Direct numerical simulations (DNSs) are used to investigate the drag-reducing performance of superhydrophobic surfaces (SHSs) in turbulent channel flow. SHSs combine surface roughness with hydrophobicity and can, in some cases, support a shear-free air–water interface. Slip velocities, wall shear stresses and Reynolds stresses are considered for a variety of SHS microfeature geometry configurations at a friction Reynolds number of Reτ ≈ 180. For the largest microfeature spacing studied, an average slip velocity over 75% of the bulk velocity is obtained, and the wall shear stress reduction is found to be nearly 40%. The simulation results suggest that the mean velocity profile near the superhydrophobic wall continues to scale with the wall shear stress but is offset by a slip velocity that increases with increasing microfeature spacing.

Chaotic mixing of fluids in slow flows is ubiquitous but incompletely understood. However, relatively simple experiments provide a wealth of information regarding mixing mechanisms and indicate the need for complementary theoretical developments in dynamical systems. In this work we presnt a versatile cavity flow apparatus, capable of producing a variety of two-dimensional velocity fields, and use it to conduct a detailed experimental study of mixing in low-Reynolds-number flows. Since the goal is detailed understanding, only two time-periodic co-rotating flows induced by wall motions are considered: one continuous and the other discontinuous. Both types of flows produce exponential growth of intermaterial area, as expected from chaotic flows, and a mixture of islands and chaotic regions. A procedure for identifying periodic points and determining their movements is presented as well as how to make meaningful comparisons between periodic flows. We observe that periodic points move very much as a planetary system; planets (hyperbolic points) have moons (elliptic points) with twice the period of the planets; furthermore the spatial arrangement of periodic points becomes symmetric at regular time intervals. Detailed analyses reveal complex behaviour: birth, bifurcation, and collapse of islands; formation and periodic motion of coherent structures, such as islands and large-scale folds. However, the richness and complexity of the results obtained indicate that these two-dimensional time-periodic systems are far from completely understood and that other wall motions might deserve a similar level of scrutiny.

To investigate the effects of the nozzle-exit conditions on jet flow and sound fields, large-eddy simulations of an isothermal Mach 0.9 jet issued from a convergent-straight nozzle are performed at a diameter-based Reynolds number of $1\times 10^{6}$. The simulations feature near-wall adaptive mesh refinement, synthetic turbulence and wall modelling inside the nozzle. This leads to fully turbulent nozzle-exit boundary layers and results in significant improvements for the flow field and sound predictions compared with those obtained from the typical approach based on laminar flow in the nozzle. The far-field pressure spectra for the turbulent jet match companion experimental measurements, which use a boundary-layer trip to ensure a turbulent nozzle-exit boundary layer to within 0.5 dB for all relevant angles and frequencies. By contrast, the initially laminar jet results in greater high-frequency noise. For both initially laminar and turbulent jets, decomposition of the radiated noise into azimuthal Fourier modes is performed, and the results show similar azimuthal characteristics for the two jets. The axisymmetric mode is the dominant source of sound at the peak radiation angles and frequencies. The first three azimuthal modes recover more than 97 % of the total acoustic energy at these angles and more than 65 % (i.e. error less than 2 dB) for all angles. For the main azimuthal modes, linear stability analysis of the near-nozzle mean-velocity profiles is conducted in both jets. The analysis suggests that the differences in radiated noise between the initially laminar and turbulent jets are related to the differences in growth rate of the Kelvin–Helmholtz mode in the near-nozzle region.

We consider a neutrally buoyant and initially uncharged drop in a second liquid subjected to a uniform electric field. Both liquids are taken to be leaky dielectrics. The jump in electrical properties creates an electric stress balanced by hydrodynamic and capillary stresses. Assuming creeping flow conditions and axisymmetry of the problem, the electric and flow fields are solved numerically withboundary integral techniques. The system is characterized by the physical property ratios R (resistivities), Q (permitivities) and λ (dynamic viscosities). Depending on these parameters, the drop deforms into a prolate or an oblate spheroid. The relative importance of the electric stress and of the drop/medium interfacial tension is measured by the dimensionless electric capillary number, Cae. For λ = 1, we present a survey of the various behaviours obtained for a wide range of R and Q. We delineate regions in the (R,Q)-plane in which the drop either attains a steady shape under any field strength or reaches a fold-point instability past a critical Cae. We identify the latter with linear instability of the steady shape to axisymmetric disturbances. Various break-up modes are identified, as well as more complex behaviours such as bifurcations and transition from unstable to stable solution branches. We also show how the viscosity contrast can stabilize the drop or advance break-up in the different situations encountered for λ = 1.

