We study the viscous-fingering instability in a radial Hele-Shaw cell in which the top boundary has been replaced by a thin elastic sheet. The introduction of wall elasticity delays the onset of the fingering instability to much larger values of the injection flow rate. Furthermore, when the instability develops, the fingers that form on the expanding air–liquid interface are short and stubby, in contrast with the highly branched patterns observed in rigid-walled cells (Pihler-Puzović et al., Phys. Rev. Lett., vol. 108, 2012, 074502). We report the outcome of a comprehensive experimental study of this problem and compare the experimental observations to the predictions from a theoretical model that is based on the solution of the Reynolds lubrication equations, coupled to the Föppl–von-Kármán equations which describe the deformation of the elastic sheet. We perform a linear stability analysis to study the evolution of small-amplitude non-axisymmetric perturbations to the time-evolving base flow. We then derive a simplified model by exploiting the observations (i) that the non-axisymmetric perturbations to the sheet are very small and (ii) that perturbations to the flow occur predominantly in a small wedge-shaped region ahead of the air–liquid interface. This allows us to identify the various physical mechanisms by which viscous fingering is weakened (or even suppressed) by the presence of wall elasticity. We show that the theoretical predictions for the growth rate of small-amplitude perturbations are in good agreement with experimental observations for injection flow rates that are slightly larger than the critical flow rate required for the onset of the instability. We also characterize the large-amplitude fingering patterns that develop at larger injection flow rates. We show that the wavenumber of these patterns is still well predicted by the linear stability analysis, and that the length of the fingers is set by the local geometry of the compliant cell.

An analysis is presented of the motion of a turbulent thermal in an unstratified environment. Although based upon the entrainment hypothesis introduced by G. I. Taylor (see Morton, Taylor & Turner 1956), the analysis differs from previous work in that it is not limited to small density differences between the thermal and its surroundings. Also, the influence of the virtual mass of the unsteadily moving fluid, ignored by previous investigators, is included and shown to be of significance for any density difference.

Calculations of the temporal variations of size, velocity and density are presented in non-dimensional form for thermals with initial density ratios covering the practically attainable range. It is shown a posteriori that losses of momentum and buoyancy to a wake are probably of negligible influence in any real case.

Experiments were conducted in two curved rectangular ducts of different radii and the results compared with flow in a straight duct. The turbulent boundary layers developing on the curved walls were examined for mean flow properties. The law of the wall takes a modified form and the usual logarithmic portion applies over a very small range of the profiles on the inner walls and a slightly larger range on the outer walls. Similarity based on a defect law did not exist for any part of the flow examined.

A viscous/inviscid interaction method is described and has been used to calculate flows around four distinctly different airfoils as a function of angle of attack. It comprises an inviscid-flow method based on conformal mapping, a boundary-layer procedure based on the numerical solution of differential equations and an algebraic eddy viscosity. The results are in close agreement with experiment up to angles close to stall. In one case, where the airfoil thickness is large, small difficulties were experienced and are described. The method is shown to be capable of obtaining results with large flow separation and quantifies the role of transition on the lift coefficient.

An experimental study has been performed on the dynamics of a large turbulent buoyanthelium plume. Two-dimensional velocity fields were measured using particle image velocimetry (PIV) while helium mass fraction was determined by planar laser-induced fluorescence (PLIF). PIV and PLIF were performed simultaneously in order to obtain velocity and mass fraction data over a plane that encompassed the plume core, the near-field mixing zones and the surrounding air. The Rayleigh–Taylor instability at the base of the plume leads to the vortex that grows to dominate the flow. This process repeats in a cyclical manner. The temporally and spatially resolved data show a strong negative correlation between density and vertical velocity, as well as a strong 90° phase lag between peaks in the vertical and horizontal velocities throughout the flow field owing to large coherent structures associated with puffing of the turbulent plume. The joint velocity an mass fraction data are used to calculate Favre-averaged statistics in addition to Reynolds-(time) averaged statistics. Unexpectedly, the difference between both the Favre-averaged and Reynolds-averaged velocities and second-order turbulent statistics is less than the uncertainty in the data throughout the flow field. A simple analysis was performed to determine the expected differences between Favre and Reynolds statistics for flows with periodic fluctuations in which the density and velocity fields are perfectly correlated, but have the phase relations as suggested by the data. The analytical results agreewith the data, showing that the Favre and Reynolds statistics will be the same to lead order. The combination of observation and simple analysis suggests that for buoyancy-dominated flows in which it can be expected that density and velocity are strongly correlated,phase relations will result in only second-order differences between Favre- and Reynolds-averaged data in spite of strong fluctuations in both density and velocity.

