectra are subject to substantial errors due to wire spatial resolution issues. These errors persist to wavelengths far beyond those which we might otherwise assume from simple isotropic assumptions of small-scale motions. The effects of hot-wire length-to-diameter ratio (l/d) are also briefly investigated. For the moderate wire Reynolds numbers investigated here, reducing l/d from 200 to 100 has a detrimental effect on measured turbulent fluctuations at a wide range of energetic scales, affecting both the broadband intensity and the energy spectra.

Stereoscopic particle image velocimetry (PIV) was used to measure all three instantaneous components of the velocity field in streamwise–spanwise planes of a turbulent boundary layer at Reτ=1060 (Reθ=2500). Datasets were obtained in the logarithmic layer and beyond. The vector fields in the log layer (z+=92 and 150) revealed signatures of vortex packets similar to those proposed by Adrian and co-workers in their PIV experiments. Groups of legs of hairpin vortices appeared to be coherently arranged in the streamwise direction. These regions also generated substantial Reynolds shear stress, sometimes as high as 40 times −uw. A feature extraction algorithm was developed to automate the identification and characterization of these packets of hairpin vortices. Identified patches contributed 28% to −uw while occupying only 4% of the total area at z+=92. At z+=150, these patches occupied 4.5% of the total area while contributing 25% to −uw. Beyond the log layer (z+=198 and 530), the spatial organization into packets is seen to break down.

Finite-amplitude solutions of plane Couette flow are discovered. They take a steady three-dimensional form. The solutions are obtained numerically by extending the bifurcation problem of a circular Couette system between co-rotating cylinders with a narrow gap to the case with zero average rotation rate.

An experimental study of the migration of dilute suspensions of particles in Poiseuille flow at Reynolds numbers $\hbox{\it Re}\,{=}\,67\hbox{--}1700$ was performed, with a few experiments performed at $\hbox{\it Re}$ up to 2400. The particles used in the majority of the experiments were neutrally buoyant spheres with diameters $d$ yielding a ratio of pipe to particle diameter in the range $D/d \,{=}\, 8\hbox{--}42$. The volume fraction of solids was less than 1% in all cases studied. The results of G. Segré & A. Silberberg (J. Fluid Mech.14, 136, 1962) have been extended to show that the tubular pinch effect in which particles accumulate on a narrow annulus is moved toward the wall as $\hbox{\it Re}$ increases. A careful comparison with asymptotic theory for Poiseuille flow in a channel was performed. Another inner annulus closer to the centre, and not predicted by this asymptotic theory, was observed at elevated $\hbox{\it Re}$. As $\hbox{\it Re}$ is increased, the distribution of particles over the cross-section of the tube at the measurement location, lying at a distance $L \doteq 310 D$ from the entrance, changes from one centred at the annulus predicted by the theory to one with the particles primarily on the inner annulus. The case of slightly non-neutrally buoyant particles was also investigated. A particle trajectory simulation based on asymptotic theory was performed to facilitate the comparison of theory and the experimental observations.

An experimental and theoretical investigation of low Reynolds number, high subsonic Mach number, compressible gas flow in channels is presented. Nitrogen, helium, and argon gases were used. The channels were microfabricated on silicon wafers and were typically 100 μm wide, 104 μm long, and ranged in depth from 0.5 to 20 μm. The Knudsen number ranged from 10-3 to 0.4. The measured friction factor was in good agreement with theoretical predictions assuming isothermal, locally fully developed, first-order, slip flow.

The problem of determining the slow viscous flow of an unbounded fluid past a single solid particle is formulated exactly as a system of linear integral equations of the first kind for a distribution of Stokeslets over the particle surface. The unknown density of Stokeslets is identical with the surface-stress force and can be obtained numerically by reducing the integral equations to a system of linear algebraic equations. This appears to be an efficient way of determining solutions for several external flows past a particle, since it requires that the matrix of the algebraic system be inverted only once for a given particle.

