Non-decaying bores are generated in a laboratory using a dam-break system with different reservoir length and depth ratios at the dam-break gate. The experimental data show the dependency of inundation depth, run-up height and flood duration on the reservoir length and the bore strength at the beach toe. Employing the method of characteristics, the relationship between the reservoir length and the bore characteristics at beach toe is obtained. Numerical simulations are carried out for a series of dam-break generated bores, extending the range of physical parameters used in the laboratory experiments. The accuracy of the numerical results is confirmed by the experimental data. Predictive formulae are then obtained for the inundation depth, run-up height and flood duration in terms of the bore characteristics at beach toe and beach slope. Finally, the minimum bore lengths at beach toe necessary to produce the maximum inundation depths and the flooding plateau are identified in the parameter space. These relations can be employed to design dam-break experiments for generating bores with a target length or duration.

New experimental results on turbulent boundary layers (TBLs) with wall suction show that it is possible to experimentally realise a turbulent asymptotic suction boundary layer (TASBL), i.e. a boundary layer which becomes independent of the streamwise location and with the suction rate as the only control parameter. Turbulent asymptotic suction boundary layers show a mean-velocity profile with a large logarithmic region and without a clear wake region. If outer-scaling is adopted, using the free stream velocity and the boundary layer thickness as characteristic velocity and length scale, respectively, a single log-law describes the logarithmic region of all the measured TASBLs independently from the suction rate. Streamwise velocity profiles were measured with different hot-wire probe sizes, in order to account for and correct for probe-filtering effects. It emerges that wall suction is responsible for strong damping of the velocity fluctuations, with a decrease of the near-wall peak of the velocity-variance profile ranging from $50\,\%$ to $65\,\%$ when compared with a canonical zero-pressure gradient TBL at comparable Reynolds number. The analysis of the power spectral density maps suggests that the decrease in the turbulent activity can be explained by increased stability of the near-wall streaks.

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).

The stratified inner-shelf response to surf-zone-generated transient rip currents (TRC) is examined using idealized simulations with uniform initial thermal stratification $\mathrm {d}{T_0}/\mathrm {d}{z}$, with initial temperature T0 and vertical coordinate z, or initial squared buoyancy frequency $N_0^2$, varying from unstratified to highly stratified ($0.75^\circ \text {C}\ \text {m}^{-1}$). The TRC-induced depth-integrated cross-shore eddy kinetic energy flux is independent of $\mathrm {d}{T_0}/\mathrm {d}{z}$ and decays to near zero within $4L_{SZ}$ (surf-zone width $L_{SZ}=100$ m). Cross-shore inhomogeneous TRC mixing causes shoreward broadening isotherms, driving a near-field inner-shelf overturning circulation and a far-field geostrophic along-shore velocity that strengthen with $\mathrm {d}{T_0}/\mathrm {d}{z}$. TRC mixing mostly (90 %) increases background potential energy (BPE) and also available (APE, 10 %) potential energy, driving inner-shelf mean circulation. The specific BPE zero-crossing depth $d_{s}$ collapses the near-field $3$-layer cross-shore velocity, using an intrusive gravity current scaling $(d_{s} N_0)$. The approximately steady exchange flow exports low-stratification fluid ($N/N_0\approx 1/2$) at a depth $z/d_{s}\approx -1$, re-supplying the TRC region with stratified fluid from above/below, similar to localized mixing laboratory experiments. Offshore of $\approx 5L_{SZ}$, a self-similar far-field intrusion with characteristic isotherm slope $d_{s}/L_{R}$ (Rossby deformation radius $L_{R}\sim d_{s} f/N_0$) is in approximate geostrophic balance with the non-dimensional along-shore velocity. Inner-shelf near-field and far-field horizontal length scales vary as $x/d_{s}$ and $x/L_{R}$, respectively. The length scale $d_{s}$ is related to the work performed by TRC mixing using an idealized well-mixed wedge geometry. Idealized analytical scalings are qualitatively consistent with modelled BPE and APE distributions. Thus, the self-similar stratified inner-shelf response to TRC-driven mixing depends on key dimensional parameters $N_0$ and $d_{s}$.

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 $1\,{:}\,3$.

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.

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.

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.

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.

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 $Re$.

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.

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.

Based on the linear shallow-water equations, new analytical solutions are derived for trapped waves over a ridge with a hyperbolic-cosine squared cross-sectional profile which may be used to idealize many real-world ocean ridges. In the new analytical formulation, the free surface of the trapped waves is described using the combination of the first and second kinds of the associated Legendre functions, which is further analysed to reveal the existence of both symmetrical and anti-symmetrical trapped waves on the ridge under consideration. New algebraic equations are also derived to depict the wave dispersion relationships, allowing explicit quantification of their sensitivity to the topographic profile. Furthermore, a ray-tracing method is applied to interpret the propagation paths of trapped waves over the ridge and better understand the excitation mechanisms. Finally, an extensively validated Boussinesq wave model is used to conduct numerical experiments for trapped waves induced by tsunamis. The numerical predictions are consistent with the new analytical solutions, which effectively confirms the validity of the new analytical framework for trapped waves over a more general type of oceanic ridges.

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.

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.

