A large eddy simulation (LES) is used to estimate a reliable horizontal turbulent diffusion coefficient, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K_{{h}}$, in a convective mixed layer (CML). The introduction of a passive scalar field with a fixed horizontal gradient at a given time enables $K_{{h}}$ estimation as a function of height, based on the simulated turbulent horizontal scalar flux. Here $K_{{h}}$ is found to be of the order of $100\ {\mathrm{m}}^2\ {\mathrm{s}}^{-1}$ for a typical terrestrial atmospheric CML. It is shown to scale by the product of the CML convective velocity, $w_{*}$, and its depth, $h$. Here $K_{{h}}$ is characterized by a vertical profile in the CML: it is large near both the bottom and top of the CML, where horizontal flows associated with convection are large. The equation pertaining to the temporal rate of change of a horizontal scalar flux suggests that $K_{{h}}$ is determined by a balance between production and pressure correlation at a fully developed stage. Pressure correlation near the bottom of the CML is localized in convergence zones near the boundaries of convective cells and becomes large within an eddy turnover time, $h/w_{*}$, after the introduction of the passive scalar field.

Linear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich & Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.

We investigate the linear stability threshold of a dielectric liquid subjected to unipolar injection in a two-dimensional rectangular enclosure with rigid boundaries. A finite element formulation transforms the set of linear partial differential equations that governs the system into a set of algebraic equations. The resulting system poses an eigenvalue problem. We calculate the linear stability threshold, as well as the velocity field and charge density distribution, as a function of the aspect ratio of the domain. The stability parameter as a function of the aspect ratio describes paths of symmetry-breaking bifurcation. The symmetry properties of the different linear modes determine whether these paths cross each other or not. The resulting structure has important consequences in the nonlinear behaviour of the system after the bifurcation points.

A quasi-cylindrical approximation is used to analyse the axisymmetric swirling flow of a liquid with a hollow air core in the chamber of a pressure swirl atomizer. The liquid is injected into the chamber with an azimuthal velocity component through a number of slots at the periphery of one end of the chamber, and flows out as an annular sheet through a central orifice at the other end, following a conical convergence of the chamber wall. An effective inlet condition is used to model the effects of the slots and the boundary layer that develops at the nearby endwall of the chamber. An analysis is presented of the structure of the liquid sheet at the end of the exit orifice, where the flow becomes critical in the sense that upstream propagation of long-wave perturbations ceases to be possible. This analysis leads to a boundary condition at the end of the orifice that is an extension of the condition of maximum flux used with irrotational models of the flow. As is well known, the radial pressure gradient induced by the swirling flow in the bulk of the chamber causes the overpressure that drives the liquid towards the exit orifice, and also leads to Ekman pumping in the boundary layers of reduced azimuthal velocity at the convergent wall of the chamber and at the wall opposite to the exit orifice. The numerical results confirm the important role played by the boundary layers. They make the thickness of the liquid sheet at the end of the orifice larger than predicted by irrotational models, and at the same time tend to decrease the overpressure required to pass a given flow rate through the chamber, because the large axial velocity in the boundary layers takes care of part of the flow rate. The thickness of the boundary layers increases when the atomizer constant (the inverse of a swirl number, proportional to the flow rate scaled with the radius of the exit orifice and the circulation around the air core) decreases. A minimum value of this parameter is found below which the layer of reduced azimuthal velocity around the air core prevents the pressure from increasing and steadily driving the flow through the exit orifice. The effects of other parameters not accounted for by irrotational models are also analysed in terms of their influence on the boundary layers.

The behaviour of inertial waves in a rotating spherical container, filled with homogeneous fluid, is here investigated by means of a three-dimensional ray tracing algorithm, in a linear, inviscid framework. In particular, the classical, two-dimensional association between regular modes and periodic trajectories is addressed here for the first time in a fully three-dimensional setting. Three-dimensional, repelling periodic trajectories are found and classified on the basis of the associated frequency and spatial structure, although associated frequencies are hardly reconcilable to Bryan’s (Proc. R. Soc. Lond., vol. 45, 1889, pp. 42–45) classical solutions for inertial waves in the sphere. The normalized squared frequency $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\omega ^2 = 1/2$ appears to divide the frequency range into two different trajectory regimes, where critical latitudes play a different role. Chaotic orbits are not found, as expected, while invariant, non-domain-filling orbits (whispering gallery modes) constitute the majority of the trajectories in the sphere. From ray tracing alone, the wavefield is still far from being completely reconstructed, and a study performed in such a simplified setting is clearly far from any realistic application, however, it appears that three-dimensional ray dynamics constitutes a valid approach to infer information on the spectrum and regularity properties of a system, and is then able to bring new insight in a variety of fundamental problems of geophysical and astrophysical relevance, once its power and limitations have been recognized.

