The wake of a rotating circular cylinder in a free stream is investigated for Reynolds numbers $\mathit{Re}\leqslant 400$ and non-dimensional rotation rates of $\alpha \leqslant 2. 5$. Two aspects are considered. The first is the transition from a steady flow to unsteady flow characterized by periodic vortex shedding. The two-dimensional computations show that the onset of unsteady flow is delayed to higher Reynolds numbers as the rotation rate is increased, and vortex shedding is suppressed for $\alpha \geqslant 2. 1$ for all Reynolds numbers in the parameter space investigated. The second aspect investigated is the transition from two-dimensional to three-dimensional flow using linear stability analysis. It is shown that at low rotation rates of $\alpha \leqslant 1$, the three-dimensional transition scenario is similar to that of the non-rotating cylinder. However, at higher rotation rates, the three-dimensional scenario becomes increasingly complex, with three new modes identified that bifurcate from the unsteady flow, and two modes that bifurcate from the steady flow. Curves of marginal stability for all of the modes are presented in a parameter space map, the defining characteristics for each mode presented, and the physical mechanisms of instability are discussed.

Singular behaviour of a rarefied gas on a planar boundary is clarified on the basis of the Boltzmann equation. The thermal transpiration between two parallel plates is taken as a specific example. First, the flow velocity is shown to behave like $x\ln x$ in the vicinity of the boundary, where $x$ is a distance from the boundary. This implies a logarithmic divergence of the flow velocity gradient as $x\rightarrow 0$. Then, such a spatial singularity is shown to induce a similar singularity of the velocity distribution function (VDF) with respect to ${\zeta }_{n} $ on the boundary, where ${\zeta }_{n} $ is a normal component of the molecular velocity to the boundary. Moreover, the spatial singularity is shown to be quantitatively related to the discontinuity of the VDF on the boundary at ${\zeta }_{n} = 0$. These macroscopic and microscopic singularities should be observed generally in a rarefied gas on a planar boundary.

Most aquatic vertebrates swim by passing a bending wave down their bodies, a swimming mode known as undulatory propulsion. Except for very elongated swimmers like eels and lampreys, these animals have generally evolved to a similar shape: an anterior streamlined region of large volume separated from a caudal fin by a caudal peduncle of reduced cross-section. However, the link between this particular shape and the hydrodynamical constraints remains to be explored. Here, this question is addressed by seeking the optimal design for undulatory swimmers with an evolutionary algorithm. Animals of varying elliptic cross-section are considered whose motions are prescribed by arbitrary periodic curvature laws. In the elongated-body limit, reactive and resistive forces can be formulated at any cross-section, allowing the recoil motion and the mean swimming speed of a given animal to be calculated. A bi-objective optimization problem then consists of finding body shapes and corresponding motions associated with the lowest energetic costs, the highest stride lengths (which is a dimensionless measure of swimming speed) or any trade-offs between the two. For biologically relevant parameters, this optimization calculation yields two distinct ‘species’: one specialized in economical swimming and the other in large stride lengths. By comparing the attributes and performance of these numerically obtained swimmers with data on undulatory-swimming animals, it is argued that evolution is consistent with the selection of species with low energetic costs.

We investigate numerically the transient linear growth of three-dimensional (3D) perturbations in a homogeneous time-evolving mixing layer in order to identify which perturbations are optimal in terms of their kinetic energy gain over a finite, predetermined time interval. We model the mixing layer with an initial parallel velocity distribution $\mathbi{U}(y)= {U}_{0} \tanh (y/ d)\mathbi{e}_{x}$ with Reynolds number $Re= {U}_{0} d/ \nu = 1000$, where $\nu $ is the kinematic viscosity of the fluid. We consider a range of time intervals on both a constant ‘frozen’ base flow and a time-dependent two-dimensional (2D) flow associated with the growth and nonlinear saturation of two wavelengths of the most-unstable eigenmode of linear theory of the initial parallel velocity distribution, which rolls up into two classical Rayleigh instabilities commonly referred to as Kelvin–Helmholtz (KH) billows, which eventually pair to form a larger vortex. For short times, the most-amplified perturbations on the frozen $\tanh $ profile are inherently 3D, and are most appropriately described as oblique wave ‘OL’ perturbations which grow through a combination of the Orr and lift-up mechanisms, while for longer times, the optimal perturbations are 2D and similar to the KH normal mode, with a slight enhancement of gain. For the time-evolving KH base flow, OL perturbations continue to dominate over sufficiently short time intervals. However, for longer time intervals which involve substantial evolution of the primary KH billows, two broad classes of inherently 3D linear optimal perturbation arise, associated at low wavenumbers with the well-known core-centred elliptical translative instability, and at higher wavenumbers with the braid-centred hyperbolic instability. The hyperbolic perturbation is relatively inefficient in exploiting the gain of the OL perturbations, and so only dominates the smaller wavenumber (ultimately) core-centred perturbations when the time evolution of the base flow or the start time of the optimization interval does not allow the OL perturbations much opportunity to grow. When the OL perturbations can grow, they initially grow in the braid, and then trigger an elliptical core-centred perturbation by a strong coupling with the primary KH billow. If the optimization time interval includes pairing of the primary billows, the secondary elliptical perturbations are strongly suppressed during the pairing event, due to the significant disruption of the primary billow cores during pairing.