The evolution of the boundary layer vortex sheet on a rotating and translating accelerating circular cylinder at Reynolds numbers of 10 000 and 20 000 is investigated using planar particle image velocimetry. The vortex sheet is decomposed into contributions resulting from translation and rotation as well as from local and far-field vorticity. Their individual development is explored to understand the overall time history of the boundary layer as well as its evolution at the unsteady separation point. The boundary layer vortex sheet distribution changes considerably throughout the motion as well as between different kinematic cases. The same is observed for the vortex sheet strength at the unsteady separation point. A non-dimensional parameter is proposed which removes the effect of rotation rate, instantaneous velocity and shed vorticity accumulating in the far field. It was found that this was successful at collapsing the vortex sheet strength at the unsteady separation point during cylinder motion as well as for the individual kinematic test cases investigated. This confirms that cylinder kinematics and far-field vorticity are driving factors contributing to the development of the unsteady boundary layer and its strength at the separation point.

Elastohydrodynamic lubrication, or simply soft lubrication, refers to the motion of deformable objects near a boundary lubricated by a fluid, and is one of the key physical mechanisms to minimise friction and wear in natural and engineered systems. Hence, it is of particular interest to relate the thickness of the lubricant layer to the entrainment (sliding/rolling) velocity, the mechanical loading exerted onto the contacting elements and the properties of the elastic boundary. In this work, we provide an overview of the various regimes of soft lubrication for two-dimensional cylinders in lubricated contact with compliant walls. We discuss the limits of small and large entrainment velocity, which are equivalent to large and small elastic deformations, as the cylinder moves near thick or thin elastic layers. The analysis focusses on thin elastic coatings, both compressible and incompressible, for which analytical scaling laws are not yet available in the regime of large deformations. By analysing the elastohydrodynamic boundary layers that appear at the edge of the contact, we establish the missing scaling laws – including prefactors. As such, we offer a rather complete overview of the physically relevant limits of soft lubrication.

Stereoscopic particle image velocimetry measurements of open-channel flows over streamwise-orientated triangular-shaped ridges were used to explore interactions between ridge-induced secondary currents (SCs) and turbulence. Terms in the double-averaged (in space and in time) momentum and energy conservation equations were analysed for a range of ridge spacings ($s$) between 0.4 and 4.0 flow depths ($H$). The double-averaged equations neatly partition momentum and energy fluxes into turbulence and SC contributions, making them well suited to this study. The obtained data indicate that for a range of $s/H$ between 0.4 and 2.0, the normalised momentum and energy fluxes due to SCs approximately collapse when plotted as functions of $(z-d)/s$, where $z$ is the vertical coordinate and $d$ a constant that aligns the elevations of SC cell centres. The SCs controlled the shape of the mean velocity distribution with the vertical gradient of the double-averaged streamwise velocity found to be inversely proportional to $s$ near the elevations of SC cell centres. Partitioning the total kinetic energy into double-mean (DMKE), dispersive (DKE) and turbulent (SATKE) components and considering the balance equation for each component indicated that at the elevation of SC cell centres the production rate of SATKE via exchange with DMKE was comparable in magnitude to the production rate via exchange with DKE (due to SCs). For all ridge spacings, SATKE was reduced compared to a no-ridge benchmark case due to suppression of very-large-scale turbulent motions by the SCs. Finally, it is demonstrated that energy is supplied to SCs by turbulence.

Geometric focusing of monochromatic internal waves is a well-known linear mechanism that provides a pathway to smaller length scales and hence energy dissipation, especially when repeated reflection from inclined boundaries focuses these waves onto internal wave attractors (IWAs). While simple attractors have been well documented both theoretically and in idealized laboratory settings, the small aspect ratio and complex shape of the ocean has made it difficult to detect the narrow frequency bands where attractors may form. The greatest obstruction however is the restricted spatiotemporal resolution of measurements of the ocean's internal wave field. Moored instruments provide a detailed temporal but poor spatial resolution, except along line-segments spanning part of the water column only. The ability to measure a fully resolved two-dimensional section through a three-dimensional field, as in present-day laboratory experiments, is missing in the ocean. This prohibits the search for IWAs in ocean data, especially if this concerns a multi-frequency wave field. Mimicking an ocean observation set-up, our experimental study aims at detecting IWA signatures from laboratory observations along a vertical line-segment. We perform small and large amplitude forcing experiments at a fixed wavenumber with both monochromatic and broadband frequency components. We examine energy intensification at dissipative length scales at frequencies known to support attractors. Importantly, our laboratory experiments indicate that even in the presence of multiple forcing frequencies, or for large-amplitude monochromatic forcing (where beams become unstable to Triadic Resonant Instability), focusing reflections due to attractors are the predominant mechanism for transferring energy to dissipative scales.

We propose a model and numerical method for the propulsion of rectangular and rhombic lattices of flapping plates at $O$(10–100) Reynolds numbers in incompressible flow. The numerical method uses an adaptive mesh to mitigate singularities at the plates’ edges. We establish convergence rates and find good numerical accuracy in a test problem: Laplace's equation in the region exterior to a plate. We then use the method to establish benchmark results for a single flapping plate, including vortex wake characteristics and Froude efficiency over ranges of flapping amplitude, frequency and Strouhal number. As a prelude to a study of propulsive efficiency in Part 2 (J. Fluid Mech., vol. 915, 2021, A21), we study a key ingredient: the time-averaged input power in lattices of plates. Scaling laws for the mean input power are estimated in the limits of small and large streamwise spacings, using steady flow models with small-gap and Poiseuille-like flows between the plates respectively in the two limits. For both lattice types, the mean input power saturates as the lateral spacing becomes large (and thrust occurs). At small lateral spacings, the rhombic lattices’ input power becomes much larger when the plates overlap. The time-averaged input power in flapping lattices agrees qualitatively with the steady models.