We determine solutions of the Euler equation representing isolated vortices (monopoles, dipoles) in an infinite domain, for arbitrary values of energy, circulation, angular momentum and impulse. A linear relationship between vorticity and stream function is assumed inside the vortex (while the flow is irrotational outside). The emergence of these solutions in a turbulent flow is justified by the statistical mechanics of continuous vorticity fields. The additional restriction of mixing to a ‘maximum-entropy bubble’, due to kinetic constraints, is assumed. The linear relationship between vorticity and stream function is obtained from the statistical theory in the limit of strong mixing (when constraints are weak). In this limit, maximizing entropy becomes equivalent to a kind of enstrophy minimization. New stability criteria are investigated and imply in particular that, in most cases, the vorticity must be continuous (or slightly discontinuous) at the vortex boundary. Then, the vortex radius is automatically determined by the integral constraints and we can obtain a classification of isolated vortices such as monopoles and dipoles (rotating or translating) in terms of a single control parameter. This article generalizes the classification obtained in a bounded domain by Chavanis & Sommeria (1996).

Flow-visualization studies in transitional pipe flow are used to reveal the mechanism responsible for the self-sustenance of a turbulent equilibrium puff. The upstream laminar fluid continuously enters the relatively-slower-moving turbulent puff around the pipe centre. The passage of this high-speed laminar plug flow past the slower fluid that resides near the wall at the upstream interface leads to the shedding of a train of three-dimensional wake-like vortices near the wall. A helical motion near the upstream interface is associated with the vortex-shedding process. The remainder of the puff is a cone of turbulence filled with these wake-like vortices that are decaying slowly; the prominent feature of the decay region is the longitudinal vortices that are apparently undergoing stretching. No toroidal vortex has been observed in the instantaneous flow field at the upstream interface of an individual puff. On the other hand, the wake-like vortices reported here have not been observed before because their three-dimensional and random nature does not allow detection by an ensemble-averaging that is not phase-referenced appropriately.

From hot-wire anemometer measurements in active-grid wind-tunnel turbulence we have determined the Reynolds number dependence of the velocity derivative moments, the mean-squared pressure gradient, $\chi$, and the normalized acceleration variance, $a_0$, over the Reynolds number range $100\,{\leq}\,R_\lambda\,{\leq}\,900$. The values of $\chi$ and $a_0$ were obtained from the fourth-order velocity structure functions. The derivative moments show power-law dependence on Reynolds number and the exponent is the same with or without shear. In particular, we find the derivative kurtosis, $K_{\partial u/\partial x }\,{\sim}\,R_\lambda^{0.39}$, and there is no evidence of the transition that has been observed in this quantity in some recent work. We find that at high Reynolds numbers, $\chi$ and $a_0$ tend to values similar to those obtained by direct particle tracking measurements and by direct numerical simulation. However, at lower Reynolds number our estimates of $\chi$ and $a_0$ appear to be affected by the evaluation technique which imposes strict requirements on local homogeneity and isotropy.

Several experiments have suggested that streamwise vortices, with their accompany-ing low-momentum streaks in a turbulent boundary layer have a characteristic spanwise wavelength of approximately λz+ = 100. Here a mechanism is proposed which selects a comparable spanwise wavelength and produces counter-rotating streamwise vortices in a turbulent boundary layer. Examining the equations which describe the deviation of the velocity field from its time-average, it is found that a resonance (Benney & Gustavsson 1981), is associated with the mean-velocity profile. As an integral part of this resonance, there is a mean secondary flow which has a spanwise wavelength λz+ = 90 and whose velocities exhibit a streamwise vortex structure similar to those observed.

