We study the dynamics and the statistics of dilute suspensions of gyrotactic swimmers, a model for many aquatic motile microorganisms. By means of extensive numerical simulations of the Navier–Stokes equations at different Reynolds numbers, we investigate preferential sampling and small-scale clustering as a function of the swimming (stability and speed) and shape parameters, considering in particular the limits of spherical and rod-like particles. While spherical swimmers preferentially sample local downwelling flow, for elongated swimmers we observe a transition from downwelling to upwelling regions at sufficiently high swimming speed. The spatial distribution of both spherical and elongated swimmers is found to be fractal at small scales in a wide range of swimming parameters. The direct comparison between the different shapes shows that spherical swimmers are more clusterized at small stability and speed numbers, while for large values of the parameters elongated cells concentrate more. The relevance of our results for phytoplankton swimming in the ocean is briefly discussed.

The nonlinear evolution of an interface between a perfect conducting liquid and a perfect dielectric gas subject to periodic electrostatic forcing is studied under the long-wave approximation. It is shown that inertial thin films become unstable to finite-wavelength Faraday modes at the onset, prior to the long-wave pillaring instability reported in the lubrication limit. It is further shown that the pillaring-mode instability is subcritical in nature, with the interface approaching either the top or the bottom wall, depending on the liquid–gas holdup. On the other hand, the Faraday modes exhibit subharmonic or harmonic oscillations that nonlinearly saturate to standing waves at low forcing amplitudes. Unlike the pillaring mode, wherein the interface approaches the wall, Faraday modes may exhibit saturated standing waves when the instability is subcritical. At higher forcing amplitudes, the interface may approach either wall, again depending on the liquid–gas holdup. It is also shown that a gravitationally unstable configuration of such thin films, under the long-wave approximation, cannot be stabilized by periodic electrostatic forcing, unlike mechanical Faraday forcing. In this case, it is observed that the interface exhibits oscillatory sliding behaviour, approaching the wall in an ‘earthworm-like’ motion.

Experimental evidence is provided to demonstrate that the upstream-travelling waves in two jets screeching in the A1 and A2 modes are not free-stream acoustic waves, but rather waves with support within the jet. Proper orthogonal decomposition is used to educe the coherent fluctuations associated with jet screech from a set of randomly sampled velocity fields. A streamwise Fourier transform is then used to isolate components with positive and negative phase speeds. The component with negative phase speed is shown, by comparison with a vortex-sheet model, to resemble the upstream-travelling jet wave first studied by Tam & Hu (J. Fluid Mech., vol. 201, 1989, pp. 447–483). It is further demonstrated that screech tones are only observed over the frequency range where this upstream-travelling wave is propagative.

Designing effective control for complex three-dimensional flow fields proves to be non-trivial. Often, intuitive control strategies lead to suboptimal control. To navigate the control space, we use a linear parabolized stability analysis to guide the design of a control scheme for a trailing vortex flow field aft of a NACA0012 half-wing at an angle of attack $\unicode[STIX]{x1D6FC}=5^{\circ }$ and a chord-based Reynolds number $Re=1000$. The stability results show that the unstable mode with the smallest growth rate (fifth wake mode) provides a pathway to excite a vortex instability, whereas the principal unstable mode does not. Inspired by this finding, we perform direct numerical simulations that excite each mode with body forces matching the shape function from the stability analysis. Relative to the uncontrolled case, the controlled flows show increased attenuation of circulation and peak streamwise vorticity, with the fifth-mode-based control set-up outperforming the principal-mode-based set-up. From these results, we conclude that a rudimentary linear stability analysis can provide key insights into the underlying physics and help engineers design effective physics-based flow control strategies.

