Improved models for the movement of fluid by cilia are presented. A theory which models the cilia of an organism by an array of flexible long slender bodies distributed over and attached at one end to a plane surface is developed. The slender bodies are constrained to move in similar patterns to the cilia of the microorganisms Opalina, Paramecium and Pleurobrachia.

The velocity field is represented by a distribution of force singularities (Stokes flow) along the centre-line of each slender body. Contributions to the velocity field from all the cilia distributed over the plane are summed, to give a streaming effect which in turn implies propulsion of the organism. From this we have been able to model the mean velocity field through the cilia sublayer for the three organisms. We find that, in a frame of reference situated in the organism, the velocity near the surface of the organism is very small – up to one half the length of the cilium – but it increases rapidly to near the velocity of propulsion from then on. This is because of the beating pattern of the cilia; they beat in a near rigid-body rotation during the effective (‘power’) stroke, but during the recovery stroke move close to the wall. Backflow (‘reflux’) is found to occur in the organisms exhibiting antiplectic metachronism (i.e. Paramecium and Pleurobrachia). The occurrence of gradient reversal, but not backflow, has recently been confirmed experimentally (Sleigh & Aiello 1971).

Other important physical values that are obtained from this analysis are the force, bending moment about the base of a cilium and the rate of working. It is found, for antiplectic metachronism, that the force exerted by a cilium in the direction of propulsion is large and positive during the effective stroke whereas it is small and negative during the recovery stroke. However, the duration of the recovery stroke is longer than the effective stroke so the force exerted over one cycle of a ciliary beat is very small. The bending moment follows a similar pattern to the component of force in the direction of propulsion, being larger in the effective stroke for antiplectic metachronism. In symplectic metachronism (i.e. Opalina) the force and bending moment are largest in magnitude when the bending wave is propagated along the cilium. The rate of working indicates that more energy is consumed in the effective stroke for Paramecium and Pleurobrachia than in the recovery stroke, whereas in Opalina it is found to be large during the propagation of the bending wave.

The global linear stability of incompressible, two-dimensional shear flows is investigated under the assumptions that far-field pressure feedback between distant points in the flow field is negligible and that the basic flow is only weakly non-parallel, i.e. that its streamwise development is slow on the scale of a typical instability wavelength. This implies the general study of the temporal evolution of global modes, which are time-harmonic solutions of the linear disturbance equations, subject to homogeneous boundary conditions in all space directions. Flow domains of both doubly infinite and semi-infinite streamwise extent are considered and complete solutions are obtained within the framework of asymptotically matched WKBJ approximations. In both cases the global eigenfrequency is given, to leading order in the WKBJ parameter, by the absolute frequency ω0(Xt) at the dominant turning point Xt of the WKBJ approximation, while its quantization is provided by the connection of solutions across Xt. Within the context of the present analysis, global modes can therefore only become time-amplified or self-excited if the basic flow contains a region of absolute instability.

Mean flow profiles, skin friction, and integral parameters for boundary layers developing naturally over a wide variety of fully aerodynamically rough surfaces are presented and discussed. The momentum thickness Reynolds number Reθ extends to values in excess of 47000 and, unlike previous work, a very wide range of the ratio of roughness element height to boundary-layer depth is covered (0.03 < h/δ > 0.5). Comparisons are made with some classical formulations based on the assumption of a universal two-parameter form for the mean velocity profile, and also with other recent measurements. It is shown that appropriately re-written versions of the former can be used to collapse all the data, irrespective of the nature of the roughness, unless the surface is very rough, meaning that the typical roughness element height exceeds some 50% of the boundary-layer momentum thickness, corresponding to about .

