This paper presents an infinite-series solution to the creeping viscous motion of a fluid through low- and moderate-aspect-ratio pores. The flow field is divided into two simply bounded regions: a cylindrical volume bounded by the walls of the pore and the entrance and exit planes, and an infinite half-space outside the pore. Analytic solutions are first obtained in each region for unknown functions representing arbitrary axial and radial velocity profiles at the pore entrance (exit). These unknown functions are then determined by matching the normal and tangential stress at the pore opening.

The results indicate that the velocity profile approaches to within 1·5 per cent of a Poiseuille profile after a short entrance distance of half the pore radius. In the far field the solution matches exactly the streamline pattern for a flow through an orifice of zero thickness obtained by Sampson (1891). The pressure drop across the pore exhibits linear dependence on the aspect ratio and is closely approximated (less than one per cent error) by a simple algebraic expression.

A detailed experimental study of developing turbulent flow in a rectangular duct was made using a laser-Doppler anemometer. The purposes of the work were to obtain data of value to fluid mechanicists, particularly those interested in the development and testing of mathematical turbulence models, and to evaluate the performance of the anemometer. For the first purpose, contours of axial mean velocity and turbulence intensity were measured in the developing flow, and all three mean velocity components and five of the six Reynolds stresses were obtained in the nearly fully developed flow.

The symmetry of the present flow appears to be better than that of previous measurements and the range of measurements is more extensive. In addition, the laser-Doppler anemometer has the potential advantage, particularly in the measurement of secondary velocities, of avoiding probe interference.

Physiological pumps produce flows by alternate contraction and expansion of the vessel. When muscles start to squeeze its wall the valve at the upstream end is closed and that at the downstream end is opened, and the fluid is pumped out in the downstream direction. These systems can be modelled by a semi-infinite pipe with one end closed by a compliant membrane which prevents only axial motion of the fluid, leaving radial motion completely unrestricted. In the present paper an exact similar solution of the Navier–Stokes equation for unsteady flow is a semi-infinite contracting or expanding circular pipe is calculated and reveals the following characteristics of this type of flow. In a contracting pipe the effects of viscosity are limited to a thin boundary layer attached to the wall, which becomes thinner for higher Reynolds numbers. In an expanding pipe the flow adjacent to the wall is highly retarded and eventually reverses at Reynolds numbers above a critical value. The pressure gradient along the axis of pipe is favourable for a contracting wall, while it is adverse for an expanding wall in most cases. These solutions are valid down to the state of a completely collapsed pipe, since the nonlinearity is retained in full. The results of the present theory may be applied to the unsteady flow produced by a certain class of forced contractions and expansions of a valved vein or a thin bronchial tube.

The problem of subgrid modelling, that is, of representing energy transfers from large to small eddies in terms of the large eddies only, must arise in any large eddy simulation, whether the equations of motion are open or direct (unaveraged) or closed (averaged). Models for closed calculations are derived from classical closures, and these are used to determine the effect of filter shape, grid-scale spectrum and grid-scale anisotropy on the effective eddy viscosity: the Leonard or resolvable-scale stress is calculated separately and is found to account for 14% of the total drain in a typical high Reynolds number case.

The validity of using these eddy viscosities in an open calculation is considered. It is concluded that this is not unreasonable, but that the simulation would be much improved if the gross drain could be separated into net drain and backscatter.

Hot-film measurements of the streamwise velocity component were carried out in a fully developed turbulent water-channel flow for three different Reynolds numbers (13800, 34600 and 48900). The results for the first four statistical moments complement and extend the results from previous studies of turbulent channel flow. The VITA variance technique waa employed to detect deterministic events in the streamwise velocity. It waa demonstrated that the VITA technique has a band-pass-filter character. The number of events detected was found to decrerrae exponentially with the threshold level and the events occupy a wide range of timescales. This makes it impossible to define one unique frequency of occurrence or one unique duration of the events. However, by using this technique information was obtained on the amplitude and timescale distributions of the events. The chmacteristic features of the conditional iverages were found to be related to the skewness and flatness factors.

