In the present study, compressible low-Reynolds-number flow past a stationary isolated sphere was investigated by direct numerical simulations of the Navier–Stokes equations using a body-fitted grid with high-order schemes. The Reynolds number based on free-stream quantities and the diameter of the sphere was set to be between 250 and 1000, and the free-stream Mach number was set to be between 0.3 and 2.0. As a result, it was clarified that the wake of the sphere is significantly stabilized as the Mach number increases, particularly at the Mach number greater than or equal to 0.95, but turbulent kinetic energy at the higher Mach numbers conditions is higher than that at the lower Mach numbers conditions of similar flow regimes. A rapid extension of the length of the recirculation region was observed under the transitional condition between the steady and unsteady flows. The drag coefficient increases as the Mach number increases mainly in the transonic regime and its increment is almost due to the increment in the pressure component. In addition, the increment in the drag coefficient is approximately a function of the Mach number and independent of the Reynolds number in the continuum regime. Moreover, the effect of the Mach and Reynolds numbers on the flow properties such as the drag coefficient and flow regime can approximately be characterized by the position of the separation point.

We investigate theoretically, on the basis of the steady Stokes equations for a viscous incompressible fluid, the flow induced by a stokeslet located on the centre axis of two coaxially positioned rigid disks. The stokeslet is directed along the centre axis. No-slip boundary conditions are assumed to hold at the surfaces of the disks. We perform the calculation of the associated Green's function in large parts analytically, reducing the spatial evaluation of the flow field to one-dimensional integrations amenable to numerical treatment. To this end, we formulate the solution of the hydrodynamic problem for the viscous flow surrounding the two disks as a mixed boundary-value problem, which we then reduce to a system of four dual integral equations. We show the existence of viscous toroidal eddies arising in the fluid domain bounded by the two disks, manifested in the plane containing the centre axis through adjacent counter-rotating eddies. Additionally, we probe the effect of the confining disks on the slow dynamics of a point-like particle by evaluating the hydrodynamic mobility function associated with axial motion. Thereupon, we assess the appropriateness of the commonly employed superposition approximation and discuss its validity and applicability as a function of the geometrical properties of the system. Additionally, we complement our semi-analytical approach by finite-element computer simulations, which reveals a good agreement. Our results may find applications in guiding the design of microparticle-based sensing devices and electrokinetic transport in small-scale capacitors.

The Ginzburg–Landau equation (GLE) can phenomenologically model several key features of non-equilibrium systems including those in fluid mechanics. Its validity in real flows, however, remains questionable. Here, we show that the linear GLE can be formulated such that it has the same Wentzel–Kramers–Brillouin (WKB) approximation as for the linear global stability problem in open shear flows. We use the GLE to model the linear global modes of three different wakes and find that it can accurately capture the linear growth rate and frequency to first order in the WKB approximation. Furthermore, we find that it can also provide the shapes of the direct and adjoint eigenvectors and the regions of maximal structural sensitivity. The proposed model requires only the basic flow as input, but gives robust predictions and is computationally inexpensive. As well as opening up new possibilities for GLE-based control strategies, the proposed model makes accurate stability calculations possible, even for some computationally intractable open shear flows.

Reconfiguration events in turbulent mixed convection, i.e. the superposition of thermal and forced flow contributions, at the two different Richardson numbers $Ri=1.5$ and $Ri=3.7$ and similar Rayleigh numbers of $Ra \approx 10^8$ are investigated with tomographic particle image velocimetry in combination with local temperature measurements. For both cases, the three-dimensional velocity fields reflect diagonally aligned large-scale circulations (LSC) switching their alignment by rotating their axes around a pivot located at the centre of the LSC, while the temperatures perform a translation movement of the structures in agreement with earlier temperature-based investigations. For the high $Ri$ case, the switching process of the observed spontaneous reconfigurations is induced by a reversing thermal flow contribution while the forced flow contribution is constant. Furthermore, it is shown that a secondary roll structure, which drives the reconfiguration process in Rayleigh–Bénard convection, also exists in mixed convection. However, in the latter, the flow reversals are triggered by different structures which accumulate and release their kinetic energy according to a proper orthogonal decomposition analysis. In contrast, for the low $Ri$ case, the structure formation during continuous reconfigurations is governed by a Taylor- or Görtler-type instability. This means that the forced convection substantially affects the reconfiguration mechanism of these structures. Therefore, the reconfigurations cannot be described by a simple superposition of structures associated with the two flow contributions as for the high $Ri$.

