The aim of this paper is to make the formation of liquid plugs as difficult as possible in liquid partially filling a horizontal rectangular tube in a downward gravity field by setting the walls to have differing contact angles. Manning et al.'s method (J. Fluid Mech., vol. 682, 2011, pp. 397–414), extended from Concus–Finn theory, is applied to the existence of capillary plugs in rectangular tubes. The critical Bond numbers ($B_c$) determining the existence of capillary plugs in a rectangular tube are studied for different settings of the non-uniform contact angles, and the influence of the aspect ratio (defined as the width-to-height ratio) of the rectangular cross-section on $B_c$ is examined. Compared to the maximum and minimum of $B_c$ reached for uniform contact angles, the maximum of $B_c$ is higher, which is attained for the bottom contact angle $\gamma _2=135^{\circ }$, the top contact angle $\gamma _4=45^{\circ }$, and the side contact angles $\gamma _1=\gamma _3=90^{\circ }$; while the minimum is considerably lowered to zero, which is reached for $\gamma _1=\gamma _2=45^{\circ }$ and $\gamma _3=\gamma _4=135^{\circ }$. The aspect ratio of the rectangle has no influence on the maximum and minimum $B_c$ for a tube with walls of differing contact angles. There is only one non-occluded liquid topology in a square, while two topologies may occur in a rectangle with aspect ratio 2, and the transition between the two topologies is accompanied by a kink of the curve of $B_c$. Optimization of the non-uniform contact angles can facilitate or effectively block the capillary plugs in rectangular tubes regardless of the aspect ratios.

It is well known that perforation of a flat plate reduces its drag when exposed to a flow. However, studies have shown an opposite effect in the case of cylinders. Such a counterintuitive result can have significant consequences on the momentum modelling often used for wind turbine performance predictions, where increased porosity is intrinsically linked to lower drag. Here, a study of the drag of various types of porous cylinders, bars and plates under steady laminar inflow is presented. It is shown that, for most cases, the drag decreases with increased porosity. Only special types of perforations can increase the drag on both cylinders and bars, either by enhancing the effect of the rear half of the models or by organizing the wake structures. These rare occurrences are not relevant to wind turbine modelling, which indicates that current momentum models exhibit the qualitatively correct behaviour.

In most cases, the dominant mechanism of energy dissipation for a bubble in volume oscillations is the thermal energy exchanged with the liquid. The process is subtle and its precise description a matter of some complexity. These features have prevented its ready incorporation in many applications, which forcedly have to rely on the rather inaccurate polytropic pressure–volume relation. This paper develops two approximate models of the thermal interaction, formulated in terms of ordinary differential equations, which can be readily added to standard Rayleigh–Plesset-type formulations at a modest computational cost.

This experimental study addresses the longstanding question of why inception of attached cavitation on curved surfaces or hydrofoils at incidence is relatively insensitive to the concentration of free-stream nuclei. High-speed imaging and high-resolution particle image velocimetry measurements examine cavitation inception on three curved surfaces with varying pressure minima followed by regions with adverse pressure gradients. When these pressure gradients either thicken the boundary layer or cause local flow separation, thin $(50\text {--}60\ \mathrm {\mu }\textrm {m})$ low-momentum zones form close to the wall. Microbubbles trapped in these regions are generated initially from the collapse of intermittent attachment of travelling bubble cavitation. These bubbles migrate slowly upstream for a few milliseconds either under the influence of the adverse pressure gradients when the flow remains attached or carried by the recirculating flow when the boundary layer is separated. Their speed is only 2 %–4 % of the free-stream velocity, and their trajectories are erratic, indicating near-dynamic equilibrium. Owing to the low local pressure, their diameter increases by two to four times by non-condensable gas diffusion, from $10$ to $30\ \mathrm {\mu }\textrm {m}$ to the thickness of the low-momentum zone. At that time, either they are swept downstream by the free-stream flow or they become nuclei for new attached cavitation events. When the new patches collapse, new microbubbles form and the process repeats itself frequently, and independently of the free-stream nuclei. These phenomena do not occur when the adverse pressure gradients are too mild to create low-momentum zones with sufficient thickness to facilitate the slow upstream migration and growth.