The technique was tested successfully against the analytic solutions for spheroids in uniform and simple shear flows, and was then applied to two problems involving the motion of finite circular cylinders: (i) a cylinder translating parallel to its axis, for which the local stress force distribution and the drag were determined; and (ii) the equivalent axis ratio of a freely suspended cylinder, which was calculated by determining the couple on a stationary cylinder placed symmetrically in two different simple shear flows. The numerical results were found to be consistent with the asymptotic analysis of Cox (1970, 1971) and in excellent agreement with the experiments of Anczurowski & Mason (1968), but not with those of Harris & Pittman (1975).

We consider the evolution of packets of water waves that travel predominantly in one direction, but in which the wave amplitudes are modulated slowly in both horizontal directions. Two separate models are discussed, depending on whether or not the waves are long in comparison with the fluid depth. These models are two-dimensional generalizations of the Korteweg-de Vries equation (for long waves) and the cubic nonlinear Schrödinger equation (for short waves). In either case, we find that the two-dimensional evolution of the wave packets depends fundamentally on the dimensionless surface tension and fluid depth. In particular, for the long waves, one-dimensional (KdV) solitons become unstable with respect to even longer transverse perturbations when the surface-tension parameter becomes large enough, i.e. in very thin sheets of water. Two-dimensional long waves (‘lumps’) that decay algebraically in all horizontal directions and interact like solitons exist only when the one-dimensional solitons are found to be unstable.

The most dramatic consequence of surface tension and depth, however, occurs for capillary-type waves in sufficiently deep water. Here a packet of waves that are everywhere small (but not infinitesimal) and modulated in both horizontal dimensions can ‘focus’ in a finite time, producing a region in which the wave amplitudes are finite. This nonlinear instability should be stronger and more apparent than the linear instabilities examined to date; it should be readily observable.

Another feature of the evolution of short wave packets in two dimensions is that all one-dimensional solitons are unstable with respect to long transverse perturbations. Finally, we identify some exact similarity solutions to the evolution equations.

A comprehensive study is reported of the Lagrangian statistics of velocity, acceleration, dissipation and related quantities, in isotropic turbulence. High-resolution direct numerical simulations are performed on 643 and 1283 grids, resulting in Taylor-scale Reynolds numbers Rλ in the range 38-93. The low-wavenumber modes of the velocity field are forced so that the turbulence is statistically stationary. Using an accurate numerical scheme, of order 4000 fluid particles are tracked through the computed flow field, and hence time series of Lagrangian velocity and velocity gradients are obtained.

The results reported include: velocity and acceleration autocorrelations and spectra; probability density functions (p.d.f.'s) and moments of Lagrangian velocity increments; and p.d.f.'s, correlation functions and spectra of dissipation and other velocity-gradient invariants. It is found that the acceleration variance (normalized by the Kolmogorov scales) increases as R½λ - a much stronger dependence than predicted by the refined Kolmogorov hypotheses. At small time lags, the Lagrangian velocity increments are distinctly non-Gaussian with, for example, flatness factors in excess of 10. The enstrophy (vorticity squared) is found to be more intermittent than dissipation, having a standard-deviation-to-mean ratio of about 1.5 (compared to 1.0 for dissipation). The acceleration vector rotates on a timescale about twice the Kolmogorov scale, while the timescales of acceleration magnitude, dissipation and enstrophy appear to scale with the Lagrangian velocity timescale.

We present experimental force and power measurements demonstrating that the power required to propel an actively swimming, streamlined, fish-like body is significantly smaller than the power needed to tow the body straight and rigid at the same speed U. The data have been obtained through accurate force and motion measurements on a laboratory fish-like robotic mechanism, 1.2 m long, covered with a flexible skin and equipped with a tail fin, at Reynolds numbers up to 106, with turbulence stimulation. The lateral motion of the body is in the form of a travelling wave with wavelength λ and varying amplitude along the length, smoothly increasing from the front to the tail end. A parametric investigation shows sensitivity of drag reduction to the non-dimensional frequency (Strouhal number), amplitude of body oscillation and wavelength λ, and angle of attack and phase angle of the tail fin. A necessary condition for drag reduction is that the phase speed of the body wave be greater than the forward speed U. Power estimates using an inviscid numerical scheme compare favourably with the experimental data. The method employs a boundary-integral method for arbitrary flexible body geometry and motions, while the wake shed from the fish-like form is modelled by an evolving desingularized dipole sheet.

