The effect of surface curvature on the development of a two-dimensional wall jet was investigated experimentally. A comparison was made between a wall jet flowing around a circular cylinder and its plane equivalent. Velocity surveys and surface pressure measurements in the curved wall jet suggest the existence of two primary regions of interest. The first region, ranging from the end of the potential core to an approximate angular position of θ=120°, is characterized by a constant surface pressure and a self similarity of the mean flow. The second region is marked by an adverse pressure gradient leading to separation around θ=230°. The rate of spread of this flow, even in the initial region, is much higher than in the plane wall jet and so are the levels of turbulence and Reynolds stress. The dominant lengthscale in this flow is the radius of curvature R and the dominant velocity scale is the square root of the kinematic jet momentum divided by the radius of curvature. Entrainment of ambient fluid which causes the jet to adhere to the curved surface is also the main reason for its separation which is preceded by a rapid rate of spread of the flow leading to the failure of the boundary-layer approximation.

A numerical and experimental study of buoyancy-driven flow in the annulus between two horizontal coaxial cylinders at Rayleigh numbers approaching and exceeding the critical values is presented. The stability of the flow is investigated using linear theory and the energy method. Theoretical predictions of the critical Rayleigh number for onset of secondary flows are obtained for a wide range of radius ratio R and are verified by comparison with results of previous experimental studies. A subcritical Rayleigh number which provides a necessary condition for global flow stability is also determined. The three-dimensional transient equations of fluid flow and heat transfer are solved to study the manifestation of instabilities within annuli having impermeable endwalls, which are encountered in various applications. For the first time, a thorough examination of the development of spiral vortex secondary flow within a moderate gap annulus and its interaction with the primary flow is performed for air. Simulations are conducted to investigate factors influencing the size and number of post-transitional vortex cells. The evolution of stable three-dimensional flow and temperature fields with increasing Rayleigh number in a large gap annulus is also studied. The distinct flow structures which coexist in the large gap annulus at high Rayleigh numbers preceding transition to oscillatory flow, including transverse vortices at the end walls which have not been previously identified, are established numerically and experimentally. The solutions for the large-gap annulus are compared to those for the moderate-gap case to clarify fundamental differences in behaviour. Heat transfer results in the form of local Nusselt number distributions are presented for both the moderate- and large-gap cases. Results from a series of experiments performed with air to obtain data for validation of the numerical scheme and further information on the flow stability are presented. Additionally, the change from a crescent-shaped flow pattern to a unicellular pattern with centre of rotation at the top of the annulus is investigated numerically and experimentally for a Prandtl number of 100. Excellent agreement between the numerical and experimental results is shown for both Prandtl numbers studied. The present work provides, for the first time, quantitative three-dimensional descriptions of spiral convection within a moderate-gap annulus containing air, flow structures preceding oscillation in a large-gap annulus for air, and unicellular flow development in a large-gap annulus for large Prandtl number fluids.

The inertial migration of a small rigid sphere translating parallel to the walls within a channel flow at large channel Reynolds numbers is investigated. The method of matched asymptotic expansions is used to solve the equations governing the disturbance flow past a particle at small particle Reynolds number and to evaluate the lift. Both neutrally and non-neutrally buoyant particles are considered. The wall-induced inertia is significant in the thin layers near the walls where the lift is close to that calculated for linear shear flow, bounded by a single wall. In the major portion of the flow, excluding near-wall layers, the wall effect can be neglected, and the outer flow past a sphere can be treated as unbounded parabolic shear flow. The effect of the curvature of the unperturbed velocity profile is significant, and the lift differs from the values corresponding to a linear shear flow even at large Reynolds numbers.

The onset of transition in a boundary layer is dependent on the initialization and interaction of disturbances in a laminar flow. Here, theory and full Navier–Stokes simulations focus on the transient period just after disturbances enter the boundary layer. The temporal evolution of disturbances within a boundary layer is investigated by examining a series of initial value problems. In each instance, the complete spectra (i.e. the discrete and the continuum) are included so that the solutions can be completely arbitrary. Both numerical and analytical solutions of the linearized Navier–Stokes equations subject to the arbitrary initial conditions are presented. The temporal evolution of disturbances during the transient period are compared with the spatial evolution of the same disturbances and a strong correlation between the two approaches is demonstrated indicating that the theory may be used for the transient period of disturbance evolution. The theory and simulations demonstrate that strong amplification of the disturbances can occur as a result of the inclusion of the continuum in the prediction of disturbance evolution. The results further show that any approach proposed for use in bypass boundary layer transition must include the transient growth that results from the continuum. Finally, although a connection between temporal and spatial evolution in the transient period has been demonstrated, a theoretical basis as an explanation for this connection remains the focus of additional study.

