Air-induced drag reduction was investigated on a 12.9 m long flat plate test model at a free stream speed of $6. 3~\mathrm{m} ~{\mathrm{s} }^{- 1} $. Measurements of the local skin friction, phase velocity profiles (liquid and gas) and void fraction profiles were acquired at downstream distances to 11.5 m, which yielded downstream-distance-based Reynolds numbers above 80 million. Air was injected within the boundary layer behind a 13 mm backward facing step (BFS) while the incoming boundary layer was perturbed with vortex generators in various configurations immediately upstream of the BFS. Measurements confirmed that air layer drag reduction (ALDR) is sensitive to upstream disturbances, but a clean boundary layer separation line (i.e. the BFS) reduces such sensitivity. Empirical scaling of the experimental data was investigated for: (a) the critical air flux required to establish ALDR; (b) void fraction profiles; and (c) the interfacial velocity profiles. A scaling of the critical air flux for ALDR was developed from balancing shear-induced lift forces and buoyancy forces on a single bubble within a shear flow. The resulting scaling successfully collapses ALDR results from the current and past studies over a range of flow conditions and test model configurations. The interfacial velocity and void fraction profiles were acquired and scaled within the bubble drag reduction (BDR), ALDR and transitional ALDR regimes. The BDR interfacial velocity profile revealed that there was slip between phases. The ALDR results showed that the air layer thickness was nominally three-quarters of the total volumetric flux (per unit span) of air injected divided by the free stream speed. Furthermore, the air layer had an average void fraction of 0.75 and a velocity of approximately 0.2 times the free stream speed. Beyond the air layer was a bubbly mixture that scaled in a similar fashion to the BDR results. Transitional ALDR results indicate that this regime was comprised of intermittent generation and subsequent fragmentation of an air layer, with the resulting drag reduction determined by the fraction of time that an air layer was present.

A new single equation for the surface elevation of a travelling water wave in an incompressible, inviscid, irrotational fluid is derived. This new equation is derived without approximation from Euler’s equations, valid for both a one- and two-dimensional travelling-wave surface. We show that this new formulation can be used to efficiently derive higher-order Stokes-wave approximations, and pose that this new formulation provides a useful framework for further investigation of travelling water waves.

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still, that is, it is stagnant in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadratic ones with arbitrary positive coefficients.

The effects of polymer additives on decaying isotropic turbulence are numerically investigated using a hybrid approach consisting of Brownian dynamics simulations for an enormous number of dumbbells (of the order of 10 billion, $O(1{0}^{10} )$) and direct numerical simulations of turbulence making full use of large-scale parallel computations. Reduction of the energy dissipation rate and modification of the kinetic energy spectrum in the dissipation range scale were observed when the reaction term due to the polymer additives was incorporated into the equation of motion for the solvent fluid. An increase in the polymer concentration or Weissenberg number ${W}_{i} $ yielded significant modifications of the turbulence statistics at small scales, such as a suppression of the local energy dissipation fluctuations. A power-law decay of the kinetic energy spectrum $E(k, t)\sim {k}^{- 4. 7} $ was observed in the wavenumber range below the Kolmogorov length scale when ${W}_{i} = 25$. The generation of intense vortices was suppressed by the polymer additives, consistent with previous studies using the constitutive equations. The field structures of the trace of the polymer stress depended on the intensity of its fluctuation: sheet-like structures were observed for the intermediate intensity region and filamentary structures were observed for the intense region. The results obtained with few polymers and large replicas could approximate those with many polymers and smaller replicas as far as the large-scale statistics were concerned.

Simulations of precipitating convection would typically use a non-Boussinesq dynamical core such as the anelastic equations, and would incorporate water substance in all of its phases: vapour, liquid and ice. Furthermore, the liquid water phase would be separated into cloud water (small droplets suspended in air) and rain water (larger droplets that fall). Depending on environmental conditions, the moist convection may organize itself on multiple length and time scales. Here we investigate the question, what is the minimal representation of water substance and dynamics that still reproduces the basic regimes of turbulent convective organization? The simplified models investigated here use a Boussinesq atmosphere with bulk cloud physics involving equations for water vapour and rain water only. As a first test of the minimal models, we investigate organization or lack thereof on relatively small length scales of approximately 100 km and time scales of a few days. It is demonstrated that the minimal models produce either unorganized (‘scattered’) or organized convection in appropriate environmental conditions, depending on the environmental wind shear. For the case of organized convection, the models qualitatively capture features of propagating squall lines that are observed in nature and in more comprehensive cloud resolving models, such as tilted rain water profiles, low-altitude cold pools and propagation speed corresponding to the maximum of the horizontally averaged, horizontal velocity.

