The permanent precession of a baroclinic geophysical vortex is reproduced, under the quasi-geostrophic approximation, using three potential vorticity anomaly modes in spherical geometry. The potential vorticity modes involve the spherical Bessel functions of the first kind $\text{j}_{l}(\unicode[STIX]{x1D70C})$ and the spherical harmonics $\text{Y}_{l}^{m}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$, where $l$ is the degree, $m$ is the order, and $(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ are the spherical coordinates. The vortex precession is interpreted as the horizontal and circular advection by a large-amplitude background flow associated with the spherical mode $c_{0}\text{j}_{0}(\unicode[STIX]{x1D70C})$ of the small-amplitude zonal mode $c_{2,0}\text{j}_{2}(\unicode[STIX]{x1D70C})\text{Y}_{2}^{0}(\unicode[STIX]{x1D703})$ tilted by a small-amplitude mode $c_{2,1}\text{j}_{2}(\unicode[STIX]{x1D70C})\text{Y}_{2}^{1}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$, where $\{c_{0},c_{2,0},c_{2,1}\}$ are constant potential vorticity modal amplitudes. An approximate time-dependent, closed-form solution for the potential vorticity anomaly is given. In this solution the motion of the potential vorticity field is periodic but not rigid. The vortex precession frequency $\unicode[STIX]{x1D714}_{0}$ depends linearly on the amplitudes $c_{0}$ and $c_{2,0}$ of the modal components of order 0, while the slope of the precessing axis depends on the ratio between the modal amplitude $c_{2,1}$ and $\unicode[STIX]{x1D714}_{0}$.

In this paper, we present the two-dimensional (2-D) energy cross-spectrum of the streamwise velocity ($u$) component and use it to test the notion of self-similarity in turbulent boundary layers. The primary focus is on the cross-spectrum ($\unicode[STIX]{x1D6F7}_{cross}^{w}$) measured across the logarithmic ($z_{o}$) and near-wall ($z_{r}$) wall-normal locations, providing the energy distribution across the range of streamwise ($\unicode[STIX]{x1D706}_{x}$) and spanwise ($\unicode[STIX]{x1D706}_{y}$) wavelengths (or length scales) that are coherent across the wall-normal distance. $\unicode[STIX]{x1D6F7}_{cross}^{w}$ may thus be interpreted as a wall-filtered subset of the full 2-D $u$-spectrum ($\unicode[STIX]{x1D6F7}$), the latter providing information on all coexisting eddies at $z_{o}$. To this end, datasets comprising synchronized two-point $u$-signals at $z_{o}$ and $z_{r}$, across the friction Reynolds number range $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3}){-}O(10^{4})$, are analysed. The published direct numerical simulation (DNS) dataset of Sillero et al. (Phys. Fluids, vol. 26 (10), 2014, 105109) is considered for low-$Re_{\unicode[STIX]{x1D70F}}$ analysis, while the high-$Re_{\unicode[STIX]{x1D70F}}$ dataset is obtained by conducting synchronous multipoint hot-wire measurements. High-$Re_{\unicode[STIX]{x1D70F}}$ cross-spectra reveal that the wall-attached large scales follow a $\unicode[STIX]{x1D706}_{y}/z_{o}\sim \unicode[STIX]{x1D706}_{x}/z_{o}$ relationship more closely than seen for $\unicode[STIX]{x1D6F7}$, where this self-similar trend is obscured by coexisting scales. The present analysis reaffirms that a self-similar structure, conforming to Townsend’s attached eddy hypothesis, is ingrained in the flow.