The motions resulting when a linear, stable salt gradient is heated uniformly and at a steady rate from below are investigated theoretically and by laboratory experiment. A convecting, growing layer is first formed whose depth, temperature and salinity differences from the fluid above, are all increasing as t½. The way in which these quantities depend on the salinity gradient and heating rate is also predicted, and verified experimentally. A stability criterion is then developed which describes the breakdown of the diffusive boundary layer ahead of the advancing front, and leads to an expression for the thickness of the bottom layer when a second layer forms above it. The predicted form of dependence of layer thickness on the given parameters is again borne out by the experiments.

The complex hydrodynamics of water entry by a spinning sphere are investigated experimentally for low Froude numbers. Standard billiard balls are shot down at the free surface with controlled spin around one horizontal axis. High-speed digital video sequences reveal unique hydrodynamic phenomena which vary with spin rate and impact velocity. As anticipated, the spinning motion induces a lift force on the sphere and thus causes significant curvature in the trajectory of the object along its descent, similar to a curveball pitch in baseball. However, the splash and cavity dynamics are highly altered for the spinning case compared to impact of a sphere without spin. As spin rate increases, the splash curtain and cavity form and collapse asymmetrically with a persistent wedge of fluid emerging across the centre of the cavity. The wedge is formed as the sphere drags fluid along the surface, due to the no-slip condition; the wedge crosses the cavity in the same time it takes the sphere to rotate one-half a revolution. The spin rate relaxation time plateaus to a constant for tangential velocities above half the translational velocity of the sphere. Non-dimensional time to pinch off scales with Froude number as does the depth of pinch-off; however, a clear mass ratio dependence is noted in the depth to pinch off data. A force model is used to evaluate the lift and drag forces on the sphere after impact; resulting forces follow similar trends to those found for spinning spheres in oncoming flow, but are altered as a result of the subsurface air cavity. Images of the cavity and splash evolution, as well as force data, are presented for a range of spin rates and impact speeds; the influence of sphere density and diameter are also considered.

The dynamics of an inverted flag are investigated experimentally in order to find the conditions under which self-excited flapping can occur. In contrast to a typical flag with a fixed leading edge and a free trailing edge, the inverted flag of our study has a free leading edge and a fixed trailing edge. The behaviour of the inverted flag can be classified into three regimes based on its non-dimensional bending stiffness scaled by flow velocity and flag length. Two quasi-steady regimes, straight mode and fully deflected mode, are observed, and a limit-cycle flapping mode with large amplitude appears between the two quasi-steady regimes. Bistable states are found in both straight to flapping mode transition and flapping to deflected mode transition. The effect of mass ratio, relative magnitude of flag inertia and fluid inertia, on the non-dimensional bending stiffness range for flapping is negligible, unlike the instability of the typical flag. Because of the unsteady fluid force, a flapping sheet can produce elastic strain energy several times larger than a sheet of the deformed mode, improving the conversion of fluid kinetic energy to elastic strain energy. According to the analysis of the leading-edge vortex formation process, the time scale of optimal vortex formation correlates with efficient conversion to elastic strain energy during bending.

Thrust performance and wake structure were investigated for a rigid rectangular panel pitching about its leading edge in a free stream. For ReC = O(104), thrust coefficient was found to depend primarily on Strouhal number St and the aspect ratio of the panel AR. Propulsive efficiency was sensitive to aspect ratio only for AR less than 0.83; however, the magnitude of the peak efficiency of a given panel with variation in Strouhal number varied inversely with the amplitude to span ratio A/S, while the Strouhal number of optimum efficiency increased with increasing A/S. Peak efficiencies between 9% and 21% were measured. Wake structures corresponding to a subset of the thrust measurements were investigated using dye visualization and digital particle image velocimetry. In general, the wakes divided into two oblique jets; however, when operating at or near peak efficiency, the near wake in many cases represented a Kármán vortex street with the signs of the vortices reversed. The three-dimensional structure of the wakes was investigated in detail for AR = 0.54, A/S = 0.31 and ReC = 640. Three distinct wake structures were observed with variation in Strouhal number. For approximately 0.20 < St < 0.25, the main constituent of the wake was a horseshoe vortex shed by the tips and trailing edge of the panel. Streamwise variation in the circulation of the streamwise horseshoe legs was consistent with a spanwise shear layer bridging them. For St > 0.25, a reorganization of some of the spanwise vorticity yielded a bifurcating wake formed by trains of vortex rings connected to the tips of the horseshoes. For St > 0.5, an additional structure formed from a perturbation of the streamwise leg which caused a spanwise expansion. The wake model paradigm established here is robust with variation in Reynolds number and is consistent with structures observed for a wide variety of unsteady flows. Movies are available with the online version of the paper.