Upon review of past experimental results and theoretical efforts it is apparent that the mechanism by which combustion noise is generated is not well understood. A theory of combustion noise is developed in this paper which follows rigorously from the principles of fluid mechanics. Lighthill's approach, used in his studies of aerodynamic noise, is closely followed in the present work. The sound radiated from open, turbulent flames is found to depend strongly upon the structure of such flames; at present their structure is not well known. However, meaningful bounds and scaling rules for the sound power output and spectral content are derived based upon the present limited knowledge. A framework is developed which explains past experimental work and the origin of combustion noise.

The wind-driven stably stratified mid-latitude oceanic surface turbulent boundary layer is computationally simulated in the presence of a specified surface gravity-wave field. The gravity waves have broad wavenumber and frequency spectra typical of measured conditions in near-equilibrium with the mean wind speed. The simulation model is based on (i) an asymptotic theory for the conservative dynamical effects of waves on the wave-averaged boundary-layer currents and (ii) a boundary-layer forcing by a stochastic representation of the impulses and energy fluxes in a field of breaking waves. The wave influences are shown to be profound on both the mean current profile and turbulent statistics compared to a simulation without these wave influences and forced by an equivalent mean surface stress. As expected from previous studies with partial combinations of these wave influences, Langmuir circulations due to the wave-averaged vortex force make vertical eddy fluxes of momentum and material concentration much more efficient and non-local (i.e. with negative eddy viscosity near the surface), and they combine with the breakers to increase the turbulent energy and dissipation rate. They also combine in an unexpected positive feedback in which breaker-generated vorticity seeds the creation of a new Langmuir circulation and instigates a deep strong intermittent downwelling jet that penetrates through the boundary layer and increases the material entrainment rate at the base of the layer. These wave effects on the boundary layer are greater for smaller wave ages and higher mean wind speeds.

We present a stochastic analysis of a data set consisting of 1.25 × 107 samples of the local velocity measured in the turbulent region of a round free jet. We find evidence that the statistics of the longitudinal velocity increment v(r) can be described as a Markov process. This new approach to characterize small-scale turbulence leads to a Fokker–Planck equation for the r-evolution of the probability density function (p.d.f.) of v(r). This equation for p(v, r) is completely determined by two coefficients D1(v, r) and D2(v, r) (drift and diffusion coefficient, respectively). It is shown how these coefficients can be estimated directly from the experimental data without using any assumptions or models for the underlying stochastic process. The solutions of the resulting Fokker–Planck equation are compared with experimentally determined probability density functions. It is shown that the Fokker–Planck equation describes the measured p.d.f.(s) correctly, including intermittency effects. Furthermore, knowledge of the Fokker–Planck equation also allows the joint probability density of N increments on N different scales p(v1, r1, …, vN, rN) to be determined.

Direct numerical simulations of the motion of two- and three-dimensional buoyant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a finite difference/front tracking method that allows a fully deformable interface between the bubbles and the ambient fluid and the inclusion of surface tension. The governing parameters are selected such that the average rise Reynolds number is O(1) and deformations of the bubbles are small. The rise velocity of a regular array of three-dimensional bubbles at different volume fractions agrees relatively well with the prediction of Sangani (1988) for Stokes flow. A regular array of two- and three-dimensional bubbles, however, is an unstable configuration and the breakup, and the subsequent bubble–bubble interactions take place by ‘drafting, kissing, and tumbling’. A comparison between a finite Reynolds number two-dimensional simulation with sixteen bubbles and a Stokes flow simulation shows that the finite Reynolds number array breaks up much faster. It is found that a freely evolving array of two-dimensional bubbles rises faster than a regular array and simulations with different numbers of two-dimensional bubbles (1–49) show that the rise velocity increases slowly with the size of the system. Computations of four and eight three-dimensional bubbles per period also show a slight increase in the average rise velocity compared to a regular array. The difference between two- and three-dimensional bubbles is discussed.

