Sequestration of carbon dioxide by injecting it into the deep subsurface is critical to successful mitigation of climate change by reducing anthropogenic emissions of greenhouse gases into the atmosphere. To achieve this we must understand how CO2 moves in the subsurface. Many interesting fluid mechanics problems emerge. Szulczewski, Hesse & Juanes (J. Fluid Mech., vol. 736, 2013, pp. 287–315) focus on one critical aspect, namely the dissolution of CO2 into the fluid resident in the subsurface and the flow dynamics that ensue. Even for this single problem, an elegant analysis identifies seven behavioural regimes that control the amount and timing of dissolution.

We derive the local induction approximation (LIA) for a quantum vortex filament in the arclength coordinate frame where the tangent vector is the unknown function. The equation for the tangent vector to the filament is then converted to a potential form, which ends up being a type of nonlinear Schrödinger equation that governs the tangential LIA model (T-LIA). Such a formulation was previously derived by Umeki for the standard fluid model (in the absence of superfluid friction terms). While it is challenging to generalize many of the exact solutions found for the standard LIA to the quantum LIA model, we demonstrate that the T-LIA model facilitates this generalization nicely. Indeed, under the T-LIA model, we are able to construct a variety of solutions. The Hasimoto solution related to elastica is one of the fundamental solutions present for the standard fluid model; however, using the T-LIA model, we are able to demonstrate the existence of such a solution, thereby extending the Hasimoto solution to the superfluid case. In the zero-temperature limit, purely self-similar solutions are shown to exist for the T-LIA model. As the superfluid warms (so that the influence of the normal flow is no longer negligible), the analogue to the self-similar solution is a new class of solutions, which depend on the similarity variable as well as a time-dependent additive scaling. In other words, the self-similar structures gradually deform as the magnitude of the normal-fluid velocity increases, which makes complete physical sense. When dealing with small deviations from the central axis of alignment, we can describe such solutions analytically. There exists a family of helical vortex filaments in the presence of a normal fluid impinging on the vortex, in complete agreement with the previously studied results for the LIA model. Finally, a number of soliton solutions are shown to exist in different regimes of the T-LIA model.

We investigate the relevance of Stokes waves for the propagation of ocean swell in the absence of strong currents. By providing estimates for the depth of the near-surface layer to which the main effects of a Stokes flow are confined, we show that wind-generated uniform wave trains can be modelled as Stokes waves over a fictitious flat bed, immersed in the water. Throughout the lower parts of this layer the deviations of the flow from a pure current are negligible.

We present an experimental study of turbulent thermal convection with smooth and rough surface plates in various combinations. A total of five cells were used in the experiments. Both the global $\mathit{Nu}$ and the $\mathit{Nu}$ for each plate (or the associated boundary layer) are measured. The results reveal that the smooth plates are insensitive to the surface (rough or smooth) and boundary conditions (i.e. nominally constant temperature or constant flux) of the other plate of the same cell. The heat transport properties of the rough plates, on the other hand, depend not only on the nature of the plate at the opposite side of the cell, but also on the boundary condition of that plate. It thus appears that, at the present level of experimental resolution, the smooth plate can influence the rough plate, but cannot be influenced by either the rough or the smooth plates. It is further found that the scaling of $\mathit{Nu}$ with $\mathit{Ra}$ for all of the smooth plates is consistent with the classical $1/ 3$ exponent. But the scaling exponent for the global $\mathit{Nu}$ for the cell with both plates being smooth is definitely less than $1/ 3$ (this result itself is consistent with all previous studies at comparable parameter range). The discrepancy between the $\mathit{Nu}$ behaviour at the whole-cell and individual-plate levels is not understood and deserves further investigation.

In the unsteady process of shock reflection off convexly curved surfaces, the Reynolds number can have an influence on the development of the irregular reflection pattern. Time-resolved visualizations of the reflection process and high-resolution numerical simulation are used in this investigation to quantify this influence, which manifests itself in a delayed growth of the shock pattern with decreasing Reynolds number. In order to conduct reliable and unambiguous measurements, the present study concentrates on observing the development of the established irregular reflection pattern rather than attempting to determine the transition point directly. It can be seen that the influence of the Reynolds number is highly nonlinear and that changes of two orders of magnitude or more are required to produce a reliably measurable difference in the triple point trajectories, which is considerably more than what has so far been reported in the literature. The results allow one to make inferences regarding the transition process and they help to clarify previously reported discrepancies between predicted and experimentally determined transition angles.

