A study was made of the transition of a two-dimensional jet. In the region where laminar flow becomes unstable, two kinds of sinusoidal velocity fluctuation have been found; one is symmetrical and the other is anti-symmetrical with respect to the centre line of the jet. The fluctuations grow exponentially at first and develop into turbulence without being accompanied by abrupt bursts or turbulent spots.

The response characteristics of laminar jets to artificial external excitation were investigated in detail by using sound as an exciting agent. The effect of excitation was seen to be most remarkable when the frequency of excitation coincides with that of self-excited sinusoidal fluctuations.

Numerical solutions of equation of small disturbances superposed on laminar flow were obtained assuming the Reynolds number as infinity. Theoretical eigenvalues and eigenfunctions are in good agreement with experimental results, thus verifying the existence of a region of linear disturbance in the two-dimensional jet.

Moderate-amplitude axisymmetric oscillations of incompressible inviscid drops and bubbles are studied using a Poincaré–Lindstedt expansion technique. The corrections to the drop shape and velocity potential caused by mode coupling at second order in amplitude are predicted for two-, three- and four-lobed motions. The frequency of oscillation is found to decrease with the square of the amplitude; this result compares well with experiments and numerical calculations for drops undergoing two-lobed oscillations.

The dynamic response of an initially spherical capsule subject to different externally imposed flows is examined. The neo-Hookean and Skalak et al. (Biophys. J., vol. 13 (1973), pp. 245–264) constitutive laws are used for the description of the membrane mechanics, assuming negligible bending resistance. The viscosity ratio between the interior and exterior fluids of the capsule is taken to be unity and creeping-flow conditions are assumed to prevail. The capillary number $\varepsilon $ is the basic dimensionless number of the problem, which measures the relative importance of viscous and elastic forces. The boundary-element method is used with bi-cubic B-splines as basis functions in order to discretize the capsule surface by a structured mesh. This guarantees continuity of second derivatives with respect to the position of the Lagrangian particles used for tracking the location of the interface at each time step and improves the accuracy of the method. For simple shear flow and hyperbolic flow, an interval in $\varepsilon $ is identified within which stable equilibrium shapes are obtained. For smaller values of $\varepsilon $, steady shapes are briefly captured, but they soon become unstable owing to the development of compressive tensions in the membrane near the equator that cause the capsule to buckle. The post-buckling state of the capsule is conjectured to exhibit small folds around the equator similar to those reported by Walter et al. Colloid Polymer Sci. Vol. 278 (2001), pp. 123–132 for polysiloxane microcapsules. For large values of $\varepsilon $, beyond the interval of stability, the membrane has two tips along the direction of elongation where the deformation is most severe, and no equilibrium shapes could be identified. For both regions outside the interval of stability, the membrane model is not appropriate and bending resistance is essential to obtain realistic capsule shapes. This pattern persists for the two constitutive laws that were used, with the Skalak et al. law producing a wider stability interval than the neo-Hookean law owing to its strain hardening nature.

Surface-tension-driven Bénard (Marangoni) convection in liquid layers heated from below can exhibit a long-wavelength primary instability that differs from the more familiar hexagonal instability associated with Bénard. This long-wavelength instability is predicted to be significant in microgravity and for thin liquid layers. The instability is studied experimentally in terrestrial gravity for silicone oil layers 0.007 to 0.027 cm thick on a conducting plate. For shallow liquid depths (<.017 cm for 0.102 cm2 s−1 viscosity liquid), the system evolves to a strongly deformed long-wavelength state which can take the form of a localized depression (‘dry spot’) or a localized elevation (‘high spot’), depending on the thickness and thermal conductivity of the gas layer above the liquid. For slightly thicker liquid depths (0.017–0.024 cm), the formation of a dry spot induces the formation of hexagons. For even thicker liquid depths (>0.024 cm), the system forms only the hexagonal convection cells. A two-layer nonlinear theory is developed to account properly for the effect of deformation on the interface temperature profile. Experimental results for the long-wavelength instability are compared to our two-layer theory and to a one-layer theory that accounts for the upper gas layer solely with a heat transfer coefficient. The two-layer model better describes the onset of instability and also predicts the formation of localized elevations, which the one-layer model does not predict. A weakly nonlinear analysis shows that the bifurcation is subcritical. Solving for steady states of the system shows that the subcritical pitchfork bifurcation curve never turns over to a stable branch. Numerical simulations also predict a subcritical instability and yield long-wavelength states that qualitatively agree with the experiments. The observations agree with the onset prediction of the two-layer model, except for very thin liquid layers; this deviation from theory may arise from small non-uniformities in the experiment. Theoretical analysis shows that a small non-uniformity in heating produces a large steady-state deformation (seen in the experiment) that becomes more pronounced with increasing temperature difference across the liquid. This steady-state deformation becomes unstable to the long-wavelength instability at a smaller temperature difference than that at which the undeformed state becomes unstable in the absence of non-uniformity.

