The structure and evolution of homogeneous stably stratified sheared turbulence have been investigated through direct numerical simulation. In these simulations the primary dimensionless parameter is the Richardson number which measures the relative importance of stratification and mean shear.

For Richardson numbers less than the transition value the Reynolds stress and vertical density flux are down-gradient. Some of the vertical kinetic energy gained indirectly through production is expended in creating potential energy. Included in this shear-dominated regime is the stationary Richardson number at which the turbulent kinetic energy is constant in time although the spectra are evolving. At low dimensionless shear rate the stationary Richardson number increases with increasing Reynolds number.

At the transition Richardson number the maximum anisotropy and energy partition are achieved. For larger Richardson numbers potential energy is released into vertical kinetic energy and the vertical density flux becomes counter-gradient. The associated production reversal enhances the decay rate of the turbulent kinetic energy.

The effects of other dimensionless parameters have been investigated. After initial transients the developed flow is rather insensitive to the presence of significant initial potential energy. An increase in the Schmidt number increases the effect of stable stratification, e.g. the counter-gradient vertical density flux occurs earlier.

In the shear dominated case the down-gradient fluxes are produced by the pumping of fluid through coherent hairpin-shaped vorticity. In the buoyancy dominated flow the counter-gradient fluid parcels induce helical vorticity structures as they move toward a position of neutral buoyancy.

The three-dimensional flow past two identical spherical bubbles moving side by side in a viscous fluid is studied numerically by solving the full Navier–Stokes equations. The bubble surface is assumed to be clean so that the outer flow obeys a zero-shear-stress condition. The present study describes the interaction between the two bubbles over a wide range of Reynolds number ($0.02\,{\le}\,Re\,{\le}\,500$, $Re$ being based on the bubble diameter and rise velocity), and separation $S$ ($2.25\,{\le}\,S\,{\le}\,20$, S being the distance between the bubble centres normalized by the bubble radius). The flow structure, the vorticity field, the sign of the interaction force and the magnitude of the drag and lift forces are analysed; in particular the latter are compared with analytical expressions available in the potential flow limit and in the limit of low-but-finite Reynolds number. This study sheds light on the role of the vorticity generated at the bubble surface in the interaction process. When vorticity remains confined in a boundary layer whose thickness is small compared to the distance between the two bubbles, the interaction is dominated by the irrotational mechanism that results in an attractive transverse force. In contrast, when viscous effects are sufficiently large, the vorticity field about each bubble interacts with that existing about the other bubble, resulting in a repulsive transverse force. Computational results combined with available high-Reynolds-number theory provide empirical expressions for the drag and lift forces in the moderate-to-large Reynolds number regime. They show that the transverse force changes sign for a critical Reynolds number whose value depends on the separation. Using these computational results it is shown that, depending on their initial separation, freely moving bubbles may either reach a stable equilibrium separation or move apart from each other up to infinity.

Gudmundsson and Colonius (J. Fluid Mech., vol. 689, 2011, pp. 97–128) have recently shown that the average evolution of low-frequency, low-azimuthal modal large-scale structures in the near field of subsonic jets are remarkably well predicted as linear instability waves of the turbulent mean flow using parabolized stability equations. In this work, we extend this modelling technique to an isothermal and a moderately heated Mach 1.5 jet for which the mean flow fields are obtained from a high-fidelity large-eddy simulation database. The latter affords a rigourous and extensive validation of the model, which had only been pursued earlier with more limited experimental data. A filter based on proper orthogonal decomposition is applied to the data to extract the most energetic coherent components. These components display a distinct wavepacket character, and agree fairly well with the parabolized stability equations model predictions in terms of near-field pressure and flow velocity. We next apply a Kirchhoff surface acoustic propagation technique to the near-field pressure model and obtain an encouraging match for far-field noise levels in the peak aft direction. The results suggest that linear wavepackets in the turbulence are responsible for the loudest portion of the supersonic jet acoustic field.