Shallow flow past successive cavities is characterized by means of high-image-density particle image velocimetry (PIV). Highly coherent, self-sustained oscillations arise due to coupling between the inherent instability of the separated shear layer along the opening of each sequential cavity; and a gravity standing wave mode within each cavity. The globally coupled nature of the flow structure is evident through dominance of the same spectral component in the undulating vorticity layers along each of the successive cavities and the wall pressure fluctuations within the cavities. Unlike coupled phenomena associated with flow past a single cavity, optimal coupling for successive cavities requires a defined phase shift between the gravity standing wave patterns in adjacent cavities and, furthermore, an overall phase shift of the undulating shear layer along the cavity openings. The magnitudes of these phase shifts depend on the mode of the gravity standing wave in each cavity, i.e. longitudinal or transverse mode, which is respectively aligned with or normal to the main stream. Such phase shifts result in corresponding displacements of patterns of phase-referenced vorticity concentrations along the cavity openings and changes in the timing of impingement of these concentrations upon the downstream corners of successive cavities. All of the foregoing aspects are related to the unsteady recirculation flow within the cavity, the time-dependent streamline topology, and concentrations of Reynolds stress along the cavity opening.

We propose that the collision rates of non-spherical particles settling in a turbulent environment are significantly higher than those of spherical particles of the same mass and volume. The theoretical argument is based on the dependence of the particle drag force on the particle orientation, thus varying gravitational settling velocities, which can remain different until contact due to the particle inertia. Therefore, non-spherical particles can collide with large relative velocities. Direct numerical simulations (DNS) of streamwise decaying isotropic turbulence seeded with small, heavy, rotationally symmetric ellipsoids of five different aspect ratios are performed to confirm these arguments. The motion of 21 million ellipsoids is tracked by a Lagrangian particle solver assuming creeping flow conditions and neglecting the influence of the particles on the flow. We find that ellipsoids collide considerably more often than spherical particles of the same volume and mass due to a drastically increased mean relative velocity at contact.

Measurements are presented of the structural response and wake of a two-degree-of-freedom (2-DOF) pivoted cylinder undergoing streamwise vortex-induced vibrations (VIV), which were carried out using particle-image velocimetry (PIV). The results are compared with those of previous studies performed in the same experimental facility examining a cylinder free to move only in the streamwise direction (1-DOF). The aim of this study is to examine to what extent the results of previous work on streamwise-only VIV can be extrapolated to the more practical, multi-DOF case. The response regimes measured for the 1- and 2-DOF cases are similar, containing two response branches separated by a low-amplitude region. The first branch is characterised by negligible transverse motion and the appearance of both alternate and symmetric vortex shedding. The two wake modes compete in an unsteady manner; however, the competition does not appear to have a significant effect on either the streamwise or transverse motion. Comparison of the phase-averaged vorticity fields acquired in the second response branch also indicates that the additional DOF does not alter the vortex-shedding process. However, the additional DOF affects the cylinder-wake system in other ways; for the 1-DOF case the second branch can appear in three different forms (each associated with a different wake mode), while for the 2-DOF case the second branch only exists in one form, and does not exhibit hysteresis. The cylinder follows a figure-of-eight trajectory throughout the lock-in range. The phase angle between the streamwise and transverse motion decreases linearly with reduced velocity. This work highlights the similarities and differences between the fluid–structure interaction and wake dynamics associated with 1- and 2-DOF cylinders throughout the streamwise response regime, which has not received attention to date.

