Free-falling objects impacting onto water pools experience a very high initial impact force, greatest at the moment when breaking through the free surface. Many have intuitively wondered whether throwing another object in front of an important object (like oneself) before impacting the water surface may reduce this high impact force. Here, we test this idea experimentally by allowing two spheres to consecutively enter the water and measuring the forces on the trailing sphere. We find that the impact acceleration reduction on the trailing sphere depends on the dynamics of the cavity created by the first sphere and the relative timing of the second sphere impact. These combined effects are captured by the non-dimensional ‘Matryoshka’ number, which classifies the observed phenomena into four major regimes. In three of these regimes, we find that the impact acceleration on the second sphere is reduced by up to 78 % relative to impact on a quiescent water surface. Surprisingly, in one of the regimes the force on the trailing sphere is dramatically increased by more than 400 % in the worst case observed. We explain how the various stages of cavity evolution result in the observed alterations in impact force in this multi-body water entry problem.

We examine the effect on near-wall turbulence of displacing the apparent, virtual origins perceived by different components of the overlying flow. This mechanism is commonly reported for drag-altering textured surfaces of small size. For the particular case of riblets, Luchini et al. (J. Fluid Mech., vol. 228, 1991, pp. 87–109) proposed that their effect on the overlying flow could be reduced to an offset between the origins perceived by the streamwise and spanwise velocities, with the latter being the origin perceived by turbulence. Later results, particularly in the context of superhydrophobic surfaces, suggest that this effect is not determined by the apparent origins of the tangential velocities alone, but also by the one for the wall-normal velocity. To investigate this, the present paper focuses on direct simulations of turbulent channels imposing different virtual origins for all three velocity components using Robin, slip-like boundary conditions, and also using opposition control. Our simulation results support that the relevant parameter is the offset between the virtual origins perceived by the mean flow and turbulence. When using Robin, slip-like boundary conditions, the virtual origin for the mean flow is determined by the streamwise slip length. Meanwhile, the virtual origin for turbulence results from the combined effect of the wall-normal and spanwise slip lengths. The slip experienced by the streamwise velocity fluctuations, in turn, has a negligible effect on the virtual origin for turbulence, and hence the drag, at least in the regime of drag reduction. This suggests that the origin perceived by the quasi-streamwise vortices, which induce the cross-flow velocities at the surface, is key in determining the virtual origin for turbulence, while that perceived by the near-wall streaks, which are associated with the streamwise velocity fluctuations, plays a secondary role. In this framework, the changes in turbulent quantities typically reported in the flow-control literature are shown to be merely a result of the choice of origin, and are absent when using as origin the one experienced by turbulence. Other than this shift in origin, we demonstrate that turbulence thus remains essentially smooth-wall-like. A simple expression can predict the virtual origin for turbulence in this regime. The effect can also be reproduced a priori by introducing the virtual origins into a smooth-wall eddy-viscosity framework.

Synthetic jet actuators remain coveted components in flow control applications as the convection of vortex rings induces a redistribution of the momentum in a boundary layer although the net mass flux remains zero. The ability to predict the trajectory of these vortical structures, and subsequently the jet, is critical for efficient and targeted usage of such actuators. In this investigation, a synthetic jet is issued into a zero pressure gradient turbulent boundary layer from rectangular orifices with aspect ratios 3, 6 and 12 over a range of actuation frequencies and velocity ratios, with the flow field captured through particle image velocimetry measurements. An assessment of the trajectories culminates in scaling characteristics which encapsulate the aspect ratio, the momentum ratio between the jet and the cross-flow and the Strouhal number. However, the scaling mechanism is found to be sensitive to constraints which include the interaction between successive vortex pairs and the various regions of the flow field where the degree of interaction of the jet with the cross-flow changes. By redefining the threshold between the near-field, transitioning and far-field regions and accounting for vortex interaction, we observe the relevant non-dimensional parameters describing the scaling to vary. Additionally, together with cross-flow properties, variation in the orifice dimensions and actuation parameters enable the exact form of the Strouhal number for each combination of constraints to be determined. Under conditions of no vortex interaction, synthetic jets are found to follow identical trajectories provided the ratio of total momentum between the jet and the cross-flow remains the same, irrespective of the actuation frequency or Reynolds number of the incoming flow.