We consider the problem of nonlinear resonant interactions of interfacial waves with the presence of a linear interfacial instability in an inviscid two-fluid stratified flow through a horizontal channel. The resonant triad consists of a (linearly) unstable wave and two stable waves, one of which has a wavelength that can be much longer than that of the unstable component. Of special interest is the development of the long wave by energy transfer from the base flow due to the coupled effect of nonlinear resonance and interfacial instability. By use of the method of multiple scales, we derive the interaction equations which govern the time evolution of the amplitudes of the interacting waves including the effect of interfacial instability. The solution of the evolution equations shows that depending on the flow conditions, the (stable) long wave can achieve a bi-exponential growth rate through the resonant interaction with the unstable wave. Moreover, the unstable wave can grow unboundedly even when the nonlinear self-interaction effect is included, as do the stable waves in the associated resonant triad. For the verification of the theoretical analysis and the practical application involving a broadbanded spectrum of waves, we develop an effective direct simulation method, based on a high-order pseudo-spectral approach, which accounts for nonlinear interactions of interfacial waves up to an arbitrary high order. The direct numerical simulations compare well with the theoretical analysis for all of the characteristic flows considered, and agree qualitatively with the experimental observation of slug development near the entrance of two-phase flow into a pipe.

We consider the segregation of spheres of equal size and different density flowing over an inclined plane, theoretically and computationally by means of distinct element method (DEM) simulations. In the first part of the work, we study the settling of a single higher-density particle in the flow of otherwise identical particles. We show that the motion of the high-density tracer particle can be understood in terms of the buoyancy and drag forces acting on it. The buoyancy force is given by Archimedes principle, with an effective volume associated with the particle, which depends upon the local packing fraction, $\phi $. The buoyancy arises primarily from normal forces acting on the particle, and tangential forces have a negligible contribution. The drag force on a sphere of diameter $d$ sinking with a velocity $v$ in a granular medium of apparent viscosity $\eta $ is given by a modified Stokes law, ${F}_{d} = c\pi \eta dv$. The coefficient ($c$) is found to decrease with packing fraction. In the second part of the work, we consider the case of binary granular mixtures of particles of the same size but differing in density. A continuum model for segregation is presented, based on the single-particle results. The number fraction profile for the heavy particles at equilibrium is obtained in terms of the effective temperature, defined by a fluctuation–dissipation relation. The model predicts the equilibrium number fraction profiles at different inclination angles and for different mass ratios of the particles, which match the DEM results very well. Finally, a complete model for the theoretical prediction of the flow and number fraction profiles for a mixture of particles of different density is presented, which combines the segregation model with a model for the rheology of mixtures. The model predictions agree quite well with the simulation results.

The lift-generating capabilities of a translating and passively pitching rectangular plate are assessed experimentally. The plate pitch dynamics are generated by a rotational spring, and a genetic algorithm isolates a set of spring parameters maximizing the average lift. Our experiments identified a range of parameters that produce kinematic trajectories associated with optimal lift production. The stroke length and the dynamic response of the spring at the driving frequency are revealed to play crucial roles in the generation of such trajectories. Measurements taken with digital particle image velocimetry are used to analyse the results.

The time evolution of pressure statistics in freely decaying homogeneous isotropic turbulence (HIT) is investigated via eddy-damped quasi-normal Markovian (EDQNM) computations. The present results show that the time decay rate of pressure-based statistical quantities, such as pressure variance and pressure gradient variance, are sensitive to the breakdown of permanence of large eddies. New formulae for the associated time-decay exponents are proposed, which extend previous relations proposed in Lesieur, Ossia & Metais (Phys. Fluids, vol. 11, 1999, p. 1535). Particular attention is paid to finite-Reynolds-number (FRN) effects on the pressure spectrum and pressure statistics. The results also suggest that $R{e}_{\lambda } = O(1{0}^{4} )$ must be considered to observe a one-decade inertial range in the pressure spectrum with Kolmogorov $- 7/ 3$ scaling. This threshold value is larger than almost all existing direct numerical simulation (DNS) and experimental data, justifying the discussion about other possible scaling laws. The $- 5/ 3$ slope reported in some DNS data is also recovered by the EDQNM model, but it is observed to be a low-Reynolds-number effect. Another important result is that FRN effects yield a departure from asymptotic theoretical behaviours which appear similar to some effects attributed to intermittency by most authors. This is exemplified by the ratio between pressure-based and velocity-based Taylor microscales. Therefore, high-Reynolds-number DNS or experiments such that $R{e}_{\lambda } = O(1{0}^{4} )$ would be required in order to remove FRN effects and to analyse pure intermittency effects.