Experiments by Huppert (1982) have demonstrated that a finite volume of fluid placed on an inclined plane will elongate into a thin sheet of fluid as it slides down the plane. If the fluid is initially placed uniformly across the plane, the sheet retains its two-dimensionality for some time, but when it has become sufficiently long and thin, the leading edge develops a spanwise instability. A similarity solution for this motion was derived by Huppert, without taking account of the edge regions where surface tension is important. When these regions are examined, it is found that the conditions at the edges can be satisfied, but only when the singularity associated with the moving contact line is removed. When the sheet is sufficiently elongated, the profile of the free surface shows an upward bulge near the leading edge. It is suggested that the observed instability of the shape of the leading edge is a result of the dynamics of the fluid in this bulge. The related problem of a ridge of fluid sliding down the plane is examined and found to be linearly unstable. The spanwise lengthscale of this instability is, however, dependent on the width of the channel occupied by the fluid, which is at variance with the observed nature of the instability. Preliminary numerical solutions for the nonlinear development of a small disturbance to the position of a straight leading edge show the incipient development of a finger of fluid with a width that does not depend on the channel size, in accordance with the observations.

The flow patterns in the steady, viscous flow in a cylinder with a rotating bottom and a free surface are investigated by a combination of topological and numerical methods. Assuming the flow is axisymmetric, we derive a list of possible bifurcations of streamline structures on varying two parameters, the Reynolds number and the aspect ratio of the cylinder. Using this theory we systematically perform numerical simulations to obtain the bifurcation diagram. The stability limit for steady flow is found and established as a Hopf bifurcation. We compare with the experiments by Spohn, Mory & Hopfinger (1993) and find both similarities and differences.

The lateral migration properties of a rigid spherical particle suspended in a pressure-driven flow through channels with square cross-sections were investigated numerically, in the range of Reynolds numbers ($Re$) from 20 to 1000. The flow field around the particle was computed by the immersed boundary method to calculate the lateral forces exerted on the particle and its trajectories, starting from various initial positions. The numerical simulation showed that eight equilibrium positions of the particle are present at the centres of the channel faces and near the corners of the channel cross-section. The equilibrium positions at the centres of the channel faces are always stable, whereas the equilibrium positions at the corners are unstable until $Re$ exceeds a certain critical value, $Re_{c}$. At $Re\approx Re_{c}$, additional equilibrium positions appear on a heteroclinic orbit that joins the channel face and corner equilibrium positions, and the lateral forces along the heteroclinic orbit are very small. As $Re$ increases, the channel face equilibrium positions are shifted towards the channel wall at first, and then shifted away from the channel wall. The channel corner equilibrium positions exhibit a monotonic shift towards the channel corner with increasing $Re$. Migration behaviours of the particle in the cross-section are also predicted for various values of $Re$. These numerical results account for the experimental observations of particle distributions in the cross-section of micro and millimetre scale channels, including the characteristic alignment and focusing of the particles, the absence of the corner equilibrium positions at low $Re$ and the progressive shift of the equilibrium positions with $Re$.

The wake meandering characteristics of four different wind turbine designs with diameters ranging from a few centimetres (wind tunnel scale) to a hundred metres (utility scale) are investigated using large-eddy simulation with the turbine blades and nacelle parametrised using a new actuator surface model. Different velocity fields and meandering behaviours are observed at near-wake locations. At far-wake locations, on the other hand, the mean velocity deficit profiles begin to collapse when scaled by the centreline velocity deficit based on the incoming wind speed at turbine hub height, suggesting far-wake similarity across scales. The turbine-added turbulence kinetic energy profiles are shown to also nearly collapse with each other in the far wake when normalised using a velocity scale defined by the thrust on the turbine rotor. Moreover, we show that at far-wake locations, the simulated flow fields for all four turbine designs exhibit similar wake meandering characteristics in terms of (1) a Strouhal number independent of rotor designs of different sizes and (2) the distributions of wake meandering wavelengths and amplitudes when normalised by the rotor diameter and a length scale defined by the turbine thrust respectively.