We perform a global stability analysis of a flow over an open cavity, characterized by a Reynolds number, based on the upstream velocity and the cavity length, of $7500$. We compute base flows and unstable global modes of the flow for different Mach numbers ranging from $0$ to $0. 9$. In the incompressible regime ($M= 0$), we show that the flow is subject to global instabilities due to Kelvin–Helmholtz instabilities in the shear layer, which become strengthened by a hydrodynamic pressure feedback. The influence of the boundary-layer thickness and of the length-to-depth ratio of the cavity on these shear-layer modes has been investigated. In the compressible regime ($M\gt 0$), we have shown that all unstable global modes are continuously connected to the incompressible shear-layer modes as $M\rightarrow 0$. These shear-layer modes correspond to the beginning of branches of global modes, whose frequencies evolve (as a function of the Mach number), in accordance with the feedback aeroacoustic mechanism (Rossiter, Tech. Rep. Aero. Res. Counc. R. & M., 1964). We have also identified branches of global modes behaving in agreement with acoustic resonance mechanisms (East, J. Sound Vib., vol. 3, 1966, pp. 277–287; Tam, J. Sound Vib., vol. 49, 1976, pp. 353–364; Koch, AIAA J., vol. 43, 2005, pp. 2342–2349). At the intersections between both types of branches, the growth rate of the global modes is seen to display a local maximum. Along the aeroacoustic feedback branches, the number of vortical structures in the shear layer is kept constant, while the pressure pattern inside the cavity is conserved along the acoustic resonance branches. We show that both the feedback aeroacoustic and acoustic resonance mechanisms are at play over the entire subsonic regime, from $M= 0$ to $M= 0. 9$. At low Mach numbers, we suggest that it is still the feedback aeroacoustic mechanism that selects the frequency, even though the fundamental acoustic resonance mode is also important due to enhancing the response. At higher Mach numbers, we observe that the pressure pattern of the acoustic resonance modes (fundamental acoustic modes, first longitudinal acoustic modes, first longitudinal-depth acoustic modes) inside the cavity determines the directivity of the radiated noise. Links with experimental results are finally discussed.

Previous studies investigating the effect of aspect ratio ($\mathit{AR}$) for insect-like regimes have reported seemingly different trends in aerodynamic forces, however no detailed flow observations have been made. In this study, the effect of $\mathit{AR}$ and Reynolds number on the flow structures over insect-like wings is explored using a numerical model of an altered fruit fly wing revolving at a constant angular velocity. Increasing the Reynolds number for an $\mathit{AR}$ of 2.91 resulted in the development of a dual leading-edge vortex (LEV) structure, however increasing $\mathit{AR}$ at a fixed Reynolds number generated the same flow structures. This result shows that the effects of Reynolds number and $\mathit{AR}$ are linked. We present an alternative scaling using wing span as the characteristic length to decouple the effects of Reynolds number from those of $\mathit{AR}$. This results in a span-based Reynolds number, which can be used to independently describe the development of the LEV. Indeed, universal behaviour was found for various parameters using this scaling. The effect of $\mathit{AR}$ on the vortex structures and aerodynamic forces was then assessed at different span-based Reynolds numbers. Scaling the flow using the wing span was found to apply when a strong spanwise velocity is present on the leeward side of the wing and therefore may prove to be useful for similar studies involving flapping or rotating wings at high angles of attack.