Finite-amplitude hydromagnetic Rossby waves in the magnetostrophic regime are studied. We consider the slow mode, which travels in the opposite direction to the hydrodynamic or fast mode, in the presence of a toroidal magnetic field and zonal flow by means of quasi-geostrophic models for thick spherical shells. The weakly nonlinear long waves are derived asymptotically using a reductive perturbation method. The problem at the first order is found to obey a second-order ordinary differential equation, leading to a hypergeometric equation for a Malkus field and a confluent Heun equation for an electrical wire field, and is non-singular when the wave speed approaches the mean flow. Investigating its neutral non-singular eigensolutions for different basic states, we find the evolution is described by the Korteweg–de Vries equation. This implies that the nonlinear slow wave forms solitons and solitary waves. These may take the form of a coherent eddy, such as a single anticyclone. We speculate on the relation of the anticyclone to the asymmetric gyre seen in the Earth's fluid core, and in state-of-the-art dynamo direct numerical simulations.

We study singular jets from the collapse of drop-impact craters, when the drop and pool are of different immiscible liquids. The fastest jets emerge from a dimple at the bottom of the rebounding crater, when no bubble is pinched off. The parameter space is considerably more complex than for identical liquids, revealing intricate compound-dimple shapes. In contrast to the universal capillary–inertial drop pinch-off regime, where the neck radius scales as $R\sim t^{2/3}$, for a purely inertial air dimple the collapse has $R \sim t^{1/2}$. The bottom dimple dynamics is not self-similar but possesses memory effects, being sensitive to initial and boundary conditions. Sequence of capillary waves can therefore mould the air dimple into different collapse shapes, such as bamboo-like and telescopic forms. The finest jets are only $12\ \mathrm {\mu }\textrm {m}$ in diameter and the normalized jetting speeds are up to one order of magnitude larger than for jets from bursting bubbles. We study the cross-over between the two power laws approaching the singularity. The singular jets show the earliest cross-over into the inertial regime. The fastest jets can pinch off a toroidal micro-bubble from the cusp at the base of the jet.

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Under the assumption that the vorticity is a negative constant whose absolute value is sufficiently large, we construct a solution with the following properties. The corresponding flow is unidirectional at infinity and has a solitary wave of elevation as its upper boundary; under this unidirectional flow, there is a bounded domain adjacent to the bottom, which surrounds an interior stagnation point and is divided into two subdomains with opposite directions of flow by a critical level curve connecting two stagnation points on the bottom.

When pushed out of a syringe, polymer solutions form droplets attached by long and slender cylindrical filaments whose diameter decreases exponentially with time before eventually breaking. In the last stages of this process, a striking feature is the self-similarity of the interface shape near the end of the filament. This means that shapes at different times, if properly rescaled, collapse onto a single universal shape. A theoretical description based on the Oldroyd-B model was recently shown to disagree with existing experimental results. By revisiting these measurements and analysing the interface profiles of very diluted polyethylene oxide solutions at high temporal and spatial resolution, we show that they are very well described by the model.

We investigate the late-time Richtmyer–Meshkov instability (RMI) growth of sinuous perturbations on an air/sulphur hexafluoride interface (Atwood number, $A \sim 0.67$) subjected to a Mach 1.2 planar shock wave at Los Alamos National Laboratory's vertical shock tube facility. Interface perturbations are established using a novel membraneless technique where cross-flowing air and SF$_6$ separated by an oscillating splitter plate create a perturbed density interface. The interface formed has multi-modal features and residual small perturbations, however, a dominant mode is still noticeable. The late-time perturbation growths scale with $ka_0$ initial conditions (where $k$ is the wavenumber and $a_0$ is the initial amplitude of the dominant mode) as measured at the pre-shock interface. Past nonlinear models based on potential-flow theory, heuristic/interpolation approaches, Padé approximants and numerical simulations are evaluated against present experimental results. Accounting for an explicit $ka_0$ dependence in Sadot et al.'s (Phys. Rev. Lett., vol. 80, issue 8, 1998, pp. 1654–1657) model, we propose an empirical rational function that captures the asymptotic behaviour of perturbation growth for a broad range of initial conditions ($0.30 \leq ka_0 \leq 0.86$). The onset of mixing transition and its initial condition dependence are investigated with respect to the minimum state criterion ($Re = 1.6 \times 10^5$) for unsteady flows by Zhou (Phys. Plasmas, vol. 14, 2007, 082701). Earlier mixing transitions for higher $ka_0$ initial conditions are noted from local and global Reynolds number estimates which are corroborated by the existence of an inertial sub-range and formation of mixing regions indicating the physical significance of the minimum state criterion in RMI flows. The transition is accompanied by the increasing teapot-like appearance of joint probability density functions of $p$–$q$ (invariants of the reduced velocity gradient tensor), establishing the technique as a useful tool for turbulence detection in two-dimensional diagnostics.