A solution to the dispersion of small particles suspended in a turbulent fluid is presented, based on the approximation proposed by Phythian for the dispersion of fluid points in an incompressible random fluid. Motion is considered in a frame moving with the mean velocity of the fluid, the forces acting on the particle being taken as gravity and a fluid drag assumed linear in the particle velocity relative to that of the fluid. The probability distribution of the fluid velocity field in this frame is taken as Gaussian, homogeneous, isotropic, stationary and of zero mean. It is shown that, in the absence of gravity, the long-time particle diffusion coefficient is in general greater than that of the fluid, approaching with increasing particle relaxation time a value consistent with the particle being in an Eulerian frame of reference. The effect of gravity is consistent with Yudine's effect of crossing trajectories, reducing unequally the particle diffusion in directions normal to and parallel to the direction of the gravitational field. To characterize the effect of flow and gravity on particle diffusion it has been found useful to use a Froude number defined in terms of the turbulent intensity rather than the mean velocity. Depending upon the value of this number, it is found that the particle integral time scale may initially decrease with increasing particle relaxation time though it eventually rises and approaches the particle relaxation time. It is finally shown how this analysis may be extended to include the extra forces generated by the fluid and particle accelerations.

Waves of spanwise velocity imposed at the walls of a plane turbulent channel flow are studied by direct numerical simulations. We consider sinusoidal waves of spanwise velocity which vary in time and are modulated in space along the streamwise direction. The phase speed may be null, positive or negative, so that the waves may be either stationary or travelling forward or backward in the direction of the mean flow. Such a forcing includes as particular cases two known techniques for reducing friction drag: the oscillating wall technique (a travelling wave with infinite phase speed) and the recently proposed steady distribution of spanwise velocity (a wave with zero phase speed). The travelling waves alter the friction drag significantly. Waves which slowly travel forward produce a large reduction of drag that can relaminarize the flow at low values of the Reynolds number. Faster waves yield a totally different outcome, i.e. drag increase (DI). Even faster waves produce a drag reduction (DR) effect again. Backward-travelling waves instead lead to DR at any speed. The travelling waves, when they reduce drag, operate in similar fashion to the oscillating wall, with an improved energetic efficiency. DI is observed when the waves travel at a speed comparable with that of the convecting near-wall turbulence structures. A diagram illustrating the different flow behaviours is presented.

Scattering of waves by vertical barriers in infinite-depth water has received much attention due to the ability to solve many of these problems exactly. However, the same problems in finite depth require the use of approximation methods. In this paper we present an accurate method of solving these problems based on a Galerkin approximation. We will show how highly accurate complementary bounds can be computed with relative ease for many scattering problems involving vertical barriers in finite depth and also for a sloshing problem involving a vertical barrier in a rectangular tank.

An experimental and theoretical investigation has been carried out to understand the tonal noise generation mechanism on aerofoils at moderate Reynolds number. Experiments were conducted on a NACA0012 aerofoil section in a low-turbulence closed working section wind tunnel. Narrow band acoustic tones were observed up to 40 dB above background noise. The ladder structure of these tones was eliminated by modifying the tunnel to approximate to anechoic conditions. High-resolution flow velocity measurements have been made with a three-component laser-Doppler anemometer (LDA) which have revealed the presence of strongly amplified boundary-layer instabilities in a region of separated shear flow just upstream of the pressure surface trailing edge, which match the frequency of the acoustic tones. Flow visualization experiments have shown these instabilities to roll up to form a regular Kármán-type vortex street.