The dynamical properties of a fluid, occupying the space between two concentric rotating spheres, are considered, attention being focused on the case where the angular velocities of the spheres are only slightly different and the Reynolds number R of the flow is large. It is found that the flow properties differ inside and outside a cylinder [Cscr ], circumscribing the inner sphere and having its generators parallel to the axis of rotation. Outside [Cscr ] the fluid rotates as if rigid with the angular velocity of the outer sphere. Inside [Cscr ] the fluid rotates with an angular velocity intermediate to the angular velocities of the two spheres and determined by the condition that the flux of fluid into the boundary layer of the faster-rotating sphere is equal to the flux out of the boundary layer of the slower-rotating sphere at the same distance from the axis. The return of fluid is effected by a shear layer near [Cscr ] and we show that it has a complicated structure for it can be divided into three separate layers, two outer ones, of thickness $\sim R^{-\frac{2}{7}}$ and ∼R−¼, and an inner layer of thickness $\sim R^{-\frac{1}{3}}$.

Theory and modelling remain central to improving our understanding of undulatory and oscillatory swimming. Simple models based on added mass can help to give great insight into the mechanics of undulatory swimming, as demonstrated by animals such as eels, stingrays and knifefish. To understand the swimming of oscillatory swimmers such as tuna and dolphins, models need to consider both added mass forces and circulatory forces. For all types of swimming, experiments and theory agree that the most important velocity scale is the characteristic lateral velocity of the tail motion rather than the swimming speed, which erases to a large extent the difference between results obtained in a tethered mode, compared to those obtained using a free swimming condition. There is no one-to-one connection between the integrated swimming performance and the details of the wake structure, in that similar levels of efficiency can occur with very different wake structures. Flexibility and viscous effects play crucial roles in determining the efficiency, and for isolated propulsors changing the profile shape can significantly improve both thrust and efficiency. Also, combined heave and pitch motions with an appropriate phase difference are essential to achieve high performance. Reducing the aspect ratio will always reduce thrust and efficiency, but its effects are now reasonably well understood. Planform shape can have an important mitigating influence, as do non-sinusoidal gaits and intermittent actuation.

Direct numerical simulation and inviscid linear analysis are used to study the interaction of a normal shock wave with an isotropic turbulent field of vorticity and entropy fluctuations. The role of the upstream entropy fluctuations is emphasized. The upstream correlation between the vorticity and entropy fluctuations is shown to strongly influence the evolution of the turbulence across the shock. Negative upstream correlation between u′ and T′ is seen to enhance the amplification of the turbulence kinetic energy, vorticity and thermodynamic fluctuations across the shock wave. Positive upstream correlation has a suppressing effect. An explanation based on the relative effects of bulk compression and baroclinic torque is proposed, and a scaling law is derived for the evolution of vorticity fluctuations across the shock. The validity of Morkovin's hypothesis across a shock wave is examined. Linear analysis is used to suggest that shock-front oscillation would invalidate the relation between urms and Trms, as expressed by the hypothesis.

