Drops impacting at low velocities onto a pool surface can stretch out thin hemispherical sheets of air between the drop and the pool. These air sheets can remain intact until they reach submicron thicknesses, at which point they rupture to form a myriad of microbubbles. By impacting a higher-viscosity drop onto a lower-viscosity pool, we have explored new geometries of such air films. In this way we are able to maintain stable air layers which can wrap around the entire drop to form repeatable antibubbles, i.e. spherical air layers bounded by inner and outer liquid masses. Furthermore, for the most viscous drops they enter the pool trailing a viscous thread reaching all the way to the pinch-off nozzle. The air sheet can also wrap around this thread and remain stable over an extended period of time to form a cylindrical air sheet. We study the parameter regime where these structures appear and their subsequent breakup. The stability of these thin cylindrical air sheets is inconsistent with inviscid stability theory, suggesting stabilization by lubrication forces within the submicron air layer. We use interferometry to measure the air-layer thickness versus depth along the cylindrical air sheet and around the drop. The air film is thickest above the equator of the drop, but thinner below the drop and up along the air cylinder. Based on microbubble volumes, the thickness of the cylindrical air layer becomes less than 100 nm before it ruptures.

The dynamic behaviour of vortex cavitation on marine propellers may cause inboard noise and vibration, but is not well understood. The main goal of the present study is to experimentally analyse the dynamics of an isolated tip vortex cavity generated at the tip of a wing of elliptical planform. Detailed high-speed video shadowgraphy was used to determine the cavity deformations in combination with force and sound measurements. The cavity deformations can be divided in different modes, each of which show a distinct dispersion relation between frequency and wavenumber. The dispersion relations show good agreement with an analytical formulation. Finally, experimental support is given to the hypothesis that the resonance frequency of the cavity volume variation is related to a zero group velocity.

A modelling methodology to reproduce the experimental measurements of a turbulent flow in the presence of symmetry is presented. The flow is a three-dimensional wake generated by an axisymmetric body. We show that the dynamics of the turbulent wake flow can be assimilated by a nonlinear two-dimensional Langevin equation, the deterministic part of which accounts for the broken symmetries that occur in the laminar and transitional regimes at low Reynolds numbers and the stochastic part of which accounts for the turbulent fluctuations. Comparison between theoretical and experimental results allows the extraction of the model parameters.

The fluid dynamics of the atmosphere and oceans is to a large extent controlled by the slow evolution of a scalar field called ‘potential vorticity’ (PV), with relatively fast motions such as inertia-gravity waves playing only a minor role. This state of affairs is commonly referred to as ‘balance’. Potential vorticity is a special scalar field which is materially conserved in the absence of diabatic effects and dissipation, effects that are generally weak in the atmosphere and oceans. Moreover, in a balanced flow, PV induces the entire fluid motion and its thermodynamic structure (Hoskins et al., Q. J. R. Meteorol. Soc., vol. 111, 1985, pp. 877–946). While exact balance is generally not achievable, it is now well established that balance holds to a high degree of accuracy in rapidly rotating and strongly stratified flows. Such flows are characterised by both a small Rossby number, $\mathit{Ro}\equiv |{\it\zeta}|_{max}/f$, and a small Froude number, $\mathit{Fr}\equiv |{\bf\omega}_{h}|_{max}/N$, where ${\it\zeta}$ and ${\bf\omega}_{h}$ are the relative vertical and horizontal vorticity components, while $f$ and $N$ are the Coriolis and buoyancy frequencies. In fact, balance can even be a good approximation when $\mathit{Fr}\lesssim \mathit{Ro}\sim \mathit{O}(1)$. In this study, we examine how balance depends specifically on Prandtl’s ratio, $f/N$, in unforced freely evolving turbulence. We examine a wide variety of turbulent flows, at a mature and complex stage of their evolution, making use of the fully non-hydrostatic equations under the Boussinesq and incompressible approximations. We perform numerical simulations at exceptionally high resolution in order to carefully assess the degree to which balance holds, and to determine when it breaks down. For this purpose, it proves most useful to employ an invariant PV-based Rossby number ${\it\varepsilon}$, together with $f/N$. For a given ${\it\varepsilon}$, our key finding is that – for at least tens of characteristic vortex rotation periods – the flow is insensitive to $f/N$ for all values for which the flow remains statically stable (typically $f/N\lesssim 1$). Only the vertical velocity varies in proportion to $f/N$, in line with quasi-geostrophic (QG) scaling for which $\mathit{Fr}^{2}\ll \mathit{Ro}\ll 1$. We also find that as ${\it\varepsilon}$ increases towards unity, the maximum $f/N$ attainable decreases towards 0. No statically stable flows occur for ${\it\varepsilon}\gtrsim 1$. For all stable flows, balance is found to hold to a remarkably high degree: as measured by an energy norm, imbalance never exceeds more than a few per cent of the balance, even in flows where $\mathit{Ro}>1$. The vertical velocity $w$ remains a tiny fraction of the horizontal velocity $\boldsymbol{u}_{h}$, even when $w$ is dominantly balanced. Finally, typical vertical to horizontal scale ratios $H/L$ remain close to $f/N$, as found previously in QG turbulence for which $\mathit{Fr}\sim \mathit{Ro}\ll 1$.