This paper studies the transition to three-dimensional flow in the wake of a cylinder immersed in a free stream, where the cylinder is externally forced to continuously rotate about its axis and to linearly oscillate in the streamwise direction. Floquet stability analysis is used to assess the stability of the nominal two-dimensional flows at a Reynolds number $Re=100$ and rotation rate $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D714}D/U=3$ to three-dimensional perturbations, as a function of the amplitude and frequency of the linear oscillations. Two modes of instability are found, distinguished by their spatial structure, temporal behaviour and apparent mechanism. The first mode has a shorter wavelength in the spanwise direction and appears to be linked to a centrifugal instability in the layer of fluid near the rotating body. The second mode has a longer wavelength and is linked to an instability of the vortex cores in the wake that is subharmonic, leading to a period doubling. Either mode can be stable while the other is unstable, depending primarily on the frequency of the oscillation of the cylinder. This indicates that either mode can control the transition to a three-dimensional flow. The results are compared to the fully three-dimensional simulation results of a rotating cylinder elastically mounted and free to oscillate in the streamwise direction from Bourguet & Lo Jacono (J. Fluid Mech., vol. 781, 2015, pp. 127–165), and appear to be able to explain the surprising switching of the observed spanwise wavelength in that flow as a change in the dominant mode, and therefore mechanism, of instability.

A structure capable of substantially amplifying water waves over a broad range of frequencies at selected locations is proposed. The structure consists of a small number of C-shaped cylinders in a line array, with the cylinder properties graded along the array. Using linear potential-flow theory, it is shown that the energy carried by a plane incident wave is amplified within specified cylinders for wavelengths comparable to the array length and for a range of incident directions. Transfer-matrix analysis is used to attribute the large amplifications to excitation of local Rayleigh–Bloch waves and gradual slowing down of their group velocity along the array.

We numerically study the impact of a compound drop on a hydrophobic substrate using a ternary-fluid diffuse-interface method, aiming to understand how the presence of the inner droplet affects the spreading dynamics and maximal spreading of the compound drop. First, it is interesting to see that the numerical results for an impacting pure drop agree well with the universal rescaling of maximal spreading ratio proposed by Lee et al. (J. Fluid Mech., vol. 786, 2016, R4). Second, two flow regimes have been identified for an impacting compound drop: namely jammed spreading and joint rim formation. The maximal spreading ratio of the compound drop is found to depend on the volume fraction of the inner droplet $\unicode[STIX]{x1D6FC}$, the surface tension ratio $\unicode[STIX]{x1D6FE}$, the Weber number and the flow regime. Moreover, we propose a universal rescaling of maximal spreading ratio for compound drops, by integrating the one for pure drops with a corrected Weber number that takes $\unicode[STIX]{x1D6FC}$, $\unicode[STIX]{x1D6FE}$ and the flow regime into account. The predictions of the universal rescaling are in good agreement with the numerical results for impacting compound drops.

The motion of a free-moving plate atop turbulent thermal convection exhibits diverse dynamics that displays characteristics of both deterministic and chaotic motions. Early experiments performed by Zhong & Zhang (Phys. Rev. E, vol. 75 (5), 2007, 055301) found an oscillatory and a trapped state existing for a plate floating on convective fluid in a rectangular tank. They proposed a piecewise smooth physical model (ZZ model) that successfully captures this transition of states. However, their model was deterministic and therefore could not describe the stochastic behaviours. In this study, we combine the ZZ model with a novel approach that models the stochastic aspects through a variational inequality structure. With the powerful mathematical tools for stochastic variational inequalities, the properties of the Markov process and corresponding Kolmogorov equations could be studied both numerically and analytically. Moreover, this framework also allows one to compute the transition probabilities. Our present work captures the stochastic aspects of the two aforementioned boundary–fluid coupling states, predicts the stochastic behaviours and shows excellent qualitative and quantitative agreements with the experimental data.

We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of mixed convective and stably stratified fluids. We demonstrate that in the limit of large Rayleigh number ($Ra\in [4\times 10^{7},10^{9}]$) and large stratification (Brunt–Väisälä frequencies $f_{N}\gg f_{c}$, where $f_{c}$ is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region as described in Lecoanet & Quataert (Mon. Not. R. Astron. Soc., vol. 430 (3), 2013, pp. 2363–2376). Specifically, we demonstrate that the wave energy flux spectrum scales like $k_{\bot }^{4}\,f^{-13/2}$ for weakly damped waves (with $k_{\bot }$ and $f$ the waves’ horizontal wavenumbers and frequencies, respectively), and that the total wave energy flux decays with $z$, the distance from the convective region, like $z^{-13/8}$.