In recent years a new paradigm has emerged in linear stability theory due to the recognition of the importance of non-normality in the Orr–Sommerfeld equation as derived from the method of normal modes. For single-fluid flows it has been shown that it is possible for the kinetic energy of certain stable mode combinations to grow transiently before decaying to zero. We look again at the linear stability of two-fluid plane Poiseuille flow in two dimensions, concentrating on transient growth and its dependence on the viscosity and depth ratio. The procedure is to solve the stability equations numerically and consider disturbances defined as a sum of the least stable eigenmodes (not just the least stable interfacial mode). It is found that the variational method used to find maximum growth cannot be based upon the kinetic energy of the flow only and that interface deflection must be included in the formulation. We show which modes are necessary for inclusion in the disturbance expression and find that the interfacial mode does not make a significant contribution to possible energy growth. We examine the magnitude of maximum growth and the nature of the disturbances that lead to this growth. The linear energy rate equation shows that at moderate Reynolds numbers the mechanism responsible for the largest two-fluid growth is transfer of energy from the basic flow via the Reynolds stresses. The energy transfer is facilitated by streamline tilting that can be seen at the channel walls or at the interface. A similar effect has been found in single-fluid plane Poiseuille flow.

Since the availability of data from direct numerical simulation (DNS) of turbulence, researchers have utilized the joint PDFs of invariants of the velocity gradient tensor to study the geometry of small-scale motions of turbulence. However, the joint PDFs only give an instantaneous static representation of the properties of fluid particles and dynamical Lagrangian information cannot be extracted. In this paper, the Lagrangian evolution of the invariants of the velocity gradient tensor is studied using conditional mean trajectories (CMT). These CMT are derived using the concept of the conditional mean time rate of change of invariants calculated from a numerical simulation of isotropic turbulence. The study of the CMT in the invariant space (RA, QA) of the velocity-gradient tensor, invariant space (RS, QS) of the rate-of-strain tensor, and invariant space (RW, QW) of the rate-of-rotation tensor show that the mean evolution in the (Σ, QW) phase plane, where Σ is the vortex stretching, is cyclic with a characteristic period similar to that found by Martin et al. (1998) in the cyclic mean evolution of the CMT in the (RA, QA) phase plane. Conditional mean trajectories in the (Σ, QW) phase plane suggest that the initial reduction of QW in regions of high QW is due to viscous diffusion and that vorticity contraction only plays a secondary role subsequent to this initial decay. It is also found that in regions of the flow with small values of QW, the local values of QW do not begin to increase, even in the presence of self-stretching, until a certain self-stretching rate threshold is reached, i.e. when Σ≈0.25 〈QW〉1/2. This study also shows that in regions where the kinematic vorticity number (as defined by Truesdell 1954) is low, the local value of dissipation tends to increase in the mean as observed from a Lagrangian frame of reference. However, in regions where the kinematic vorticity number is high, the local value of enstrophy tends to decrease. From the CMT in the (−QS, RS phase plane, it is also deduced that for large values of dissipation, there is a tendency for fluid particles to evolve towards having a positive local value of the intermediate principal rate of strain.

Gravitational settling of dense particles through density interfaces is common in many environmental and engineering flow situations, yet very little research has been done to understand the mechanics of particle–stratification interactions. To this end, a detailed experimental study was carried out to investigate the settling of solid spherical particles through density interfaces. In these experiments, the solid particles first descended through a deep homogeneous layer, entered a thick pycnocline and then descended to another denser homogeneous layer. It was found that the stratification has a significant impact on the settling of particles in the approximate parameter range 1.5<Re1<15, where Re1=U1dp/v is the Reynolds number based on the particle entry velocity U1 to the stratified layer, dp is the particle diameter and v is the kinematic viscosity of the fluid. In the above parameter range, the particles tend to drag lighter fluid from the upper layer into the stratified region, thus increasing the drag on them substantially and decelerating them within the stratified layer. In the Froude number Fr1=U1/Ndp range investigated, 3<Fr1<10, where N is the buoyancy frequency of the stratified layer, the drag coefficient was found to be an order of magnitude larger than its homogeneous-fluid counterpart. The internal-wave contribution to the drag was small compared to that of fluid dragged into the stratified layer, but substantial internal-wave activity could be detected after the fluid dragged from the lighter layer (the caudal fluid) detached from the particle.

The minimum velocity of the solid particle within the stratified layer was found to be given by Umin/U1= 5.5×10−2Fr9/101, occurring on a time scale tmin/ (d2p/v)= 1.4×102Re−1.71, where tmin was measured relative to the time of the particle's entry into the stratified region. Outside the parameter range 1.5<Re1<15, the drag on the sphere in the density-stratified layer could be approximated to that in a homogeneous fluid, whence the bringing of lighter fluid into the stratified layer as a tail behind the descending particle was found to be negligible.

We discuss laboratory experiments with a continuous source or sink of fluid in a two-layer rotating environment which produces anticyclonic and cyclonic vortices, respectively. Experiments were carried out with a sloping bottom in order to simulate the β-effect and they were conducted for different values for the source/sink flow rate Q and the Coriolis parameter f. The Rossby number Ro of these vortices was small but finite and the flow was expected to be quasi-geostrophic. The qualitative behaviour of the anticyclonic and cyclonic vortices was generally similar, but it depended on the flow rate. For low flow rates, a single vortex formed at the source and extended to the west. At higher flow rates, the vortex broke free from the source and moved to the west; this vortex was then followed sequentially by other vortices behaving similarly. The westward velocity U of these vortices was calculated and compared with the speed Us of a linear topographic Rossby wave. For multiple vortices the westward velocities were greater than Us while for a single vortex produced by a low flow rate the velocity was less than Us. Significant asymmetry between the anticyclonic and cyclonic vortices was observed in the transition zone from single to multiple vortices which implies that ageostrophic effects were still present in the flow.