Liquid rings can be generated in the Leidenfrost state using liquid oxygen of low boiling point ($- 183~\textdegree \mathrm{C} $) and high magnetic susceptibility, allowing one to ‘sculpt’ the liquid into a ring shape using an annular magnet. When the magnetic field is turned off, the ring shrinks back into a puddle with a constant acceleration. A potential flow approach accurately describes the dynamics of closure with an equation reminiscent of the Rayleigh–Plesset equation for the collapse of transient cavities.

The breakup of droplets due to creeping motion in a confined microchannel geometry is studied using three-dimensional numerical simulations. Analogously to unconfined droplets, there exist two distinct breakup phases: (i) a quasi-steady droplet deformation driven by the externally applied flow; and (ii) a surface-tension-driven three-dimensional rapid pinching that is independent of the externally applied flow. In the first phase, the droplet relaxes back to its original shape if the externally applied flow stops; if the second phase is reached, the droplet will always break. Also analogously to unconfined droplets, there exist two distinct critical conditions: (i) one that determines whether the droplet reaches the second phase and breaks, or it reaches a steady shape and does not break; and (ii) one that determines when the rapid autonomous pinching starts. We analyse the second phase using stop–flow simulations, which reveal that the mechanism responsible for the autonomous breakup is similar to the end-pinching mechanism for unconfined droplets reported in the literature: the rapid pinching starts when, in the channel mid-plane, the curvature at the neck becomes larger than the curvature everywhere else. The same critical condition is observed in simulations in which we do not stop the flow: the breakup dynamics and the neck thickness corresponding to the crossover of curvatures are similar in both cases. This critical neck thickness depends strongly on the aspect ratio, and, unlike unconfined flows, depends only weakly on the capillary number and the viscosity contrast between the fluids inside and outside the droplet.

The transient dynamics of the linearized Euler–Boussinesq equations governing parallel stratified shear flows is presented and analysed. Solutions are expressed as integral superpositions of generalized eigenfunctions associated with the continuous-spectrum component of the Taylor–Goldstein linear stability operator, and reveal intrinsic dynamics not captured by its discrete-spectrum counterpart. It is shown how continuous-spectrum perturbations are generally characterized by non-normal energy growth and decay with algebraic asymptotic behaviour in either time or space. This behaviour is captured by explicit long-time/far-field expressions from rigorous asymptotic analysis, and it is illustrated with direct numerical simulations of the whole (non-Boussinesq) stratified Euler system. These results can be helpful in understanding recent numerical observations for parallel and non-parallel perturbed stratified shear flows.

The dynamics on the laminar–turbulent separatrix is investigated numerically for boundary-layer flows in the subcritical regime. Constant homogeneous suction is applied at the wall, resulting in a parallel asymptotic suction boundary layer (ASBL). When the numerical domain is sufficiently extended in the spanwise direction, the coherent structures found by edge tracking are invariably localized and their dynamics shows bursts that drive a remarkable regular or irregular spanwise dynamics. Depending on the parameters, the asymptotic dynamics on the edge can be either periodic in time or chaotic. A clear mechanism for the regeneration of streaks and streamwise vortices emerges in all cases and is investigated in detail.

We prove that periodic solutions of the steady Prandtl equations do not exist on a stationary boundary. On a moving boundary, there are no solutions with a monotone velocity profile.

In this paper we study the self-propulsion of a dumbbell micro-robot submerged in a viscous fluid. The micro-robot consists of two rigid spherical beads connected by a rod or a spring; the rod/spring length changes periodically. The constant density of each sphere differs from the density of the fluid, while the whole micro-robot has neutral buoyancy. An effective oscillating gravity field is created via rigid-body oscillations of the fluid. Our calculations show that the micro-robot undertakes both translational and rotational motion. Using an asymptotic procedure containing a two-time method and a distinguished limit, we obtain analytic expressions for the averaged self-propulsion velocity and averaged angular velocity. The important special case of zero angular velocity represents rectilinear self-propulsion with constant velocity. In particular, we have shown that: (a) no unidirectional oscillations of a fluid result in self-propulsion; and (b) for the oscillations of a fluid in two directions rectilinear motion of a micro-robot can be achieved.

Quantum turbulence in thermal counterflow of superfluid ${\text{} }^{4} \mathrm{He} $ is studied at length scales comparable to the mean distance $\ell $ between quantized vortices. The Lagrangian dynamics of solid deuterium particles, of radius ${R}_{p} $ about one order of magnitude smaller than $\ell $, is analysed in a planar section of the experimental volume by using the particle tracking velocimetry technique. We show that the average amplitude of the acceleration of the particles seems to increase as the temperature decreases and applied heat flux increases and this can be explained by exploiting the two-fluid model of superfluid ${\text{} }^{4} \mathrm{He} $. We also report that, at the probed length scales, the normalized distribution of the acceleration of the particles appears to follow an unexpected classical-like behaviour.