A classic lift decomposition (Von Kármán & Sears, J. Aeronaut. Sci., vol. 5, 1938, pp. 379–390) is conducted on potential flow simulations of a near-ground pitching hydrofoil. It is discovered that previously observed stable and unstable equilibrium altitudes are generated by a balance between positive wake-induced lift and negative quasi-steady lift while the added mass lift does not play a role. Using both simulations and experiments, detailed analyses of each lift component's near-ground behaviour provide further physical insights. When applied to three-dimensional pitching hydrofoils the lift decomposition reveals that the disappearance of equilibrium altitudes for ${A{\kern-4pt}R}\ {\rm (aspect\ ratio)} <1.5$ occurs due to the magnitude of the quasi-steady lift outweighing the magnitude of the wake-induced lift at all ground distances. Scaling laws for the quasi-steady lift, wake-induced lift and the stable equilibrium altitude are discovered. A simple scaling law for the lift of a steady foil in ground effect is derived. This scaling shows that both circulation enhancement and the velocity induced at a foil's leading edge by the bound vortex of its ground image foil are the essential physics to understand steady ground effect. The scaling laws for unsteady pitching foils can predict the equilibrium altitude to within $20\,\%$ of its value when $St\ {\rm (Strouhal\ number)} < 0.45$. For $St \ge 0.45$ there is a wake instability effect, not accounted for in the scaling relations, that significantly alters the wake-induced lift. These results not only provide key physical insights and scaling laws for steady and unsteady ground effects, but also for two schooling hydrofoils in a side-by-side formation with an out-of-phase synchronization.

We report an experimental investigation of the heat transport and flow field in a rectangular Rayleigh–Bénard convection (RBC) cell with two immiscible fluids: silicone oil and glycerol. The global heat transport of the system is divided into three ranges corresponding to the different flow structures formed in the glycerol layer. In range I, the glycerol layer is dominated by conduction, and no plume is formed over the interface. In range II, cellular rolls are formed in the glycerol layer and the horizontal motion of rolls causes an oscillation of temperature in the interface. In range III, the cellular pattern is time-independent, and the interface forms a group of wavelets with wave numbers consistent with the mode of the cellular pattern. In lower-thin glycerol, the Nusselt (Nu) grows from conduction to convection through an oscillating subcritical bifurcation at critical Rayleigh number $Ra_c$. The value of $Ra_c$ in the present work is smaller than the theoretical prediction of both-rigid boundaries and greater than the prediction of one-rigid and one-free boundaries. In the upper-thick silicone oil layer, $Nu$ increases with increasing $Ra$, but it is smaller than that of traditional RBC. For the silicone oil layer in two-layer RBC, the hot plumes emitting over the liquid–liquid interface showed different shape and different velocity from cold plumes emitting from the top rigid plate. This implies that the velocity boundary condition strongly influences the flow structure in turbulent convection.

Eukaryotic swimming cells such as spermatozoa, algae or protozoa use flagella or cilia to move in viscous fluids. The motion of their flexible appendages in the surrounding fluid induces propulsive forces that balance viscous drag on the cells and lead to a directed swimming motion. Here, we use our recently built database of cell motility (BOSO-Micro) to investigate the extent to which the shapes of eukaryotic swimming cells may be optimal from a hydrodynamic standpoint. We first examine the morphology of flexible flagella undergoing waving deformation and show that their amplitude-to-wavelength ratio is near that predicted theoretically to optimise the propulsive efficiency of active filaments. Next, we consider ciliates, for which locomotion is induced by the collective beating of short cilia covering their surface. We show that the aspect ratios of ciliates are close to that predicted to minimise the viscous drag of the cell body. Both results strongly suggest a key role played by hydrodynamic constraints, in particular viscous drag, in shaping eukaryotic swimming cells.