In this study, we connect concepts that have been recently developed in thermoacoustics, specifically (i) high-order spectral perturbation theory, (ii) symmetry-induced degenerate thermoacoustic modes, (iii) intrinsic thermoacoustic modes and (iv) exceptional points. Their connection helps gain physical insight into the behaviour of the thermoacoustic spectrum when parameters of the system are varied. First, we extend high-order adjoint-based perturbation theory of thermoacoustic modes to the degenerate case. We provide explicit formulae for the calculation of the eigenvalue corrections to any order. These formulae are valid for self-adjoint, non-self-adjoint or even non-normal systems; therefore, they can be applied to a large range of problems, including fluid dynamics. Second, by analysing the expansion coefficients of the eigenvalue corrections as a function of a parameter of interest, we accurately estimate the radius of convergence of the power series. Third, we connect the existence of a finite radius of convergence to the existence of singularities in parameter space. We identify these singularities as exceptional points, which correspond to defective thermoacoustic eigenvalues, with infinite sensitivity to infinitesimal changes in the parameters. At an exceptional point, two eigenvalues and their associated eigenvectors coalesce. Close to an exceptional point, strong veering of the eigenvalue trajectories is observed. As demonstrated in recent work, exceptional points naturally arise in thermoacoustic systems due to the interaction between modes of acoustic and intrinsic origin. The role of exceptional points in thermoacoustic systems sheds new light on the physics and sensitivity of thermoacoustic stability, which can be leveraged for passive control by small design modifications.

In this paper, a semi-analytical model based on linear potential flow theory and an eigenfunction expansion method is developed to study wave scattering by an array of structured cylinders in water of finite depth. Each cylinder is formed by a closely spaced array of thin vertical plates, between which fluid can flow, extending through the depth. In order to consider the wave attenuation and energy dissipation in narrow gaps between the thin vertical plates, a damping mechanism is introduced at the surface of the fluid occupied by the structured cylinders. In addition to a direct calculation of the energy dissipation, an indirect method based on Kochin functions is derived with the employment of energy identities. The present model is shown to be in excellent agreement with both the published data and those obtained by using different methods. The validated model is then applied to study the effect of a pair of structured cylinders on wave focusing/blocking, scattered far-field amplitude and wave power dissipation. Results show that wave focusing/blocking can be achieved by the appropriate choice of plate alignment. The structured cylinders hold profound potential for wave power extraction.

We report on the experimental investigation of the large-scale instantaneous flow structures in turbulent Taylor–Couette flow using tomographic particle image velocimetry. The results indicate three distinct regimes for counter-rotating flow within a shear Reynolds number range of $11\,000 < Re_S < 47\,000$. Close to only inner cylinder rotation, large-scale structures are aligned in the azimuthal direction, similar to Taylor vortices. Near the point of only outer cylinder rotation, we observe columnar vortical structures in the axial direction, which are associated with small Rossby numbers. This is the first time such columnar structures are reported in a fully turbulent Taylor–Couette flow. A transition between these two regimes is observed around the point of exact counter-rotation, where the instantaneous azimuthal structures are inclined with respect to the walls. Furthermore, it is shown that the reported transitions in the turbulent flow structure modify the angular momentum transport, thereby affecting the torque scaling.

We measure the Kelvin–Helmholtz instability in between a layer of a diamagnetic fluid flowing in a channel and a layer of ferrofluid resting on top. When the diamagnetic fluid exceeds a critical flow velocity the interface in between both fluids becomes unstable and waves develop. It has been predicted by Sutyrin & Taktarov (J. Appl. Math. Mech., vol. 39, 1975, pp. 520–524) that a homogeneous magnetic field, oriented horizontally, stabilizes the liquid interface. To test this prediction we apply in a closed flow channel a local periodic perturbation of the interface by magnetic or mechanic means. From the measured growth and decay rates of the interface undulations we determine the critical flow velocity for various driving frequencies and applied magnetic fields. In this way we confirm quantitatively the stabilizing effect of the horizontal field. Moreover we measure the dispersion relation of the interfacial waves.

