A rapid method for the prediction of flow separation results from an approximate solution of the equations of motion; a single empirical factor is required. The equations are integrated by a modified ‘inner and outer solutions’ technique developed recently for laminar boundary layers, the criterion for separation being obtained as a simple formula applying directly to the separation position. At Reynolds numbers of the order of 106, the criterion is $C_p(xdC_p|dx)^{\frac {1}{2}} = 0 \cdot 39 (10^{-6}R)^{\frac {1}{10}},$ when d2p/dx2 [ges ] 0 and Cp [les ] 4/7; the coefficient 0·39 is replaced by 0·35 when d2p/dx2 < 0.

The prediction of the pressure rise to separation is likely to be from 0 to 10% too low, which puts it second in accuracy to those methods, such as Maskell's (1951), which utilize the Ludweig-Tillmann skin friction law. However, the convenience of the method makes the present error acceptable for many applications, while a greater accuracy should be attainable from an improved allowance for the quantity d2p/dx2.

The main derivation is for arbitrary pressure distributions, while an extension leads to the pressure distribution which just maintains zero skin friction throughout the region of pressure rise.

The concept of a turbulent inner layer with zero wall stress is put forward, and it is deduced that in the neighbourhood of the wall the velocity is proportional to the square root of the distance from the wall.

The effect of rotary Brownian motion on the rheology of a dilute suspension of rigid spheroids in shear flow is considered for various limiting cases of the particle aspect ratio r and dimensionless shear rate γ/D. As a preliminary the probability distribution function is calculated for the orientation of a single, axisymmetric particle in steady shear flow, assuming small particle Reynolds number. The result for the case of weak-shear flows, γ/D [Lt ] 1, has been known for many years. After briefly reviewing this limiting case, we present expressions for the case of strong shear where (r3 + r−3) [Lt ] γ/D, and for an intermediate regime relevant for extreme aspect ratios where 1 [Lt ] γ/D [Lt ] (r3 + r−3). The bulk stress is then calculated for these cases, as well as the case of nearly spherical particles r ∼ 1, which has not hitherto been discussed in detail. Various non-Newtonian features of the suspension rheology are discussed in terms of prior continuum mechanical and experimental results.

The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value, above which perturbations may grow, the asymptotic solution is used as a guide in the choice of appropriate length and time scales for slow variations in the amplitude A of a non-linear two-dimensional perturbation wave. It is found that suitable time and space variables are εt and ε½(x+a1rt), where t is the time, x the distance in the direction of flow, ε the growth rate of linearized theory and (−a1r) the group velocity. By the method of multiple scales, A is found to satisfy a non-linear parabolic differential equation, a generalization of the time-dependent equation of earlier work. Initial conditions are given by the asymptotic solution of linearized theory.

A model to explain the low-frequency unsteadiness found in shock-induced separation is proposed for cases in which the flow is reattaching downstream. It is based on the properties of fluid entrainment in the mixing layer generated downstream of the separation shock whose low-frequency motions are related to successive contractions and dilatations of the separated bubble. The main aerodynamic parameters on which the process depends are presented. This model is consistent with experimental observations obtained by particle image velocimetry (PIV) in a Mach 2.3 oblique shock wave/turbulent boundary layer interaction, as well as with several different configurations reported in the literature for Mach numbers ranging from 0 to 5.

On the basis of previous work by the author, equations are derived describing one-dimensional unsteady flow in bubble-fluid mixtures. Attention is subsequently focused on pressure waves of small and moderate amplitude propagating through the mixture. Four characteristic lengths occur, namely, wavelength, amplitude, bubble diameter and inter-bubble distance. The significance of their relative magnitudes for the theory is discussed. It appears that for high gas content the dispersion is weak and then the conservation of mass and momentum lead to equations similar to the Boussinesq equations, describing long dispersive waves of finite amplitude on a fluid of finite depth. For waves propagating in one direction only, the corresponding equation is similar to the Korteweg–de Vries equation.

It is shown that for mixtures of low gas content the frequency dispersion is in most cases not small. Finally, solutions of the Korteweg–de Vries equation representing cnoidal and solitary waves in a bubble–liquid mixture are given explicitly.

We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikov's technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate.