The formation and pinch-off of non-axisymmetric vortex rings is considered experimentally. Vortex rings are generated using a non-circular piston–cylinder arrangement, and the resulting velocity fields are measured using digital particle image velocimetry. Three different nozzle geometries are considered: an elliptical nozzle with an aspect ratio of two, an elliptical nozzle with an aspect ratio of four and an oval nozzle constructed from tangent circular arcs. The formation of vortices from the three nozzles is analysed by means of the vorticity and circulation, as well as by investigation of the Lagrangian coherent structures in the flow. The results indicate that, in all three nozzles, the maximum circulation the vortex can attain is determined by the equivalent diameter of the nozzle: the diameter of a circular nozzle of identical cross-sectional area. In addition, the time at which the vortex rings pinch off is found to be constant along the nozzle contours, and independent of relative variations in the local curvature. A formation number for this class of vortex rings is defined based on the equivalent diameter of the nozzle, and the formation number for vortex rings of the three geometries considered is found to lie in the range of 3–4. The implications of the relative shape and local curvature independence of the formation number on the study and modelling of naturally occurring vortex rings such as those that appear in biological flows is discussed.

The problem of oceanic gap-traversing boundary currents, such as the Kuroshio current crossing the Luzon Strait or the Gulf Stream traversing the mouth of the Gulf of Mexico, is considered. Systems such as these are known to admit two dominant states: leaping across the gap or penetrating into the gap forming a loop current. Which state the system will assume and when transitions between states will occur are open problems. Sheremet (J. Phys. Oceanogr., vol. 31, 2001, pp. 1247–1259) proposed, based on idealized barotropic numerical results, that variation in the current’s inertia is responsible for these transitions and that the system admits multiple states. Generalized versions of these results have been confirmed by barotropic rotating-table experiments (Sheremet & Kuehl, J. Phys. Oceanogr., vol. 37, 2007, 1488–1495; Kuehl & Sheremet,J. Mar. Res., vol. 67, 2009, pp. 25–42). However, the typical structure of oceanic boundary currents, such as the Gulf Stream or Kuroshio, consists of an upper-layer intensified flow riding atop a weakly circulating lower layer. To more accurately address this oceanic situation, the present work extends the above findings by considering two-layer rotating table experiments. The flow is driven by pumping water through sponges and vertical seals, creating a Sverdrup interior circulation in the upper layer which impinges on a ridge where a boundary current is formed. The $\beta $ effect is incorporated in both layers by a sloping rigid lid as well as a sloping bottom and the flow is visualized with the particle image velocimetry method. The experimental set-up is found to produce boundary currents consistent with theory. The existence of multiple states and hysteresis, characterized by a cusp topology of solutions, is found to be robust to stratification and various properties of the two-layer system are explored.

A local available potential energy (APE) density useful as suitable diagnostic tool in turbulent stratified flows is considered under the Boussinesq approximation. The local APE is positive, and in the limit of infinitesimal perturbation from an equilibrium state recovers the Lorenz energy cycle definition of APE. In a turbulent stratified flow, the APE can be Reynolds-decomposed into non-trivial mean and turbulent components, which are connected to the mean and turbulent kinetic energy by suitably defined fluxes. We show that the turbulent buoyancy flux $\overline{w'b'}$ and the rate of production of turbulent APE coincide only under very special circumstances. The framework is applied to derive some global bounds on the mixing efficiency of some representative flows.

A systematic study of the effects of cell geometry on the dynamics of large-scale flows in turbulent thermal convection is carried out in horizontal cylindrical cells of different lengths filled with water. Four different flow modes are identified with increasing aspect ratio $\Gamma $. For small aspect ratios ($\Gamma \leq 0.16$), the flow is highly confined in a thin disc-like cell with a quasi-two-dimensional (quasi-2D) large-scale circulation (LSC) in the circular plane of the cell. For larger aspect ratios ($\Gamma >0.16$), we observe periodic switching of the angular orientation $\theta $ of the rotation plane of LSC between the two longest diagonals of the cell. The sides of the container along which the LSC oscillates changes at a critical aspect ratio $\Gamma _{c}\simeq 0.82$. The measured switching period is equal to the LSC turnover time for $\Gamma \leq \Gamma _c$, shows a sharp increase at $\Gamma _{c}$ and decays exponentially to the LSC turnover time with increasing $\Gamma $. For $\Gamma \geq 1.3$, a periodic rocking of LSC along the long axis of the cylinder is also observed. The measured probability density function $P(\theta )$ of the LSC orientation $\theta $ peaks at the two diagonal positions, and its shape is described by a phenomenological model proposed by Brown & Ahlers (Phys. Fluids, vol. 20, 2008b, 075101; J. Fluid Mech., vol. 638, 2009, pp. 383–400). Using this model, we describe the dynamics of the LSC orientation $\theta $ by stochastic motion in a double-well potential. The potential is predicted from a model in which the sidewall shape produces an orientation-dependent pressure on the LSC. This model also captures key features of the four flow modes. The experiment reveals an interesting array of rich dynamics of LSC in the horizontal cylinders, which are very different from those observed in the upright cylindrical convection cells. The success of the model for both upright and horizontal cylinders suggests that it can be applied to different geometries.