This paper is concerned with the boundary-layer separation in subsonic and transonic flows caused by a two-dimensional isolated wall roughness. The process of the separation is analysed by means of two approaches: the direct numerical simulation (DNS) of the flow using the Navier–Stokes equations, and the numerical solution of the triple-deck equations. Since the triple-deck theory relies on the assumption that the Reynolds number ($\mathit{Re}$) is large, we performed the Navier–Stokes calculations at $\mathit{Re}=4\times 10^{5}$ based on the distance of the roughness element from the leading edge of the flat plate. This $\mathit{Re}$ is also relevant for aeronautical applications. Two sets of calculation were conducted with the free-stream Mach number $\mathit{Ma}_{\infty }=0.5$ and $\mathit{Ma}_{\infty }=0.87$. We used different roughness element heights, some of which were large enough to cause a well-developed separation region behind the roughness. We found that the two approaches generally compare well with one another in terms of wall shear stress, longitudinal pressure gradient and detachment/reattachment points of the separation bubbles (when present). The main differences were found in proximity to the centre of the roughness element, where the wall shear stress and longitudinal pressure gradient predicted by the triple-deck theory are noticeably different from those predicted by DNS. In addition, DNS predicts slightly longer separation regions.

A slow uniform flow of a rarefied gas past a sphere with a uniform temperature is considered. The steady behaviour of the gas is investigated on the basis of the Boltzmann equation by a systematic asymptotic analysis for small Mach numbers in the case where the Knudsen number is finite. Introducing a slowly varying solution whose length scale of variation is much larger than the sphere dimension, the fluid-dynamic-type equations describing the overall behaviour of the gas in the far region are derived. Then, the solution in the near region which varies on the scale of the sphere size, described by the linearised Boltzmann equation, and the solution in the far region, described by the fluid-dynamic-type equations, are sought in the form of a Mach number expansion up to the second order, in a way that they are joined in the intermediate overlapping region. As a result, the drag is derived up to the second order of the Mach number, which formally extends the linear drag obtained by Takata et al. (Phys. Fluids A, vol. 5, 1993, pp. 716–737) to a weakly nonlinear case. Numerical results for the drag on the basis of the Bhatnagar–Gross–Krook (BGK) model are also presented.

The spectral model of Perry et al. (J. Fluid Mech., vol. 165, 1986, pp. 163–199) predicts that the integral length scale varies very slowly with distance to the wall in the intermediate layer. The only way for the integral length scale’s variation to be more realistic while keeping with the Townsend–Perry attached eddy spectrum is to add a new wavenumber range to the model at wavenumbers smaller than that spectrum. This necessary addition can also account for the high-Reynolds-number outer peak of the turbulent kinetic energy in the intermediate layer. An analytic expression is obtained for this outer peak in agreement with extremely high-Reynolds-number data by Hultmark et al. (Phys. Rev. Lett., vol. 108, 2012, 094501; J. Fluid Mech., vol. 728, 2013, pp. 376–395). Townsend’s (The Structure of Turbulent Shear Flows, 1976, Cambridge University Press) production–dissipation balance and the finding of Dallas et al. (Phys. Rev. E, vol. 80, 2009, 046306) that, in the intermediate layer, the eddy turnover time scales with skin friction velocity and distance to the wall implies that the logarithmic derivative of the mean flow has an outer peak at the same location as the turbulent kinetic energy. This is seen in the data of Hultmark et al. (Phys. Rev. Lett., vol. 108, 2012, 094501; J. Fluid Mech., vol. 728, 2013, pp. 376–395). The same approach also predicts that the logarithmic derivative of the mean flow has a logarithmic decay at distances to the wall larger than the position of the outer peak. This qualitative prediction is also supported by the aforementioned data.