An investigation of the near-field flow structure of coaxial jets with large outer to inner velocity ratio ru has been conducted. Since in all cases ru>1, the outer jet dominates the near-field flow structure. Two flow regimes are identified depending on whether ru is larger or smaller than a critical value ruc. When ru<ruc, the fast annular jet periodically pinches the central, slow jet near the end of the inner potential cone. The pinching frequency corresponds to the outer-jet mode. The length of the inner potential cone is strongly dependent on ru and behaves like A/ru, where A depends weakly on the initial conditions. When ru>ruc, the inner potential cone is truncated and is followed by an unsteady recirculation bubble with low-frequency oscillation.

The transition from one regime to another is explained by a simple model whose ingredients are the turbulent entrainment rate, governed by the outer-jet mixing layers and mass conservation. This model satisfactorily predicts the dependence of the inner potential cone length on ru and the critical velocity ratio ruc. The recirculation bubble has a wake-type instability. It oscillates at a low frequency and a large amplitude compared to the Kelvin–Helmholtz mode. Angular cross-correlations in the plane parallel to the jet outlet show moreover that this oscillation displays an azimuthal precession such that the rotation time of the phase of the oscillation equals the oscillation period. These salient features are discussed in the framework of the nonlinear delayed saturation (NLDS) model.

In this paper we present an experimental study of the long-surface-wave instability that can develop when a granular material flows down a rough inclined plane. The threshold and the dispersion relation of the instability are precisely measured by imposing a controlled perturbation at the entrance of the flow and measuring its evolution down the slope. The results are compared with the prediction of a linear stability analysis conducted in the framework of the depth-averaged or Saint-Venant equations. We show that when the friction law proposed in Pouliquen (1999a) is introduced in the Saint-Venant equations, the theory is able to predict quantitatively the stability threshold and the phase velocity of the waves but fails in predicting the observed cutoff frequency. The instability is shown to be of the same nature as the long-wave instability observed in classical fluids but with characteristics that can dramatically differ due to the specificity of the granular rheology.

The interaction of a turbulent eddy with a semi-infinite poroelastic edge is examined with respect to the effects of both elasticity and porosity on the efficiency of aerodynamic noise generation. The scattering problem is solved using the Wiener–Hopf technique to identify the scaling dependence of the resulting aerodynamic noise on plate and flow properties, including the dependence on a characteristic flow velocity $U$. Special attention is paid to the limiting cases of porous-rigid and impermeable–elastic plate conditions. Asymptotic analysis of these special cases reveals parametric limits where the far-field acoustic power scales like ${U}^{6} $ for a porous edge, and a new finite range of ${U}^{7} $ behaviour is found for an elastic edge, to be compared with the well-known ${U}^{5} $ dependence for a rigid impermeable edge. Further numerical results attempt to address how trailing-edge noise may be mitigated by porosity and flexibility and seek to deepen the understanding of how owls hunt in acoustic stealth.

We investigate the changes to a fully developed turbulent boundary layer caused by the presence of a two-dimensional moving wave of wavelength L = 2π/k and amplitude a. Attention is focused on small slopes, ak, and small wave speeds, c, so that the linear perturbations are calculated as asymptotic sequences in the limit (u* + c)/UB(L) → 0 (u* is the unperturbed friction velocity and UB(L) is the approach-flow mean velocity at height L). The perturbations can then be described by an extension of the four-layer asymptotic structure developed by Hunt, Leibovich & Richards (1988) to calculate the changes to a boundary layer passing over a low hill.

When (u* + c)/UB(L) is small, the matched height, zm (the height where UB equals c), lies within an inner surface layer, where the perturbation Reynolds shear stress varies only slowly. Solutions across the matched height are then constructed by considering an equation for the shear stress. The importance of the shear-stress perturbation at the matched height implies that the inviscid theory of Miles (1957) is inappropriate in this parameter range. The perturbations above the inner surface layer are not directly influenced by the matched height and the region of reversed flow below zm: they are similar to the perturbations due to a static undulation, but the ‘effective roughness length’ that determines the shape of the unperturbed velocity profile is modified to zm = z0 exp (kc/u*).