It is shown that the fundamental features of both thermal instabilities and the corresponding nonlinear convection in rapidly rotating spherical systems (in the range of the Taylor number 109 < T < 1012) are determined by the fluid properties characterized by the size of the Prandtl number. Coefficients of the asymptotic power law for the onset of convection at large Taylor number are estimated in the range of the Prandtl number 0.1 ≤ Pr ≤ 100. For fluids of moderately small Prandtl number, a new type of convective instability in the form of prograde spiralling drifting columnar rolls is discovered. The linear columnar rolls extend spirally from near latitude 60° to the equatorial region, and each spans azimuthally approximately five wavelengths with the inclination angle between a spirally elongated roll and the radial direction exceeding 45°. As a consequence, the radial lengthscale of the linear roll becomes comparable with the azimuthal lengthscale. A particularly significant finding is the connection between the new instability and the predominantly axisymmetric convection. Though non-axisymmetric motions are preferred at the onset of convection, the nonlinear convection (at the Rayleigh number of the order of (R—Rc)/Rc = O(0.1)) bifurcating supercritically from the spiralling mode is primarily dominated by the component of the axisymmetric zonal flow, which contains nearly 90% of the total kinetic energy. For fluids of moderately large Prandtl numbers, thermal instabilities at the onset of convection are concentrated in a cylindrical annulus coaxial with the axis of rotation; the position of the convection cylinder is strongly dependent on the size of the Prandtl number. The associated nonlinear convection consists of predominantly non-axisymmetric columnar rolls together with a superimposed weak mean flow that contains less than 10% of the total kinetic energy at (R—Rc)/Rc = O(0.1). A double-layer structure of the temperature field (with respect to the basic state) forms as a result of strong nonlinear interactions between the nonlinear flow and the temperature field. It is also demonstrated that the aspect ratio of the spherical shell does not substantially influence the fundamental properties of convection.

The flow rate of liquid metals is commonly measured by electromagnetic flowmeters. In these the fluid moves through a region of transverse magnetic field, inducing a potential difference between two electrodes on the walls of the pipe. The ratio of signal to flow rate is dependent on the velocity profile, and this is affected by electromagnetic forces.

In this paper the ultimate steady velocity profile and its associated pressure gradient and induced potential are calculated for the case of laminar flow in a circular pipe whose walls are conducting but without contact resistance. Laminar flow is encouraged by a transverse field. When the fluid conductivity and field strength are sufficiently high, boundary layers occur with a thickness inversely proportional to normal field intensity. The induced potential difference is then 0·926 of the value corresponding to the case of uniform velocity if the walls are non-conducting.

The distance the fluid must travel after entering the transverse field before the steady state is reached is next estimated by a Rayleigh approximation. The inlet velocity is taken to be uniform and effects which occur at the edge of the field are neglected. The process falls into two stages, first a boundary-layer growth and then an adjustment of the velocity away from the walls, occupying a much greater length of pipe. The entry length is shorter than it is in the case of flow in a rectangular pipe, but is still too long for appreciable distortion of the velocity profile to occur within practical flowmeters except at low flow rates. The pressure drop associated with the adjustment of the velocity profile is found to be independent of field strength, if this is high, and about one-eighth of the drop which occurs in the non-conducting case.

Experiments are described in which steady-state pressure gradients and induced potential differences were measured in mercury flowing along Perspex pipes of 0·5 and 0·25 in. bore in transverse fields up to 14500 gauss. The results confirmed the steady-state theory within the limitations of experimental accuracy and the assumption in the theory of high conductivity and an intense field. The experiments also covered the entry region in many cases, and showed that the theoretical entry lengths were correct in order of magnitude but over-estimated. However, the exact entry condition was uncertain, and steady readings were difficult to obtain in the entry region.

A turbulent flame–wall interaction (FWI) configuration is studied using three-dimensional direct numerical simulation (DNS) and detailed chemical kinetics. The simulations are used to investigate the effects of the wall turbulent boundary layer (i) on the structure of a hydrogen–air premixed flame, (ii) on its near-wall propagation characteristics and (iii) on the spatial and temporal patterns of the convective wall heat flux. Results show that the local flame thickness and propagation speed vary between the core flow and the boundary layer, resulting in a regime change from flamelet near the channel centreline to a thickened flame at the wall. This finding has strong implications for the modelling of turbulent combustion using Reynolds-averaged Navier–Stokes or large-eddy simulation techniques. Moreover, the DNS results suggest that the near-wall coherent turbulent structures play an important role on the convective wall heat transfer by pushing the hot reactive zone towards the cold solid surface. At the wall, exothermic radical recombination reactions become important, and are responsible for approximately 70% of the overall heat release rate at the wall. Spectral analysis of the convective wall heat flux provides an unambiguous picture of its spatial and temporal patterns, previously unobserved, that is directly related to the spatial and temporal characteristic scalings of the coherent near-wall turbulent structures.