The interaction of a planar shock wave ($M\approx 1.2$) with an $\text{SF}_{6}$ polygonal inhomogeneity surrounded by air is experimentally investigated. Six polygons including a square, two types of rectangle, two types of triangle, and a diamond are generated by the soap film technique developed in our previous work, in which thin pins are used as angular vertexes to avoid the pressure singularities caused by the surface tension. The evolutions of the shock-accelerated $\text{SF}_{6}$ polygons are captured by a high-speed schlieren system from which wave systems and the interface characteristics can be clearly identified. Both regular and irregular refraction phenomena are observed outside the volume, and more complex wave patterns, including transmitted shock, refracted shock, Mach stem and the interactions between them, are found inside the volume. Two typical irregular refraction phenomena (free precursor refraction, FPR, and free precursor von Neumann refraction, FNR) are observed and analysed, and the transition from FPR to FNR is found, providing the experimental evidence for the transition between different wave patterns numerically found in the literature. Combined with our previous work (Zhai et al., J. Fluid Mech., vol. 757, 2014, pp. 800–816), the reciprocal transitions between FPR and FNR are experimentally confirmed. The velocities and trajectories of the triple points are further measured and it is found that the motions of the triple points are self-similar or pseudo-stationary. Besides the shock dynamics phenomena, the evolutions of these shocked heavy polygonal volumes, which are quite different from the light ones, are captured and found to be closely related to their initial shapes. Specifically, for square and rectangular geometries, the different width–height ratios result in different behaviours of shock–shock interaction inside the volume, and subsequently different features for the outward jet and the interface. Quantitatively, the time-variations of the interface scales, such as the width and the normalized displacements of the edges, are obtained and compared with those from previous work. The comparison illustrates the superiority of the interface formation method and the significant effect of the initial interface shape on the interface features. Furthermore, the characteristics of the vortex core, including the velocity and vortex spacing, are experimentally measured, and the vortex velocity is compared with those from some circulation models to check the validity of the models. The results in the present work enrich understanding of the shock refraction phenomenon and the database of research into Richtmyer–Meshkov instability (RMI).

Results of point-particle direct numerical simulations of downward gas–solid flow in smooth and rough vertical channels are presented. Two-way coupling and inter-particle collisions are included. The rough walls are shaped as fixed layers of tiny spherical particles with diameter much smaller than the viscous wall unit. The turbulence attenuation induced by the free solid particles in the gas flow is shown to be enhanced with increasing wall roughness. The so-called feedback force, the force exerted by the free particles on the gas, is decomposed into three contributions: the domain average of the mean feedback force, the non-uniform part of the mean feedback force and the fluctuating part of the feedback force. Since the non-uniformity of the mean feedback force increases with wall roughness, the effect of the non-uniform part of the mean feedback force is investigated in detail. For both smooth and rough walls, the non-uniform part of the mean feedback force is shown to contribute significantly to the particle-induced turbulence attenuation.

This paper presents a mathematical model for predicting the propagation of circumferential waves (acoustic, entropy and vorticity waves) through an annular nozzle. Combustion chambers in modern aero-engines are typically annular, and so a model for circumferential waves is essential for understanding and predicting both combustion noise and thermoacoustic instabilities for such geometries. The linearised Euler equations are solved using the Magnus expansion to obtain the reflection and transmission coefficients of the annular nozzle for acoustic, entropy and vorticity perturbations. Predictions which account for flow physics, such as a non-zero mean flow angle and the generation of vorticity noise, are obtained for the first time. Results are compared with two numerical methods, showing that the mathematical model is able to predict the transmission and reflection of waves for both compact and non-compact frequencies. The model is used to prove one particularly interesting and relevant feature of annular geometries: the generation of a vorticity wave by the acceleration of a circumferential entropy wave. It is shown that this phenomenon originates from the baroclinic torque in the vorticity equation.