Analytical expressions for the flow field as well as for the effective slip length of a shear flow over a surface with periodic rectangular grooves are derived. The primary fluid is in the Cassie state with the grooves being filled with a secondary immiscible fluid. The coupling of the two fluids is reflected in a locally varying slip distribution along the fluid–fluid interface, which models the effect of the secondary fluid on the outer flow. The obtained closed-form analytical expressions for the flow field and effective slip length of the primary fluid explicitly contain the influence of the viscosities of the two fluids as well as the magnitude of the local slip, which is a function of the surface geometry. They agree well with results from numerical computations of the full geometry. The analytical expressions allow an investigation of the influence of the viscous stresses inside the secondary fluid for arbitrary geometries of the rectangular grooves. For classic superhydrophobic surfaces, the deviations in the effective slip length compared to the case of inviscid gas flow are pointed out. Another important finding with respect to an accurate modelling of flow over microstructured surfaces is that not only the effective slip length, but also the local slip length of a grooved surface, is anisotropic.

Associated with Taylor’s classical analysis of scalar solute dispersion in the laminar flow of a solvent in a straight pipe, this work explores the approach towards transverse uniformity of concentration distribution. Mei’s homogenization technique is extended to find solutions for the concentration transport. Chatwin’s result for the approach to longitudinal normality is recovered in terms of the mean concentration over the cross-section. The asymmetrical structure of the concentration cloud and the transverse variation of the concentration distribution are concretely illustrated for the initial stage. The rate of approach to uniformity is shown to be much slower than that to normality. When the longitudinal normality of mean concentration is well established, the maximum transverse concentration difference remains near one-half of the centroid concentration of the cloud. A time scale up to $10 R^2/D$ ($R$ is the radius of the pipe and $D$ is the molecular diffusivity) is suggested to characterize the transition to transverse uniformity, in contrast to the time scale of $0.1 R^2/D$ estimated by Taylor for the initial stage of dispersion, and that of $1.0 R^2/D$ by Chatwin for longitudinal normality.

This paper gives the most-complete characterization to date of the optical aberrations imposed on a laser beam propagated through a subsonic, compressible, turbulent boundary layer in a zero-pressure gradient environment, over a range of boundary-layer thicknesses, oblique propagation angles and Mach numbers. This characterization is based on optical measurements using optical-wavefront-sensing instruments that have only become available in the last decade. The optical characterization includes and discusses in detail: optical-wavefront spectra, convective velocities of optically active large-scale structures and correlation functions in both streamwise and cross-stream directions, as well as root-mean-square optical path difference levels for different apertures. The scaling law based on the extended strong Reynolds analogy is derived and is shown to successfully collapse optical data collected in a number of facilities. Anisotropy of aero-optical distortions for different oblique viewing angles was experimentally quantified and is discussed.

Using the Hamiltonian approach of Krasitskii (J. Fluid Mech., vol. 272, 1994, pp. 1–20), we derive a variant of the Zakharov equation in which the wave-induced mean surface elevation and the surface potential of the wave-induced mean flow are represented as separate variables governed by separate evolution equations. The kernel function of this new variant is simpler, and in particular also well defined in the uniform-wave-train limit for waves on finite depth. This form of the Zakharov equation may be advantageous in some applications. One example is the derivation of nonlinear Schrödinger equations in the narrow-band limit, where the handling of the mean flow and mean surface is significantly simpler than when starting from the original Zakharov equation. In this paper we have used the alternative form of the Zakharov equation to derive a Hamiltonian nonlinear Schrödinger equation for directional waves on arbitrary depth, valid to one order higher in bandwidth than the Hamiltonian equation recently presented by Craig, Guyenne and Sulem (Wave Motion, vol. 47, 2010, pp. 552–563).