The dependence on initial conditions of the three-dimensional algebraic spatial instability of the Blasius boundary layer is examined by a recently developed method of receptivity analysis based on the upstream integration of adjoint equations. This method allows us to determine optimal perturbations, i.e. initial perturbations that maximize the energy growth, even in the wavenumber range where the problem is not amenable to a mode analysis, and thus to complement a previous paper in which the small-wavenumber regime was described.

Measurements are presented of the mean and fluctuating pressure field acting on a two-dimensional square cylinder in uniform and turbulent flows. The addition of turbulence to the flow is shown to raise the base pressure and reduce the drag of the body. It is suggested that this is attributable to the manner in which the increased turbulence intensity thickens the shear layers, which causes them to be deflected by the downstream corners of the body and results in the downstream movement of the vortex formation region. The strength of the vortex shedding is shown to be reduced as the intensity of the incident turbulence is increased.

Several aspects of the growth and departure of bubbles from a submerged needle are considered. A simple model shows the existence of two different growth regimes according to whether the gas flow rate into the bubble is smaller or greater than a critical value. These conclusions are refined by means of a boundary-integral potential-flow calculation that gives results in remarkable agreement with experiment. It is shown that bubbles growing in a liquid flowing parallel to the needle may detach with a considerably smaller radius than in a quiescent liquid. The study also demonstrates the critical role played by the gas flow resistance in the needle. A considerable control on the rate and size of bubble production can be achieved by a careful consideration of this parameter. The effect is particularly noticeable in the case of small bubbles, which are the most difficult ones to produce in practice.

An analysis is given of a secondary instability that obtains in a wide class of wall-bounded parallel shear flows, including plane Poiseuille flow, plane Couette flow, flat-plate boundary layers, and pipe Poiseuille flow. In these flows it is shown that two-dimensional finite-amplitude waves are (exponentially) unstable to infinitesimal three-dimensional disturbances. This secondary instability seems to be the prototype of transitional instability in these flows in that it has the characteristic (convective) timescales observed in the typical transitions. In the case of plane Poiseuille flow, two-dimensional nonlinear equilibria and quasi-equilibria exist, and the stability of the secondary flow is determined by a three-dimensional linear eigenvalue calculation. In flows without equilibria (e.g. pipe flow), a time-dependent stability analysis is performed by direct spectral numerical calculation of the incompressible three-dimensional Navier–Stokes equations.

The energetics and vorticity dynamics of the instability are discussed. It is shown that the two-dimensional wave mediates the transfer of energy from the mean flow to the three-dimensional perturbation but does not directly provide energy to the disturbance. The instability is of an inviscid character as it persists to high Reynolds numbers and grows on convective timescales. Maximum vorticity (inflexion-point) arguments predict some features of the instability like phase-locking of the two-dimensional and three-dimensional waves, but they do not explain its essential three-dimensionality. The inviscid vorticity dynamics of the instability shows that vortext-stretching and tilting effects are both required to explain the persistent exponential growth. The instability is not centrifugal in nature.

The three-dimensional instability requires that a threshold two-dimensional amplitude be achieved (about 1% of the centreline velocity in plane Poiseuille flow): the growth rates are relatively insensitive to amplitude for moderate two-dimensional amplitudes. With moderate two-dimensional amplitudes, the critical Reynolds numbers for substantial three-dimensional growth are about 1000 in plane Poiseuille and Couette flows and several thousand in pipe Poiseuille flow. The asymptotic (as R approaches infinity) growth rate in plane Poiseuille flow is approximately 0·15h/U0, where h is the half-channel width and U0 is the centreline velocity.

It is possible to make some progress identifying experimental features of transitional spot structure with aspects of the nonlinear two-dimensional/linear three-dimensional instability. The principal excitation of the eigenfunction of the three-dimensional (growing) disturbance is localized within a given periodicity length (in both the stream and cross-stream directions) near the vorticity maxima of the two-dimensional flow; its planwise structure corresponds to that of observed streaks in early transitional spots; its vortical structure resembles that of a streamwise vortex lifting off the wall. As the three-dimensional disturbance grows to finite amplitude, the flows become chaotic with statistical structure similar to that observed experimentally in moderate-Reynolds-number turbulent shear flows.