The interaction of two laser-induced bubbles in bulk water is investigated. The strength and direction of the emerging liquid jets can be controlled by adjusting the relative bubble positions, the time difference between bubble generation, and the laser pulse energies determining the bubble sizes. Experimental and numerical studies are performed for millimetre-sized bubble pairs. Taking bubbles of equal energy, a maximum jet velocity is found for close anti-phase bubbles, i.e. when the second bubble is produced at the maximum volume of the first one and the bubble walls are almost touching and not merging. Under these conditions, one bubble produces a fast jet with a peak velocity of about $150~\text{m}~\text{s}^{-1}$ that reaches a distance into the surrounding liquid of at least three times the maximum bubble radius. Collapse of the other bubble results in a slow jet of large mass that rapidly converts into a ring vortex. Correspondingly, the interaction with adjacent structures is dominated either by localized jet impact or by shear stresses extending over a larger area. Furthermore, interactions between micrometre-sized bubble pairs are investigated numerically to understand and predict how the effects of the physical parameters on bubble dynamics would change when the bubbles become smaller. The results are discussed with respect to micropumping and opto-injection.

The flow of an electrically conducting fluid in an array of square ducts, separated by arbitrary thickness conducting walls, subject to an applied magnetic field is studied. The analytical solution presented here is valid for thick walls and is based on the homogeneous solution obtained by Shercliff (Math. Proc. Camb. Phil. Soc., vol. 49 (01), 1953, pp. 136–144). Arrangements of ducts arise in a number of applications, most notably in fusion blankets, where liquid metal is used both as coolant and for tritium generation purposes. Analytical solutions, such as those presented here, provide insight into the physics and important benchmarking and validation data for computational magnetohydrodynamics (MHD), as well as providing approximate flow parameters for 1D systems codes. It is well known that arrays of such ducts with conducting walls exhibit varying degrees of coupling, significantly affecting the flow. An important practical example is the so-called Madarame problem (Madarame et al., Fusion Technol., vol. 8, 1985, pp. 264–269). In this work analytical results are derived for the relevant hydrodynamic and magnetic parameters for a single duct with thick walls analogous to the Hunt II case. These results are then extended to an array of such ducts stacked in the direction of the applied magnetic field. It is seen that there is a significant coupling affect, resulting in modifications to pressure drop and velocity profile. In certain circumstances, counter-current flow can occur as a result of the MHD effects, even to the point where the mean flow is reversed. Such phenomena are likely to have significant detrimental effects on both heat and mass transfer in fusion applications. The dependence of this coupling on parameters such as conductivities, wall thickness and Hartmann number is studied.

We derive an equation that relates the evolution in time of the maximum of the horizontal fluid velocity at the surface of an irrotational deep-water plunging or spilling breaker to the first component of the pressure gradient at the surface. The approach applies to overhanging wave profiles, up to breaking time.

We present an internal-flow multiscale method (‘unsteady-IMM’) for compressible, time-varying/unsteady flow problems in nano-confined high-aspect-ratio geometries. The IMM is a hybrid molecular–continuum method that provides accurate flow predictions at macroscopic scales because local microscopic corrections to the continuum-fluid formulation are generated by spatially and temporally distributed molecular simulations. Exploiting separation in both time and length scales enables orders of magnitude computational savings, far greater than seen in other hybrid methods. We apply the unsteady-IMM to a converging–diverging channel flow problem with various time- and length-scale separations. Comparisons are made with a full molecular simulation wherever possible; the level of accuracy of the hybrid solution is excellent in most cases. We demonstrate that the sensitivity of the accuracy of a solution to the macro–micro time-stepping, as well as the computational speed-up over a full molecular simulation, is dependent on the degree of scale separation that exists in a problem. For the largest channel lengths considered in this paper, a speed-up of six orders of magnitude has been obtained, compared with a notional full molecular simulation.