A new mechanism for tonal noise generation has been proposed, based on the growth of Tollmien–Schlichting (T–S) instability waves strongly amplified by inflectional profiles in the separating laminar shear layer on the pressure surface of the aerofoil. The growth of fixed frequency, spatially growing boundary-layer instability waves propagating over the aerofoil pressure surface has been calculated using experimentally obtained boundary-layer characteristics. The effect of boundary-layer separation has been incorporated into the model. Frequency selection and prediction of T–S waves are in remarkably good agreement with experimental data.

The inviscid stability of swirling flows with mean velocity profiles similar to that obtained by Batchelor (1964) for a trailing vortex from an aircraft is studied with respect to infinitesimal non-axisymmetric disturbances. The flow is characterized by a swirl parameter q involving the ratio of the magnitude of the maximum swirl velocity to that of the maximum axial velocity. It is found that, as the swirl is continuously increased from zero, the disturbances die out quickly for a small value of q if n = 1 (n is the azimuthal wavenumber of the Fourier disturbance of type exp{i(αx + nϕ − αct)}); but for negative values of n, the amplification rate increases and then decreases, falling to negative values at q slightly greater than 1·5 for n = −1. The maximum amplification rate increases for increasingly negative n up to n = −6 (the highest mode investigated), and corresponds to q ≃ 0·85. The applicability of these results to attempts at destabilizing vortices is briefly discussed.

Liquid flow in a baffled stirred reactor vessel driven by a six-blade disk impeller has been investigated experimentally. Laser-slit photography provided an overview of the flows which were quantified by measurements of velocity characteristics, obtained with a laser-Doppler anemometer, for an impeller rotational speed of 300 r.p.m. and for three impeller clearances from the bottom of the vessel. The mean flow results show an inclination of the impeller stream and the formation of ring vortices above and below the impeller, which depend on the clearance; the flow was strongly three-dimensional with large regions having circumferential velocities in a direction opposite to that of the impeller rotation. Impeller-induced torque measurements show that the Power number is invariant with clearance for turbulent-flow Reynolds numbers ([ges ] 40000) and increases with impeller diameter. The flow structure was controlled mainly by convection and pressure forces with turbulent mixing important in the impeller region.

The response of lean (ϕ ≤ 0.4) premixed hydrogen flames to maintained homogeneous isotropic turbulence is investigated using detailed numerical simulation in an idealised three-dimensional configuration over a range of Karlovitz numbers from 10 to 1562. In particular, a focus is placed on turbulence sufficiently intense that the flames can no longer be considered to be in the thin reaction burning regime. This transition to the so-called distributed burning regime is characterised through a number of diagnostics, and the relative roles of molecular and turbulent mixing processes are examined. The phenomenology and statistics of these flames are contrasted with a distributed thermonuclear flame from a related astrophysical study.

Stereoscopic particle image velocimetry (PIV) measurements are made in streamwise–spanwise and inclined cross-stream planes (inclined at $45^\circ$ and $135^\circ$ to the principal flow direction) of a turbulent boundary layer at moderate Reynolds number ($\hbox{\it Re}_\tau\,{\sim} 1100$). Two-point spatial velocity correlations computed using the PIV data reveal results that are consistent with an earlier study in which packets of hairpin vortices were identified by a feature-detection algorithm in the log region, but not in the outer wake region. Both streamwise–streamwise ($R_{\hbox{\scriptsize\it uu}}$) and streamwise–wall-normal ($R_{uw}$) correlations are significant for streamwise displacements of more than 1500 wall units. Zero crossing data for the streamwise fluctuating component $u$ reveal that streamwise strips between zero crossings of 1500 wall units or longer occur more frequently for negative $u$ than positive $u$, suggesting that long streamwise correlations in $R_{\hbox{\scriptsize\it uu}}$ are dominated by slower streamwise structures. Additional analysis of $R_{ww}$ correlations suggests that the long streamwise slow-moving regions contain discrete zones of strong upwash over extended streamwise distances, as might occur within packets of angled hairpin vortices. At a wall-normal location outside of the log region ($z/\delta \,{=}\, 0.5$), the correlations are shorter in the streamwise direction and broader in the spanwise direction. Correlations in the inclined cross-stream plane data reveal good agreement with the streamwise–spanwise plane. $R_{\hbox{\scriptsize\it uu}}$ in the $45^\circ$ plane is more elongated along the in-plane wall-normal direction than in the $135^\circ$ plane, which is consistent with the presence of hairpin packets with a low-speed region lifting away from the wall.