Turbulent flows driven by a vertically invariant body force were proven to become exactly two-dimensional (2-D) above a critical rotation rate, using upper bound theory. This transition in dimensionality of a turbulent flow has key consequences for the energy dissipation rate. However, its location in parameter space is not provided by the bounding procedure. To determine this precise threshold between exactly two-dimensional and partially three-dimensional (3-D) flows, we perform a linear stability analysis over a fully turbulent 2-D base state. This requires integrating numerically a quasi-2-D set of equations over thousands of turnover times, to accurately average the growth rate of the 3-D perturbations over the statistics of the turbulent 2-D base flow. We leverage the capabilities of modern graphics processing units to achieve this task, which allows us to investigate the parameter space up to $Re=10^5$. At the Reynolds numbers typical of 3-D direct numerical simulations and laboratory experiments, $Re\in [10^2, 5\times 10^3]$, the turbulent 2-D flow becomes unstable to 3-D motion through a centrifugal-type instability. However, at even higher Reynolds numbers, another instability takes over. A candidate mechanism for the latter instability is the parametric excitation of inertial waves by the modulated 2-D flow, a phenomenon that we illustrate with an oscillatory 2-D Kolmogorov flow.

We study the effects of nozzle geometry on the dynamics of thin fluid films flowing down a vertical cylindrical fibre. Recent experiments show that varying the nozzle diameter can lead to different flow regimes and droplet characteristics in the film. Using a weighted residual modelling approach, we develop a system of coupled equations that account for inertia, surface tension effects, gravity and a film stabilization mechanism to describe both near-nozzle fluid structures and downstream bead dynamics. We report good agreement between the predicted droplet properties and the experimental data.

We address the fluid–structure interaction of flexible fin models oscillating in a water flow. Here, we investigate in particular the dependence of hydrodynamic force distributions on fin geometry and flapping frequency. For this purpose, we employ state-of-the-art techniques in pressure evaluation to describe fluid force maps with high temporal and spatial resolution on the deforming surfaces of the hydrofoils. Particle tracking velocimetry is used to measure the three-dimensional fluid velocity field, and the hydrodynamic stress tensor is subsequently calculated based on the Navier–Stokes equation. The shape and kinematics of the fin-like foils are linked to their ability to generate propulsive thrust efficiently, as well as the accumulation of external contact forces and the resulting internal tension throughout a flapping cycle.

Prior mathematical work of Constantin & Iyer (Commun. Pure Appl. Maths, vol. 61, 2008, pp. 330–345; Ann. Appl. Probab., vol. 21, 2011, pp. 1466–1492) has shown that incompressible Navier–Stokes solutions possess infinitely many stochastic Lagrangian conservation laws for vorticity, backward in time, which generalize the invariants of Cauchy (Sciences mathématiques et physique, vol. I, 1815, pp. 33–73) for smooth Euler solutions. We reformulate this theory for the case of wall-bounded flows by appealing to the Kuz'min (Phys. Lett. A, vol. 96, 1983, pp. 88–90)–Oseledets (Russ. Math. Surv., vol. 44, 1989, p. 210) representation of Navier–Stokes dynamics, in terms of the vortex-momentum density associated to a continuous distribution of infinitesimal vortex rings. The Constantin–Iyer theory provides an exact representation for vorticity at any interior point as an average over stochastic vorticity contributions transported from the wall. We point out relations of this Lagrangian formulation with the Eulerian theory of Lighthill (Boundary layer theory. In Laminar Boundary Layers (ed. L. Rosenhead), 1963, pp. 46–113)–Morton (Geophys. Astrophys. Fluid Dyn., vol. 28, 1984, pp. 277–308) for vorticity generation at solid walls, and also with a statistical result of Taylor (Proc. R. Soc. Lond. A, vol. 135, 1932, pp. 685–702)–Huggins (J. Low Temp. Phys., vol. 96, 1994, pp. 317–346), which connects dissipative drag with organized cross-stream motion of vorticity and which is closely analogous to the ‘Josephson–Anderson relation’ for quantum superfluids. We elaborate a Monte Carlo numerical Lagrangian scheme to calculate the stochastic Cauchy invariants and their statistics, given the Eulerian space–time velocity field. The method is validated using an online database of a turbulent channel-flow simulation (Graham et al., J. Turbul., vol. 17, 2016, pp. 181–215), where conservation of the mean Cauchy invariant is verified for two selected buffer-layer events corresponding to an ‘ejection’ and a ‘sweep’. The variances of the stochastic Cauchy invariants grow exponentially backward in time, however, revealing Lagrangian chaos of the stochastic trajectories undergoing both fluid advection and viscous diffusion.