We use a multiple-scale expansion to average the wave action balance equation over an ensemble of sea-surface velocity fields characteristic of the ocean mesoscale and submesoscale. Assuming that the statistical properties of the flow are stationary and homogeneous, we derive an expression for a diffusivity tensor of surface-wave action density. The small parameter in this expansion is the ratio of surface current speed to gravity wave group speed. For isotropic currents, the action diffusivity is expressed in terms of the kinetic energy spectrum of the flow. A Helmholtz decomposition of the sea-surface currents into solenoidal (vortical) and potential (divergent) components shows that, to leading order, the potential component of the surface velocity field has no effect on the diffusivity of wave action: only the vortical component of the sea-surface velocity results in diffusion of surface-wave action. We validate our analytic results for the action diffusivity by Monte Carlo ray-tracing simulations through an ensemble of stochastic velocity fields.

The thermal protection system in atmospheric-entry and hypersonic-cruise vehicles are oftentimes designed to ablate during their operation, thus injecting a mixture of various gases with distinct properties into the boundary layer. Such outgassing affects the propagation of instabilities within the boundary layer that ultimately originates the transition to turbulence. This work uses linear stability theory, in combination with the e$^{N}$ method, to establish the underlying reason for the experimentally observed advancement/delay of transition in sharp slender hypersonic cones, when injecting lighter/heavier gases. Contrary to the current understanding and experimental correlations, this numerical analysis suggests that such a behaviour is not linked to the isolated effect of the injected gas’ molar weight, but to its combination with the blowing discontinuity, porosity and the appearance of a shocklet, consequence of the injected gas composition. The shocklet constitutes a density gradient that acts on second-mode instabilities like a thermoacoustic impedance.

When a lightning bolt darts across the sky, the thunderclap that reaches our ears a few seconds later is an example of a fluid dynamical shock: a wave across which flow properties such as pressure and density change almost discontinuously. In compressible fluids these shocks are associated with high-energy supersonic flows and so require specialist equipment to realise in steady state. But in granular media, shocks occur much more readily and at flow speeds easily obtainable in the laboratory. In the featured article, Khan et al. (J. Fluid Mech., vol. 884, 2020, R4) exploit this to explore a remarkable range of steady and oscillatory shocks and shock interactions, which demonstrate many of the unique rheological complexities of granular flow.

Using especially designed laboratory experiments, we demonstrate that the flow-driven deformation of sufficiently porous, wall-mounted, flexible plates can exhibit positive Vogel exponent $V$, i.e. drag proportional to the $(2+V)$ power of the incoming flow velocity. High-resolution force balance, planar particle image velocimetry and particle tracking velocimetry are used to measure the drag force, flow characteristics and plate bending. For a flexible plate with relatively high porosity given by an array of regularly spaced square openings, we derive a simple analytical argument that accounts for the sub-quadratic trends of the drag in a range of flow velocities spanning one order of magnitude. There, the drag experienced by the structure is modulated by the contributions of the local structure containing an open area. The effective approach velocity for each of these sections appears to increase monotonically with increased structure deformation due to the reduced effect of local wakes produced by adjacent areas. The uncovered aerodynamic behaviour may help to understand the complex flow–structure interaction of perforated structures in nature and engineering.

Polymer solutions are often injected in porous media for applications such as oil recovery and groundwater remediation. As the fluid navigates the tortuous pore space, elastic stresses build up, causing the flow to become unstable at sufficiently large injection rates. However, it is poorly understood how the spatial and temporal characteristics of this unstable flow depend on pore space geometry, which can vary widely between different porous media. We investigate this dependence by systematically varying the spacing between pore constrictions in a one-dimensional ordered array. We find that when the pore spacing is large, unstable eddies form upstream of each constriction, similar to observations of an isolated constriction. By contrast, when the pore spacing is sufficiently small, the flow in the different pores exhibits a surprising bistability, stochastically switching between two distinct unstable flow states. We hypothesize that this unusual behaviour arises from the interplay between elongation and relaxation of polymers as they are advected through the pore space. Consistent with this idea, we find that the flow state in a given pore persists for long times; moreover, flow states are strongly correlated between neighbouring pores. Thus, the characteristics of unstable flow are not determined just by injection conditions and the geometry of the individual pores, but also depend on the spacing between pores. Ultimately, these results help to elucidate the rich array of behaviours that can arise in polymer solution flow through porous media.