The solutions for the perturbations to the boundary layer are used to calculate the growth rate of waves, which is determined at leading order by the asymmetric pressure perturbation induced by the thickening of the perturbed boundary layer on the leeside of the wave crest. At first order in (u* + c)/UB(L), however, there are three new effects which, numerically, contribute significantly to the growth rate, namely: the asymmetries in both the normal and shear Reynolds stresses associated with the leeside thickening of the boundary layer, and asymmetric perturbations induced by the varying surface velocity associated with the fluid motion in the wave; further asymmetries induced by the variation in the surface roughness along the wave may also be important.

A new method using a matched asymptotic expansions technique is presented for obtaining the Stokes flow solution for a rigid sphere of radius a moving uniformly in a direction parallel to a fixed infinite plane wall when the minimum clearance ea between the sphere and the plane is very much less than a. An ‘inner’ solution is constructed valid for the region in the neighbourhood of the nearest points of the sphere and the plane where the velocity gradients and pressure are large; in this region the leading term of the asymptotic expansion of the solution satisfies the equations of lubrication theory. A matching ‘outer’ solution is constructed which is valid in the remainder of the fluid where velocity gradients are moderate but it is possible to assume that ε = 0. The forces and couples acting on the sphere and the plane are shown to be of the form (α0+α1ε) log ε + β0 + O(ε) where α0, α1 and β0 are constants which have been determined explicitly.

The Rayleigh number at which steady convective flow changes to time-dependent flow is determined experimentally for several fluids with Prandtl numbers from 1 to 104. The time dependence is of two forms: (i) a slow tilting of the cell boundary, with time scale of the vertical thermal diffusion time, (ii) an oscillation with a faster time scale determined by the orbit time of the fluid around the cell. The nature of this oscillation is one of hot (or cold) spots advected with the original cellular motion. At a fixed point in the fluid this produces a time periodic oscillation of the temperature. A discrete change of slope of the heat flux curve accompanies this transition. As the Rayleigh number is increased, transition to disorder is seen to result from an increase in the frequency and number of these oscillations.

Turbulence measurements for rough-wall boundary layers are presented and compared to those for a smooth wall. The rough-wall experiments were made on a woven mesh surface at Reynolds numbers approximately equal to those for the smooth wall. Fully rough conditions were achieved. The present work focuses on turbulence structure, as documented through spectra of the fluctuating velocity components, swirl strength, and two-point auto- and cross-correlations of the fluctuating velocity and swirl. The present results are in good agreement, both qualitatively and quantitatively, with the turbulence structure for smooth-wall boundary layers documented in the literature. The boundary layer is characterized by packets of hairpin vortices which induce low-speed regions with regular spanwise spacing. The same types of structure are observed for the rough- and smooth-wall flows. When the measured quantities are normalized using outer variables, some differences are observed, but quantitative similarity, in large part, holds. The present results support and help to explain the previously documented outer-region similarity in turbulence statistics between smooth- and rough-wall boundary layers.

We investigate air entrainment and bubble statistics in three-dimensional breaking waves through novel direct numerical simulations of the two-phase air–water flow, resolving the length scales relevant for the bubble formation problem, the capillary length and the Hinze scale. The dissipation due to breaking is found to be in good agreement with previous experimental observations and inertial scaling arguments. The air entrainment properties and bubble size statistics are investigated for various initial characteristic wave slopes. For radii larger than the Hinze scale, the bubble size distribution, can be described by $N(r,t)=B(V_{0}/2{\rm\pi})({\it\varepsilon}(t-{\rm\Delta}{\it\tau})/Wg)r^{-10/3}r_{m}^{-2/3}$ during the active breaking stages, where ${\it\varepsilon}(t-{\rm\Delta}{\it\tau})$ is the time-dependent turbulent dissipation rate, with ${\rm\Delta}{\it\tau}$ the collapse time of the initial air pocket entrained by the breaking wave, $W$ a weighted vertical velocity of the bubble plume, $r_{m}$ the maximum bubble radius, $g$ gravity, $V_{0}$ the initial volume of air entrained, $r$ the bubble radius and $B$ a dimensionless constant. The active breaking time-averaged bubble size distribution is described by $\bar{N}(r)=B(1/2{\rm\pi})({\it\epsilon}_{l}L_{c}/Wg{\it\rho})r^{-10/3}r_{m}^{-2/3}$, where ${\it\epsilon}_{l}$ is the wave dissipation rate per unit length of breaking crest, ${\it\rho}$ the water density and $L_{c}$ the length of breaking crest. Finally, the averaged total volume of entrained air, $\bar{V}$, per breaking event can be simply related to ${\it\epsilon}_{l}$ by $\bar{V}=B({\it\epsilon}_{l}L_{c}/Wg{\it\rho})$, which leads to a relationship for a characteristic slope, $S$, of $\bar{V}\propto S^{5/2}$. We propose a phenomenological turbulent bubble break-up model based on earlier models and the balance between mechanical dissipation and work done against buoyancy forces. The model is consistent with the numerical results and existing experimental results.