Liquids can invade fibrous porous media when the fibres are either wettable or soluble, and the infiltration rate can differ depending on spatial distribution of fibres as well as liquid properties. With continuing developments in dissolution-driven release mechanisms of porous drugs and chemical pattern formations, the understanding of how liquids spontaneously infiltrate into soluble fibrous media is strongly called for. Here we show that unlike capillarity-driven insoluble wicking (exhibiting diffusive growth of wetting distance with time), the wicking distance in soluble porous media grows linearly with time as dominated by liquid viscosity rather than surface tension. Such soluble wicking is highly sensitive to flow orientation relative to fibre alignment, so that it arises only in the crosswise direction while being strongly inhibited in the lengthwise direction. We present a theoretical model to explain the experimentally measured wicking rates in soluble porous media.

Two mean-field models for polyelectrolytes in simultaneous electric field and pressure-driven flow field were developed and compared. The models predict migration perpendicular to the anti-parallel or parallel fields, where the migration is caused by electrohydrodynamic interactions calculated using either a short- or long-range approximation. Inputs for the mean-field models were determined from Brownian dynamics simulations in a simple shear flow. Both models qualitatively reproduce experimental observations of DNA focusing as reported in previous publications. Specifically, it is observed that combination of the shear and electric fields leads to polyelectrolyte motion in the direction transverse to the flow and electric field direction, which in turn leads to concentration of the polyelectrolyte in the centre of a microfluidic channel. Furthermore, both models predict that there is an optimal strength of electric field that leads to the narrowest distribution profile of the polyelectrolyte in the centre of the channel. The analysis suggests that this is due to dispersion induced by the electrohydrodynamic interactions. However, quantitative disagreement between the model predictions and the experimental data indicates that further progress in the model development is needed.

We report an experimental study of the Prandtl-number effects in quasi-two-dimensional (quasi-2-D) Rayleigh–Bénard convection. The experiments were conducted in four rectangular convection cells over the Prandtl-number range of $11.7 \leqslant Pr \leqslant 650.7$ and over the Rayleigh-number range of $6.0\times 10^8 \leqslant Ra \leqslant 3.0\times 10^{10}$. Flow visualization reveals that, as $Pr$ increases from 11.7 to 145.7, thermal plumes pass through the central region much less frequently and their self-organized large-scale motion is more confined along the periphery of the convection cell. The large-scale flow is found to break down for higher $Pr$, resulting in a regime transition in the Reynolds number $Re$. For the $Pr$ range with a large-scale flow of system size, the $Re$ number, Nusselt number $Nu$ and local temperature fluctuations were investigated systematically. It is found that $Re$ scales as $Re \sim Ra^{0.58}Pr^{-0.82}$ in the present geometry, which suggests that it is in line with the behaviour in the 2-D configuration. On the other hand, the measured $Nu(Ra, Pr)$ relation $Nu \sim Ra^{0.289}Pr^{-0.02}$ tends to be compatible with the finding in a three-dimensional (3-D) system. For the temperature fluctuations in the cell centre and near the sidewall, they exhibit distinct $Ra$-dependent scalings that could not be accounted for with existing theories, but their $Pr$ dependences for $Pr \lesssim 50$ are in agreement with the predictions by Grossmann & Lohse (Phys. Fluids, vol. 16, 2004, pp. 4462–4472). These results enrich our understanding of quasi-2-D thermal convection, and its similarities and differences compared to 2-D and 3-D systems.