Supercritical fluid injection using a single round injector into a quiescent atmosphere at subcritical and supercritical conditions was studied experimentally with particular attention paid to supercritical-into-subcritical injection and the reassertion of surface tension. The entire system was binary since the surrounding atmosphere consisted of an inert gas of a different composition than that of the injected fluid. Average densities and density gradients were quantified and a method was applied to quantify the resulting drop formation due to the disintegration of the jet based on the experimental conditions. The evolution of drop size with distance from the injector was identified.

We develop a dynamic model for suspensions of negatively buoyant particles on an incline. Our model includes settling due to gravity and resuspension of particles by shear-induced migration. We consider the case where the particles settle onto the solid substrate and two distinct fronts form: a faster liquid and a slower particle front. The resulting transport equations for the liquid and the particles are of hyperbolic type and we study the dilute limit for which we compute exact solutions. We also carry out systematic laboratory experiments, focusing on the motion of the two fronts. We show that the dynamic model predictions for small to moderate values of the particle volume fraction and the inclination angle of the solid substrate agree well with the experimental data.

The results presented here examine the quasi-geostrophic dynamics of a point vortex structure with one upper-layer vortex and two identical bottom-layer vortices in a two-layer fluid. The problem of three vortices in a barotropic fluid is known to be integrable. This fundamental result is also valid in a stratified fluid, in particular a two-layer one. In this case, unlike the barotropic situation, vortices belonging to the same layer or to different layers interact according to different formulae. Previously, this occurrence has been poorly investigated. In the present work, the existence conditions for stable stationary (translational and rotational) collinear two-layer configurations of three vortices are obtained. Small disturbances of stationary configurations lead to periodic oscillations of the vortices about their undisturbed shapes. These oscillations occur along elliptical orbits up to the second order of the Hamiltonian expansion. Analytical expressions for the parameters of the corresponding ellipses and for oscillation frequencies are obtained. In the case of finite disturbances, vortex motion becomes more complicated. In this case we have made a classification of all possible movements, by analysing phase portraits in trilinear coordinates and by computing numerically the characteristic trajectories of the absolute and relative vortex motions.

We study fluid-particle motion in the velocity field induced by a quasi-stationary point vortex structure consisting of one upper-layer vortex and two identical vortices in the bottom layer of a rotating two-layer fluid. The regular regimes are investigated, and the possibility of chaotic regimes (chaotic advection) under the effect of quite small non-stationary disturbances of stationary configurations has been shown. Examples of different scenarios are given for the origin and development of chaos. We analyse the role played by the stochastic layer in the processes of mixing and in the capture of fluid particles within a vortex area. We also study the influence of stratification on these effects. It is shown that regular and chaotic advection situations exhibit significant differences in the two layers.

Capillary instability of a Newtonian liquid torus suspended in an immiscible Newtonian medium is computed using a Cahn–Hilliard diffuse-interface model. The main differences between the torus and a straight thread are the presence of an axial curvature and an external flow field caused by the retraction of the torus. We show that the capillary wave initially grows linearly as on a straight thread. The axial curvature decreases the growth rate of the capillary waves while the external flow enhances it. Breakup depends on the competition of two time scales: one for torus retraction and the other for neck pinch-off. The outcome is determined by the initial amplitude of the disturbance, the thickness of the torus relative to its circumference, and the torus-to-medium viscosity ratio. The linearly dominant mode may not persist till nonlinear growth and breakup. The numerical results are generally consistent with experimental observations.

The interaction between isotropic turbulence and a normal shock wave is investigated through a series of direct numerical simulations at different Reynolds numbers and mean and turbulent Mach numbers. The computed data are compared to experiments and linear theory, showing that the amplification of turbulence kinetic energy across a shock wave is described well using linearized dynamics. The post-shock anisotropy of the turbulence, however, is qualitatively different from that predicted by linear analysis. The jumps in mean density and pressure are lower than the non-turbulent Rankine–Hugoniot results by a factor of the square of the turbulence intensity. It is shown that the dissipative scales of turbulence return to isotropy within about 10 convected Kolmogorov time scales, a distance that becomes very small at high Reynolds numbers. Special attention is paid to the ‘broken shock’ regime of intense turbulence, where the shock can be locally replaced by smooth compressions. Grid convergence of the probability density function of the shock jumps proves that this effect is physical, and not an artefact of the numerical scheme.