Premixed H2/O2/N2 flames propagating in two-dimensional turbulence have been studied using direct numerical simulations (DNS: simulations in which all fluid and thermochemical scales are fully resolved). Simulations include realistic chemical kinetics and molecular transport over a range of equivalence ratios Φ (Φ = 0.35, 0.5, 0.7, 1.0, 1.3). The validity of the flamelet assumption for premixed turbulent flames is checked by comparing DNS data and results obtained for steady strained premixed flames with the same chemistry (flamelet ‘library’). This comparison shows that flamelet libraries overestimate the influence of stretch on flame structure. Results are also compared with earlier zero-chemistry (flame sheet) and one-step chemistry simulations. Consistent with the simpler models, the turbulent flame with realistic chemistry aligns preferentially with extensive strain rates in the tangent plane and flame curvature probability density functions are close to symmetric with near-zero means. For very lean flames it is also found that the local flame structure correlates with curvature as predicted by DNS based on simple chemistry. However, for richer flames, by contrast to simple-chemistry results with non-unity Lewis numbers (ratio of thermal to species diffusivity), local flame structure does not correlate with curvature but rather with tangential strain rate. Turbulent straining results in substantial thinning of the flame relative to the steady unstrained laminar case. Heat-release and H2O2 contours remain thin and connected (‘flamelet-like’) while species including H-atom and OH are more diffuse. Peak OH concentration occurs well behind the peak heat-release zone when the flame temperature is high (of the order of 2800 K). For cooler and leaner flames (about 1600 K and for an equivalence ratio below 0.5) the OH radical is concentrated near the reaction zone and the maximum OH level provides an estimate of the local flamelet speed as assumed by Becker et al. (1990).

This paper presents a closed-form solution for the inertial lift force acting on a small rigid sphere that translates parallel to a flat wall in a linear shear flow. The results provide connections between results derived by other workers for various limiting cases. An analytical form for the lift force is derived in the limit of large separations. Some new results are presented for the disturbance flow created by a small rigid sphere translating through an unbounded linear shear flow.

A fundamental understanding of nonlinear oscillations of a viscous liquid drop is needed in diverse areas of science and technology. In this paper, the moderate- to large-amplitude axisymmetric oscillations of a viscous liquid drop, which is immersed in dynamically inactive surroundings, are analysed by solving the free boundary problem comprised of the Navier–Stokes system and appropriate interfacial conditions at the drop–ambient fluid interface. The means are the Galerkin/finite-element technique, an implicit predictor-corrector method, and Newton's method for solving the resulting system of nonlinear algebraic equations. Attention is focused here on oscillations of drops that are released from an initial static deformation. Two dimensionless groups govern such nonlinear oscillations: a Reynolds number, Re, and some measure of the initial drop deformation. Accuracy is attested by demonstrating that (i) the drop volume remains virtually constant, (ii) dynamic response to small-and moderate-amplitude disturbances agrees with linear and perturbation theories, and (iii) large-amplitude oscillations compare well with the few published predictions made with the marker-and-cell method and experiments. The new results show that viscous drops that are released from an initially two-lobed configuration spend less time in prolate form than inviscid drops, in agreement with experiments. Moreover, the frequency of oscillation of viscous drops released from such initially two-lobed configurations decreases with the square of the initial amplitude of deformation as Re gets large for moderate-amplitude oscillations, but the change becomes less dramatic as Re falls and/or the initial amplitude of deformation rises. The rate at which these oscillations are damped during the first period rises as initial drop deformation increases; thereafter the damping rate is lower but remains virtually time-independent regardless of Re or the initial amplitude of deformation. The new results also show that finite viscosity has a much bigger effect on mode coupling phenomena and, in particular, on resonant mode interactions than might be anticipated based on results of computations incorporating only an infinitesimal amount of viscosity.

A new theoretical approach for turbulent flows based on Lie-group analysis is presented. It unifies a large set of ‘solutions’ for the mean velocity of stationary parallel turbulent shear flows. These results are not solutions in the classical sense but instead are defined by the maximum number of possible symmetries, only restricted by the flow geometry and other external constraints. The approach is derived from the Reynolds-averaged Navier–Stokes equations, the fluctuation equations, and the velocity product equations, which are the dyad product of the velocity fluctuations with the equations for the velocity fluctuations. The results include the logarithmic law of the wall, an algebraic law, the viscous sublayer, the linear region in the centre of a Couette flow and in the centre of a rotating channel flow, and a new exponential mean velocity profile not previously reported that is found in the mid-wake region of high Reynolds number flat-plate boundary layers. The algebraic scaling law is confirmed in both the centre and the near-wall regions in both experimental and DNS data of turbulent channel flows. In the case of the logarithmic law of the wall, the scaling with the distance from the wall arises as a result of the analysis and has not been assumed in the derivation. All solutions are consistent with the similarity of the velocity product equations to arbitrary order. A method to derive the mean velocity profiles directly from the two-point correlation equations is shown.