A small-disturbance model for a steady, two-dimensional, inviscid and transonic flow of a single-phase real gas around a thin airfoil is presented. The approach explores the nonlinear interactions among near-sonic speed of the flow, small thickness ratio of the airfoil and upstream properties of the fluid. The gas thermodynamic properties are related by a general equation of state. Information about thermodynamic modelling of the gas is lumped into one similarity parameter, $K_G$, related to the fundamental derivative of gas dynamics. The flow field is described by a modified transonic small-disturbance problem. The theory applies to any working fluid of interest. Model problems are derived for steam flows described by the perfect, van der Waals, virial and Redlich–Kwong gas equations of state. Predictions are compared according to the various gas models under various free-stream operating conditions from low subcritical to high supercritical thermodynamic states to gain insights into the sensitivity of the small-disturbance problem solution to thermodynamic modelling of the gas. Results show that transonic flows are independent of gas modelling at low subcritical thermodynamic conditions. However, at near-critical and supercritical thermodynamic conditions, transonic flow behaviour is significantly sensitive to gas modelling and variations of $K_G$. The upstream flow critical Mach number increases as the flow approaches thermodynamic critical state and a wider range of upstream Mach numbers can be found where pressure drag is zero. However, at supercritical conditions, $K_G$ increases, resulting in lower critical Mach numbers and higher pressure drags.

Recently, in Zhang et al. (Phys. Rev. Lett., vol. 124, 2020, 084505), it was found that, in rapidly rotating turbulent Rayleigh–Bénard convection in slender cylindrical containers (with diameter-to-height aspect ratio $\varGamma =1/2$) filled with a small-Prandtl-number fluid (${Pr}\approx 0.8$), the large-scale circulation is suppressed and a boundary zonal flow (BZF) develops near the sidewall, characterized by a bimodal probability density function of the temperature, cyclonic fluid motion and anticyclonic drift of the flow pattern (with respect to the rotating frame). This BZF carries a disproportionate amount (${>}60\,\%$) of the total heat transport for ${Pr} < 1$, but decreases rather abruptly for larger ${Pr}$ to approximately $35\,\%$. In this work, we show that the BZF is robust and appears in rapidly rotating turbulent Rayleigh–Bénard convection in containers of different $\varGamma$ and over a broad range of ${Pr}$ and ${Ra}$. Direct numerical simulations for Prandtl number $0.1 \leq {\textit {Pr}} \leq 12.3$, Rayleigh number $10^7 \leq {Ra} \leq 5\times 10^{9}$, inverse Ekman number $10^{5} \leq 1/{\textit {Ek}} \leq 10^{7}$ and $\varGamma = 1/3$, 1/2, 3/4, 1 and 2 show that the BZF width $\delta _0$ scales with the Rayleigh number ${Ra}$ and Ekman number ${\textit {Ek}}$ as $\delta _0/H \sim \varGamma ^{0} Pr^{\{-1/4, 0\}} {Ra}^{1/4} {\textit {Ek}}^{2/3}$ ($\{{\textit {Pr}}<1, {\textit {Pr}}>1\}$) and with the drift frequency scales as $\omega /\varOmega \sim \varGamma ^{0} Pr^{-4/3} {Ra}\,{\textit {Ek}}^{5/3}$, where $H$ is the cell height and $\varOmega$ the angular rotation rate. The mode number of the BZF is 1 for $\varGamma \lesssim 1$ and $2 \varGamma$ for $\varGamma = \{1,2\}$ independent of ${Ra}$ and ${Pr}$. The BZF is quite reminiscent of wall mode states in rotating convection.

The Rayleigh–Taylor instability of a linear viscoelastic fluid overlying a passive gas is considered, where, under neutral conditions, the key dimensionless groups are the Bond number and the Weissenberg number. The branching behaviour upon instability to sinusoidal disturbances is determined by weak nonlinear analysis with the Bond number advanced from its critical value at neutral stability. It is shown that the solutions emanating from the critical state either branch out supercritically to steady waves at predictable wavelengths or break up subcritically with a wavelength having a single node. The nonlinear analysis leads to the counterintuitive observation that Rayleigh–Taylor instability of a viscoelastic fluid in a laterally unbounded layer must always result in saturated steady waves. The analysis also shows that the subcritical breakup in a viscoelastic fluid can only occur if the layer is laterally bounded below a critical horizontal width. If the special case of an infinitely deep viscoelastic layer is considered, a simple expression is obtained from which the transition between steady saturated waves and subcritical behaviour can be determined in terms of the leading dimensionless groups. This expression reveals that the supercritical saturation of the free surface is due to the influence of the normal elastic stresses, while the subcritical rupture of the free surface is attributed to the influence of capillary effects. In short, depending upon the magnitude of the scaled shear modulus, there exists a wavenumber at which a transition from saturated waves to subcritical breakup occurs.

