Turbulent boundary layer skin friction in liquid flows may be reduced when bubbles are present near the surface on which the boundary layer forms. Prior experimental studies of this phenomenon reached downstream-distance-based Reynolds numbers ($Re_{x}$) of several million, but potential applications may occur at $Re_{x}$ orders of magnitude higher. This paper presents results for $Re_{x}$ as high as 210 million from skin-friction drag-reduction experiments conducted in the USA Navy's William B. Morgan Large Cavitation Channel (LCC). Here, a near-zero-pressure-gradient flat-plate turbulent boundary layer was generated on a 12.9 m long hydraulically smooth flat plate that spanned the 3 m wide test section. The test surface faced downward and air was injected at volumetric rates as high as 0.38 m$^{3}$ s$^{-1}$ through one of two flush-mounted 40 $\mu$m sintered-metal strips that nearly spanned the test model at upstream and downstream locations. Spatially and temporally averaged shear stress and bubble-image-based measurements are reported here for nominal test speeds of 6, 12 and 18 m s$^{-1}$. The mean bubble diameter was $\sim$300 $\mu$m. At the lowest test speed and highest air injection rate, buoyancy pushed the air bubbles to the plate surface where they coalesced to form a nearly continuous gas film that persisted to the end of the plate with near-100% skin-friction drag reduction. At the higher two flow speeds, the bubbles generally remained distinct and skin-friction drag reduction was observed when the bubbly mixture was closer to the plate surface than 300 wall units of the boundary-layer flow without air injection, even when the bubble diameter was more than 100 of these wall units. Skin-friction drag reduction was lost when the near-wall shear induced the bubbles to migrate from the plate surface. This bubble-migration phenomenon limited the persistence of bubble-induced skin-friction drag reduction to the first few metres downstream of the air injector in the current experiments.

We propose and numerically implement a constitutive framework for granular media that allows the material to traverse through its many common phases during the flow process. When dense, the material is treated as a pressure-sensitive elasto-viscoplastic solid obeying a yield criterion and a plastic flow rule given by the ${\it\mu}(I)$ inertial rheology of granular materials. When the free volume exceeds a critical level, the material is deemed to separate and is treated as disconnected, stress-free media. A material point method (MPM) procedure is written for the simulation of this model and many demonstrations are provided in different geometries, which highlight the ability of the numerical model to handle transitions through dense and disconnected states. By using the MPM framework, extremely large strains and nonlinear deformations, which are common in granular flows, are representable. The method is verified numerically and its physical predictions are validated against many known experimental phenomena, such as Beverloo’s scaling in silo flows, jointed power-law scaling of the run-out distance in granular-column-collapse problems, and various known behaviours in inclined chute flows.

An investigation has been made of the flow over axisymmetric spiked bodies at a Mach number of 6·8. For some ranges of the ratio of spike length to body diameter the flow was found to be unsteady. The effect of the shape of the body nose on this unsteadiness was investigated and an explanation of the mechanism of the oscillation is given.

Phoretic self-propulsion is a unique example of force- and torque-free motion on small scales. The classical framework describing the flow field around a particle swimming by self-diffusiophoresis neglects the advection of the solute field by the flow and assumes that the chemical interaction layer is thin compared to the particle size. In this paper we quantify and characterize the effect of solute advection on the phoretic swimming of a sphere. We first rigorously derive the regime of validity of the thin-interaction-layer assumption at finite values of the Péclet number (${Pe}$). Under this assumption, we solve computationally the flow around Janus phoretic particles and examine the impact of solute advection on propulsion and the flow created by the particle. We demonstrate that although advection always leads to a decrease of the swimming speed and flow stresslet at high values of the Péclet number, an increase can be obtained at intermediate values of ${Pe}$. This possible enhancement of swimming depends critically on the nature of the chemical interactions between the solute and the surface. We then derive an asymptotic analysis of the problem at small ${Pe}$ which allows us to rationalize our computational results. Our computational and theoretical analysis is accompanied by a parallel study of the influence of reactive effects at the surface of the particle (Damköhler number) on swimming.