We report experimental investigations of the dynamics of the large-scale circulation (LSC) in turbulent Rayleigh–Bénard convection at high Prandtl number $\mathit{Pr}= 19. 4$ (and also $\mathit{Pr}= 7. 8$) and Rayleigh number $\mathit{Ra}$ varying from $8. 3\times 1{0}^{8} $ to $2. 9\times 1{0}^{11} $ in a cylindrical convection cell with aspect ratio unity. The dynamics of the LSC is measured using the multithermal probe technique. Both the sinusoidal-fitting (SF) and the temperature-extrema-extraction (TEE) methods are used to analyse the properties of the LSC. It is found that the LSC in high-$\mathit{Pr}$ regime remains a single-roll structure. The azimuthal motion of the LSC is a diffusive process, which is the same as those for $\mathit{Pr}$ around 1. However, the azimuthal diffusion of the LSC, characterized by the angular speed $\Omega $ is almost two orders of magnitude smaller when compared with that in water. The non-dimensional time-averaged amplitude of the angular speed $\langle \vert \Omega \vert \rangle {T}_{d} $ (${T}_{d} = {L}^{2} / \kappa $ is the thermal diffusion time) of the LSC at the mid-height of the convection cell increases with $\mathit{Ra}$ as a power law, which is $\langle \vert \Omega \vert \rangle {T}_{d} \propto {\mathit{Ra}}^{0. 36\pm 0. 01} $. The $\mathit{Re}$ number based on the oscillation frequency of the LSC is found to scale with $\mathit{Ra}$ as $\mathit{Re}= 0. 13{\mathit{Ra}}^{0. 43\pm 0. 01} $. It is also found that the normalized flow strength $\langle \delta \rangle / \mrm{\Delta} T\times \mathit{Ra}/ \mathit{Pr}\propto {\mathit{Re}}^{1. 5\pm 0. 1} $, with the exponent in good agreement with that predicted by Brown & Ahlers (Phys. Fluids, vol. 20, 2008, p. 075101). A wealth of dynamical features of the LSC, such as the cessations, flow reversals, flow mode transitions, torsional and sloshing oscillations are observed in the high-$\mathit{Pr}$ regime as well.

We argue that a method developed by Ångström (Ann. Phys. Chem., vol. 114, 1861, pp. 513–530) to measure the thermal conductivity of solids can be adapted to determine the effective diffusivity of a large-scale magnetic field in a turbulent electrically conducting fluid. The method consists of applying an oscillatory source and measuring the steady-state response. We illustrate this method in a two-dimensional system. This geometry is chosen because it is possible to compare the results with independent methods that are restricted to two-dimensional flows. We describe two variants of this method: one (the ‘turbulent Ångström method’) that is better suited to laboratory experiments and a second (the ‘method of oscillatory sines’) that is effective for numerical experiments. We show that, if correctly implemented, all methods agree. Based on these results we argue that these methods can be extended to three-dimensional numerical simulations and laboratory experiments.

A slender flexible body immersed in sheared cross-flow may exhibit vortex-induced vibrations (VIVs) involving a wide range of excited frequencies and structural wavenumbers. The mechanisms of broadband VIVs of a cylindrical tensioned beam of length-to-diameter aspect ratio 200 placed in shear flow, with an exponentially varying profile along the span, are investigated by means of direct numerical simulation. The Reynolds number is equal to 330 based on the maximum velocity, for comparison with previous work on narrowband vibrations in linear shear flow. The flow is found to excite the structure at a number of different locations under a condition of wake–body synchronization, or lock-in. Broadband responses are associated with a distributed occurrence of the lock-in condition along the span, as opposed to the localized lock-in regions limited to the high inflow velocity zone, reported for narrowband vibrations in sheared current. Despite the instantaneously multi-frequency nature of broadband responses, the lock-in phenomenon remains a locally mono-frequency event, since the vortex formation is generally synchronized with a single vibration frequency at a given location. The spanwise distribution of the excitation zones induces travelling structural waves moving in both directions; this contrasts with the narrowband case where the direction of propagation toward decreasing inflow velocity is preferred. A generalization of the mechanism of phase-locking between the in-line and cross-flow responses is proposed for broadband VIVs under the lock-in condition. A spanwise drift of the in-line/cross-flow phase difference is identified for the high-wavenumber vibration components; this drift is related to the strong travelling wave character of the corresponding structural waves.