Analytical solutions of barotropic, quasi-geostrophic vortices over an axisymmetric bottom topography are presented. The solutions are based on independent azimuthal modes adapted to the shape of the topography. Modes 0 (circular monopoles) and 1 (asymmetric dipoles) are evaluated for different topographic profiles that represent either submarine mountains or valleys. The interior fields are matched with an exterior flow with streamlines enclosing the vortices, so the structures remain trapped over the topographic feature. The solutions are steady in a reference frame attached to the rotation of the vortices around the topography. The main features of trapped vortices as a function of the topographic parameters, such as the structure, strength and angular speed, are discussed through numerical simulations initialised with theoretical vorticity fields. The model results reproduce reasonably well the analytical solutions when the topographic effects are strong enough to inhibit the dipole self-propagation. In contrast, very intense dipolar modes may escape from the influence of the topography.

We report the results of an experimental investigation into the decay of turbulence in plane Couette–Poiseuille flow using ‘quench’ experiments where the flow laminarises after a sudden reduction in Reynolds number $Re$. Specifically, we study the velocity field in the streamwise–spanwise plane. We show that the spanwise velocity containing rolls decays faster than the streamwise velocity, which displays elongated regions of higher or lower velocity called streaks. At final Reynolds numbers above $425$, the decay of streaks displays two stages: first a slow decay when rolls are present and secondly a more rapid decay of streaks alone. The difference in behaviour results from the regeneration of streaks by rolls, called the lift-up effect. We define the turbulent fraction as the portion of the flow containing turbulence and this is estimated by thresholding the spanwise velocity component. It decreases linearly with time in the whole range of final $Re$. The corresponding decay slope increases linearly with final $Re$. The extrapolated value at which this decay slope vanishes is $Re_{a_z}\approx 656\pm 10$, close to $Re_g\approx 670$ at which turbulence is self-sustained. The decay of the energy computed from the spanwise velocity component is found to be exponential. The corresponding decay rate increases linearly with $Re$, with an extrapolated vanishing value at $Re_{A_z}\approx 688\pm 10$. This value is also close to the value at which the turbulence is self-sustained, showing that valuable information on the transition can be obtained over a wide range of $Re$.

Place a droplet of mineral oil on water and the oil will spread to cover the water surface in a thin film. In this paper we study the everted problem: an aqueous droplet deposited onto a deep layer of silicone oil. As it is energetically favourable for the oil phase to spread to cover the droplet surface completely, the droplet is ultimately engulfed in the oil layer. We present a detailed study of engulfment dynamics, from the instant the droplet impacts the oil surface until it finally sediments into the less dense oil. We study a broad range of droplet sizes (micrometric to millimetric) and oil kinematic viscosities ($10^2$–$10^5$ cSt), corresponding to a viscosity-dominated parameter regime. We find that droplet engulfment involves the rapid submersion of the droplet driven by capillary forces in the oil surface, followed by the much slower peeling of the droplet from the interface, to which it is adhered via a thin cloaking layer of oil formed during the earlier stage. The later peeling stage is driven by a combination of geometric constraints at the apparent contact line and gravity pulling on the droplet. Gravitational effects are therefore essential to complete engulfment, even for micrometric droplets. Furthermore, the opposing effects of geometry and gravity result in the longest engulfment times for droplets of intermediate size. Experiments at fixed droplet size reveal a power law dependence of engulfment time on oil kinematic viscosity, which we argue reflects the dynamical formation of the oil cloaking layer.