Computations of channel flow with rough walls comprising staggered arrays of cubes having various plan area densities are presented and discussed. The cube height h is 12.5% of the channel half-depth and Reynolds numbers (uτh/ν) are typically around 700 – well into the fully rough regime. A direct numerical simulation technique, using an immersed boundary method for the obstacles, was employed with typically 35 million cells. It is shown that the surface drag is predominantly form drag, which is greatest at an area coverage around 15%. The height variation of the axial pressure force across the obstacles weakens significantly as the area coverage decreases, but is always largest near the top of the obstacles. Mean flow velocity and pressure data allow precise determination of the zero-plane displacement (defined as the height at which the axial surface drag force acts) and this leads to noticeably better fits to the log-law region than can be obtained by using the zero-plane displacement merely as a fitting parameter. There are consequent implications for the value of von Kármán's constant. As the effective roughness of the surface increases, it is also shown that there are significant changes to the structure of the turbulence field around the bottom boundary of the inertial sublayer. In distinct contrast to two-dimensional roughness (longitudinal or transverse bars), increasing the area density of this three-dimensional roughness leads to a monotonic decrease in normalized vertical stress around the top of the roughness elements. Normalized turbulence stresses in the outer part of the flows are nonetheless very similar to those in smooth-wall flows.

The effect of sloping boundaries on thermal convection is studied theoretically and in the laboratory in the context of a model in which fluid is contained in a differentially heated rectangular box of small aspect ratio (depth/length), inclined at an angle δ to the vertical. Like its two limiting cases, Bénard convection and convection in the vertical slot, a basic state which exists for low Rayleigh numbers becomes unstable as this parameter is increased. The types of instability and indeed the manner in which the motions become turbulent depend crucially on δ. In our work with water the following general picture of the primary instabilities applies:

1. For 90° > δ > 10° with the bottom plate hotter, the instabilities are stationary longitudinal convectively driven rolls with axes oriented up the slope. Near δ = 10° there is an upper and lower Rayleigh number cut off. If the Rayleigh number is too small diffusion damps the instabilities, but if it is too large they are damped by the development of a stable upslope temperature gradient in the mean flow.

2. For 10° > δ > −10° (negative angles imply a hotter upper plate), transverse travelling waves oriented across the slope are the first instabilities of the mean flow. They obtain their kinetic energy via the working of the upslope buoyancy force.

3. For - 10° > δ > −85° longitudinal modes are again observed. These are rather curious in that they may exist when the stratification $-\hat{g}\cdot\nabla T $ is everywhere positive. The necessary energy for these modes comes out of the mean velocity field and out of the mean available potential energy.

Agreement between the stability theory and the experiments is generally quite good over the whole range of δ, considering the approximations involved in finding a suitable basic flow solution.

For Rayleigh numbers less than ∼ 106 turbulence is only possible for positive angles. For 85° > δ > 20° the development of unsteadiness involves the occurrence and the breaking of wavy longitudinal vortices in a manner reminiscent of the development of turbulence in cylindrical Couette flow.

Bellhouse et al. (1973) have developed a high-efficiency membrane oxygenator which utilizes pulsatile flow through furrowed channels to achieve high mass transfer rates. We present numerical solutions of the time-dependent two-dimensional Navier–Stokes equations in order to show the structure of the flow. Experimental observations which support this work are presented in a companion paper (Stephanoff, Sobey & Bellhouse 1980).