Pairs of nested vortex rings, one with coils, are evolved numerically to compare their topological numbers to those of recent experiments reported in Scheeler et al. (Science, vol. 357, 2017, pp. 487–491). Included are the twist $Tw$, writhe $Wr$ and self-linking ${\mathcal{L}}_{S}$ numbers, plus centreline helicities ${\mathcal{H}}_{c}$. The questions are: can the experimental numbers be validated and do these numbers have roles in the dynamics of the global helicities ${\mathcal{H}}$ and enstrophies $Z$ with respect to cascades? Topological analysis of the experiments $t=0$ analytic centreline vortex trajectories validates only the writhe measurements, not their values of $Tw$ and ${\mathcal{L}}_{S}$, which obey $Tw\lesssim {\mathcal{L}}_{S}=m\gg Wr$ for $m$-coil rings. Not $Tw\ll Wr$. To suggest why the large twists do not contribute to ${\mathcal{H}}$, it is noted that the mapping of the coiled rings onto the mesh is to a first approximation a single pair of Clebsch potentials, whose self-helicity ${\mathcal{H}}_{S}\equiv 0$. Numerical rings with circulations $\unicode[STIX]{x1D6E4}$, including some single rings, show small initial helicities with ${\mathcal{H}}(0)\approx {\mathcal{H}}_{c}\sim (\text{1 to 2})Wr\unicode[STIX]{x1D6E4}^{2}$$\ll {\mathcal{L}}_{S}\unicode[STIX]{x1D6E4}^{2}$. For time and velocity scales that are consistent with the experiments, as the coils evolve, their $Tw$, $Wr$, ${\mathcal{L}}_{S}$ numbers and their helicities are nearly static until reconnection. Nonetheless, $Wr$ and $Tw$ retain important complementary roles in the dynamics of the global helicity ${\mathcal{H}}$ and enstrophy $Z$, with the evolution of the torsion $\unicode[STIX]{x1D70F}(s)$ profile showing the beginnings of a cascade to small scales.

The question of whether significant subvolumes of a turbulent flow can be identified by automatic means, independently of a priori assumptions, is addressed using the example of two-dimensional decaying turbulence. Significance is defined as influence on the future evolution of the flow, and the problem is cast as an unsupervised machine ‘game’ in which the rules are the Navier–Stokes equations. It is shown that significance is an intermittent quantity in this particular flow, and that, in accordance with previous intuition, its most significant features are vortices, while the least significant ones are dominated by strain. Subject to cost considerations, the method should be applicable to more general turbulent flows.

A new model for the drag force on a two-dimensional flat plate of arbitrary porosity, oriented normal to the free stream, is introduced. The model is an extension of that introduced by Koo & James (J. Fluid Mech., vol. 60(3), 1973, pp. 513–538), where the performance at low porosities is improved by including a base-suction term. The additional drag due to the base suction is calculated implicitly using momentum theory, which makes the model self-contained. The model predictions exhibit convincing agreement with experimental observations over a wide range of porosities, including the solid case, as long as shedding is absent or suppressed.

High-fidelity experiments of Richtmyer–Meshkov instability on a single-mode air/$\text{SF}_{6}$ interface are carried out at weak shock conditions. The soap-film technique is extended to create single-mode gaseous interfaces which are free of small-wavelength perturbations, diffusion layers and three-dimensionality. The interfacial morphologies captured show that the instability evolution evidently involves the smallest experimental uncertainty among all existing results. The performances of the impulsive model and other nonlinear models are thoroughly examined through temporal variations of the perturbation amplitude. The individual growth of bubbles or spikes demonstrates that all nonlinear models can provide a reliable forecast of bubble development, but only the model of Zhang & Guo (J. Fluid Mech., vol. 786, 2016, pp. 47–61) can reasonably predict spike development. The distinct images of the interface morphology obtained also provide a rare opportunity to extract interface contours such that a spectral analysis of the interfacial contours can be performed, which realizes the first direct validation of the high-order nonlinear models of Zhang & Sohn (Phys. Fluids, vol. 9, 1997, pp. 1106–1124) and Vandenboomgaerde et al. (Phys. Fluids, vol. 14 (3), 2002, pp. 1111–1122) in terms of the fundamental mode and high-order harmonics. It is found that both models show a very good and almost identical accuracy in predicting the first two modes. However, the model of Zhang & Sohn (1997) becomes much more accurate in modelling the third-order harmonics due to the fewer simplifications used.