In this article, we introduce techniques to build a reduced-order model of a fluid system that accurately predicts the dynamics of a flow from local wall measurements. This is particularly difficult in the case of noise amplifiers where the upstream noise environment, triggering the flow via a receptivity process, is not known. A system identification approach, rather than a classical Galerkin technique, is used to extract the model from time-synchronous velocity snapshots and wall shear-stress measurements. The technique will be illustrated for the case of a transitional flat-plate boundary layer, where the snapshots of the flow are obtained from direct numerical simulations. Particular attention is directed to limiting the processed data to data that would be readily available in experiments, thus making the technique applicable to an experimental set-up. The proposed approach combines a reduction of the degrees of freedom of the system by a projection of the velocity snapshots onto a proper orthogonal decomposition basis combined with a system identification technique to obtain a state-space model. This model is then used in a feedforward control set-up to significantly reduce the kinetic energy of the perturbation field and thus successfully delay transition.

Vortex stretching and compression, which cause enstrophy production by inviscid processes, are investigated near the turbulent/non-turbulent (T/NT) interface in a planar jet by using a direct numerical simulation (DNS). The enstrophy production is investigated by analysing the relationship among a vorticity vector, strain-rate eigenvectors and strain-rate eigenvalues. The statistics are calculated individually for three different interface orientations. The vorticity near the T/NT interface is oriented in the tangential direction to the interface. The enstrophy production is affected by the interface orientation because the intensity of vortex stretching depends on the interface orientation, and the alignment between the vorticity vector and the strain-rate eigenvectors is confined by the interface. The enstrophy production near the T/NT interface is analysed by considering the motion of turbulent fluid relative to that of the interface. The results show that the alignment between the interface and the strain-rate eigenvectors changes depending on the velocity field near the T/NT interface. When the turbulent fluid moves toward the T/NT interface, the enstrophy is generated by vortex stretching without being greatly affected by vortex compression. In contrast, when the turbulent fluid relatively moves away from the T/NT interface, large enstrophy reduction frequently occurs by vortex compression. Thus, it is shown that the velocity field near the T/NT interface affects the enstrophy production near the interface through the alignment between the vorticity and the strain-rate eigenvectors.

Recent directional solidification experiments with aqueous suspensions of alumina particles (Anderson & Worster, Langmuir, vol. 28 (48), 2012, pp. 16512–16523) motivate a model for freezing colloidal suspensions that builds upon a theoretical framework developed by Rempel et al. (J. Fluid Mech., vol. 498, 2004, pp. 227–244) in the context of freezing soils. Ice segregates from the suspension at slow freezing rates into discrete horizontal layers of particle-free ice, known as ice lenses. A portion of the particles is trapped between ice lenses, while the remainder are pushed ahead, forming a layer of fully compacted particles separated from the bulk suspension by a sharp compaction front. By dynamically modelling the compaction front, the growth kinetics of the ice lenses are fully coupled to the viscous flow through the evolving compacted layer. We examine the periodic states that develop at fixed freezing rates in a constant, uniform temperature gradient, and compare the results against experimental observations. Congruent with the experiments, three periodic regimes are identified. At low freezing rates, a regular periodic sequence of ice lenses is obtained; predictions for the compacted layer thickness and ice-lens characteristics as a function of freezing rate are consistent with experiments. At intermediate freezing rates, multiple modes of periodic ice lenses occur with a significantly diminished compacted layer. When the cohesion between the compacted particles is sufficiently strong, a sequence of mode-doubling bifurcations lead to chaos, which may explain the disordered ice lenses observed experimentally. Finally, beyond a critical freezing rate, the regime for sustained ice-lens growth breaks down. This breakdown is consistent with the emergence of a distinct regime of ice segregation found experimentally, which exhibits a periodic, banded structure that is qualitatively distinct from ice lenses.