Steady flow through a furrowed channel will separate provided the Reynolds number is sufficiently large. The effect of varying the Reynolds number and the geometric parameters is given and comparisons with solutions calculated using the modern boundary-layer theory of Smith (1976) show excellent agreement. Unsteady flow solutions are given as the physical and geometric parameters are varied. The structure of the flow patterns leads to an explanation of the high efficiency of the devices of Bellhouse.

Numerical simulations of turbulent polymer solutions using the FENE-P model are used to characterize the action of polymers on turbulence in drag-reduced flows. The energetics of turbulence is investigated by correlating the work done by polymers on the flow with turbulent structures. Polymers are found to store and to release energy to the flow in a well-organized manner. The storage of energy occurs around near-wall vortices as has been anticipated for a long time. Quite unexpectedly, coherent release of energy is observed in the very near-wall region. Large fluctuations of polymer work are shown to re-energize decaying streamwise velocity fluctuations in high-speed streaks just above the viscous sublayer. These distinct behaviours are used to propose an autonomous regeneration cycle of polymer wall turbulence, in the spirit of Jiménez & Pinelli (1999).

The unifying theory of scaling in thermal convection (Grossmann & Lohse, J. Fluid. Mech., vol. 407, 2000, pp. 27–56; henceforth the GL theory) suggests that there are no pure power laws for the Nusselt and Reynolds numbers as function of the Rayleigh and Prandtl numbers in the experimentally accessible parameter regime. In Grossmann & Lohse (Phys. Rev. Lett., vol. 86, 2001, pp. 3316–3319) the dimensionless parameters of the theory were fitted to 155 experimental data points by Ahlers & Xu (Phys. Rev. Lett., vol. 86, 2001, pp. 3320–3323) in the regime $3\times 1{0}^{7} \leq \mathit{Ra}\leq 3\times 1{0}^{9} $ and $4\leq \mathit{Pr}\leq 34$ and Grossmann & Lohse (Phys. Rev. E, vol. 66, 2002, p. 016305) used the experimental data point from Qiu & Tong (Phys. Rev. E, vol. 64, 2001, p. 036304) and the fact that $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ is independent of the parameter $a$, which relates the dimensionless kinetic boundary thickness with the square root of the wind Reynolds number, to fix the Reynolds number dependence. Meanwhile the theory is, on the one hand, well-confirmed through various new experiments and numerical simulations; on the other hand, these new data points provide the basis for an updated fit in a much larger parameter space. Here we pick four well-established (and sufficiently distant) $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ data points and show that the resulting $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ function is in agreement with almost all established experimental and numerical data up to the ultimate regime of thermal convection, whose onset also follows from the theory. One extra $\mathit{Re}(\mathit{Ra}, \mathit{Pr})$ data point is used to fix $\mathit{Re}(\mathit{Ra}, \mathit{Pr})$. As $\mathit{Re}$ can depend on the definition and the aspect ratio, the transformation properties of the GL equations are discussed in order to show how the GL coefficients can easily be adapted to new Reynolds number data while keeping $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ unchanged.

This work determines the pressure–velocity relation of bubble flow in polygonal capillaries. The liquid pressure drop needed to drive a long bubble at a given velocity U is solved by an integral method. In this method, the pressure drop is shown to balance the drag of the bubble, which is determined by the films at the two ends of the bubble. Using the liquid-film results of Part 1 (Wong, Radke & Morris 1995), we find that the drag scales as Ca2/3 in the limit Ca → 0 (Ca μU/σ, where μ is the liquid viscosity and σ the surface tension). Thus, the pressure drop also scales as Ca2/3. The proportionality constant for six different polygonal capillaries is roughly the same and is about a third that for the circular capillary.

The liquid in a polygonal capillary flows by pushing the bubble (plug flow) and by bypassing the bubble through corner channels (corner flow). The resistance to the plug flow comes mainly from the drag of the bubble. Thus, the plug flow obeys the nonlinear pressure–velocity relation of the bubble. Corner flow, however, is chiefly unidirectional because the bubble is long. The ratio of plug to corner flow varies with liquid flow rate Q (made dimensionless by σa2/μ, where a is the radius of the largest inscribed sphere). The two flows are equal at a critical flow rate Qc, whose value depends strongly on capillary geometry and bubble length. For the six polygonal capillaries studied, Qc [Lt ] 10−6. For Qc [Lt ] Q [Lt ] 1, the plug flow dominates, and the gradient in liquid pressure varies with Q2/3. For Q [Lt ] Qc, the corner flow dominates, and the pressure gradient varies linearly with Q. A transition at such low flow rates is unexpected and partly explains the complex rheology of foam flow in porous media.