We use an online database of a turbulent channel-flow simulation at $Re_\tau =1000$ (Graham et al. J. Turbul., vol. 17, issue 2, 2016, pp. 181–215) to determine the origin of vorticity in the near-wall buffer layer. Following an experimental study of Sheng et al. (J. Fluid Mech., vol. 633, 2009, pp.17–60), we identify typical ‘ejection’ and ‘sweep’ events in the buffer layer by local minima/maxima of the wall stress. In contrast to their conjecture, however, we find that vortex lifting from the wall is not a discrete event requiring $\sim$1 viscous time and $\sim$10 wall units, but is instead a distributed process over a space–time region at least $1\sim 2$ orders of magnitude larger in extent. To reach this conclusion, we exploit a rigorous mathematical theory of vorticity dynamics for Navier–Stokes solutions, in terms of stochastic Lagrangian flows and stochastic Cauchy invariants, conserved on average backward in time. This theory yields exact expressions for vorticity inside the flow domain in terms of vorticity at the wall, as transported by viscous diffusion and by nonlinear advection, stretching and rotation. We show that Lagrangian chaos observed in the buffer layer can be reconciled with saturated vorticity magnitude by ‘virtual reconnection’: although the Eulerian vorticity field in the viscous sublayer has a single sign of spanwise component, opposite signs of Lagrangian vorticity evolve by rotation and cancel by viscous destruction. Our analysis reveals many unifying features of classical fluids and quantum superfluids. We argue that ‘bundles’ of quantized vortices in superfluid turbulence will also exhibit stochastic Lagrangian dynamics and satisfy stochastic conservation laws resulting from particle relabelling symmetry.

We consider the stability of an elastic membrane on the bottom of a uniform horizontal flow of an inviscid and incompressible fluid of finite depth with free surface. The membrane is simply supported at the leading and the trailing edges which attach it to the two parts of the horizontal rigid floor. The membrane has an infinite span in the direction perpendicular to the direction of the flow and a finite length in the direction of the flow. For the membrane of infinite length we derive a full dispersion relation that is valid for arbitrary depth of the fluid layer and find conditions for the flutter of the membrane due to emission of surface gravity waves. We describe this radiation-induced instability by means of the perturbation theory of the roots of the dispersion relation and the concept of negative energy waves and discuss its relation to the anomalous Doppler effect.

Efficient solution-adaptive simulations are conducted by a Cartesian cut-cell method to analyse the flow field of prolate ellipsoids in uniform flow. The parameter space defined by Reynolds numbers $1 \leq Re \leq 100$, aspect ratios $1 \leq \beta \leq 8$ and inclination angles $0^\circ \leq \phi \leq 90^\circ$ is covered by approximately 4400 simulations. Flow visualizations and skin friction distributions are presented for selected configurations. Aspect ratios $1\leq \beta \lesssim 3$ are identified as transitional geometries to fibres. For $\beta \gtrsim 3$, the flow topology is qualitatively unaffected by higher aspect ratios. If the major axis is aligned with the free stream, i.e. $\phi = 0^\circ$, the ellipsoids are slender bodies, whereas for $\phi = 90^\circ$ a bluff body flow is observed. Conditions for the onset of flow separation are reported. The data base is used to determine correlations for drag, lift and torque. The correlations are incorporated into dynamic equations for ellipsoidal Lagrangian models and limitations are discussed.

Recent technological advances have led to a novel class of microfluidic devices which can be rapidly fabricated by printing a fluid onto a solid substrate with flows generated passively via surface tension. The nonlinear dependence between flow and the heights of the conduits, however, prevent straightforward calculation of the resulting dynamics. In this paper we use matched asymptotic expansions to predict how flow through these devices can be tuned by changing their geometry. We begin with the simple ‘dumbbell’ configuration in which two fluid drops with different sizes are connected by a long, thin and narrow conduit. We calculate the time scale required for one drop to drain into the other and how this depends both on the geometry of the pinned contact line and volume of fluid deposited into the drops. Our model therefore provides the mechanistic basis to design conduits with a particular fluid flux and/or shear stress, which are often key experimental constraints. Our asymptotic predictions are shown to be in excellent agreement with numerical simulations even for moderate aspect ratios (the ratio of conduit width to length). Next, we show how our results for the simple dumbbell configuration can be extended to predict the flow through networks of conduits with multiple drops and nodes, and hence may assist in their design and implementation. This new mathematical framework has the potential to increase the use of surface tension driven microfluidics across a wide range of disciplines as it allows alternate designs to be rapidly assessed.