The slat represents an important airframe noise source as it extends over almost the entire aircraft wingspan. Most studies of slat noise consider idealized geometries. However, for practical applications, several elements are installed on its cove, such as bulb seals to avoid direct contact with the main wing surface. Previous investigations of an unswept and untapered MD30P30N airfoil reported that the flow dynamics and the corresponding acoustic noise are very sensitive to the presence and location of the bulb seal. For certain locations a second recirculation bubble is created inside the slat cove and the acoustic narrowband peaks are intensified. The present paper shows that the two-bubble topology promotes the recirculation of turbulence within the slat cove. Spectral proper orthogonal decomposition analysis based on the radiated pressure intensity is used to identify the flow structures responsible for sound generation. Even though the recirculating turbulence is mostly incoherent, it interacts with the coherent Kelvin–Helmholtz vortices in the initial part of the mixing layer. Then, vortex merging and straining lead to the formation of complex vortex clusters. Our results show that the origin and evolution of these clusters are consistent with Rossiter's mechanism responsible for the narrowband peaks. The enhanced recirculation accelerates the cluster evolution leading to wider clusters and lower-frequency Rossiter modes.

In geo- and astrophysics, low Prandtl number convective flows often interact with magnetic fields. Although a static magnetic field acts as a stabilizing force on such flow fields, we find that self-organized convective flow structures reach an optimal state where the heat transport significantly increases and convective velocities reach the theoretical free-fall limit, i.e. the maximum possible velocity a fluid parcel can achieve when its potential buoyant energy is fully converted into kinetic energy. Our measurements show that the application of a static magnetic field leads to an anisotropic, highly ordered flow structure and a decrease of the turbulent fluctuations. When the magnetic field strength is increased beyond the optimum state, Hartmann braking becomes dominant and leads to a reduction of the heat and momentum transport. The results are relevant for the understanding of magneto-hydrodynamic convective flows in planetary cores and stellar interiors in regions with strong toroidal magnetic fields oriented perpendicular to temperature gradients.

Converging flows of viscoplastic fluids, driven steadily through wedges and axisymmetric cones, are studied analytically and numerically. When the yield stress is relatively large, the bulk of the fluid flows plastically apart from within thin layers where the fluid is strongly sheared in order to achieve no slip at the boundary. Conversely, when the yield stress is relatively small, the motion is viscously dominated with weak corrections to the velocity and stress fields due to viscoplastic effects. For both regimes, viscoplasticity induces a weak angular velocity, directed away from the boundaries, and purely radial flow is not possible. The structure of the flow is calculated using asymptotic methods, confirmed by finite element numerical simulations. Flows of both Bingham and Herschel–Bulkley fluids are analysed, and both planar and axisymmetric geometries are considered. Although these cases differ in their details, they share the same qualitative structure. In particular, the viscoplastic boundary layers that emerge when the yield stress is relatively large, ensure not only that no slip is enforced, but also, through an intermediate matching layer, that the shear rates remain bounded.

The swift deformations of flagella and cilia are crucial for locomotion and fluid transport on the micron scale. Most hydrodynamic models of flagellar and ciliary flows assume the zero Reynolds number limit and model the flow using Stokes equations. Recent work has demonstrated that this quasi-steady approximation breaks down at increasing distances from the cilia. Here, we use optical tweezer-based velocimetry to measure the flow velocity with high temporal accuracy, and to reconstruct the entire unsteady flow field around beating cilia. We report both the steady and the unsteady component of the ciliary flow and compare them with the solutions to both the Stokes and the Navier–Stokes equations. Our experimental measurements of the velocity and vorticity fields are in agreement with the numerical solution to the Navier–Stokes equations and show significant differences with the solution to the Stokes equations. We characterize the phase difference between the flow oscillations and the oscillations of the ciliary motion and evidence a significant anisotropic phase lag. We show that this phase lag presents the spatiotemporal characteristics of the unsteady Stokes equations and that the flow field around beating cilia is well represented by the fundamental solution to the unsteady Stokes equations: the oscillet.

This paper develops second-order theory for narrow-banded surface gravity wavepackets experiencing a sudden depth transition based on a Stokes and multiple-scales expansion. As a wavepacket travels over a sudden depth transition, additional wavepackets are generated that propagate freely obeying the linear dispersion relation and arise at both first and second order in wave steepness in a Stokes expansion. In the region near the top of the depth transition, the resulting transient processes play a crucial role. At second order in wave steepness, free and bound waves coexist with different phases. Their different speeds of travel result in a local peak a certain distance after the depth transition. This distance depends on the water depth $h_s$ relative to the carrier wavelength on the shallower side $\lambda _{0s}$. We validate our theory through comparison with fully nonlinear numerical simulations. Experimental validation is provided in a companion paper (Li et al, J. Fluid Mech., 2021, 915, A72). We conjecture that the combination of the local transient peak at second order and the magnitude of the linear free waves provides the explanation for the rogue waves observed after a sudden depth transition reported in a significant number of papers and reviewed in Trulsen etal (J. Fluid Mech., vol. 882, 2020, R2).