The structure of a self-preserving turbulent plane jet exhausting into a slow-moving parallel airstream is studied. The investigation includes results of turbulence measurements and the structure is compared with that of a self-preserving plane wake. The results show that self-preservation is established at a distance of about thirty jet widths downstream of the jet nozzle and that, in the self-preserving region of the jet, the distributions of the turbulent intensities and shear stress across the jet are very similar to those found in the plane wake. The distribution of the intermittency factor, however, is found to be more like that found in an axi-symmetric jet than in a plane wake. The turbulent energy balance also shows important differences to that of the wake flow. The unsteady irrotational flow outside the turbulent shear layer is investigated and it is found that the experimental results agree with the predictions of the theories of Phillips (1955) and Stewart (1956). Some comments are also made on the eddy structure and the applicability of the simple theories of turbulence.

This paper reports the result of direct simulations of fluid–particle motions in two dimensions. We solve the initial value problem for the sedimentation of circular and elliptical particles in a vertical channel. The fluid motion is computed from the Navier–Stokes equations for moderate Reynolds numbers in the hundreds. The particles are moved according to the equations of motion of a rigid body under the action of gravity and hydrodynamic forces arising from the motion of the fluid. The solutions are as exact as our finite-element calculations will allow. As the Reynolds number is increased to 600, a circular particle can be said to experience five different regimes of motion: steady motion with and without overshoot and weak, strong and irregular oscillations. An elliptic particle always turn its long axis perpendicular to the fall, and drifts to the centreline of the channel during sedimentation. Steady drift, damped oscillation and periodic oscillation of the particle are observed for different ranges of the Reynolds number. For two particles which interact while settling, a steady staggered structure, a periodic wake-action regime and an active drafting–kissing–tumbling scenario are realized at increasing Reynolds numbers. The non-linear effects of particle–fluid, particle–wall and interparticle interactions are analysed, and the mechanisms controlling the simulated flows are shown to be lubrication, turning couples on long bodies, steady and unsteady wakes and wake interactions. The results are compared to experimental and theoretical results previously published.

A reasonably uniform mean temperature gradient has been superimposed upon a nearly homogeneous turbulent shear flow in a wind tunnel. The overheat is small enough to have negligible effect on the turbulence. Away from the wind-tunnel entrance, the transverse statistical homogeneity is good and the temperature fluctuations and their integral scales grow monotonically like the corresponding velocity fluctuations (Harris, Graham & Corrsin 1977). Measurements of several moments, one- and two-point correlation functions, spectra, integral scales, microscales, probability densities, and joint probability densities of the turbulent velocities, temperature fluctuations, and temperature-velocity products are reported. The heat-transport characteristics are much like those of momentum transport, with the turbulent Prandtl number nearly 1. The temperature fluctuation is better correlated with the streamwise than the transverse velocity component, and the cross-component D12 of the turbulent diffusivity tensor has sign opposite to and about twice the magnitude of the diagonal component D22. Some resemblance of directional properties (relative magnitudes of correlation functions, integral scales, microscales) of the temperature with those of the streamwise velocity is also observed. Comparisons of the present data with measurements in the inner part of a heated boundary layer and a fully turbulent pipe flow (x2/d = 0·25) show comparable magnitudes of temperature-velocity correlation coefficients, turbulent Prandtl numbers and ratios of turbulent diffusivities, and show similar shapes of two-point correlation functions.

It has been suggested that hairpin vortices may play a key role in developing and sustaining the turbulence process in the near-wall region of turbulent boundary layers. To examine this suggestion, a study was done of the hairpin vortices generated by the interaction of a hemisphere protuberancee within a developing laminar boundary layer. Under the proper conditions, hairpin vortices are shed extremely periodically, which allows detailed examination of their behaviour. Shedding characteristics of the hemispheres were determined using hot-film-anemometry techniques. The flow patterns created by the presence of the hairpin vortices have been documented using flow visualization and hot-film-anemometry techniques, and cross-compared with the patterns observed in the near-wall of a fully turbulent boundary layer. In general, it has been observed that many of the visual patterns observed in the near-wall region of a turbulent boundary layer can also be observed in the wake of the hairpin-shedding hemisphere, which appears supportive of the importance of hairpin vortices in the near-wall turbulence production process. Furthermore, velocity measurements indicate the presence of strong inflexional profiles just downstream of the hairpin-vortex generation region which evolve into fuller profiles with downstream distance, eventually developing a remarkable similarity to a turbulent-boundary-layer velocity profile.

Lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at moderate Reynolds numbers, on flows in simple cubic, face-centred cubic and random arrays of spheres. The drag force on the spheres, and hence the permeability of the arrays, is calculated as a function of the Reynolds number at solid volume fractions up to the close-packed limits of the arrays. At Reynolds numbers up to O(102), the non-dimensional drag force has a more complex dependence on the Reynolds number and the solid volume fraction than suggested by the well-known Ergun correlation, particularly at solid volume fractions smaller than those that can be achieved in physical experiments. However, good agreement is found between the simulations and Ergun's correlation at solid volume fractions approaching the close-packed limit. For ordered arrays, the drag force is further complicated by its dependence on the direction of the flow relative to the axes of the arrays, even though in the absence of fluid inertia the permeability is isotropic. Visualizations of the flows are used to help interpret the numerical results. For random arrays, the transition to unsteady flow and the effect of moderate Reynolds numbers on hydrodynamic dispersion are discussed.

In order to understand the rheological and transport properties of a suspension of swimming micro-organisms, it is necessary to analyse the fluid-dynamical interaction of pairs of such swimming cells. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, referred to as a squirmer. The centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). The effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The interaction of two squirmers is calculated analytically for the limits of small and large separations and is also calculated numerically using a boundary-element method. The analytical and the numerical results for the translational–rotational velocities and for the stresslet of two squirmers correspond very well. We sought to generate a database for an interacting pair of squirmers from which one can easily predict the motion of a collection of squirmers. The behaviour of two interacting squirmers is discussed phenomenologically, too. The results for the trajectories of two squirmers show that first the squirmers attract each other, then they change their orientation dramatically when they are in near contact and finally they separate from each other. The effect of bottom-heaviness is considerable. Restricting the trajectories to two dimensions is shown to give misleading results. Some movies of interacting squirmers are available with the online version of the paper.

The three-dimensional flow around a spherical bubble moving steadily in a viscous linear shear flow is studied numerically by solving the full Navier–Stokes equations. The bubble surface is assumed to be clean so that the outer flow obeys a zero-shear-stress condition and does not induce any rotation of the bubble. The main goal of the present study is to provide a complete description of the lift force experienced by the bubble and of the mechanisms responsible for this force over a wide range of Reynolds number (0.1[les ]Re[les ]500, Re being based on the bubble diameter) and shear rate (0[les ]Sr[les ]1, Sr being the ratio between the velocity difference across the bubble and the relative velocity). For that purpose the structure of the flow field, the influence of the Reynolds number on the streamwise vorticity field and the distribution of the tangential velocities at the surface of the bubble are first studied in detail. It is shown that the latter distribution which plays a central role in the production of the lift force is dramatically dependent on viscous effects. The numerical results concerning the lift coefficient reveal very different behaviours at low and high Reynolds numbers. These two asymptotic regimes shed light on the respective roles played by the vorticity produced at the bubble surface and by that contained in the undisturbed flow. At low Reynolds number it is found that the lift coefficient depends strongly on both the Reynolds number and the shear rate. In contrast, for moderate to high Reynolds numbers these dependences are found to be very weak. The numerical values obtained for the lift coefficient agree very well with available asymptotic results in the low- and high-Reynolds-number limits. The range of validity of these asymptotic solutions is specified by varying the characteristic parameters of the problem and examining the corresponding evolution of the lift coefficient. The numerical results are also used for obtaining empirical correlations useful for practical calculations at finite Reynolds number. The transient behaviour of the lift force is then examined. It is found that, starting from the undisturbed flow, the value of the lift force at short time differs from its steady value, even when the Reynolds number is high, because the vorticity field needs a finite time to reach its steady distribution. This finding is confirmed by an analytical derivation of the initial value of the lift coefficient in an inviscid shear flow. Finally, a specific investigation of the evolution of the lift and drag coefficients with the shear rate at high Reynolds number is carried out. It is found that when the shear rate becomes large, i.e. Sr=O(1), a small but consistent decrease of the lift coefficient occurs while a very significant increase of the drag coefficient, essentially produced by the modifications of the pressure distribution, is observed. Some of the foregoing results are used to show that the well-known equality between the added mass coefficient and the lift coefficient holds only in the limit of weak shears and nearly steady flows.