This paper examines the kinematic behaviour of the reduced velocity gradient tensor (VGT), $\tilde{\unicode[STIX]{x1D608}}_{ij}$, which is defined as a $2\times 2$ block, from a single interrogation plane, of the full VGT $\unicode[STIX]{x1D608}_{ij}=\partial u_{i}/\partial x_{j}$. Direct numerical simulation data from the fully developed turbulent region of a nominally two-dimensional mixing layer are used in order to examine the extent to which information on the full VGT can be derived from the reduced VGT. It is shown that the reduced VGT is able to reveal significantly more information about regions of the flow in which strain rate is dominant over rotation. It is thus possible to use the assumptions of homogeneity and isotropy to place bounds on the first two statistical moments (and their covariance) of the eigenvalues of the reduced strain-rate tensor (the symmetric part of the reduced VGT) which in turn relate to the turbulent strain rates. These bounds are shown to be dependent upon the kurtosis of $\partial u_{1}/\partial x_{1}$ and another variable defined from the constituents of the reduced VGT. The kurtosis is observed to be minimised on the centreline of the mixing layer and thus tighter bounds are possible at the centre of the mixing layer than at the periphery. Nevertheless, these bounds are observed to hold for the entirety of the mixing layer, despite departures from local isotropy. The interrogation plane from which the reduced VGT is formed is observed not to affect the joint probability density functions (p.d.f.s) between the strain-rate eigenvalues and the reduced strain-rate eigenvalues despite the fact that this shear flow has a significant mean shear in the cross-stream direction. Further, it is found that the projection of the eigenframe of the strain-rate tensor onto the interrogation plane of the reduced VGT is also independent of the plane that is chosen, validating the approach of bounding the full VGT using the assumption of local isotropy.

Flow control over a NACA 0012 airfoil is carried out using a dielectric barrier discharge (DBD) plasma actuator at the Reynolds number of 20 000. Here, the plasma actuator is placed over the pressure (lower) side of the airfoil near the trailing edge, which produces a wall jet against the free stream. This reverse flow creates a quasi-steady recirculation region, reducing the velocity over the pressure side of the airfoil. On the other hand, the air over the suction (upper) side of the airfoil is drawn by the recirculation, increasing its velocity. Measured phase-averaged vorticity and velocity fields also indicate that the recirculation region created by the plasma actuator over the pressure surface modifies the near-wake dynamics. These flow modifications around the airfoil lead to an increase in the lift coefficient, which is similar to the effect of a mechanical Gurney flap. This configuration of DBD plasma actuators, which is investigated for the first time in this study, is therefore called a virtual Gurney flap. The purpose of this investigation is to understand the mechanism of lift enhancement by virtual Gurney flaps by carefully studying the global flow behaviour over the airfoil. First, the recirculation region draws the air from the suction surface around the trailing edge. The upper shear layer then interacts with the opposite-signed shear layer from the pressure surface, creating a stronger vortex shedding from the airfoil. Secondly, the recirculation region created by a DBD plasma actuator over the pressure surface displaces the positive shear layer away from the airfoil, thereby shifting the near-wake region downwards. The virtual Gurney flap also changes the dynamics of laminar separation bubbles and associated vortical structures by accelerating laminar-to-turbulent transition through the Kelvin–Helmholtz instability mechanism. In particular, the separation point and the start of transition are advanced. The reattachment point also moves upstream with plasma control, although it is slightly delayed at a large angle of attack.

In boundary-layer flows, one may reduce skin-friction drag by delaying the onset of laminar-to-turbulent transition via the attenuation of small-amplitude Tollmien–Schlichting (TS) waves. In this work, we use numerical simulations and experiments to compare the robustness of adaptive and model-based techniques for reducing the growth of two-dimensional TS disturbances. In numerical simulations, the optimal linear quadratic Gaussian (LQG) regulator shows the best performance under the conditions it was designed for. However, it is found that the performance deteriorates linearly with the drift of the Reynolds number from its nominal value. As a result, an order-of-magnitude loss of performance is observed when applying the computation-based LQG controller in wind-tunnel experiments. In contrast, it is shown that the adaptive filtered-X least-mean-squares (FXLMS) algorithm is able to maintain an essentially constant performance for significant deviations of the nominal values of the disturbance amplitude and Reynolds number.