A data-driven framework using snapshots of an uncontrolled flow is proposed to identify, and subsequently demonstrate, effective control strategies for different objectives in supersonic impinging jets. The open-loop, feed-forward control approach, based on a dynamic mode decomposition reduced-order model (DMD-ROM), computes forcing receptivity in an economical manner by projecting flow and actuator-specific forcing snapshots onto a reduced subspace and then evolving the dynamics forwards in time. Since it effectively determines a linear response around the unsteady flow in the time domain, the method differs materially from typical techniques that use steady basic states, such as stability or input–output approaches that employ linearized Navier–Stokes operators in the frequency domain. The method presented naturally accounts for factors inherent to the snapshot basis, including configuration complexity and flow parameters such as Reynolds number. Furthermore, gain metrics calculated in the reduced subspace facilitate rapid assessments of flow sensitivities to a wide range of forcing parameters, from which optimal actuator inputs may be selected and results confirmed in scale-resolved simulations or experiments. The DMD-ROM approach is demonstrated from two different perspectives. The first concerns asymptotic feedback resonance, where the effects of harmonic pressure forcing are estimated and verified with nonlinear simulations using a blowing–suction actuator. The second examines time-local behaviour within critical feedback events, where the phase of actuation becomes important. For this, a conditional space–time mode is used to identify the optimal forcing phase that minimizes convective instability growth within the resonance cycle.

Canopy flows in the atmospheric surface layer play important economic and ecological roles, governing the dispersion of passive scalars in the environment. The interaction of high-velocity fluid and large-scale surface-mounted obstacles in canopy flows produces drag and causes intense, inhomogeneous and anisotropic turbulence. In this work, we focus on the turbulent dispersion of passive scalars by studying the ‘pair dispersion’ – a statistical measure of relative motion between particles. We analyse the results of a three-dimensional particle tracking velocimetry experiment in a wind-tunnel canopy flow, focusing on small scales. We confirm the existence of local isotropy of pair dispersion at scales smaller than a characteristic shear length scale $L_\varGamma =(\epsilon /\varGamma ^3)^{1/2}$, where $\epsilon$ and $\varGamma$ are the mean dissipation rate and shear rate, respectively. Furthermore, we show that pair dispersion in this locally isotropic regime is a scale-dependent super-diffusive process, similar to what occurs in homogeneous isotropic turbulent flows. In addition, we measure the pair relative velocity correlation function, showing that its de-correlation occurs in the locally isotropic regime, and discuss the implications of this observation for modelling pair dispersion. Thus, our study extends the fundamental understanding of turbulent pair dispersion to the anisotropic inhomogeneous turbulent canopy flow, bringing valuable information for modelling scalar dispersion in the atmospheric surface layer.

The impact of fluid drops on solid substrates is a cardinal fluid dynamics phenomenon intrinsically related to many fields. Although these impacting objects are very often non-spherical and non-Newtonian, previous studies have mainly focused on spherical Newtonian drops. As a result, both shape and rheological effects on the drop-spreading dynamics remain largely unexplored. In the present work we use a mixed approach combining experiments with multiphase three-dimensional numerical simulations to extend the work reported by Luu & Forterre (J. Fluid Mech., vol. 632, 2009, pp. 301–327) by highlighting the fundamental role of shape in the normal impact of viscoplastic drops. Such complex fluids are highly common in various industrial domains and ideally behave either like a rigid body or a shear-rate-dependent liquid, according to the stress solicitation. Spherical, prolate, cylindrical and prismatic drops are considered. The results show that, under negligible capillary effects, the impacting kinetic energy of the drop is dissipated through viscoplastic effects during the spreading process, giving rise to three flow regimes: (i) inertio-viscous, (ii) inertio-plastic, and (iii) mixed inertio-visco-plastic. These regimes are deeply affected by the drop initial aspect ratio, which in turn reveals the possibility of using drop shape to control spreading. The physical mechanisms driving the considered phenomenon are underlined by energy budget analyses and scaling laws. The results are summarised in a two-dimensional diagram linking the drop maximum spreading, minimum height and final shape with different spreading regimes through a single dimensionless parameter, here called the impact number.

While capillary imbibition in tubes or porous materials has been studied extensively in the past, less attention has been paid to imbibition into a swellable porous material. However, swelling is commonly observed when a polymeric network, such as the cellulose composing paper fibres or sponges, absorbs a solvent. The incompressibility of the fluid leads to an elastic expansion of the polymeric matrix. In a porous material, swelling can affect the geometry of the pores, thus affecting the capillary flow. To describe this complex problem, we propose a model experiment, namely the capillary imbibition in a model pore composed of two parallel and stretched elastomeric fibres. In this configuration, one can observe both the progression of a capillary meniscus and the swelling of the fibres. We show that swelling enables a capillary imbibition for fibres placed further apart than the critical distance existing for non-swelling fibres. In this swelling-dominated regime, we identify a new imbibition dynamic at constant velocity which we rationalize using a linear poro-elastic theory. Finally, we describe the elastocapillary collapse of our model pore which is observed when capillary forces overcome the restoring tension force within the fibres.