We report a series of experiments in which a cylinder, with a vertical axis, is moved back and forth along a long narrow channel containing fresh water at Reynolds numbers $Re=3220\text {--}13\,102$. We examine the mixing of a cloud of dye along the channel by the oscillatory motion of the cylinder. Using light attenuation techniques to measure the time evolution of the concentration of dye along the channel, we find that at early times the concentration profile collapses to a Gaussian profile with dispersivity, $D=(2.4\pm 0.5) fdW$, where $f$ is the frequency of the cylinder oscillation, $d$ is the diameter of the cylinder and $W$ is the width of the channel, respectively. For times much longer than $L^2/D$, with $L$ being the length of the channel, the concentration becomes progressively more uniform over the whole length of the channel, and we show that the long-time non-uniform component decays with time dependence $\exp (- 4{\rm \pi} ^2 Dt / L^2)$. We consider the implications of these experiments for the dispersal of viral aerosols along poorly ventilated corridors, with implications for infection transmission in hospitals and public buildings.

Wave shape (i.e. skewness or asymmetry) plays an important role in beach morphology evolution, remote sensing and ship safety. The wind's influence on ocean waves has been extensively studied theoretically in the context of growth, but most theories are phase averaged and cannot predict wave shape. Most laboratory and numerical studies similarly focus on wave growth. A few laboratory experiments have demonstrated that wind can change wave shape, and two-phase numerical simulations have also noted wind-induced wave-shape changes. However, the wind's effect on wave shape is poorly understood, and no theory for it exists. For weakly nonlinear waves, wave-shape parameters are the phase of the first harmonic relative to the primary frequency (or harmonic phase HP, zero for a Stokes wave) and relative amplitude of the first harmonic to the primary wave. Here, surface pressure profiles (denoted Jeffreys, Miles and generalized Miles) are prescribed based on wind–wave generation theories. Theoretical solutions are derived for quasi-periodic progressive waves and the wind-induced changes to the HP, relative harmonic amplitude, as well as the already known phase speed changes and growth rates. The wave-shape parameters depend upon the chosen surface pressure profile, pressure magnitude and phase relative to the wave profile and non-dimensional depth. Wave asymmetry is linked to the non-dimensional growth rate. Atmospheric large eddy simulations constrain pressure profile parameters. The HP predictions are qualitatively consistent with laboratory observations. This theory, together with the HP and relative harmonic amplitude observables, can provide insight into the actual wave surface pressure profile.

In a three-layer system, weakly nonlinear theory predicts that breathers exist under certain conditions which, under the Boussinesq approximation, include symmetric stratifications in which the density jump across each interface is the same and the upper and lower layer thicknesses are equal and less than 9/26 of the total water depth. The existence and characteristics of fully nonlinear breathers in this symmetric stratification are poorly understood. Therefore, this study investigates fully nonlinear breathers in a three-layer symmetric stratification in order to clarify their characteristics by making direct comparisons between numerical simulation results and theoretical solutions. A normalization of the breather profiles is introduced using theoretical solutions of a breather and a new energy scale is proposed to evaluate their potential and kinetic energy. We apply fully nonlinear and strongly dispersive internal wave equations in a three-layer system using a variational principle. The computational results show that the larger the amplitude, the shorter the length of the envelope of breathers, which agrees with the theoretical solution. However, breathers based on the theoretical solutions cannot progress without deformation and decay due to the emission of short small-amplitude internal waves. Furthermore we demonstrate that the shedding of larger amplitude waves occurs, and the amplitude of the envelope decays more strongly when the density interface crosses the critical depth where the ratio of the upper layer thickness and the total water depth is 9/26 suggesting a limiting amplitude for fully nonlinear breathers.

A rigorous derivation and validation for linear fluid–structure-interaction (FSI) equations for a rigid-body motion problem is performed in an Eulerian framework. We show that the ‘added stiffness’ terms arising in the formulation of Fanion et al. (Revue Européenne des Éléments Finis, vol. 9, issue 6–7, 2000, pp. 681–708) vanish at the FSI interface in a first-order approximation and can be neglected when considering the growth of infinitesimal disturbances. Several numerical tests with rigid-body motion are performed to show the validity of the derived formulation by comparing the time evolution between the linear and nonlinear equations when the base flow is perturbed by identical small-amplitude perturbations. In all cases both the growth rate and angular frequency of the instability matches within $0.1\,\%$ accuracy. The derived formulation is used to investigate the phenomenon of symmetry breaking for a rotating cylinder with an attached splitter plate. The results show that the onset of symmetry breaking can be explained by the existence of a zero frequency linearly unstable mode of the coupled FSI system. Finally, the structural sensitivity of the least stable eigenvalue is studied for an oscillating cylinder, which is found to change significantly when the fluid and structural frequencies are close to resonance.