We perform direct numerical simulations (DNS) of emulsions in homogeneous isotropic turbulence using a pseudopotential lattice-Boltzmann (PP-LB) method. Improving on previous literature by minimizing droplet dissolution and spurious currents, we show that the PP-LB technique is capable of long stable simulations in certain parameter regions. Varying the dispersed-phase volume fraction $\unicode[STIX]{x1D719}$, we demonstrate that droplet breakup extracts kinetic energy from the larger scales while injecting energy into the smaller scales, increasingly with higher $\unicode[STIX]{x1D719}$, with approximately the Hinze scale (Hinze, AIChE J., vol. 1 (3), 1955, pp. 289–295) separating the two effects. A generalization of the Hinze scale is proposed, which applies both to dense and dilute suspensions, including cases where there is a deviation from the $k^{-5/3}$ inertial range scaling and where coalescence becomes dominant. This is done using the Weber number spectrum $We(k)$, constructed from the multiphase kinetic energy spectrum $E(k)$, which indicates the critical droplet scale at which $We\approx 1$. This scale roughly separates coalescence and breakup dynamics as it closely corresponds to the transition of the droplet size ($d$) distribution into a $d^{-10/3}$ scaling (Garrett et al., J. Phys. Oceanogr., vol. 30 (9), 2000, pp. 2163–2171; Deane & Stokes, Nature, vol. 418 (6900), 2002, p. 839). We show the need to maintain a separation of the turbulence forcing scale and domain size to prevent the formation of large connected regions of the dispersed phase. For the first time, we show that turbulent emulsions evolve into a quasi-equilibrium cycle of alternating coalescence and breakup dominated processes. Studying the system in its state-space comprising kinetic energy $E_{k}$, enstrophy $\unicode[STIX]{x1D714}^{2}$ and the droplet number density $N_{d}$, we find that their dynamics resemble limit cycles with a time delay. Extreme values in the evolution of $E_{k}$ are manifested in the evolution of $\unicode[STIX]{x1D714}^{2}$ and $N_{d}$ with a delay of ${\sim}0.3{\mathcal{T}}$ and ${\sim}0.9{\mathcal{T}}$ respectively (with ${\mathcal{T}}$ the large eddy timescale). Lastly, we also show that flow topology of turbulence in an emulsion is significantly more different from single-phase turbulence than previously thought. In particular, vortex compression and axial straining mechanisms increase in the droplet phase.

Vortical structures were identified and characterized using velocity fields of turbulent wall-bounded flows. Two direct numerical simulation data sets of fully developed channel flow at Reτ = 934 obtained by del Álamo et al. (J. Fluid Mech., vol. 500, 2004, p. 135) and Reτ = 590 obtained by Moser, Kim & Mansour (Phys. Fluids, vol. 11, 1999, p. 943) as well as dual-plane particle image velocimetry data at z+ = 110 in a zero-pressure-gradient turbulent boundary layer at Reτ = 1160 obtained by Ganapathisubramani, Longmire & Marusic (Phys. Fluids, vol. 18, 2006, 055105) were employed. The three-dimensional swirling strength based on the local velocity gradient tensor was employed to identify vortex core locations. The real eigenvector of the tensor was used both to refine the identification algorithm and to determine the orientation of each vortex core. The identification method allowed cores of nearly all orientations to be analysed. Circulation of each vortical structure was calculated using the vorticity vector projected onto the real eigenvector direction. Various population distributions were then computed at different wall-normal locations including core size, orientation, circulation and propagation velocity. The mean radius of the cores identified was found to increase with increasing wall-normal distance, and the mean circulation increases approximately quadratically with eddy radius. Orientations of cores with stronger circulation were distributed over a much narrower range than those for vortices with weaker circulation and were consistent with legs, necks and heads of forward-leaning hairpin structures.