We study the equilibrium of two phases following gravity segregation under the influence of capillary heterogeneity. Such processes are important in a number of porous media applications, e.g. determining reservoir composition, secondary migration, gravity drainage enhanced oil recovery and CO$_{2}$ storage in aquifers. Solutions are derived for three-dimensional saturation distribution $S_{w}(x,y,z)$ and given as an analytical formula apart from a constant $P_{c}^{0}$ which is determined by numerical integration. The first solution assumes hydrostatic pressure and applies to cases without capillary entry pressure ($P_{c}(S_{w}=1)=0$). The solution can be used for validation of numerical simulations and we show a close match for a number of cases. A second analytical solution is derived, extending the first, to cases of random log-normally distributed permeability fields. A formula for ensemble average saturation solution is presented and a comparison to solutions of various realizations is discussed. When capillary entry pressure is present, the solution based on hydrostatic pressure may be inaccurate due to entry pressure trapping which occurs when regions of $S_{w}=1$ are present. Using numerical simulation, we extend the solution to include estimations of entry pressure trapping for a range of parameters and show its applicability. The comparison of analytical and numerical results helps illustrate and draw insight on the trapping mechanism.

The vortex-induced vibration of a spring-mounted, damped, rigid circular cylinder, immersed in a Newtonian viscous flow and capable of moving in the direction orthogonal to the unperturbed flow is investigated for Reynolds numbers $Re$ in the vicinity of the onset of unsteadiness ($15\leqslant Re\leqslant 60$) using the incompressible linearised Navier–Stokes equations. In a first step, we solve the linear problem considering an imposed harmonic motion of the cylinder. Results are interpreted in terms of the mechanical impedance, i.e. the ratio between the vertical force coefficient and the cylinder velocity, which is represented as function of the Reynolds number and the driving frequency. Considering the energy transfer between the cylinder and the fluid, we show that impedance results provide a simple criterion allowing the prediction of the onset of instability of the coupled fluid-elastic structure case. A global stability analysis of the fully coupled fluid/cylinder system is then performed. The instability thresholds obtained by this second approach are found to be in perfect agreement with the predictions of the impedance-based criterion. A theoretical argument, based on asymptotic developments, is then provided to give a prediction of eigenvalues of the coupled problem, as well as to characterise the region of instability beyond the threshold as function of the reduced velocity $U^{\ast }$, the dimensionless mass $m^{\ast }$ and the Reynolds number. The influence of the damping parameter $\unicode[STIX]{x1D6FE}$ on the instability region is also explored.

This article presents a revised formulation of the generation and transport of vorticity at generalised fluid–fluid interfaces, substantially extending the work of Brøns et al. (J. Fluid Mech., vol. 758, 2014, pp. 63–93). Importantly, the formulation is effectively expressed in terms of the conservation of vorticity, and the latter is shown to hold for arbitrary deformation and normal motion of the interface; previously, vorticity conservation had only been demonstrated for stationary interfaces. The present formulation also affords a simple physical description of the generation of vorticity in incompressible, Newtonian flows: the only mechanism by which vorticity may be generated on an interface is the inviscid relative acceleration of fluid elements on each side of the interface, due to pressure gradients or body forces. Viscous forces act to transfer circulation between the vortex sheet representing the interface slip velocity, and the fluid interior, but do not create vorticity on the interface. Several representative example flows are considered and interpreted under the proposed framework, illustrating the generation, transport and, importantly, the conservation of vorticity within these flows.