The purpose of this paper is to study the basic principle of fish propulsion. As a simplified model, the two-dimensional potential flow over a waving plate of finite chord is treated. The solid plate, assumed to be flexible and thin, is capable of performing the motion which consists of a progressing wave of given wavelength and phase velocity along the chord, the envelope of the wave train being an arbitrary function of the distance from the leading edge. The problem is solved by applying the general theory for oscillating deformable airfoils. The thrust, power required, and the energy imparted to the wake are calculated, and the propulsive efficiency is also evaluated. As a numerical example, the waving motion with linearly varying amplitude is carried out in detail. Finally, the basic mechanism of swimming is elucidated by applying the principle of action and reaction.

Hot-wire and flow-visualization studies have been carried out in three air jets subjected to pure-tone acoustic excitation, and the instability, vortex roll-up and transition as well as jet response to the controlled excitation have been investigated. The centreline fluctuation intensity can be enhanced by inducing stable vortex pairing to a level much higher than even that at the ‘preferred mode’, but can also be suppressed below the unexcited level under certain conditions of excitation. The conditions most favourable to vortex pairing were determined as a function of the excitation Strouhal number, the Reynolds number (ReD), and the initial shear-layer state, i.e. laminar or turbulent. It is shown that the rolled-up vortex rings undergo pairing under two distinct conditions of excitation: ‘the shear layer mode’ when the Strouhal number based on the initial shear-layer momentum thickness (Stθ) is about 0·012, and ‘the jet column mode’ when the Strouhal number based on the jet diameter (StD) is about 0·85. The former involves pairing of the near-exit thin vortex rings when the initial boundary layer is laminar, irrespective of the value of StD. The latter involves pairing of the thick vortex rings at x/D ≅ 1·75, irrespective of Stθ or whether the initial boundary layer is laminar or turbulent. For laminar exit boundary layer, pairing is found to be stable, i.e., occurring regularly in space and time, for ReD < 5 × 104, but becomes intermittent with increasing ReD or fluctuation intensity in the initial boundary layer.

The trajectories of the vortex centres and their convection velocities during a pairing event have been recorded through phase-locked measurements. In the presence of stable vortex pairing, the time average profiles of fluctuation intensities and Reynolds stress show noticeable deviations from those in the unexcited jet. The vortex pairing phenomenon produce considerably larger excursions of the $\widetilde{uv}(t)$ signal than the time-average Reynolds stress reveals, suggesting that only certain phases of the pairing process may be important in entrainment, and production of Reynolds stress and jet noise.

A solid sphere falling through a Bingham plastic moves in a small envelope of fluid with shape that depends on the yield stress. A finite-element/Newton method is presented for solving the free-boundary problem composed of the velocity and pressure fields and the yield surfaces for creeping flow. Besides the outer surface, solid occurs as caps at the front and back of the sphere because of the stagnation points in the flow. The accuracy of solutions is ascertained by mesh refinement and by calculation of the integrals corresponding to the maximum and minimum variational principles for the problem. Large differences from the Newtonian values in the flow pattern around the sphere and in the drag coefficient are predicted, depending on the dimensionless value of the critical yield stress Yg below which the material acts as a solid. The computed flow fields differ appreciably from Stokes’ solution. The sphere will fall only when Yg is below 0.143 For yield stresses near this value, a plastic boundary layer forms next to the sphere. Boundary-layer scalings give the correct forms of the dependence of the drag coefficient and mass-transfer coefficient on yield stress for values near the critical one. The Stokes limit of zero yield stress is singular in the sense that for any small value of Yg there is a region of the flow away from the sphere where the plastic portion of the viscosity is at least as important as the Newtonian part. Calculations For the approach of the flow field to the Stokes result are in good agreement with the scalings derived from the matched asymptotic expansion valid in this limit.