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.

Laboratory experiments have shown that monopolar isolated vortices in a rotating flow undergo instabilities that result in the formation of multipolar vortex states such as dipoles and tripoles. In some cases the instability is entirely two-dimensional, with the vortices taking the form of vortex columns aligned along the direction of the ambient rotation at all times. In other cases, the vortex first passes through a highly turbulent three-dimensional state before eventually reorganizing into vortex columns. Through a series of three-dimensional numerical simulations, the roles that centrifugal instability, barotropic instability, and the bottom Ekman boundary layer play in these instabilities are investigated. Evidence is presented that the centrifugal instability can trigger the barotropic instabilities by the enhancement of vorticity gradients. It is shown that the bottom Ekman layer is not essential to these instabilities but can strongly modify their evolution.

A wave of small amplitude is considered which approaches a straight beach normally and which is partially reflected at the coastline. By assuming that the local depth is much smaller than the length of the incoming wave, the shallow water equations are used to determine the water motion. The surf zone width is assumed to be small compared to the length of the incoming wave and hence the effect of wave breaking is included only parametrically. The time development of the cohesionless bottom is described by the Exner continuity equation and by an empirical sediment transport rate formula which relates the sediment flux to the steady currents and wave stirring. It is shown that the basic-state solution, which does not depend on the longshore coordinate, may be unstable with respect to longshore bedform perturbations, so that rhythmic topographies form. The instability process is due to a positive feedback mechanism involving the incoming wave, synchronous edge waves and the bedforms. The growth of the bottom perturbations is related to the presence of steady currents caused by the interaction of the incoming wave with synchronous edge waves which in turn are excited by the incoming wave moving over the wavy bed. For natural beaches the model predicts two maxima in the amplification rate: one is related to incoming waves of low frequency, the other to wind waves. Thus two bedforms of different wavelengths can coexist in the nearshore region with longshore spacings of a few hundred and a few tens of metres, respectively. To illustrate the potential validity of the model, its results are compared with field data. The overall agreement is fairly satisfactory.

We consider the nonlinear interaction problem of surface waves with a tethered near-surface buoy. Our objective is to investigate mechanisms for nonlinear short surface wave generation in this complete coupled wave–buoy–cable dynamical system. We develop an effective numerical simulation capability coupling an efficient and high-resolution high-order spectral method for the nonlinear wave–buoy interaction problem with a robust implicit finite-difference method for the cable–buoy dynamics. The numerical scheme accounts for nonlinear wave–wave and wave–body interactions up to an arbitrary high order in the wave steepness and is able to treat extreme motions of the cable including conditions of negative cable tension. Systematic simulations show that beyond a small threshold value of the incident wave amplitude, the buoy performs chaotic motions, characterized by the snapping of the cable. The root cause of the chaotic response is the interplay between the snapping of the cable and the generation of surface waves, which provides a source of strong (radiation) damping. As a result of this interaction, the chaotic buoy motion switches between two competing modes of snapping response: one with larger average peak amplitude and lower characteristic frequency, and the other with smaller amplitude and higher frequency. The generated high-harmonic/short surface waves are greatly amplified once the chaotic motion sets in. Analyses of the radiated wave spectra show significant energy at higher frequencies which is orders of magnitude larger than can be expected from nonlinear generation under regular motion.

The time-dependent response of a floating flexible plate to an impulsively started steadily moving load defines the time taken to approach a steady-state deflection due to the load, or indeed whether such a steady state is achieved at all. The asymptotic analysis for large time reported here, for both a concentrated point load and a uniformly distributed circular load, confirms that a steady-state deflection is achieved at both subcritical and supercritical load speeds. This analysis also predicts a logarithmically growing response near the critical speed corresponding to the minimum phase speed of the hybrid waves generated, but an eventual steady-state response when the load speed moves at the shallow water wave speed. These results are supported by numerical computation.

The spatial evolution of the disturbances that lead to boundary-layer transition on a swept wedge is computed by large-eddy simulations (LES). Stationary and travelling crossflow-vortex disturbances are generated using steady and random-amplitude suction and blowing on the wedge. For a fixed initial amplitude of the stationary vortex and low-amplitude unsteady disturbances, the LES show the evolution of stationary-dominated crossflow disturbances similar to previous simulations and experiments: linear amplification is followed by vortex roll-over and doubly inflectional velocity profiles just prior to transition. A high-frequency secondary instability is associated with the double inflection points in the velocity profiles. The harmonic modes of the primary disturbance were found to be amplified, while no energy was found in any subharmonic mode. The physical phenomena were significantly different when the stationary and travelling vortices have comparable initial amplitudes: in this case, the vortex roll-over does not occur and transition is dominated by the travelling-wave component.