An analytical solution for the low-Reynolds-number flow field of a shear flow over a rectangular cavity containing a second immiscible fluid is derived. While flow of a single-phase fluid over a cavity is a standard case investigated in fluid dynamics, flow over a cavity that is filled with a second immiscible fluid has received little attention. The flow field inside the cavity is considered to define a boundary condition for the outer flow, which takes the form of a Navier slip condition with locally varying slip length. The slip-length function is determined heuristically from the related problem of lid-driven cavity flow. Based on the Stokes equations and complex analysis, it is then possible to derive a closed analytical expression for the flow field over the cavity for both the transverse and the longitudinal case. The result is a comparatively simple function, which displays the dependence of the flow field on the cavity geometry and the medium filling the cavity. The analytically computed expression agrees well with results obtained from a numerical solution of the Navier–Stokes equations.

Recent numerical simulations of dynamo action resulting from rotating convection have revealed some serious problems in applying the standard picture of mean field electrodynamics at high values of the magnetic Reynolds number, and have thereby underlined the difficulties in large-scale magnetic field generation in this regime. Here we consider kinematic dynamo processes in a rotating convective layer of Boussinesq fluid with the additional influence of a large-scale horizontal velocity shear. Incorporating the shear flow enhances the dynamo growth rate and also leads to the generation of significant magnetic fields on large scales. By the technique of spectral filtering, we analyse the modes in the velocity that are principally responsible for dynamo action, and show that the magnetic field resulting from the full flow relies crucially on a range of scales in the velocity field. Filtering the flow to provide a true separation of scales between the shear and the convective flow also leads to dynamo action; however, the magnetic field in this case has a very different structure from that generated by the full velocity field. We also show that the nature of the dynamo action is broadly similar irrespective of whether the flow in the absence of shear can support dynamo action.

Two-dimensional convection in a plane layer bounded by stress-free perfectly conducting horizontal boundaries and rotating uniformly about the vertical is considered. Time-independent spatially localized structures, called convectons, of even and odd parity are computed. The convectons are embedded within a self-generated shear layer with a compensating shear flow outside the structure. These states are organized within a bifurcation structure called slanted snaking and may be present even when periodic convection sets in supercritically. These interesting properties are traced to the presence of a conserved quantity and hence to the use of stress-free boundary conditions.

Rotating Rayleigh–Bénard convection provides a simplified dynamical analogue for many planetary and stellar fluid systems. Here, we use numerical simulations of rotating Rayleigh–Bénard convection to investigate the scaling behaviour of five quantities over a range of Rayleigh ($1{0}^{3} \lesssim \mathit{Ra}\lesssim 1{0}^{9} $), Prandtl ($1\leq \mathit{Pr}\leq 100$) and Ekman ($1{0}^{- 6} \leq E\leq \infty $) numbers. The five quantities of interest are the viscous and thermal boundary layer thicknesses, ${\delta }_{v} $ and ${\delta }_{T} $, mean temperature gradients, $\beta $, characteristic horizontal length scales, $\ell $, and flow speeds, $\mathit{Pe}$. Three parameter regimes in which different scalings apply are quantified: non-rotating, weakly rotating and rotationally constrained. In the rotationally constrained regime, all five quantities are affected by rotation. In the weakly rotating regime, ${\delta }_{T} $, $\beta $ and $\mathit{Pe}$, roughly conform to their non-rotating behaviour, but ${\delta }_{v} $ and $\ell $ are still strongly affected by the Coriolis force. A summary of scaling results is given in table 2.

This paper investigates theoretically the vertical evolution of a turbulent plume into a linearly stratified ambient fluid, by regarding it as composed of two distinct regions. In the first region, called the positive buoyant region, the plume buoyancy and the plume momentum act in the same upward direction, whereas in the second region, called the negative buoyant region, they act in opposite directions. In a first step, analytical expressions for the plume variables at the transition height (i.e. between the two regions) are obtained from one-dimensional conservation equations, using the plume entrainment model and under the Boussinesq approximation. In a second step, these variables are used in order to determine analytically the buoyancy and volume fluxes as well as the density deficit of the plume at its top. In this investigation, the transition height (denoted ${z}_{t} $) and the total plume height (denoted ${z}_{p} $) are obtained in the form of two integrals. These integrals are evaluated asymptotically in three different cases associated with particular flow regimes. Finally, the limit of the Boussinesq assumption for such flows is discussed.