Surface gravity wavepackets in intermediate water depth experiencing an abrupt depth decrease are investigated experimentally. The experiments provide validation for the second-order (in steepness) theory for narrow-banded surface gravity wavepackets experiencing a sudden depth transition derived in a companion paper (Li et al., J. Fluid Mech., 2021, 915, A71). We observe the generation of free second-order sub- and superharmonic wavepackets due to the sudden depth transition, in addition to changes to the main (first-order) wavepacket and its second-order bound waves. Locally, just after the step, this leads to the superposition of different wavepacket components. Thereafter, separation occurs because of the different group speeds of the free second-order sub- and superharmonic wavepackets compared with the main packet. Experiments show that the local superposition of waves can lead to significant amplification of wave crests near the top of a step, as predicted by theory. In addition to a step, we also experimentally examine more gradual depth changes in the form of 1 : 1 and 1 : 3 slopes to explore the limits of the theory's validity. Although we find small differences in amplitude and phase comparing these steep slopes with a step, these experiments suggest that the theoretical model derived in Part 1 for wavepackets travelling over a step is applicable to slopes steeper than $1\,{:}\,3$.

Periodic water waves generate Stokes drift as manifest from the orbits of Lagrangian particles not fully closing. Stokes drift can contribute to the transport of floating marine litter, including plastic. Previously, marine litter objects have been considered to be perfect Lagrangian tracers, travelling with the Stokes drift of the waves. However, floating marine litter objects have large ranges of sizes and densities, which potentially result in different rates of transport by waves due to the non-Lagrangian behaviour of the objects. Through a combination of theory and experiments for idealised spherical objects in deep-water waves, we show that different objects are transported at different rates depending on their size and density, and that larger buoyant objects can have increased drift compared with Lagrangian tracers. We show that the mechanism for the increased drift observed in our experiments comprises the variable submergence and the corresponding dynamic buoyancy force components in a direction perpendicular to the local water surface. This leads to an amplification of the drift of these objects compared to the Stokes drift when averaged over the wave cycle. Using an expansion in wave steepness, we derive a closed-form approximation for this increased drift, which can be included in ocean-scale models of marine litter transport.

Flow past a cylinder in the presence of side walls is investigated numerically for various combinations of Reynolds numbers ($50 \leqslant \textit {Re} \leqslant 100$) and aspect ratio ($5 \leqslant AR \leqslant 90$). Various attributes such as cellular shedding, dislocations, oblique angle of vortices and their structure near the end wall are studied. The complexity of the transitions in the wake, with $\textit {Re}$, increases with increase in $AR$. The flow is associated with a single cell wake for $AR=20$, at all the $\textit {Re}$ considered. For $AR$$=90$ the flow transitions from a one-cell to two-cell wake at $\textit {Re} \sim 50$, to a four-cell structure at $\textit {Re} \sim 55$ to a three-cell wake at $\textit {Re} \sim 58$, and then to a two-cell wake at $\textit {Re} \sim 61$. The vortices near the end wall diffuse for $Re\leqslant 58$ whereas linkages form between adjacent vortices of opposite polarity at larger $Re$. The frequency of dislocations increases with increase in $\textit {Re}$. Two types of dislocations have been identified: the fork type at relatively low Re and connected fork type at larger Re. Linkages between vortices in connected fork-type dislocations may lead to ring-like structures. The end conditions and nonlinear mechanisms play a significant role in the evolution of cellular shedding. A single cell across the span is the dominant mode of shedding in the linear regime. Compared with the linear analysis, nonlinearities result in smaller oblique angles, higher shedding frequency and larger streamwise speed of the convection of vortices.