The stability of the vertical path of a gravity- or buoyancy-driven disk of arbitrary thickness falling or rising in a viscous fluid, recently studied through direct numerical simulation by Auguste, Magnaudet & Fabre (J. Fluid Mech., vol. 719, 2013, pp. 388–405), is investigated numerically in the framework of global linear stability. The disk is allowed to translate and rotate arbitrarily and the stability analysis is carried out on the fully coupled system obtained by linearizing the Navier–Stokes equations for the fluid and Newton’s equations for the body. Three disks with different diameter-to-thickness ratios are considered: one is assumed to be infinitely thin, the other two are selected as archetypes of thin and thick cylindrical bodies, respectively. The analysis spans the whole range of body-to-fluid inertia ratios and considers Reynolds numbers (based on the fall/rise velocity and body diameter) up to $350$. It reveals that four unstable modes with an azimuthal wavenumber $m=\pm 1$ exist in each case. Three of these modes result from a Hopf bifurcation while the fourth is associated with a stationary bifurcation. Varying the body-to-fluid inertia ratio yields rich and complex stability diagrams with several branch crossings resulting in frequency jumps; destabilization/restabilization sequences are also found to take place in some subdomains. The spatial structure of the unstable modes is also examined. Analyzing differences between their real and imaginary parts (which virtually correspond to two different instants of time in the dynamics of a given mode) allows us to assess qualitatively the strength of the mutual coupling between the body and fluid. Qualitative and quantitative differences between present predictions and known results for wake instability past a fixed disk enlighten the fact that the first non-vertical regimes generally result from an intrinsic coupling between the body and fluid and not merely from the instability of the sole wake.

Planetary turbulent flows are observed to self-organize into large-scale structures such as zonal jets and coherent vortices. One of the simplest models of planetary turbulence is obtained by considering a barotropic flow on a beta-plane channel with turbulence sustained by random stirring. Nonlinear integrations of this model show that as the energy input rate of the forcing is increased, the homogeneity of the flow is broken with the emergence of non-zonal, coherent, westward propagating structures and at larger energy input rates by the emergence of zonal jets. We study the emergence of non-zonal coherent structures using a non-equilibrium statistical theory, stochastic structural stability theory (S3T, previously referred to as SSST). S3T directly models a second-order approximation to the statistical mean turbulent state and allows the identification of statistical turbulent equilibria and study of their stability. Using S3T, the bifurcation properties of the homogeneous state in barotropic beta-plane turbulence are determined. Analytic expressions for the zonal and non-zonal large-scale coherent flows that emerge as a result of structural instability are obtained. Through numerical integrations of the S3T dynamical system, it is found that the unstable structures equilibrate at finite amplitude. Numerical simulations of the nonlinear equations confirm the characteristics (scale, amplitude and phase speed) of the structures predicted by S3T.

The flow-induced vibrations of a circular cylinder, free to oscillate in the cross-flow direction and subjected to a forced rotation about its axis, are analysed by means of two- and three-dimensional numerical simulations. The impact of the symmetry breaking caused by the forced rotation on the vortex-induced vibration (VIV) mechanisms is investigated for a Reynolds number equal to $100$, based on the cylinder diameter and inflow velocity. The cylinder is found to oscillate freely up to a rotation rate (ratio between the cylinder surface and inflow velocities) close to $4$. Under forced rotation, the vibration amplitude exhibits a bell-shaped evolution as a function of the reduced velocity (inverse of the oscillator natural frequency) and reaches $1.9$ diameters, i.e. three times the maximum amplitude in the non-rotating case. The free vibrations of the rotating cylinder occur under a condition of wake–body synchronization similar to the lock-in condition driving non-rotating cylinder VIV. The largest vibration amplitudes are associated with a novel asymmetric wake pattern composed of a triplet of vortices and a single vortex shed per cycle, the ${\rm T} + {\rm S}$ pattern. In the low-frequency vibration regime, the flow exhibits another new topology, the U pattern, characterized by a transverse undulation of the spanwise vorticity layers without vortex detachment; consequently, free oscillations of the rotating cylinder may also develop in the absence of vortex shedding. The symmetry breaking due to the rotation is shown to directly impact the selection of the higher harmonics appearing in the fluid force spectra. The rotation also influences the mechanism of phasing between the force and the structural response.

We report a self-similar behaviour of solutions (obtained numerically) of the Navier–Stokes equations behind a single-blade rotor. That is, the helical vortex core in the wake of a rotating blade is self-similar as a function of its age. Profiles of vorticity and azimuthal velocity in the vortex core are characterized, their similarity variables are identified and scaling laws of these variables are given. Solutions of incompressible three-dimensional Navier–Stokes equations for Reynolds numbers up to $Re= 2000$ are considered.