The present study aims to provide a statistical analysis of turbulence in the mixing layer of a lock-exchange gravity current propagating over a 2 % slope based on large eddy simulation using a Boussinesq code. The statistics are calculated from the ensemble and spanwise averaging of 200 simulations for two time steps corresponding to the initial constant velocity slumping phase and the decelerating inertial phase. The overall energy balance and structure of the mixing layer are weakly influenced by the propagation time following the lock release. Thereby, streamwise dominated turbulence is produced by the positive buoyancy flux and subsequently converted into averaged flow through energy backscatter in the nose, whereas the current's interface takes the structure of a stratified mixing layer unstable to Kelvin–Helmholtz instabilities in the rear of the head. The dependency of the current head/body structure on the evolution of the turbulence kinetic energy (TKE) along the mixing layer is also investigated. The transition from the head to the body is associated with a peak of TKE and the flux Richardson number exceeding the stability criterion $Ri_f = 0.2$. It is furthermore observed that the turbulence intensity in all three spatial directions stabilises to satisfy $\langle u'u' \rangle = 2 \langle v'v' \rangle = 2 \langle w'w' \rangle$, where $u', v' \ \text{and} \ w'$ are respectively the streamwise, spanwise and vertical turbulent perturbations of velocity. Finally, a region of statistically stationary TKE is identified once the gradient Richardson number plateaus to a value dependent on the current's propagation approximately 5.5 lock heights backward from the front, where the depth-averaged TKE budget reduces to the balance between the contributions due to shear (P), buoyancy (B) and viscous dissipation $(\varepsilon)$ as $\langle P \rangle _d + \langle B \rangle _d - \langle \varepsilon \rangle _d \approx 0$.

Inertialess anisotropic particles in a Rayleigh–Bénard turbulent flow show maximal tumbling rates for weakly oblate shapes, in contrast with the universal behaviour observed in developed turbulence where the mean tumbling rate monotonically decreases with the particle aspect ratio. This is due to the concurrent effect of turbulent fluctuations and of a mean shear flow whose intensity, we show, is determined by the kinetic boundary layers. In Rayleigh–Bénard turbulence prolate particles align preferentially with the fluid velocity, while oblate ones orient with the temperature gradient. This analysis elucidates the link between particle angular dynamics and small-scale properties of convective turbulence and has implications for the wider class of sheared turbulent flows.

We present a series of analytically solvable axisymmetric flows on the torus geometry. For the single-component flows, we describe the propagation of sound waves for perfect fluids, as well as the viscous damping of shear and longitudinal waves for isothermal and thermal fluids. Unlike the case of planar geometry, the non-uniform curvature on a torus necessitates a distinct spectrum of eigenfrequencies and their corresponding basis functions. This has several interesting consequences, including breaking the degeneracy between even and odd modes, a lack of periodicity even in the flows of perfect fluids and the loss of Galilean invariance for flows with velocity components in the poloidal direction. For the multi-component flows, we study the equilibrium configurations and relaxation dynamics of axisymmetric fluid stripes, described using the Cahn–Hilliard equation. We find a second-order phase transition in the equilibrium location of the stripe as a function of its area ${\rm \Delta} A$. This phase transition leads to a complex dependence of the Laplace pressure on ${\rm \Delta} A$. We also derive the underdamped oscillatory dynamics as the stripes approach equilibrium. Furthermore, relaxing the assumption of axial symmetry, we derive the conditions under which the stripes become unstable. In all cases, the analytical results are confirmed numerically using a finite-difference Navier–Stokes solver.

This paper provides an experimental investigation on the internal shear layers and the edges of the uniform momentum zones (UMZs) in a turbulent pipe flow. The time-resolved stereoscopic particle image velocimetry data are acquired in the cross-section of the pipe, and span the range of Reynolds number $\textit {Re}_\tau =340\text {--}1259$. In the first part of the study, internal shear layers are detected using a three-dimensional detection method, and both their geometry as well as their fingerprint in the flow statistics are examined. Three-dimensional conditional mean flow analysis revealed a strong low-speed region beneath the average shear layers. This low-speed region is associated with positive wall-normal fluctuations, and it is accompanied by two swirling motions having opposite signs on either side in the azimuthal direction. Moreover, the shear layers are stretched by the two opposite azimuthal motions. In the second part of the study, the shear layers are treated as the continuous edges of the UMZs, which are detected using the histogram method following Adrian et al. (J. Fluid Mech., vol. 422, 2000, pp. 1–54) and de Silva et al. (J. Fluid Mech., vol. 786, 2016, pp. 309–331). For this part, two different orientation of the planes are used, i.e. the wall-normal–streamwise plane and the wall-normal–spanwise plane (cross-section of the pipe). Comparison of the detected structures shows that the shear layers mostly overlap with a UMZ edge (in either plane).