This paper is concerned with the behaviour of buoyant plumes driven by a source of bubbles. It is shown experimentally that, when a bubble plume rises through a stratified environment, fluid can be transported vertically for some distance and then some of this fluid can leave the plume and spread out horizontally at its own density level. A simple plume model which regards the plume as a single entity is discussed in order to make a first assessment of the effects of gas expansion and bubble slip velocity in this stratified case. However, the experiments reveal a more complicated plume structure in which the bubbles remain in the centre part of the plume, and only the outer part of the plume spreads out into the environment at certain levels. On the basis of these observations a double-plume model is proposed which regards the plume as being composed of two parts: an inner circular plume (which contains all the bubbles of gas) and an outer annular plume.

The paper presents the results of a detailed experimental examination of fully developed asymmetric flow between parallel planes. The asymmetry was introduced by roughening one of the planes while the other was left smooth; the ratio of the shear stresses at the two surfaces was typically about 4:1.

The main emphasis of the research has been on establishing the turbulence structure, particularly in the central region of the channel where the two dissimilar wall flows (generated by the smooth and rough surfaces) interact. Measurements have included profiles of all non-zero double and triple velocity correlations; spectra of the same correlations at several positions in the channel; skewness and flatness factors; and lateral two-point space correlations of the streamwise velocity fluctuation.

The region of greatest interaction is characterized by strong diffusional transport of turbulent shear stress and kinetic energy from the rough towards the smooth wall region, giving rise, inter alia, to an appreciable separation between the planes of zero shear stress and maximum mean velocity. The profiles of length scales of the larger-scale motion are, in contrast to the turbulent velocity field, nearly symmetric. Moreover, it appears that at high Reynolds numbers the small-scale motion may in many respects be treated as isotropic.

Direct numerical simulation (DNS) and large-eddy simulation (LES) are carried out to investigate the frequency effect of zero-net-mass-flux forcing (synthetic jet) on a generic separated flow. The selected test case is a rounded ramp at a Reynolds number based on the step height of 28 275. The incoming boundary layer is fully turbulent with Rθ=1410. The whole flow in the synthetic jet cavity is computed to ensure an accurate description of the actuator effect on the flow field. In a first step, DNS is used to validate LES of this particular flow. In a second step, the effect of a synthetic jet at two reduced frequencies of 0.5 and 4 (based on the separation length of the uncontrolled case and the free-stream velocity) is investigated using LES. It is demonstrated that, with a proper choice of the oscillating frequency, separation can be drastically reduced for a velocity ratio between the jet and the flow lower than one. The low frequency is close to the natural vortex shedding frequency. Two different modes of the synthetic jet have been identified. A vorticity-dominated mode is observed in the low-frequency forcing case for which the separation length is reduced by 54%, while an acoustic-dominated mode is identified in the high-frequency forcing case for which the separation length is increased by 43%. The decrease of the separation length in the low-frequency forcing case is correlated with an increase of the turbulent kinetic energy level and consequently with an increase of the entrainment in the separated zone. A linear inviscid stability analysis shows that the increase of the separation length in the high-frequency forcing case is due to a modification of the mean velocity profile suggested by Stanek and coworkers. The result is a lower amplification of the perturbations and consequently, a lower entrainment into the mixing layer. To our knowledge, it is the first time that Stanek's hypothesis has been assessed, thanks to numerical simulations of fully turbulent flow.

We study theoretically the slow viscous motion of a long slender drop placed in a simple shear flow, the drop having a low viscosity compared with that of the suspending fluid. As a simplifying approximation, the cross-section of the drop is taken to be circular. An equilibrium shape with the drop nearly aligned with the flow is found for all shear rates, although the equilibrium is only stable to small disturbances for shear rates below some critical value. The stable equilibria just below the critical shear rate are found to be accessible only if the shear rate is increased slowly.