The Lagrangian-mean motion of fluid particles induced by horizontally localized small-amplitude wavepackets of vertically trapped inertia–gravity waves is computed analytically, at second order in wave amplitude, and the results are supported by direct nonlinear numerical simulations. The leading-order motion is assumed to be inertia–gravity waves, which is applicable to oceanic mesoscale flows in regions where wave activity is as strong as or stronger than the balanced flow. The analytical computation is based on time-dependent asymptotic wave–mean interaction theory, and the numerical simulation uses a Galerkin-truncated $f$-plane nonlinear hydrostatic Boussinesq model that retains the barotropic mode and two baroclinic modes (vertical wavenumbers 0, $m$ and $2m$), this being the minimal set on which consistent wave–mean interactions can take place. Two novel dynamical effects are revealed: First, we find that the barotropic component robustly dominates the Lagrangian-mean flow response, which is contrary to earlier findings for the same problem. Second, we discovered a new wavepacket regime in which the baroclinic mean-flow response consists of the persistent radiation of resonantly forced secondary internal waves. The latter effect occurs in an oceanically accessible parameter regime.

This experimental study explores the physical mechanisms by which a transverse jet’s upstream shear layer can transition from being a convective instability to an absolute/global instability as the jet-to-cross-flow momentum flux ratio $J$ is reduced. As first proposed in computational studies by Iyer & Mahesh (J. Fluid Mech., vol. 790, 2016, pp. 275–307), the upstream shear layer just beyond the jet injection may be analogous to a local counter-current shear layer, which is known for a planar geometry to become absolutely unstable at a large enough counter-current shear layer velocity ratio, $R_{1}$. The present study explores this analogy for a range of transverse jet momentum flux ratios and jet-to-cross-flow density ratios $S$, for jets containing differing species concentrations (nitrogen, helium and acetone vapour) at several different jet Reynolds numbers. These studies make use of experimental data extracted from stereo particle image velocimetry as well as simultaneous stereo particle image velocimetry and acetone planar laser-induced fluorescence imaging. They provide experimental evidence for the relevance of the counter-current shear layer analogy to upstream shear layer instability transition in a nozzle-generated transverse jet.

In this study, stereo particle imaging velocimetry measurements are performed at multiple streamwise locations behind a yawed wind turbine to reveal the formation mechanisms of the counter-rotating vortex pair (CVP), and a point vortex transportation (PVT) model is proposed to reproduce the top–down asymmetric kidney-shaped wake (also referred to as a curled wake). Results indicate that the CVP formed behind a yawed wind turbine originates from the complex interactions between the hub vortex and the streamwise components of the blade tip vortices, which is fundamentally different from the case of a yawed drag disk where the hub vortex is absent. Specifically, when the yaw angle exceeds a critical value, a small part of the streamwise vorticity shed from the rotor disk edge switches its sign from negative to positive and subsequently merges with the concentrated hub vortex under mutual induction, creating a patch of positive vorticity; meanwhile, the remaining streamwise vorticity distributed along the rotor edge curls and evolves into another patch of negative vorticity. These two patches of streamwise vorticity essentially constitute the CVP. Based on the physics learnt from the experiments, the non-uniform cross-stream velocity fields are first reconstructed by a cloud of point vortices distributed along the rotor edge and a hub vortex located in the rotor centre, and subsequently used to numerically solve a simplified transportation–diffusion equation of the wake velocity deficit, which altogether constitute the PVT model. This physics-based reduced-order model is the first model capable of accurately reproducing the wake deformation behind a yawed wind turbine.

In a recent experiment on the flow between two concentric cylinders with the inner cylinder rotating and the fluid being stably stratified, Flór et al. (Phys. Fluids, vol. 30, 2018, 084103) found helical wave structures confined to the inner cylinder in an annulus with small inner-to-outer radius ratio (very large gap) in regimes where the Froude number (ratio of cylinder rotation frequency to buoyancy frequency) is less than one. These helical waves were reported to originate at the corners where the inner cylinder meets the top and bottom boundaries, and were found to be asymmetric with the lower helical wave being more intense. These observations are in marked contrast with other stratified Taylor–Couette experiments that employed much larger inner-to-outer radius ratios and much larger annulus height-to-gap ratios. Here, we present direct numerical simulations of the Navier–Stokes equations, with a Boussinesq approximation that accounts for centrifugal buoyancy effects which are normally neglected. Fixing the stratification and increasing the rotation rate of the inner cylinder (quantified by a Reynolds number), we find a sequence of bifurcations, each one introducing a new frequency, from the steady base state to a three-torus state. The instabilities are generated at the corners where the inner cylinder meets the endwalls, and the first instability is localized at the lower corner as a consequence of centrifugal buoyancy effects. We have also conducted simulations without centrifugal buoyancy and find that centrifugal buoyancy plays a crucial role in breaking the up–down reflection symmetry of the problem, capturing the most salient features of the experimental observations.