A number of new experiments have been performed on the rise of air bubbles in clean mixtures of distilled water and pure, reagent grade, glycerine covering a range of the relevant parameter, the Morton number, Mo = gv4ρ3/σ3, of 1013. Here g is the acceleration due to gravity, v the kinematic viscosity, ρ the density and σ the surface tension of the mixture. In these careful measurements several scaling regimes have been found that have not been discussed before in the extensive literature on the subject. The transitions between these regimes have been delineated and attempts made to discuss the dynamical processes that might be important in each of them.

Direct numerical simulations (DNS) are conducted for turbulent flow through pipes with three-dimensional sinusoidal roughnesses explicitly represented by body-conforming grids. The same viscous-scaled roughness geometry is first simulated at a range of different Reynolds numbers to investigate the effects of low Reynolds numbers and low $R_{0}/h$, where $R_{0}$ is the pipe radius and $h$ is the roughness height. Results for the present class of surfaces show that the Hama roughness function ${\rm\Delta}U^{+}$ is only marginally affected by low Reynolds numbers (or low $R_{0}/h$), and observations of outer-layer similarity (or lack thereof) show no signs of sensitivity to Reynolds number. Then, building on this, a systematic approach is taken to isolate the effects of roughness height $h^{+}$ and wavelength ${\it\lambda}^{+}$ in a turbulent wall-bounded flow in both transitionally rough and fully rough regimes. Current findings show that while the effective slope $\mathit{ES}$ (which for the present sinusoidal surfaces is proportional to $h^{+}/{\it\lambda}^{+}$) is an important roughness parameter, the roughness function ${\rm\Delta}U^{+}$ must also depend on some measure of the viscous roughness height. A simplistic linear–log fit clearly illustrates the strong correlation between ${\rm\Delta}U^{+}$ and both the roughness average height $k_{a}^{+}$ (which is related to $h^{+}$) and $\mathit{ES}$ for the surfaces simulated here, consistent with published literature. Various definitions of the virtual origin for rough-wall turbulent pipe flow are investigated and, for the surfaces simulated here, the hydraulic radius of the pipe appears to be the most suitable parameter, and indeed is the only virtual origin that can ever lead to collapse in the total stress. First- and second-order statistics are also analysed and collapses in the outer layer are observed for all cases, including those where the largest roughness height is a substantial proportion of the reference radius (low $R_{0}/h$). These results provide evidence that turbulent pipe flow over the present sinusoidal surfaces adheres to Townsend’s notion of outer-layer similarity, which pertains to statistics of relative motion.

A theory is given for the non-linear transfer of energy from gravity waves on water to capillary waves. When a progressive gravity wave approaches its maximum steepness it develops a sharp crest, at which the surface tension must be locally important. This gives rise to a travelling disturbance which produces a train of capillary waves ahead of the crest, i.e. on the foward face of the gravity wave. The capillary waves, once formed, then take further energy from the gravity wave through the radiation stresses, at the same time losing energy by viscosity.

The steepness of the capillary waves is calculated and found to be in substantial agreement with some observations by C. S. Cox. An approximate expression for the ripple steepness near the crest of the gravity wave is $(2 \pi|3)e^{-g|6T^\prime k^2},$ where T’ is the surface tension constant and K Is the curvature at the wave crest. The ripple steepness also varies with distance from the wave crest.

Under favourable circumstances the dissipation of energy by the capillary waves can be many times the dissipation in the gravity waves. The capillary waves may therefore play a significant role in the generation of waves by wind, in that they tend to delay the onset of breaking.