The exact time-dependent three-dimensional Navier-Stokes and temperature equations are integrated numerically to simulate stably stratified homogeneous turbulent shear flows at moderate Reynolds numbers whose horizontal mean velocity and mean temperature have uniform vertical gradients. The method uses shear-periodic boundary conditions and a combination of finite-difference and pseudospectral approximations. The gradient Richardson number Ri is varied between 0 and 1. The simulations start from isotropic Gaussian fields for velocity and temperature both having the same variances.

The simulations represent approximately the conditions of the experiment by Komori et al. (1983) who studied stably stratified flows in a water channel (molecular Prandtl number Pr = 5). In these flows internal gravity waves build up, superposed by hot cells leading to a persistent counter-gradient heat-flux (CGHF) in the vertical direction, i.e. heat is transported from lower-temperature to higher-temperature regions. Further, simulations with Pr = 0.7 for air have been carried out in order to investigate the influence of the molecular Prandtl number. In these cases, no persistent CGHF occurred. This confirms our general conclusion that the counter-gradient heat flux develops for strongly stable flows (Ri ≈ 0.5–1.0) at sufficiently large Prandtl numbers (Pr = 5). The flux is carried by hot ascending, as well as cold descending turbulent cells which form at places where the highest positive and negative temperature fluctuations initially existed. Buoyancy forces suppress vertical motions so that the cells degenerate to two-dimensional fossil turbulence. The counter-gradient heat flux acts to enforce a quasi-static equilibrium between potential and kinetic energy.

Previously derived turbulence closure models for the pressure-strain and pressure-temperature gradients in the equations for the Reynolds stress and turbulent heat flux are tested for moderate-Reynolds-number flows with strongly stable stratification (Ri = 1). These models overestimate the turbulent interactions and underestimate the buoyancy contributions. The dissipative timescale ratio for stably stratified turbulence is a strong function of the Richardson number and is inversely proportional to the molecular Prandtl number of the fluid.

Spatial ‘intermittency’ in the velocity field fine-structure of fully turbulent flow regions, first observed by Batchelor & Townsend (1949), is studied further here in grid-generated nearly isotropic turbulence and on the axis of a round jet. At large enough Reynolds numbers, appropriately filtered hot-wire anemometer signals appear intermittent as the turbulent patterns are convected past the hot wire by the mean flow. Measurements show that there is a decrease in the relative fluid volume (equal to the ‘intermittency factor’) occupied by fine-structure of given size as the turbulence Reynolds number is increased. They show also that, for a fixed Reynolds number, the relative volume is smaller for smaller fine-structure. The average linear dimension of the fine-structure regions turns out to be much larger than the sizes of fine-structure therein. At Rλ, = 110, for example, the ratio ranges from 15 to 30, decreasing with decreasing ‘eddy’ size. It appears to be approaching an asymptote with increasing Rλ.

The flatness factors and probability distributions of the first derivative, the second derivative, band-passed and high-passed velocity fluctuation signals were also measured. The turbulence Reynolds numbers Rλ ranged from 12 to 830. The flatness factors of the first and the second derivatives increase monotonically with Rλ. Those of the second derivative vary with Rλ0.25 for Rλ < 100, and with Rλ0.75 for Rλ > 300. No indication of asymptotic constant values was observed for Rλ up to the order of one thousand.

The probability distributions of velocity fluctuations and large-scale signals are nearly normal, while the small-scale signals are not. The flatness factor of the filtered band-pass velocity signal increases with increasing frequency.

At the larger Reynolds numbers, the square of the signal associated with large wave-numbers may be approximated by a log-normal probability distribution for amplitudes when probabilities fall between 0·3 and 0·95, in limited agreement with the theory of Kolmogorov (1962), Oboukhov (1962), Gurvich & Yaglom (1967).