A wave basin experiment has been performed in the MARINTEK laboratories, in one of the largest existing three-dimensional wave tanks in the world. The aim of the experiment is to investigate the effects of directional energy distribution on the statistical properties of surface gravity waves. Different degrees of directionality have been considered, starting from long-crested waves up to directional distributions with a spread of ±30° at the spectral peak. Particular attention is given to the tails of the distribution function of the surface elevation, wave heights and wave crests. Comparison with a simplified model based on second-order theory is reported. The results show that for long-crested, steep and narrow-banded waves, the second-order theory underestimates the probability of occurrence of large waves. As directional effects are included, the departure from second-order theory becomes less accentuated and the surface elevation is characterized by weak deviations from Gaussian statistics.

We consider generalizations of fluid models wherein the fluid is assumed to be incompressible, but with the viscosity depending on the pressure. We show that a natural setting for the development of such models is a class of implicit constitutive relations, which, in addition to the fluid model described here, provides a means for developing other complex models for viscoelastic fluids which cannot be set within the ambit of classical explicit constitutive relations for the stress in terms of the histories of appropriate kinematical variables.

We have investigated the effect of a solidifying crust on the dynamics and surface morphology of radial viscous-gravity currents. Liquid polyethylene glycol was admitted into the base of a tank filled with cold sucrose solution maintained at a temperature below the wax freezing point. As the radial current advanced away from the inlet, its surface solidified and deformed through a combination of folding and fracturing. For the warmest experiments, during which solidification did not occur, the radius of the current increased in proportion to the square root of time, as demonstrated both experimentally and theoretically for isothermal viscous fluids by Huppert (1982). When cooling was sufficiently rapid, solid crust formed and caused the spreading rate to decrease. A cooling model combining conduction in the wax with convection in the sucrose solution predicts the distance from the source at which the solid crust first appeared

Progressively colder experiments revealed a sequence of surface morphologies which resembled features observed on cooling lava flows and lava lakes. Flows in which crust formed very slowly developed marginal levees which contained and channelled the main portion of the current. Colder flows with more rapid crust growth formed regularly spaced surface folds, multi-armed rift structures complete with shear offsets, and bulbous lobate forms similar to pillow lavas seen under the ocean. The same transitions between modes of surface deformation were also generated by keeping the ambient water temperature constant and decreasing the extrusion rate. This demonstration that surfaces can exhibit a well-defined sequence of morphologies which depend on the underlying flow conditions offers the prospect of more successful interpretation of natural lava flows.

The unsteady behaviour in shockwave turbulent boundary layer interaction is investigated by analysing results from a large eddy simulation of a supersonic turbulent boundary layer over a compression–expansion ramp. The interaction leads to a very-low-frequency motion near the foot of the shock, with a characteristic frequency that is three orders of magnitude lower than the typical frequency of the incoming boundary layer. Wall pressure data are first analysed by means of Fourier analysis, highlighting the low-frequency phenomenon in the interaction region. Furthermore, the flow dynamics are analysed by a dynamic mode decomposition which shows the presence of a low-frequency mode associated with the pulsation of the separation bubble and accompanied by a forward–backward motion of the shock.

An experimental study was conducted to investigate the’ generation process of random small-scale turbulence in an originally laminar mixing layer. The evolutions of the two types of deterministic structures, the spanwise and streamwise vortices, were first clarified in order to determine their roles in the transition process. A scaling rule for the streamwise distance from the trailing edge of the splitter plate to the vortex merging position was found for various velocity ratios. After this streamwise lengthscale was determined, it became clear that the spanwise wavelength of the streamwise vortices doubled after the merging of the spanwise structures which nominally doubled streamwise wavelengths. The most interesting finding was that the random small-scale eddies were produced by the interactions between the merging spanwise structures and the streamwise vortices.

In this paper, the results of experiments on unsteady disturbances in the boundary-layer flow over a disk rotating in otherwise still air are presented. The flow was perturbed impulsively at a point corresponding to a Reynolds number R below the value at which transition from laminar to turbulent flow is observed. Among the frequencies excited are convectively unstable modes, which form a three-dimensional wave packet that initially convects away from the source. The wave packet consists of two families of travelling convectively unstable waves that propagate together as one packet. These two families are predicted by linear-stability theory: branch-2 modes dominate close to the source but, as the packet moves outwards into regions with higher Reynolds numbers, branch-1 modes grow preferentially and this behaviour was found in the experiment. However, the radial propagation of the trailing edge of the wave packet was observed to tend towards zero as it approaches the critical Reynolds number (about 510) for the onset of radial absolute instability. The wave packet remains convectively unstable in the circumferential direction up to this critical Reynolds number, but it is suggested that the accumulation of energy at a well-defined radius, due to the flow becoming radially absolutely unstable, causes the onset of laminar–turbulent transition. The onset of transition has been consistently observed by previous authors at an average value of 513, with only a small scatter around this value. Here, transition is also observed at about this average value, with and without artificial excitation of the boundary layer. This lack of sensitivity to the exact form of the disturbance environment is characteristic of an absolutely unstable flow, because absolute growth of disturbances can start from either noise or artificial sources to reach the same final state, which is determined by nonlinear effects.