Although the relative importance of airborne transmission of the SARS-CoV-2 virus is controversial, increasing evidence suggests that understanding airflows is important for estimation of the risk of contracting COVID-19. The data available so far indicate that indoor transmission of the virus far outstrips outdoor transmission, possibly due to longer exposure times and the decreased turbulence levels (and therefore dispersion) found indoors. In this paper we discuss the role of building ventilation on the possible pathways of airborne particles and examine the fluid mechanics of the processes involved.

We extend upon the known flow transitions in neutrally buoyant particle-laden Taylor–Couette flows by accessing higher suspension Reynolds numbers $(Re_{{susp}} \sim O(10^3))$ in a geometry with radius ratio $\eta = 0.917$ and aspect ratio $\varGamma = 21.67$. Flow transitions for several particle volume fractions ($0 \leq \phi \leq 0.40$) are investigated by means of flow visualization experiments, in a flow driven by a rotating inner cylinder. Despite higher effective ramp rates, we observe non-axisymmetric patterns, such as spirals, in the presence of particles. A novel observation in our experiments is the azimuthally localized wavy vortex flow, characterized by waviness present on a fraction of the otherwise axisymmetric Taylor vortices. The existence of this flow state suggests that in addition to the already established, destabilizing effect of particles, they may also inhibit the growth of instabilities. Flow topologies corresponding to higher-order transitions in particle-laden suspensions appear to be qualitatively similar to those observed in single-phase flows. A key difference, however, is the visible reduction in the appearance of a second, incommensurate frequency at higher particle loadings, which could have implications for the onset of chaos. Simultaneous torque measurements allow us to estimate an empirical scaling law between the Nusselt number ($Nu_{\omega }$), the Taylor number ($Ta$) and the relative viscosity ($\chi ^{e}): Nu_{\omega } \propto Ta^{0.24} \chi ^{e \, 0.41}$. The scaling exponent of $Ta$ is non-trivially independent of the particle loading. Apparently, particles do not trigger a qualitative change in the nature of angular momentum transfer between the cylinders.

Biflagellate algal cells of the genus Volvox form spherical colonies that propel themselves, vertically upwards in still fluid, by the coordinated beating of thousands of flagella, that also cause the colonies to rotate about their vertical axes. When they are swimming in a chamber of finite depth, pairs (or more) of Volvox carteri colonies were observed by Drescher et al. (Phys. Rev. Lett., vol. 102, 2009, 168101) to exhibit hydrodynamic bound states when they are close to a rigid horizontal boundary. When the boundary is above, the colonies are attracted to each other and orbit around each other in a ‘waltz’; when the boundary is below they perform more complex ‘minuet’ motions. These dances are simulated in the present paper, using a novel ‘spherical squirmer’ model of a colony in which, instead of a time-independent but $\theta$-dependent tangential velocity being imposed on the spherical surface (radius $a$; $\theta$ is the polar angle), a time-independent and uniform tangential shear stress is applied to the fluid on a sphere of radius $(1+\epsilon )a, \epsilon \ll 1$, where $\epsilon a$ represents the length of the flagella. The fluid must satisfy the no-slip condition on the sphere at radius $a$. In addition to the shear stress, the motions depend on two dimensionless parameters that describe the effect of gravity on a colony: $F_g$, proportional to the ratio of the sedimentation speed of a non-swimming colony to its swimming speed, and $G_{bh}$, that represents the fact that colonies are bottom heavy; $G_{bh}$ is the ratio of the time scale to swim a distance equal to the radius, to the time scale for gravitational reorientation of the colony's axis to the vertical when it is disturbed. In addition to reproducing both of the dancing modes, the simulations are able to determine values of $F_g$ and $G_{bh}$ for which they are stable (or not); there is reasonable agreement with the experiments. A far-field model for the minuet motions is also shown to have qualitative agreement, but does not describe some features that are reproduced in the full simulations.