Overall heat transfer and mean temperature distribution measurements have been made of turbulent thermal convection in horizontal water layers heated from below. The Nusselt number is found to be proportional to Ra0·278 in the range 2·76 × 105 < Ra < 1·05 × 108. Eight discrete heat flux transitions are found in this Rayleigh number range. An interferometric method is used to measure the mean temperature distribution for Rayleigh numbers between 3·11 × 105 and 1·86 × 107. Direct visual and photographic observations of the fluctuating interferogram patterns show that the main heat transfer mechanism is the release of thermals from the boundary layers. For relatively low Rayleigh numbers (up to 5 × 105) many of the thermals reach the opposite surface and coalesce to form large masses of relatively warm fluid near the cold surface and masses of cold fluid near the warm surface, resulting in a temperature-gradient reversal. With increasing Rayleigh numbers, fewer and fewer thermals reach the opposite bounding surface and the thermals show persistent horizontal movements near the bounding surfaces. The central region of the layer becomes an isothermal core. The mean temperature distributions for the high Rayleigh number range are found to follow a Z−2 power law over a considerable range, where Z is the distance from the bounding surface. A very limited agreement with the theoretically predicted Z−1 power law is also found.

Low-Reynolds-number, mildly curved, turbulent channel flow has been simulated by direct numerical solution of the Navier – Stokes equations. Computed velocity fields were found to be in good agreement with experimental measurements. The resulting flow fields were used to study the effects of streamline curvature by comparing the concave and convex sides of the channel. Observed effects are consistent with experimental measurements for mild curvature. The most significant difference in the turbulence statistics is in the Reynolds shear stress. This is accompanied by significant differences in the terms of the equation for Reynolds-shear-stress budget. In addition, it was found that stationary Taylor – Görtler vortices were present and that they had a significant effect on the flow by contributing to the mean Reynolds shear stress, enhancing the asymmetry of the channel, and affecting the underlying turbulence.

The hydrodynamic stability of a pure liquid undergoing steady rapid evaporation at reduced pressure is examined using linear stability analysis. Results show that the rapidly evaporating liquid is unstable to local variations in evaporation rate, local surface depressions being produced by the force exerted on the surface by the rapidly departing vapour and sustained liquid flows being driven by the resultant shear exerted on the liquid surface by the vapour. The coupling of this ‘differential vapour recoil’ mechanism to the Marangoni effect is investigated and the importance of inertial heat transfer, fluid inertia and viscous dissipation at the interface to system stability is resolved.

We present an experimental investigation of the flow structure and vorticity field in the wake of a NACA-0012 airfoil pitching sinusoidally at small amplitude and high reduced frequencies. Molecular tagging velocimetry is used to quantify the characteristics of the vortex array (circulation, peak vorticity, core size, spatial arrangement) and its downstream evolution over the first chord length as a function of reduced frequency. The measured mean and fluctuating velocity fields are used to estimate the mean force on the airfoil and explore the connection between flow structure and thrust generation.

Results show that strong concentrated vortices form very rapidly within the first wavelength of oscillation and exhibit interesting dynamics that depend on oscillation frequency. With increasing reduced frequency the transverse alignment of the vortex array changes from an orientation corresponding to velocity deficit (wake profile) to one with velocity excess (reverse Kármán street with jet profile). It is found, however, that the switch in the vortex array orientation does not coincide with the condition for crossover from drag to thrust. The mean force is estimated from a more complete control volume analysis, which takes into account the streamwise velocity fluctuations and the pressure term. Results clearly show that neglecting these terms can lead to a large overestimation of the mean force in strongly fluctuating velocity fields that are characteristic of airfoils executing highly unsteady motions. Our measurements show a decrease in the peak vorticity, as the vortices convect downstream, by an amount that is more than can be attributed to viscous diffusion. It is found that the presence of small levels of axial velocity gradients within the vortex cores, levels that can be difficult to measure experimentally, can lead to a measurable decrease in the peak vorticity even at the centre of the flow facility in a flow that is expected to be primarily two-dimensional.