The wake of a ship that sails at relatively large Froude numbers usually contains a number of components of different nature and with different heights, lengths, timings and propagation directions. We explore the possibilities of the spectrogram representation of one-point measurements of the ship wake to identify these components and to quantify their main properties. This representation, based on the short-time Fourier transform, facilitates a reliable decomposition of the wake into constituent components and makes it possible to quantify their variations in the time–space domain and the energy content of each component, from very low-frequency precursor waves up to high-frequency signals within the frequency range of typical wind-generated waves. A method for estimation of the ship speed and the distance of its sailing line from the measurement site is proposed, which only uses information available within the record of the ship wake surface elevation, but where it is assumed that the wake pattern does not deviate significantly from the classical Kelvin wake structure. The wake decomposition using the spectrogram method allows investigation of the energy content that can be attributed to each individual component of the wake. We demonstrate that the majority (60–80 %) of wake energy from strongly powered large ferries that sail at depth Froude numbers ${\sim}0.7$ is concentrated in components that are located near the edge of the wake wedge. Finally, we demonstrate that the spectrogram representation offers a convenient way to identify a specific signature of single types of ships.

Graetz and Nusselt studied heat transfer between a developed laminar fluid flow and a tube at constant wall temperature. Here, we extend the Graetz–Nusselt problem to dense fluid flows with partial wall slip. Its limits correspond to the classical problems for no-slip and no-shear flow. The amount of heat transfer is expressed by the local Nusselt number $\mathit{Nu}_{x}$, which is defined as the ratio of convective to conductive radial heat transfer. In the thermally developing regime, $\mathit{Nu}_{x}$ scales with the ratio of position $\tilde{x}=x/L$ to Graetz number $\mathit{Gz}$, i.e. $\mathit{Nu}_{x}\propto (\tilde{x}/\mathit{Gz})^{-{\it\beta}}$. Here, $L$ is the length of the heated or cooled tube section. The Graetz number $\mathit{Gz}$ corresponds to the ratio of axial advective to radial diffusive heat transport. In the case of no slip, the scaling exponent ${\it\beta}$ equals $1/3$. For no-shear flow, ${\it\beta}=1/2$. The results show that for partial slip, where the ratio of slip length $b$ to tube radius $R$ ranges from zero to infinity, ${\it\beta}$ transitions from $1/3$ to $1/2$ when $10^{-4}<b/R<10^{0}$. For partial slip, ${\it\beta}$ is a function of both position and slip length. The developed Nusselt number $\mathit{Nu}_{\infty }$ for $\tilde{x}/\mathit{Gz}>0.1$ transitions from 3.66 to 5.78, the classical limits, when $10^{-2}<b/R<10^{2}$. A mathematical and physical explanation is provided for the distinct transition points for ${\it\beta}$ and $\mathit{Nu}_{\infty }$.

The collision of a liquid drop against a small target results in the formation of a thin liquid sheet that extends radially until it reaches a maximum diameter. The subsequent retraction is due to the air–liquid surface tension. We have used a time- and space-resolved technique to measure the thickness field of this class of liquid sheet, based on the grey-level measurement of the image of a dyed liquid sheet recorded using a high-speed camera. This method enables a precise measurement of the thickness in the range $10{-}450~{\rm\mu}\text{m}$, with a temporal resolution equal to that of the camera. We have measured the evolution with time since impact, $t$, and radial position, $r$, of the thickness, $h(r,t)$, for various drop volumes and impact velocities. Two asymptotic regimes for the expansion of the sheet are evidenced. The scalings of the thickness with $t$ and $r$ measured in the two regimes are those that were predicted by Rozhkov et al. (Proc. R. Soc. Lond. A, vol. 460, 2004, pp. 2681–2704) for the short-time regime and Villermaux and Bossa (J. Fluid Mech., vol. 668, 2011, pp. 412–435) for the long-time regime, but never experimentally measured before. Interestingly, our experimental data also provide evidence for the existence of a maximum of the film thickness $h_{max}(r)$ at a radial position $r_{h_{max}}(t)$ corresponding to the cross-over of these two asymptotic regimes. The maximum moves with a constant velocity of the order of the drop impact velocity, as expected theoretically. Thanks to our visualization technique, we also provide evidence of an azimuthal thickness modulation of the liquid sheets.