Ensemble data assimilation methods integrate measurement data and computational flow models to estimate the state of fluid systems in a robust, scalable way. However, discretization errors in the dynamical and observation models lead to biased forecasts and poor estimator performance. We propose a low-rank representation for this bias, whose dynamics is modelled by data-informed, time-correlated processes. State and bias parameters are simultaneously corrected online with the ensemble Kalman filter. The proposed methodology is then applied to the problem of estimating the state of a two-dimensional flow at modest Reynolds number using an ensemble of coarse-mesh simulations and pressure measurements at the surface of an immersed body in a synthetic experiment framework. Using an ensemble size of 60, the bias-aware estimator is demonstrated to achieve at least 70 % error reduction when compared to its bias-blind counterpart. Strategies to determine the bias statistics and their impact on the estimator performance are discussed.

The beginning of laminar–turbulent transition is usually associated with a wave-like disturbance, but its evolution and role in precipitating the development of other flow structures are not well understood from a structure-based view. Nonlinear parabolized stability equations (NPSE) were solved numerically to simulate the transition of K-regime, N-regime and O-regime. However, only the K-regime transition was examined experimentally using both hydrogen bubble visualization and time-resolved tomographic particle image velocimetry (tomo-PIV). Based on the ‘NPSE visualization’ and ‘tomographic visualization’, at least four common characteristics of the generic transition process were identified: (i) inflectional regions representing high-shear layers (HSL) that develop in vertical velocity profiles, accompanied by ejection–sweep behaviours; (ii) low-speed streak (LSS) patterns, manifested in horizontal timelines, that seem to consist of several three-dimensional (3-D) waves; (iii) a warped wave front (WWF) pattern, displaying multiple folding processes, which develops adjacent to the LSS in the near-wall region, prior to the appearance of 𝛬-vortices; (iv) a coherent 3-D wave front, similar to a soliton, in the upper boundary layer, accompanied by regions of depression along the flanks of the wave. It was determined that the amplification and lift-up of a 3-D wave causes the development of the HSL, WWF and multiple folding behaviour of material surfaces, that all contribute to the development of a 𝛬-vortex. The amplified 3-D wave is hypothesized as a soliton-like coherent structure. Based on our results, a path to transition is proposed, which hypothesizes the function of the WWF in boundary-layer transition.

We present direct numerical simulations of breaking solitary waves in shallow water to quantify the energy dissipation during the active breaking time. We find that this dissipation can be predicted by an inertial model based on Taylor’s hypothesis as a function of the local wave height, depth and the beach slope. We obtain a relationship that gives the dissipation rate of a breaking wave on a shallow slope as a function of local breaking parameters. Next, we use empirical relations to relate the local wave parameters to the offshore conditions. This enables the energy dissipation to be predicted in terms of the initial conditions. We obtain good collapse of the numerical data with respect to the theoretical scaling.

A problem of an unsteady plane flow of an ideal incompressible fluid with a free boundary is considered. It is shown that the solution can be found by using a complex transport equation. In this case, the problem is linearized by means of the hodograph transform (the velocity components are chosen as independent variables). Examples of exact solutions are obtained. Various scenarios of formation of singularities on the free boundary within a finite time are considered.