Boundary-forced stratified turbulence is studied in the prototypical case of turbulent channel flow subject to stable stratification. The large-eddy simulation approach is used with a mixed subgrid model that involves a dynamic eddy viscosity component and a scale-similarity component. After an initial transient, the flow reaches a new balanced state corresponding to active wall-bounded turbulence with reduced vertical transport which, for the cases in our study with moderate-to-large levels of stratification, coexists with internal wave activity in the core of the channel. A systematic reduction of turbulence levels, density fluctuations and associated vertical transport with increasing stratification is observed. Countergradient buoyancy flux is observed in the outer region for sufficiently high stratification.

Mixing of the density field in stratified channel flow results from turbulent events generated near the boundaries that couple with the outer, more stable flow. The vertical density structure is thus of interest for analogous boundary-forced mixing situations in geophysical flows. It is found that, with increasing stratification, the mean density profile becomes sharper in the central region between the two turbulent layers at the upper and lower walls, similar to observations in field measurements as well as laboratory experiments with analogous density-mixing situations.

Channel flow is strongly inhomogeneous with alternative choices for the Richardson number. In spite of these complications, the gradient Richardson number, Rig, appears to be the important local determinant of buoyancy effects. All simulated cases show that correlation coefficients associated with vertical transport collapse from their nominal unstratified values over a narrow range, 0.15 < Rig < 0.25. The vertical turbulent Froude number, Frw, has an O(1) value across most of the channel. It is remarkable that stratified channel flow, with such a large variation of overall density difference (factor of 26) between cases, shows a relatively universal behaviour of correlation coefficients and vertical Froude number when plotted as a function of Rig. The visualizations show wavy motion in the core region where the gradient Richardson number, Rig, is large and low-speed streaks in the near-wall region, typical of unstratified channel flow, where Rig is small. It appears from the visualizations that, with increasing stratification, the region with wavy motion progressively encroaches into the zone with active turbulence; the location of Rig ≃ 0.2 roughly corresponds to the boundary between the two zones.

Measurements of turbulence energy diffusion and the spectral distributions of stress components in the core of turbulent pipe flow are presented. The results tend to confirm the proposal of Bradshaw (1967a, b) that an inertial subrange in the spectra can exist at quite modest laboratory Reynolds numbers. They also illuminate the inconsistencies in Laufer's (1954) measurements of dissipation and suggest that the fitting of a −5/3 power law to the spectra may well provide the most accurate method of determining dissipation for Re [gsim ] 105.

Helical vortices in swirl flow are studied theoretically and experimentally.

A theory of helical vortices has been developed. It includes the following results: an analytical solution describing an elementary helical vortex structure – an infinitely thin filament; a solution for axisymmetrical vortices accounting for the helical shape of vortex lines and different laws of vorticity distribution; a formula for calculation of the self-induced velocity of helical vortex rotation (precession) in a cylindrical tube; an explanation of the zone with reverse flow (recirculation zone) arising in swirl flows; and the classification of vortex structures.

The experimental study of helical vortices was carried out in a vertical hydrodynamical vortex chamber with a tangential supply of liquid through turning nozzles. Various vortex structures were formed owing to changing boundary conditions on the bottom and at the exit section of the chamber. The hypothesis of helical symmetry is confirmed for various types of swirl flow. The stationary helical vortex structures are described (most of them for the first time) the features of which agree with the results and predictions of the theoretical model developed. They are the following: a rectilinear vortex; a composite columnar vortex; helical vortices screwed on the right or on the left; a vortex with changing helical symmetry; a double helix – two entangled vortex filaments of the same sign.