Existing theoretical analyses of Faraday waves in Hele-Shaw cells rely on the Darcy approximation and assume a parabolic flow profile in the narrow direction. However, Darcy's model is known to be inaccurate when convective or unsteady inertial effects are important. In this work, we propose a gap-averaged Floquet theory accounting for inertial effects induced by the unsteady terms in the Navier–Stokes equations, a scenario that corresponds to a pulsatile flow where the fluid motion reduces to a two-dimensional oscillating Poiseuille flow, similarly to the Womersley flow in arteries. When gap-averaging the linearised Navier–Stokes equation, this results in a modified damping coefficient, which is a function of the ratio between the Stokes boundary layer thickness and the cell's gap, and whose complex value depends on the frequency of the wave response specific to each unstable parametric region. We first revisit the standard case of horizontally infinite rectangular Hele-Shaw cells by also accounting for a dynamic contact angle model. A comparison with existing experiments shows the predictive improvement brought by the present theory and points out how the standard gap-averaged model often underestimates the Faraday threshold. The analysis is then extended to the less conventional case of thin annuli. A series of dedicated experiments for this configuration highlights how Darcy's thin-gap approximation overlooks a frequency detuning that is essential to correctly predict the locations of the Faraday tongues in the frequency–amplitude parameter plane. These findings are well rationalised and captured by the present model.

We investigate how inhomogeneity influences the $k^{-5/3}$ inertial range scaling of turbulent kinetic energy spectra (with $k$ the wavenumber). For weak statistical inhomogeneity, the energy spectrum can be described as an equilibrium spectrum plus a perturbation. Theoretical arguments suggest that this latter contribution scales as $k^{-7/3}$. This prediction is assessed using direct numerical simulations of three-dimensional Kolmogorov flow.

Partially molten rock is a densely packed, melt-saturated, granular medium, but it has seldom been considered in these terms. In this paper we extend the continuum theory of partially molten rock to incorporate the physics of granular media. Our formulation includes dilatancy in a viscous constitutive law and introduces a non-local fluidity. We analyse the resulting poro-viscous–granular theory in terms of two modes of liquid–solid segregation that are observed in published torsion experiments: localisation of liquid into high-porosity sheets and radially inward liquid flow. We show that the newly incorporated granular physics brings the theory into agreement with experiments. We discuss these results in the context of grain-scale physics across the nominal jamming fraction at the high homologous temperatures relevant in geological systems.

A transcritical domain with a sharp two-phase interface may exist during the early times of liquid hydrocarbon fuel injection at supercritical pressure. Thus, two-phase dynamics are sustained before substantial heating of the liquid and drive the early three-dimensional deformation and atomisation. A recent study of a transcritical liquid jet showed distinct deformation features caused by interface thermodynamics, low surface tension and intraphase diffusive mixing. In the present work, the compressible vortex identification method $\lambda _\rho$ is used to study the vortex dynamics in a cool liquid n-decane transcritical jet surrounded by a hotter oxygen gaseous stream at supercritical pressures. The relationship between vortical structures and the liquid surface evolution is detailed, along with the vorticity generation mechanisms, including variable-density effects. The roles of hairpin and roller vortices in the early deformation of lobes, the layering and tearing of liquid sheets and the formation of fuel-rich gaseous blobs are analysed. At these high pressures, enhanced intraphase mixing and ambient gas dissolution affect the local liquid structures (i.e. lobes). Thus, liquid breakup differs from classical sub-critical atomisation. Near the interface, liquid density and viscosity drop by up to 10 % and 70 %, respectively, and the liquid is more easily affected by the vortical motion (e.g. liquid sheets wrap around vortices). Despite the variable density, compressible vorticity generation terms are smaller than the vortex stretching and tilting. Layering traps and aligns the vortices along the streamwise direction while mitigating the generation of new rollers.