Inverted flags – clamped–free elastic thin plates subjected to a fluid flowing axially and directed from the free end towards the clamped end – have been observed experimentally and computationally to exhibit large-amplitude flapping beyond a critical flow velocity. The motivation for further research on the dynamics of this system is partly due to its presence in some engineering and biological systems, and partly because of the very rich dynamics it displays. In the present paper, the goal is to develop a nonlinear analytical model for the dynamics and stability of high aspect ratio (i.e. height to length ratio) flags. The inviscid fluid flow is modelled via the quasi-steady version of Theodorsen’s unsteady aerodynamic theory, and the Polhamus leading-edge suction analogy is utilized to model flow separation effects from the free end (leading edge) at moderate angles of attack. Gear’s backward differentiation formula and a pseudo-arclength continuation technique are employed to solve the governing equations. Numerical results suggest that fluidelastic instability may be the underlying mechanism for the flapping motion of high aspect ratio heavy inverted flags. In other words, flapping may be viewed as a self-excited vibration. It was found from numerical results that the undeflected static equilibrium of the inverted flag is stable at low flow velocities, prior to the occurrence of a supercritical pitchfork bifurcation. The pitchfork bifurcation is associated with static divergence (buckling) of the flag. At higher flow velocities, past the pitchfork bifurcation, a supercritical Hopf bifurcation materializes, generating a flapping motion around the deflected static equilibrium. At even higher flow velocities, flapping motion becomes symmetric around the undeflected static equilibrium. Interestingly, it was also found that heavy flags may exhibit large-amplitude flapping right after the initial static equilibrium, provided that they are subjected to a sufficiently large disturbance. Moreover, inverted flags with a non-zero initial angle of attack were found to be less stable than their perfectly flow-aligned counterparts.

We study shear-induced migration in a dilute suspension of microswimmers (modelled as active Brownian particles or ABPs) subject to plane Poiseuille flow. For wide channels characterized by $U_{s}/HD_{r}\ll 1$, the separation between time scales characterizing the swimmer orientation dynamics (of $O(D_{r}^{-1})$) and those that characterize migration across the channel (of $O(H^{2}D_{r}/U_{s}^{2})$), allows for use of the method of multiple scales to derive a drift-diffusion equation for the swimmer concentration profile; here, $U_{s}$ is the swimming speed, $H$ is the channel half-width and $D_{r}$ is the swimmer rotary diffusivity. The steady state concentration profile is a function of the Péclet number, $Pe=U_{f}/(D_{r}H)$ ($U_{f}$ being the channel centreline velocity), and the swimmer aspect ratio $\unicode[STIX]{x1D705}$. Swimmers with $\unicode[STIX]{x1D705}\gg 1$ (with $\unicode[STIX]{x1D705}\sim O(1)$), in the regime $1\ll \text{Pe}\ll \unicode[STIX]{x1D705}^{3}$ ($Pe\sim O(1)$), migrate towards the channel walls, corresponding to a high-shear trapping behaviour. For $Pe\gg \unicode[STIX]{x1D705}^{3}$ ($Pe\gg 1$ for $\unicode[STIX]{x1D705}\sim O(1)$), however, swimmers migrate towards the centreline, corresponding to a low-shear trapping behaviour. Interestingly, within the low-shear trapping regime, swimmers with $\unicode[STIX]{x1D705}<2$ asymptote to a $Pe$-independent concentration profile for large $Pe$, while those with $\unicode[STIX]{x1D705}\geqslant 2$ exhibit a ‘centreline collapse’ for $Pe\rightarrow \infty$. The prediction of low-shear trapping, validated by Langevin simulations, is the first explanation of recent experimental observations (Barry et al., J. R. Soc. Interface, vol. 12 (112), 2015, 20150791). We organize the high-shear and low-shear trapping regimes on a $Pe{-}\unicode[STIX]{x1D705}$ plane, thereby highlighting the singular behaviour of infinite-aspect-ratio swimmers.