The steady movement of a liquid meniscus in a circular capillary tube has been examined theoretically for dynamic contact angles close to 90° with minute slippage of the liquid on the solid, thus relaxing the conventional no-slip boundary condition. The resulting flow field does not produce an unbounded force at the contact line, contrary to that with the no-slip condition. The interfacial velocity, wall stress, fluid pressure and the meniscus shape are calculated, and the significance of dynamic contact-angle measurements is discussed. A modified version of the classical Washburn equation which takes account of the meniscus also reveals the importance of slippage.

This paper presents experimental investigations of the diffraction of plane strong shocks by several cones, a cylinder, and a sphere. The diffraction pattern, in particular the loci of Mach triple points and the shape of the diffracted shocks are compared with theoretical results obtained from a diffraction theory proposed by Whitham (1957, 1958, 1959). The agreement between theory and experiment is shown to be good. Also given are extensive numerical results supplementing Whitham's papers, and theortical considerations applying Whitham's theory to very blunt bodies.

The performance of an electromagnetic flowmeter head is assessed in terms of a weight vector W such that the output voltage ∝ ∫ v. Wdτ, where v is the velocity and τ the flowmeter volume. The condition curl W = 0 with W → 0 at ∞ is shown to be necessary and sufficient for the velocity to depend only on the flow rate and not on the flow pattern. A class of such ‘ideal’ meters is described. It is shown that meters with point electrodes can never be ideal but may, with considerable complication of the magnetic field, be made immune to asymmetric velocity-profile variations if the flow is rectilinear.

A study has been made of the varicose instability of an axisymmetrical jet with a velocity distribution radially uniform at the nozzle mouth except for a laminar boundary layer at the wall. The evolutionary phenomena of instability, such as the rolling up of the cylindrical vortex layer into ring vortices, the coalescence of ring vortex pairs, and the eventual disintegration into turbulent eddies, have been investigated as a function of the Reynolds number using smoke photography, stroboscopic observation, and the light-scatter technique.

Emphasis has been placed on the wavelength with maximum growth rate. The jet is highly sensitive to sound and the effects of several types of acoustic excitation, including pure tones, have been determined.

An aerofoil leading-edge profile based on wavy (sinusoidal) protuberances/tubercles is investigated to understand the mechanisms by which they are able to reduce the noise produced through the interaction with turbulent mean flow. Numerical simulations are performed for non-lifting flat-plate aerofoils with straight and wavy leading edges (denoted by SLE and WLE, respectively) subjected to impinging turbulence that is synthetically generated in the upstream zone (free-stream Mach number of 0.24). Full three-dimensional Euler (inviscid) solutions are computed for this study thereby eliminating self-noise components. A high-order accurate finite-difference method and artefact-free boundary conditions are used in the current simulations. Various statistical analysis methods, including frequency spectra, are implemented to aid the understanding of the noise-reduction mechanisms. It is found with WLEs, unlike the SLE, that the surface pressure fluctuations along the leading edge exhibit a significant source-cutoff effect due to geometric obliqueness which leads to reduced levels of radiated sound pressure. It is also found that there exists a phase interference effect particularly prevalent between the peak and the hill centre of the WLE geometry, which contributes to the noise reduction in the mid- to high-frequency range.