The optimal energy growth of perturbations sustained by a zero pressure gradient turbulent boundary is computed using the eddy viscosity associated with the turbulent mean flow. It is found that even if all the considered turbulent mean profiles are linearly stable, they support transient energy growths. The most amplified perturbations are streamwise uniform and correspond to streamwise streaks originated by streamwise vortices. For sufficiently large Reynolds numbers two distinct peaks of the optimal growth exist, respectively scaling in inner and outer units. The optimal structures associated with the peak scaling in inner units correspond well with the most probable streaks and vortices observed in the buffer layer, and their moderate energy growth is independent of the Reynolds number. The energy growth associated with the peak scaling in outer units is larger than that of the inner peak and scales linearly with an effective turbulent Reynolds number formed with the maximum eddy viscosity and a modified Rotta–Clauser length based on the momentum thickness. The corresponding optimal perturbations consist of very large–scale structures with a spanwise wavelength of the order of 8δ. The associated optimal streaks scale in outer variables in the outer region and in wall units in the inner region of the boundary layer, in which they are proportional to the mean flow velocity. These outer streaks protrude far into the near wall region, having still 50% of their maximum amplitude at y+ = 20. The amplification of very large–scale structures appears to be a robust feature of the turbulent boundary layer: optimal perturbations with spanwise wavelengths ranging from 4δ to 15δ can all reach 80% of the overall optimal peak growth.

An analytical model is presented for sound radiation from a semi-infinite unflanged annular duct. The duct carries a jet which issues into a uniform mean flow while an inner cylindrical centre body extends downstream from the duct exit. This geometrical arrangement forms an idealized representation of a turbofan exhaust where noise propagates along the annular bypass duct, refracts through the external bypass stream and radiates to the far field. The instability wave of the vortex sheet and its interaction with the acoustic field are accounted for in an exact way in the current solution. Efficient numerical procedures are presented for evaluating near-field and far-field solutions, and these are used as the basis for a parametric study to illustrate the effect of varying the hub–tip ratio, and the ratio of jet velocity to external flow velocity. Since the ‘Kutta’ condition can be turned on and off in the current solution, this capability is used to assess the effect of vortex shedding on noise radiation. Far-field directivity patterns are presented for single modes and also for a multi-mode ‘broadband’ source model in which all cut-on modes are assumed to be present with equal modal power. Good agreement is found between analytical solutions and experimental data. Near-field pressure maps of the acoustic and instability portions of the solution are generated for selected tones.

A mathematical model is presented for the high pressures and sudden velocity changes which may occur in the impact between a region of incompressible liquid and either a solid surface or a second liquid region. The theory rests upon the well-known idea of pressure impulse, for the sudden initiation of fluid motion in incompressible fluids. We consider the impulsive pressure field which occurs when a moving fluid region collides with a fixed target, such as when an ocean wave strikes a sea wall. The boundary conditions are given for modelling liquid-solid and liquid-liquid impact problems. For a given fluid domain, and a given velocity field just before impact, the theory gives information on the peak pressure distribution, and the velocity after impact. Solutions for problems in simple domains are presented, which give insight into the peak pressures exerted by a wave breaking against a sea wall, and a wave impacting in a confined space. An example of liquid-liquid impact is also examined. Results of particular interest include a relative insensitivity to the shape of the incident wave, and an increased pressure impulse when impact occurs in a confined space. The theory predicts that energy is lost from the bulk fluid motion and we suggest that this energy can be transferred to a thin jet of liquid which is projected away from the impact region.

Periodic oscillations are found in a one-dimensional model of thermal convection. The model consists of a fluid-filled tube bent into rectangular shape and standing in a vertical plane. The fluid is heated at the centre of the lower horizontal segment and cooled at the centre of the upper horizontal segment. When a certain parameter exceeds unity, a periodic motion of the fluid is found in which the flow is always in the same direction but in which the speed varies. Inertia is unimportant for this oscillation, which depends upon the interplay between frictional and buoyancy forces.

Prandtl's boundary-layer theory is embedded as the first step in a systematic scheme of successive approximations for finding an asymptotic solution for viscous flow at large Reynolds number. The technique of inner and outer expansions is used to treat this singular-perturbation problem. Only analytic semi-infinite bodies free of separation are considered. The second approximation is analysed in detail for steady laminar flow past plane or axisymmetric solid bodies. Attention is restricted to low speeds and small temperature changes, so that the velocity field is that for an incompressible fluid, the temperature field being calculated subsequently. The additive effects are distinguished of longitudinal curvature, transverse curvature, external vorticity, external stagnation enthalpy gradient, and displacement speed. The effect of changing co-ordinates is examined, and the behaviour of the boundary-layer solution far downstream discussed. Application to specific problems will be made in subsequent papers.