We consider the application of active flow control to modify the radial pressure distribution of a single-phase wall-normal vortex. The present flow is based on the Burgers vortex model but with a no-slip boundary condition prescribed along its symmetry plane. The wall-normal vortex serves as a model for vortices that emerge upstream of turbomachinery, such as pumps. This study characterizes the baseline vortex unsteadiness through numerical simulation and dynamic mode decomposition. The insights gained from the baseline flow are used to develop an active flow control technique with rotating zero-net-mass blowing and suction with the objective of modifying the core-pressure distribution. The effectiveness of the control strategy is demonstrated by achieving a widened vortex core with increased pressure. This change in the flow field weakens the local strength of the wall-normal vortex core, potentially inhibiting the formation of hollow-core vortices, commonly encountered in liquids.

A data-driven model-free framework is introduced for the calculation of reduced-order models (ROMs) capable of accurately predicting time-mean responses to external forcings, or forcings needed for specified responses, e.g. for control, in fully turbulent flows. The framework is based on using the fluctuation–dissipation theorem (FDT) in the space of a limited number of modes obtained from dynamic mode decomposition (DMD). Use of the DMD modes as the basis functions, rather than the commonly used proper orthogonal decomposition modes, resolves a previously identified problem in applying FDT to high-dimensional non-normal turbulent flows. Employing this DMD-enhanced FDT method ($\text{FDT}_{DMD}$), a linear ROM with horizontally averaged temperature as state vector is calculated for a 3D Rayleigh–Bénard convection system at a Rayleigh number of $10^{6}$ using data obtained from direct numerical simulation. The calculated ROM performs well in various tests for this turbulent flow, suggesting $\text{FDT}_{DMD}$ as a promising method for developing ROMs for high-dimensional turbulent systems.

Ice-covered waterbodies are far from being quiescent systems. In this paper, we investigate ice-covered freshwater basins heated by solar radiation that penetrates across waters with temperatures below or near the temperature of maximum density. In this scenario, solar radiation sets a radiative buoyancy flux, $\unicode[STIX]{x1D6F7}_{r}$, that forces increments of temperature/density in the upper fluid volume, which can become gravitationally unstable and drive convection. The goal of this study is twofold. We first focus on formulating the mechanical energy budget, putting emphasis on the conversion of $\unicode[STIX]{x1D6F7}_{r}$ to available potential energy, $E_{a}$. We find that $E_{a}$ results from a competition among $\unicode[STIX]{x1D6F7}_{r}$ and the irreversible mixing controlled by the diapycnal and the laminar mixing rates, respectively. Secondly, and based on the above result, we introduce an integral formulation of the mixing efficiency to quantify the rate of mixing over the relevant time scale $\unicode[STIX]{x1D70F}$, $\unicode[STIX]{x1D702}_{c}\equiv \unicode[STIX]{x0394}E_{b,\unicode[STIX]{x1D70F}}/E_{r,\unicode[STIX]{x1D70F}}$, where $\unicode[STIX]{x0394}E_{b,\unicode[STIX]{x1D70F}}$ and $E_{r,\unicode[STIX]{x1D70F}}$ are the change of background potential energy and the time-integrated $\unicode[STIX]{x1D6F7}_{r}$ over $\unicode[STIX]{x1D70F}$. The above definition is applied to estimate $\unicode[STIX]{x1D702}_{c}$ for the first time, finding an approximate value of $\unicode[STIX]{x1D702}_{c}\approx 0.65$. This result suggests that radiatively heated ice-covered waterbodies might be subject to high mixing rates. Overall, the present work provides a framework to examine energetics and mixing in ice-covered waters.