The steady two-dimensional laminar flow around a stationary circular cylinder has been investigated via a stabilized finite-element method. The Reynolds number Re is based on the cylinder diameter and free-stream speed. The results have been presented for 6 ≤ Re ≤ 40 and the blockages between 0.000125 and 0.80. The blockage B is the ratio of the cylinder diameter to the domain width. There is large scatter in the value of Res, reported in the literature, marking the onset of the flow separation. From the present study the Res is found to be 6.29, approximately for B = 0.005. The effect of the blockage on the characteristic flow parameters is found to be insignificant for B ≤ 0.01. The bubble length, separation angle and Res exhibit non-monotonic variation with the blockage. It is for the first time that such a behaviour for the separation angle and Res is being reported. Two types of boundary conditions at the lateral walls have been studied: the slip wall and towing tank. In general for high blockage, the results from the slip boundary condition are closer to the ones for the unbounded flow. In that sense, the use of the slip boundary condition as opposed to the towing tank boundary condition on the lateral walls is advocated. The bubble length, separation angle, base suction, total drag, pressure drag, viscous drag and maximum vorticity on the cylinder surface for the steady flow are found to vary as Re, Re−0.5, Re−1, Re−0.5, Re−0.64, Re−0.60 and Re0.5, respectively. The extrapolated results for the steady flow, for higher Re, are found to match quite well with the other results from the literature.

We study the capillary rise of wetting liquids in the corners of different geometries and show that the meniscus rises without limit following the universal law: h(t)/a ≈ (γt/ηa)1/3, where γ and η stand for the surface tension and viscosity of the liquid while is the capillary length, based on the liquid density ρ and gravity g. This law is universal in the sense that it does not depend on the geometry of the corner.

The derivation of a new vectorial bedload formulation for the transport of coarse sediment by fluid flow is presented in the first part of the paper. This relation has been developed for slopes up to the angle of repose both in the streamwise and transverse directions. The pressure distribution is assumed to be hydrostatic. The bed shear stress for the onset of particle motion and mean particle velocity are obtained from the mean force balance on a particle. A new generalized Bagnold hypothesis is introduced to calculate the sediment content of the bedload layer. The new formulation possesses two innovative features. It is fully nonlinear and vectorial in nature, in addition, it behaves smoothly up to the angle of repose.

A mathematical model of the time evolution of straight river channels is presented in the second half of the paper. This study focuses on the evolution process due to bank erosion in the presence of bedload only. The bed and bank material is taken to be coarse, non-cohesive and uniform in size. The sediment continuity and the fluid momentum conservation equations describe the time evolution of the bed topography and flow field. These equations are coupled through the fluid shear stress acting on the bed. This bed shear stress distribution is predicted with the aid of a simple algebraic turbulent closure model. As regards the computation of the sediment flux, the new fully nonlinear vectorial formulation is found to perform well and renders the evolution model fully mechanistic.

The formation of an erosional front in the time development of straight river channels has been so far obscured in physical experiments. Herein, with the help of the new bedload formulation, the existence and migration speed of the front of erosion are inferred from the analysis of the sediment continuity equation.

The model successfully describes the time relaxation of an initially trapezoidal channel toward an equilibrium cross-sectional shape, as evidenced by comparison with experimental data. This equilibrium is characterized by a constant width, vanishing sediment transport in the transverse direction, and a small but non-vanishing streamwise transport rate of bed sediment.

An investigation on novel lines is made into the problem of water waves according to the perfect-fluid model, with reference to wave motions in both two and three space dimensions and with allowance for surface tension. Attention to the Hamiltonian structure of the complete nonlinear problem and the use of methods based on infinitesimal-transformation theory provide a Systematic account of symmetries inherent to the problem and of corresponding conservation laws.