When a rotating layer of fluid is heated uniformly from below and cooled from above, the onset of instability is inhibited by the rotation. The first part of this paper treats the stability problem as it was considered by Chandrasekhar (1953), but with particular emphasis on the physical interpretation of the results. It is shown that the time-dependent (overstable) motions occur because they can reduce the stabilizing effect of rotation. It is also shown that the boundary of a steady convection cell is distorted by the rotation in such a way that the wave length of the cell measured along the distorted boundary is equal to the wavelength of the non-rotating cell. This conservation of cellular wavelength is traced to the constancy of horizontal vorticity in the rotating and non-rotating systems. In the finite-amplitude investigation the analysis, which is pivoted about the linear stability problem, indicates that the fluid can become unstable to finite-amplitude disturbances before it becomes unstable to infinitesimal perturbations. The finite-amplitude motions generate a non-linear vorticity which tends to counteract the vorticity generated by the imposed constraint of rotation. Under experimental conditions the two fluids, mercury and air, which are considered in this paper, will not exhibit this finite amplitude instability. However, a fluid with a sufficiently small Prandtl number will become unstable to finite-amplitude perturbations. The special role of viscosity as an energy releasing mechanism in this problem and in the Orr-Sommerfeld problem suggests that the occurrence of a finite-amplitude instability depends on this dual role of viscosity (i.e. as an energy releasing mechanism as well as the more familiar dissipative mechanism). The relative stability criterion developed by Malkus & Veronis (1958) is used to determine the preferred type of cellular motions which can occur in the fluid. This preferred motion is a function of the Prandtl number and the Taylor number. In the case of air it is shown that overstable square cells become preferred in finite amplitude, even though steady convective motions occur at a lower Rayleigh number.

The stability of natural convection of a viscous fluid in a vertical slot having isothermal side walls of different temperatures is investigated analytically. Both the conduction and boundary-layer régimes are found to be unstable with respect to stationary disturbances in the form of multicellular secondary flows. Theoretical predictions of the critical Rayleigh number and of the form of the secondary flow are verified by experimental measurements.

The objective of this paper is to explain, in as much detail as possible, the physical mechanisms responsible for the reduction of skin friction in a microbubble-laden spatially developing turbulent boundary layer over a flat plate, for $Re_{\theta} = 1430$. Our DNS results with microbubble volume fraction ranging from $\phi_v = 0.001$ to 0.02 show that the presence of bubbles results in a local positive divergence of the fluid velocity, ${\bm \nabla} \,{\bm \cdot}\,{\bm U} > 0$, creating a positive mean velocity normal to (and away from) the wall which, in turn, reduces the mean streamwise velocity and displaces the quasi-streamwise longitudinal vortical structures away from the wall. This displacement has two main effects: (i) it increases the spanwise gaps between the wall streaks associated with the sweep events and reduces the streamwise velocity in these streaks, thus reducing the skin friction by up to 20.2% for $\phi_v = 0.02$; and (ii) it moves the location of peak Reynolds stress production away from the wall to a zone of a smaller transverse gradient of the mean streamwise velocity (i.e. smaller mean shear), thus reducing the production rate of turbulence kinetic energy and enstrophy.

Axisymmetric steady flows driven by an electric field about a deformable fluid drop suspended in an immiscible fluid are studied within the framework of the leaky dielectric model. Deformations of the drop and the flow fields are determined by solving the nonlinear free-boundary problem composed of the Navier-Stokes system governing the flow field and Laplace's system governing the electric field. The solutions are obtained by using the Galerkin finite-element method with an elliptic mesh generation scheme. Under conditions of creeping flow and vanishingly small drop deformations, the results of finite-element computations recover the asymptotic results. When drop deformations become noticeable, the asymptotic results are often found to underestimate both the flow intensity and drop deformation. By tracking solution branches in parameter space with an arc-length continuation method, curves in parameter space of the drop deformation parameter D versus the square of the dimensionless field strength E usually exhibit a turning point when E reaches a critical value Ec. Along such a family of drop shapes, steady solutions do not exist for E > Ec. The nonlinear relationship revealed computationally between D and E2 appears to be capable of providing insight into discrepancies reported in the literature between experiments and predictions based on the asymptotic theory. In some special cases with fluid conductivities closely matched, however, drop deformations are found to grow with E2 indefinitely and no critical value Ec is encountered by the corresponding solution branches. For most cases with realistic values of physical properties, the overall electrohydrodynamic behaviour is relatively insensitive to effects of finite-Reynolds-number flow. However, under extreme conditions when fluids of very low viscosities are involved, computational results illustrate a remarkable shape turnaround phenomenon: a drop with oblate deformation at low field strength can evolve into a prolate-like drop shape as the field strength is increased.