The dynamics of the interaction of a system of two thin volatile liquid droplets resting on a soft viscoelastic solid substrate are investigated theoretically. The developed model fully considers the effect of evaporative cooling and the generated Marangoni stresses due to the induced thermal gradients, while also accounting for the effect of the gas phase composition and the diffusion of vapour in the atmosphere of the droplets. Using the framework of lubrication theory, we derive evolution equations for both the droplet profile and the displacement of the elastic solid, which are solved in combination with Laplace's equation for the vapour concentration in the gas phase. A disjoining-pressure/precursor-film approach is used to describe contact-line motion. The evolution equations are solved numerically, using the finite-element method, and we present a thorough parametric analysis to investigate the physical properties and mechanisms that affect the dynamics of droplet interactions. The results show that the droplets interact through both the soft substrate and the gas phase. In the absence of thermocapillary phenomena, the combined effect of non-uniform evaporation due to the increased vapour concentration between the two droplets and elastocapillary phenomena determines whether the drop–drop interaction is attractive or repulsive. The Marangoni stresses suppress droplet attraction at the early stages of the drying process and lead to longer droplet lifetimes. For substrates with intermediate stiffness, the emergence of spontaneous symmetry breaking at late stages of evaporation is found. The rich dynamics of this complex system is explored by constructing a detailed map of the dynamic regimes.

We derive reduced models for extrusion problems where it is necessary to account for fluid compressibility. We consider the two-dimensional extensional flow of a compressible viscous fluid and discuss two specific applications: weakly compressible fluids and bubbly liquid–gas mixtures that behave as a single compressible fluid. The mathematical model we present consists of equations for conservation of mass, conservation of momentum and a closure relationship between the pressure and density. The most substantial differences between compressible extrusion problems is in the closure relationship. By integrating the conservation equations across the fluid cross-section and exploiting a slender aspect ratio, we derive reduced equations for conservation of mass and conservation of momentum in the direction of flow. The reduced system of equations relating cross-sectionally averaged quantities is closed by a relationship between the averaged pressure and density, which will differ substantially depending on the application. We demonstrate the utility of a reduced model for both the weakly compressible fluid and bubbly mixture applications; namely, in providing valuable quantitative insights without needing to solve a complicated free-boundary problem.

The spectral behaviour of random sawtooth waves propagating in the inner surf zone is investigated in this study. We show that the elevation energy spectrum exhibits a universal shape with an $\omega ^{-2}$ tendency in the inertial subrange and an exponential decay in the diffusive subrange ($\omega$ being the angular frequency). A theoretical spectrum is derived based on the similarities between sawtooth waves in the inner surf zone and Burgers wave solutions. Very good agreement is shown between this theoretical spectrum and laboratory experiments covering a large range of incident random wave conditions. Additionally, an equation describing the universal shape of the dissipation spectrum is derived. It highlights that the dissipation spectrum is nearly constant in the inertial subrange, consistent with prior laboratory observations. The findings presented in this study can be useful to improve broken wave dissipation parametrizations in stochastic spectral wave models.

We exploit the similarity between the mean momentum equation and the mean energy equation and derive transformations for mean temperature profiles in compressible wall-bounded flows. In contrast to prior studies that rely on the strong Reynolds analogy and the presumed similarity between the instantaneous and mean velocity and temperature signals, the discussion in this paper involves the Farve-averaged equations only. We establish that the compressible momentum and energy equations can be made identical to their incompressible counterparts under appropriate normalizations and coordinate transformations. Two types of transformations are explored for illustration purposes: Van Driest (VD)-type transformations and semi-local-type or Trettel–Larsson (TL)-type transformations. In our derivations, it becomes clear that VD-type velocity and temperature transformations hold exclusively within the logarithmic layer. On the other hand, TL-type transformations extend their applicability to incorporate wall-damping effects, at least in principle. Each type of transformation serves its distinct purpose and has its applicable range. However, it is noteworthy that while VD-type transformations can be assessed using measurements obtained from laboratory experiments, TL-type transformations necessitate viscosity and density information typically accessible only through numerical simulations. Finally, we justify the omission of the turbulent kinetic energy transfer term, a term that is unclosed, in the energy equation. This omission leads to closed-form temperature transformations that are valid for both adiabatic and isothermal walls.