The introduction includes an outline of relevant elements from Hamiltonian theory (§ 1.1) and a brief discussion of implications that the present findings may carry for the approximate mathematical modelling of water waves (§1.2). Details of the hydrodynamic problem are recalled in §2. Then in §3 questions about the regularity of solutions are put in perspective, and a general interpretation is expounded regarding the phenomenon of wave-breaking as the termination of smooth Hamil- tonian evolution. In §4 complete symmetry groups are given for several versions of the water-wave problem : easily understood forms of the main results are listed first in §4.1, and the systematic derivations of them are explained in §4.2. Conservation laws implied by the one-parameter subgroups of the full symmetry groups are worked out in §5, where a recent extension of Noether's theorem is applied relying on the Hamiltonian structure of the problem. The physical meanings of the conservation laws revealed in §5, to an extent abstractly there, are examined fully in §6 and various new insights into the water-wave problem are presented.

In Appendix 1 the parameterized version of the problem is considered, covering cases where the elevation of the free surface is not a single-valued function of horizontal position. I n Appendix 2 a general method for finding the symmetry groups of free-boundary problems is explained, and the exposition includes the mathematical material underlying the particular applications in §§4 and 5.

The frequency and damping rate of surface capillary-gravity waves in a bounded region depend on the conditions imposed where the free surface makes contact with the boundary. Extreme cases are when the free surface meets the boundary orthogonally, as in the case of pure gravity waves, and when the contact line remains fixed throughout the motion. An edge condition that models to some extent the dynamics associated with moving contact lines, but not contact-angle hysteresis, is given by making the slope of the free surface at contact proportional to its velocity. This model, which includes the two extreme cases, is used to obtain the frequency and damping rate of a standing wave between two parallel vertical walls. The effect of viscosity in the boundary layers on the walls is included and it is shown that the dissipation associated with the surface forces can exceed that produced by viscosity. The results are compared with those obtained from a number of experimental investigations, in which damping rates too large to be attributed to viscous action have been measured.

The linear, nonlinear and breakdown stages in the transition of localized disturbances in plane Poiseuille flow is studied by direct numerical simulations and analysis of the linearized Navier–Stokes equations. Three-dimensionality plays a key role and allows for algebraic growth of the normal vorticity through the linear lift-up mechanism. This growth primarily generates elongated structures in the streamwise direction since it is largest at low streamwise wavenumbers. For finite-amplitude disturbances such structures will be generated essentially independent of the details of the initial disturbance, since the preferred nonlinear interactions transfer energy to low streamwise wavenumbers. The nonlinear interactions also give a decrease in the spanwise scales. For the stronger initial disturbances the streamwise vorticity associated with the slightly inclined streaks was found to roll up into distinct streamwise vortices in the vicinity of which breakdown occurred. The breakdown starts with a local rapid growth of the normal velocity bringing low-speed fluid out from the wall. This phenomenon is similar to the low-velocity spikes previously observed in transition experiments. Soon thereafter a small turbulent spot is formed. This scenario represents a bypass of the regular Tollmien–Schlichting, secondary instability process. The simulations have been carried out with a sufficient spatial resolution to ensure an accurate description of all stages of the breakdown and spot formation processes. The generality of the observed processes is substantiated by use of different types of initial disturbances and by Blasius boundary-layer simulations. The present results point in the direction of universality of the observed transition mechanisms for localized disturbances in wall-bounded shear flows.

Using a data base generated by a numerical simulation, the three-dimensional coherent structures of a transitional, spatially evolving boundary layer are determined and their spatio-temporal behaviour is investigated in detail. The coherent structures are calculated by the proper orthogonal decomposition method (POD), which leads to an expansion of the flow field variables into Karhunen-Loéve eigenfunctions. It is shown that the dynamical coherent structures of the flat-plate boundary layer can be described by pairs of eigenfunctions that contain complete information on the spatial evolution of the structures. It is further demonstrated that first-order coherent structures determined by POD correspond to structures that are observed in experiments. In the region of the boundary layer where the spike signals of transition occur, higher-order coherent structures also play an essential role. By considering these higher-order structures as well as their dynamical behaviour in time, a compact description of the flow phenomena in the boundary layer can be obtained. The description of the events occurring at the spike stages of the transitional boundary layer shows, from a coherent structures point of view, striking similarities to the bursting event of fully turbulent boundary layers.