This is largely an experimental study of the interaction of buoyancy and shear forces in the free convective flow of a liquid in a rectangular cavity across which there is a uniform temperature difference, ΔT, produced by maintaining the two vertical walls at two different temperatures. The height of the cavity, H, is made larger than the width of the cavity, L, and the cavity is sufficiently long in the third dimension for the mean flow to be nearly everywhere two-dimensional. The flow is specified by three dimensionless parameters: σ, the Prandtl number; h = H|L, the aspect ratio; A = γgΔTL3|κν, the Rayleigh number. The experiments are generally restricted to h = 1–60, σ ÷ 103 and A < 108.

For A < 103 the temperature field closely satisfies Laplace's equation but a weak stable unicellular circulation is generated. The flow is vertical throughout the slot except for regions within a distance of order L from the ends.

For 103 < A < 105, large temperature gradients grow near the walls, and in an interior region a uniform vertical temperature gradient is established. The flow is similar to that near an isolated, heated, vertical plate except that the vertical growth of the wall layers is inhibited in the central part of the slot by the presence of the other layer which prevents entrainment of fluid.

Near A = 105 the interior region of the flow generates a steady secondary flow. A regular cellular pattern becomes superimposed on the basic flow to produce a ‘cats-eye’ pattern of streamlines. Near A = 106 when the secondary cell amplitude is large, a further steady cellular motion is generated in the weak shear regions between each cell.

The present study further explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.

These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyper-bolic profiles), while the body shapes cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.

To study effects of flow inhomogeneities on the dynamics of laminar flamelets in turbulent flames, with account taken of influences of the gas expansion produced by heat release, a previously developed theory of premixed flames in turbulent flows, that was based on a diffusive-thermal model in which thermal expansion was neglected, and that applied to turbulence having scales large compared with the laminar flame-thickness, is extended by eliminating the hypothesis of negligible expansion and by adding the postulate of weak-intensity turbulence. The consideration of thermal expansion motivates the formal introduction of multiple-scale methods, which should be useful in subsequent investigations. Although the hydrodynamic-instability mechanism of Landau is not considered, no restriction is imposed on the density change across the flame front, and the additional transverse convection correspondingly induced by the tilted front is described. By allowing the heat-to-reactant diffusivity ratio to differ slightly from unity, clarification is achieved of effects of phenomena such as flame stretch and the flame-relaxation mechanism traceable to transverse diffusive processes associated with flame-front curvature. By carrying the analysis to second order in the ratio of the laminar flame thickness to the turbulence scale, an equation for evolution of the flame front is derived, containing influences of transverse convection, flame relaxation and stretch. This equation explains anomalies recently observed at low frequencies in experimental data on power spectra of velocity fluctuations in turbulent flames. It also shows that, concerning the diffusive-stability properties of the laminar flame, the density change across the flame thickness produces a shift of the stability limits from those obtained in the purely diffusive-thermal model. At this second order, the turbulent correction to the flame speed involves only the mean area increase produced by wrinkling. The analysis is carried to the fourth order to demonstrate the mean-stretch and mean-curvature effects on the flame speed that occur if the diffusivity ratio differs from unity.

We present a detailed numerical study of the interaction of a weak shock wave with an isolated cylindrical gas inhomogeneity. Such interactions have been studied experimentally in an attempt to elucidate the mechanisms whereby shock waves propagating through random media enhance mixing. Our study concentrates on the early phases of the interaction process which are dominated by repeated refractions and reflections of acoustic fronts at the bubble interface. Specifically, we have reproduced two of the experiments performed by Haas & Sturtevant: a Mach 1.22 planar shock wave, moving through air, impinges on a cylindrical bubble which contains either helium or Refrigerant 22. These flows are modelled using the two-dimensional compressible Euler equations for a two-component fluid (air-helium or air–Refrigerant 22). Utilizing a novel shock-capturing scheme in conjunction with a sophisticated mesh refinement algorithm, we have been able to reproduce numerically the intricate mechanisms that were observed experimentally, e.g. transition from regular to irregular refraction, cusp formation and shock wave focusing, multi-shock and Mach shock structures, and jet formation. The level of agreement lends credibility to a number of observations that can be made using information from the simulations for which there is no experimental counterpart. Thus we can now present an updated description for the dynamics of a shock-bubble interaction which goes beyond that provided by the original experiments.