A laboratory experiment was performed to study the dynamically rich interaction of a turbulent open channel flow with a bed-mounted axial-flow hydrokinetic turbine. An acoustic Doppler velocimeter and a torque transducer were used to simultaneously measure at high temporal resolution the three velocity components of the flow at various locations upstream of the turbine and in the wake region and turbine power, respectively. Results show that for sufficiently low frequencies the instantaneous power generated by the turbine is modulated by the turbulent structure of the approach flow. The critical frequency above which the response of the turbine is decoupled from the turbulent flow structure is shown to vary linearly with the angular frequency of the rotor. The measurements elucidate the structure of the turbulent turbine wake, which is shown to persist for at least fifteen rotor diameters downstream of the rotor, and a new approach is proposed to quantify the wake recovery, based on the growth of the largest scale motions in the flow. Spectral analysis is employed to demonstrate the dominant effect of the tip vortices in the energy distribution in the near-wake region and uncover meandering motions.

Based on energy principles, we propose a statistical model to describe the bubble size probability density function of the daughter bubbles resulting from the shattering of a mother bubble of size D0 immersed in a fully developed turbulent water flow. The model shows that the bubble size p.d.f. depends not only on D0, but also on the value of the dissipation rate of turbulent kinetic energy of the underlying turbulence of the water, ε. The phenomenological model is simple, yet it predicts detailed experimental measurements of the transient bubble size p.d.f.s performed over a range of bubble sizes and dissipation rates ε in a very consistent manner. The agreement between the model and the experiments is particularly good for low and moderate bubble turbulent Weber numbers, Wet = ρΔu2(D0)D0/σ where the assumption of the binary breakup is shown to be consistent with the experimental observations. At larger values of Wet, it was found that the most probable number of daughter bubbles increases and the assumption of tertiary breakup is shown to lead to a better fit of the experimental measurements.

When the upward flow of a fluid through a bed of particles of appropriate and almost uniform size is rapid enough so that the drag on each particle is as great as the particles buoyant weight, the particles do not remain close packed and the bed is said to be fluidized. Industrial uses of fluidized beds in the chemical and petroleum industries in particular are already extensive. Uses in the atomicenergy industry are being developed.

In this paper a mathematical model which describes the phenomena on a continuum basis is deduced. With this model we find that the system is unstable to small internal disturbances. Alternatively, we find that surface waves can be propagated (with attenuation) in the composite fluid and these waves for fluidized beds with a high ratio of solids density to fluid density are stable. These results are in agreement with experiment. Hot beds, where strongly exothermic reactions may be taking place, centrifugal beds (beds fluidized within a rotating system), and electromagnetic beds (those in which the particulate phase is electrically conducting) are all shown to be unstable to small internal disturbances.

The equations derived here may also be used as approximate equations for dispersed particle two-phase flow.

Many experimenters have used oscillating grids to produce turbulence for various laboratory purposes, especially in studies of mixing, but there have been few direct measurements of the properties of the turbulence itself. In the present paper we report experiments which attempt to relate the turbulent velocity and length scales to the external parameters, the frequency and amplitude, for three forms of grid oscillated in a tank of water. Turbulent velocities have been measured in the absence of a mean flow by using a hot film moved through the fluid to provide its own mean velocity. The output is stored and analysed in a small computer, which rapidly evaluates velocity and length scale statistics from an ensemble of records. The spatial variation of these quantities with distance from the stirrer is of special interest. It agrees with results suggested by an inertial-decay theory, and with previous measurements made by Bouvard & Dumas (1967) using a different form of stirrer. A particular purpose of the work has been to ‘calibrate’ the entrainment experiments of Turner (1968), by providing absolute scales of velocity and length in the fluid near a mixing interface, for the same grid as was used in the earlier experiments. Evidence is presented which suggests that other forms of grid may not be calibrated simply by extrapolating these results.