The term ‘electroacoustic’ is used in connection with phenomena that involve the interaction of sound waves and electric fields, such as the voltage difference generated across a transducer by the passage of a sound wave. This paper is concerned with electroacoustic effects in suspensions of electrically charged colloidal particles. The existing methods for calculating electroacoustic effects in suspensions are limited either to the dilute case, or to one particular effect, namely the open-circuit voltage generated between two parallel wire probes by a locally plane sound wave, propagating perpendicular to the wires. In this paper we present a procedure for calculating any electroacoustic effects in suspensions of arbitrary concentration. The only restriction on the method is that the particles must be small compared with the sound wavelength. The procedure involves the solution of a set of differential equations – referred to here as the ‘electroacoustic equations’ – for the macroscopic pressure, velocity and electrical potential in the suspension. The derivation of these equations and the associated boundary conditions form the major part of this paper. General features of the solution are also discussed, and an application is given involving a novel electroacoustic measurement.

An axisymmetric jet at a diameter-based Reynolds number of 1.1 × 104 is computed by a large eddy simulation (LES) in order to investigate its self-similarity region. The LES combines low-dissipation numerical schemes and explicit filtering of the flow variables to relax energy through the smaller scales discretized. The computational domain extends up to 150 jet radii in the downstream direction, which is found to be large enough to discretize a part of this region. Turbulence in the self-preserving jet is characterized by evaluating explicitly from the LES fields the second- and third-order moments of velocity, the pressure–velocity correlations as well as the budgets for the turbulent kinetic energy and for its components. Reference solutions are thus obtained. They agree well with the experimental data given by Panchapakesan & Lumley (J. Fluid Mech., vol. 246, 1963, p. 197) for a jet at the same Reynolds number. The distance required to achieve self-similarity in the LES, around 120 radii from the inflow, is particularly similar to that in the experiment. The discrepancies observed with respect to the data provided by Panchapakesan & Lumley and by Hussein, Capp & George (J. Fluid Mech., vol. 258, 1994, p. 31) for a jet at a higher Reynolds number, specially regarding the turbulence diffusion and the dissipation, are discussed. They appear largely resulting from the approximations made in the experiments to estimate the quantities that cannot be measured with accuracy. The role of the pressure terms in the energy redistribution is also clarified by the LES. Moreover, the turbulent energy budget is calculated in the jet from an equation derived from the filtered compressible Navier–Stokes equations, which includes the dissipation due to the explicit filtering. This has allowed us to assess the behaviour of the LES approach based on relaxation filtering (LES-RF) from the contributions of filtering and viscosity to energy dissipation. The filtering activity is particularly shown to adjust by itself to the grid and flow properties.

The flow inside rectangular and other cavities in a wall has been investigated at low subsonic velocities using oil flow and surface static-pressure distributions. Evidence has been found of regular three-dimensional flows in cavities with large span-to-chord ratios which would normally be considered to have two-dimensional flow near their centre-lines. The dependence of the steadiness of the flow upon the cavity's span as well as its chord and depth has also been observed.

Critical speeds for the onset of Taylor vortices and for the later development of wavy vortices have been determined from torque measurements and visual observations on concentric cylinders of radius ratios R1/R2 = 0·894–0·954 for a range of values of the clearance c and length L: c/R1 = 0·0478–0·119 and L/c = 1–107. Effectively zero variation of the Taylor critical speed with annulus length was observed. The speed at the onset of wavy vortices was found to increase considerably as the annulus length was reduced and theoretical predictions are realistic only for L/c values exceeding say 40. The results were similar for all four clearance ratios examined. Preliminary measurements on eccentrically positioned cylinders with c/R1 = 0·119 showed corresponding effects.