The motion of membrane-bound objects is important in many aspects of biology and surface chemistry. Here we derive some general relations for objects moving in a surface film overlying a fluid of depth H. A solution to the problem of the drag can be obtained from a two-dimensional system of integral equations. Here we focus on the problem of an ideal needle moving edge-on (in the direction of its tip) or broadside-on (perpendicular to the direction of the tip). It is shown that in comparison to the drag on a circular disk a new scaling regime of the drag on a needle arises when the ratio between surface shear viscosity and subphase viscosity $\eta_s/\eta$ is smaller than the length of the needle.

Solitary waves with constant vorticity in water of finite depth are calculated numerically by a boundary integral equation method. Previous calculations are confirmed and extended. It is shown that there are branches of solutions which bifurcate from a uniform shear current. Some of these branches are characterized by a limiting configuration with a 120° angle at the crest of the wave. Other branches extend for arbitrary large values of the amplitude of the wave. The corresponding solutions ultimately approach closed regions of constant vorticity in contact with the bottom of the channel. A numerical scheme is presented to calculate directly these closed regions of constant vorticity. In addition, it is shown that there are branches of solutions which do not bifurcate from a uniform shear flow.

This paper presents experimental results on the wake of a propelled bluff body towed at a constant horizontal speed in a linearly stratified fluid. Three regimes of the wake have been found, depending on the angle of attack and on the ratio of drag force to propeller thrust. Most of the experiments were obtained in a first regime where a strong momentum flux is created in the wake, which can be oriented backward or forward depending on the ratio of drag force to thrust of the propeller. The velocity amplitude, wake width and Strouhal number of the wake can be predicted by defining a momentum thickness based on the drag coefficient of the bluff body and the thrust of the propeller. A second regime is obtained for a narrow band of towing velocities, with a relative width of 4%, in which the momentum flux is found to vanish. The wake is characterized by the velocity fluctuations; the scaling exponents of the velocity, vorticity and width of the wake are measured. A third regime is obtained for wakes with a small angle of attack, with a null momentum flux. The mean profile of the wake is found to be asymmetric and its amplitude and wake width are measured. Finally, the relevance of these results to the case of a real self-propelled bluff body is discussed. The presence of weak internal waves or of weak fluctuations of background velocity would lead to a wake in the regime with momentum flux, and would allow prediction of the amplitude, width and Strouhal number of the wake.

Linear refraction of waves on inhomogeneous current is known to provoke extreme waves. We investigate the effect of nonlinearity on this phenomenon, with respect to the variation of significant wave height, kurtosis and occurrence of freak waves. Monte Carlo simulations are performed employing a modified nonlinear Schrödinger equation that includes the effects of a prescribed non-potential current. We recommend that freak waves should be defined by a local criterion according to the wave distribution at each location of constant current, not by a global criterion that is either averaged over, or insensitive to, inhomogeneities of the current. Nonlinearity can reduce the modulation of significant wave height. Depending on the configuration of current and waves, the kurtosis and probability of freak waves can either grow or decrease when the wave height increases due to linear refraction. At the centre of an opposing current jet where waves are known to become large, we find that freak waves should be more rare than in the open ocean away from currents. The largest amount of freak waves on an opposing current jet is found at the jet sides where the significant wave height is small.

A laboratory experiment involving rip currents generated on a barred beach with periodic rip channels indicates that rip currents contain energetic low-frequency oscillations in the presence of steady wave forcing. An analytic model for the time-averaged flow in a rip current is presented and its linear stability characteristics are investigated to evaluate whether the rip current oscillations can be explained by a jet instability mechanism. The instability model considers spatially growing disturbances in an offshore directed, shallow water jet. The effects of variable cross-shore bathymetry, non-parallel flow, turbulent mixing, and bottom friction are included in the model. Model results show that rip currents are highly unstable and the linear stability model can predict the scales of the observed unsteady motions.