The dynamics of two rectangular, flexible plates of low aspect ratio $h/b$ (height/width= 4) was experimentally investigated in tandem arrangements under uniform flows at various Cauchy numbers $Ca=\unicode[STIX]{x1D70C}_{f}bU_{0}^{2}h^{3}/(EI)\in [15,77]$ (where $\unicode[STIX]{x1D70C}_{f}$ is the fluid density, $U_{0}$ is the incoming flow velocity, $E$ is the Young’s modulus and $I$ is the second moment of the area) and spacings $s_{x}=\unicode[STIX]{x0394}x/h=0.5$, 1 and 2. Planar particle image velocimetry (PIV) and particle tracking velocimetry (PTV) were used to characterize the surrounding flow field and oscillations of the wall-mounted structures. Results show that the oscillations of the upstream plate were dominated by its natural frequency. However, the motions of the downstream plate were significantly modulated by the induced flow and coherent motions shed from the upstream structure. Such modulation led to highly correlated motions between the plates with similar amplitude in the case $s_{x}=0.5$, to comparatively larger oscillation amplitude of the downstream plate in the intermediate separation $s_{x}=1$, and roughly decoupled interaction for $s_{x}=2$. Despite that the intensity of the oscillations of the upstream plate increased monotonically with $Ca$, this was not the case for the downstream plate at $s_{x}=1$ and 2 due to flow fluctuations, vortex shedding and large structure deformation; as a result, it exhibited a local minimum. Supported with measurements, a mathematical model was derived to quantitatively explain this behaviour.

The divergence-free time-independent velocity field has been determined so as to maximise heat transfer between two parallel plates with a constant temperature difference under the constraint of fixed total enstrophy. The present variational problem is the same as that first formulated by Hassanzadeh et al. (J. Fluid Mech., vol. 751, 2014, pp. 627–662); however, the search range for optimal states has been extended to a three-dimensional velocity field. A scaling of the Nusselt number $Nu$ with the Péclet number $Pe$ (i.e., the square root of the non-dimensionalised enstrophy with thermal diffusion time scale), $Nu\sim Pe^{2/3}$, has been found in the three-dimensional optimal states, corresponding to the asymptotic scaling with the Rayleigh number $Ra$, $Nu\sim Ra^{1/2}$, expected to appear in an ultimate state, and thus to the Taylor energy dissipation law in high-Reynolds-number turbulence. At $Pe\sim 10^{0}$, a two-dimensional array of large-scale convection rolls provides maximal heat transfer. A three-dimensional optimal solution emerges from bifurcation on the two-dimensional solution branch at $Pe\sim 10^{1}$, and the three-dimensional solution branch has been tracked up to $Pe\sim 10^{4}$ (corresponding to $Ra\approx 2.7\times 10^{6}$). At $Pe\gtrsim 10^{3}$, the optimised velocity fields consist of convection cells with hierarchical self-similar vortical structures, and the temperature fields exhibit a logarithmic-like mean profile near the walls.

This study explores the formation of circular thin-film hydraulic jumps caused by the normal impact of a jet on an infinite planar surface. For more than a century, it has been believed that all hydraulic jumps are created due to gravity. However, we show that these thin-film hydraulic jumps result from energy loss due to surface tension and viscous forces alone. We show that, at the jump, surface tension and viscous forces balance the momentum in the liquid film and gravity plays no significant role. Experiments show no dependence on the orientation of the surface and a scaling relation balancing viscous forces and surface tension collapses the experimental data. A theoretical analysis shows that the downstream transport of surface energy is the previously neglected critical ingredient in these flows, and that capillary waves play the role of gravity waves in a traditional jump in demarcating the transition from the supercritical to subcritical flow associated with these jumps.

The linear stability of a stably stratified fluid-filled cavity subject to vertical oscillations is determined via Floquet analysis. Retaining the viscous and diffusion terms in the Navier–Stokes–Boussinesq equations, with no-slip velocity boundary conditions, no-flux temperature conditions on the sidewalls and constant temperatures on the top and bottom walls, we find that the instabilities are primarily subharmonic (as is typical in many parametrically forced systems), except for in a few low-forcing-frequency ranges where the instabilities are synchronous. When the viscosity is small, the Floquet modes resemble the inviscid eigenmodes of the unforced problem, except in boundary layers. We establish scaling laws quantifying how viscosity regularizes the degeneracy associated with the inviscid idealization, and how it scales the thickness and intensity of the boundary layers. The product of boundary layer thickness and intensity remains constant with decreasing viscosity, leading to a delta distribution of vorticity on the walls in the limit of zero viscosity. This is in contrast to the zero wall vorticity in the inviscid case.