The collision of two equi-sized drops immersed in an immiscible liquid phase undergoing a shear flow in a parallel plate apparatus has been investigated over a range of capillary numbers. The drops were observed along the vorticity direction of shear flow by video enhanced contrast optical microscopy. Images of the colliding drops were processed by image analysis techniques. The distance Δy between the drop centres along the velocity gradient direction was measured as a function of time during approach, collision and separation of the two drops. It was found that Δy increases irreversibly after collision, thus providing a mechanism for drop dispersion in a concentrated system. Drop shape evolution during collision was characterized by measuring a deformation parameter and the angle made by the drop major axis with respect to the velocity gradient direction. The extent of the near-contact region when the drops are sliding on each other was also estimated. Coalescence was a rare event and was observed in the extensional quadrant of the shear flow. The experimental results show good agreement with numerical simulations recently reported in the literature.

Taylor's hypothesis of frozen flow has frequently been used to convert temporal experimental measurements into a spatial domain. This technique has led to the discovery of long meandering structures in the log-region of a turbulent boundary layer. There is some contention over whether Taylor's approximation is valid over large distances. This paper presents an experiment that compares velocity fields constructed using Taylor's approximation with those obtained from particle image velocimetry (PIV), i.e. spatial data, obtained in the logarithmic region of a turbulent boundary layer.

We consider a small droplet of water sitting on top of a heated superhydrophobic surface. A toroidal convection pattern develops in which fluid is observed to rise along the surface of the spherical droplet and to accelerate downwards in the interior towards the liquid/solid contact point. The internal dynamics arise due to the presence of a vertical temperature gradient; this leads to a gradient in surface tension which in turn drives fluid away from the contact point along the interface. We develop a solution to this thermocapillary-driven Marangoni flow analytically in terms of streamfunctions. Quantitative comparisons between analytical and experimental results, as well as effective heat transfer coefficients, are presented.

This paper presents combined theoretical and experimental studies of the two-dimensional piston-like steady-state motions of a fluid in a moonpool formed by two rectangular hulls (e.g. a dual pontoon or catamaran). Vertical harmonic excitation of the partly submerged structure in calm water is assumed. A high-precision analytically oriented linear-potential-flow method, which captures the singular behaviour of the velocity potential at the corner points of the rectangular structure, is developed. The linear steady-state results are compared with new experimental data and show generally satisfactory agreement. The influence of vortex shedding has been evaluated by using the local discrete-vortex method of Graham (1980). It was shown to be small. Thus, the discrepancy between the theory and experiment may be related to the free-surface nonlinearity.

The vortex streets produced by a flapping foil of span to chord aspect ratio of 4:1 are studied in a hydrodynamic tunnel experiment. In particular, the mechanisms giving rise to the symmetry breaking of the reverse Bénard–von Kármán (BvK) vortex street that characterizes fishlike swimming and forward flapping flight are examined. Two-dimensional particle image velocimetry (PIV) measurements in the midplane perpendicular to the span axis of the foil are used to characterize the different flow regimes. The deflection angle of the mean jet flow with respect to the horizontal observed in the average velocity field is used as a measure of the asymmetry of the vortex street. Time series of the vorticity field are used to calculate the advection velocity of the vortices with respect to the free stream, defined as the phase velocity Uphase, as well as the circulation Γ of each vortex and the spacing ξ between consecutive vortices in the near wake. The observation that the symmetry-breaking results from the formation of a dipolar structure from each couple of counter-rotating vortices shed on each flapping period serves as the starting point to build a model for the symmetry-breaking threshold. A symmetry-breaking criterion based on the relation between the phase velocity of the vortex street and an idealized self-advection velocity of two consecutive counter-rotating vortices in the near wake is established. The predicted threshold for symmetry breaking accounts well for the deflected wake regimes observed in the present experiments and may be useful to explain other experimental and numerical observations of similar deflected propulsive vortex streets reported in the literature.