The manipulation of near-wall turbulent structures in a turbulent boundary layer (TBL) is an effective way to reduce the turbulent frictional drag. This paper demonstrates the effectiveness of a novel approach for the manipulation of near-wall structures in a TBL with Reynolds number ($Re_\theta$) set to 1200. The manipulation is achieved by employing a sustainable wall-attached air-film array. The static and dynamic interface configuration of the air film can be modulated, which generates a dynamic slip boundary condition. For modulation frequencies within the TBL receptivity, this approach shows that it can effectively modify the TBL near-wall velocity/vorticity field. For a typical modulation frequency of 50 Hz, the near-wall mean streamwise velocity decreases and the wall-normal velocity increases when compared to the canonical flat plate TBL. The mean transverse vorticity is suppressed in the near-wall region and its peak is ‘pushed’ outward away from the wall. In the vicinity of modulated air-film array, the phase-locked velocity/vorticity field demonstrates harmonic motions such as a Stokes-type oscillatory motion. The distribution of shear stresses indicates suppressed momentum transfer toward the wall. Estimation of the wall skin friction via the Clauser chart method indicates a reduction of the wall skin friction up to 40 % in the downstream region of the air-film array. A control volume analysis shows that the TBL gains a significant amount of momentum over the oscillating air films, which suggests that the oscillating air film acts like a source of momentum. This pumping effect could potentially explain the observed wall skin friction reduction effect.

When the density of a gravitationally stable fluid depends on a fast diffusing scalar and a slowly diffusing scalar of opposite contribution to the stability, ‘double diffusive’ instabilities may develop and drive convection. When the slow diffuser settles under gravity, as is for instance the case for small sediment particles in water, settling-driven double-diffusive instabilities can additionally occur. Such instabilities are relevant in a variety of naturally occurring settings, such as particle-laden river discharges, or underground inflows in lakes. Inspired by the dynamics of the more traditional thermohaline double-diffusive instabilities, we ask whether large-scale ‘mean-field’ instabilities can develop as a result of sedimentary double-diffusive convection. We first apply the mean-field instability theory of Traxler et al. (J. Fluid Mech., vol. 677, 2011, pp. 530–553) to high-Prandtl-number fluids, and find that these are unstable to Radko's layering instability, yet collectively stable. We then extend the theory of Traxler et al. (2011) to include settling and study its impact on the development of the collective instability. We find that two distinct regimes exist. At low settling velocities, the double-diffusive turbulence in the fingering regime is relatively unaffected by settling and remains stable to the classical collective instability. It is, however, unstable to a new instability in which large-scale gravity waves are excited by the phase shift between the salinity and particle concentration fields. At higher settling velocities, the double-diffusive turbulence is substantially affected by settling, and becomes unstable to the classic collective instability. Our findings, validated by direct numerical simulations, reveal new opportunities to observe settling-driven layering in laboratory and field experiments.

The wavelet-based adaptive large-eddy simulation method is extended for computational modelling of compressible wall-bounded attached turbulent flows. The wavelet-threshold filtered compressible Navier–Stokes equations are derived. The unclosed terms in the governing equations are approximated by using eddy-viscosity and eddy-conductivity modelling procedures based on the anisotropic minimum-dissipation approach. The proposed filtering procedure is integrated with the adaptive anisotropic wavelet collocation method, which allows for the appropriate mesh stretching in the wall-normal direction. The performance of the method is assessed by conducting adaptive numerical simulations of fully developed supersonic flow in a plane channel with isothermal walls, which represents a well-established benchmark for wall-bounded turbulent compressible flows. The present results demonstrate both the feasibility and the effectiveness of the novel wavelet-based adaptive method in the high-speed compressible regime, showing good agreement with reference numerical solutions.

During the early stages of drop-pool impacts, an air film is temporarily entrapped between the liquid bodies. At low impact velocities, this film can become highly stretched, allowing for contact-free penetration of the drop into the pool. The elongated film never ruptures for the lowest impact velocities, resulting in drop bouncing. For higher impact velocities, the elongated film ruptures to entrain hundreds of micro-bubbles in a process known as Mesler entrainment. At even higher impact velocities, an elongated film never forms as early contact entraps a shorter disk-type film which retracts to one or few central bubbles. In this work we use numerical simulations of water drop-pool impacts along with theoretical analyses to discover a capillary transition that prevents early contact. This transition allows the drop to penetrate further into the pool and provides a pathway for the formation of elongated films. Since Mesler entrainment is only possible if early contact is prevented, we use the occurrence of transition as a criterion to provide an upper boundary for the Mesler entrainment regime. We observe from low $We$ simulations that after transition, the drop spreads on the pool surface, during which the minimum film thickness increases and the film regularizes. Interestingly, we observe the formation of kinks between the centre of the film and the spreading fronts, and find asymptotic scaling laws governing the film thickness. Lastly, by examining the role of liquid viscosity, we shed light on transition dynamics for different liquids.