We report measurements of logarithmic temperature profiles $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\varTheta (z,r) = A(r)\times \ln (z/L) + B(r)$ in the bulk of turbulent Rayleigh–Bénard convection (here $\varTheta $ is a scaled and time-averaged local temperature in the fluid, $ z$ is the vertical and $r$ the radial position, and $L$ is the sample height). Two samples had aspect ratios $\varGamma \equiv D/L = 1.00$ and 0.50 (where $D=190\ \mathrm{mm}$ is the diameter). The fluid was a fluorocarbon with a Prandtl number of $\mathit{Pr} = 12.3$. The measurements covered the Rayleigh-number range $2\times 10^{10} \lesssim \mathit{Ra} \lesssim 2\times 10^{11}$ for $\varGamma = 1.00$ and $3\times 10^{11} \lesssim \mathit{Ra} \lesssim 2\times 10^{12}$ for $\varGamma = 0.50$. In contradistinction to what had been found for $\varGamma = 0.50$ and $\mathit{Pr} = 0.78$ by Ahlers et al. (Phys. Rev. Lett., vol. 109, 2012, art. 114501; J. Fluid Mech., 2014, in press), the measurements revealed no $\mathit{Ra}$ dependence of the amplitude $A(r)$ of the logarithmic term. Within the experimental resolution, the amplitude was also found to be independent of $\varGamma $. It varied with $r$ in a manner consistent with the function $A(\xi ) = A_1/\sqrt{2\xi - \xi ^2}$, where $\xi \equiv (R-r)/R$ with $R=D/2$ and $A_1 \simeq 0.0016$. The results for $A(r)$ are smaller than those obtained from experiments and direct numerical simulations (Ahlers et al., Phys. Rev. Lett., vol. 109, 2012, art. 114501) at similar values of $\mathit{Ra}$ for $\mathit{Pr} = 0.7$ and $\varGamma = \frac{1}{2}$ by a factor that depended slightly upon $\mathit{Ra}$ but was close to $2$.

The effect of the Reynolds number on the behaviour of inertial particles in wall-bounded turbulent flows is investigated through large-scale direct numerical simulations (DNS) of particle-laden canonical channel flow spanning almost a decade in the friction Reynolds number, from $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_{\tau } = 150$ to $\mathit{Re}_{\tau } = 1000$. Lagrangian particle tracking is used to study the motion of six different particle sets, described by a Stokes number in the range $\mathit{St} = 1\text {--}1000$. At all Reynolds numbers a strong segregation in the near-wall region is observed for particles characterized by intermediate Stokes number, in the range $\mathit{St} =10\text {--}100$. The wall-normal concentration profiles of such particles collapse in inner scaling, thus suggesting the independence of the turbophoretic drift from the large-scale outer motions. This observation is also supported by the spatial organization of the suspended phase in the inner layer, which is found to be universal with the Reynolds number. The deposition rate coefficient increases with $\mathit{Re}_{\tau }$ for a given $\mathit{St}$. Suitable inner and outer scalings are proposed to collapse the deposition curves across the available ranges of Reynolds and Stokes numbers for the different deposition regimes.

Numerical simulations of the transverse dam-break waves (TDWs) produced by the sudden removal of a gate on the side of a waterway are conducted based on the shallow-water equations to find solutions to a family of water-diversion problems. The Froude numbers in the main flow identify the members of the family. The depth and discharge profiles are analysed in terms of Ritter’s similarity variable. For subcritical main flow, the waves are comprised of a supercritical flow expansion followed by a subcritical outflow. For supercritical main flow, on the other hand, the waves are analogous to the Prandtl–Meyer expansion in gas dynamics. The diversion flow rate of two-dimensional TDWs on a flat bed is 55 % greater than the one-dimensional flow rate of Ritter in the limiting case of zero main flow, and approaches the rate of Ritter in the other limit when the value of the Froude number in the main flow approaches infinity. The diversion flow rate over a weir is generally higher than the rate over a flat bed depending on the Froude number of the main flow. These numerical simulation results are consistent with laboratory observations.

The swimming behaviour of micro-organisms can be strongly influenced by the rheology of their fluid environment. In this article, we experimentally investigate the effects of shear-thinning (ST) viscosity on the swimming behaviour of an undulatory swimmer, the nematode Caenorhabditis elegans. Tracking methods are used to measure the swimmer’s kinematic data (including propulsion speed) and velocity fields. We find that ST viscosity modifies the velocity fields produced by the swimming nematode but does not modify the nematode’s speed and beating kinematics. Velocimetry data show significant enhancement in local vorticity and circulation and an increase in fluid velocity near the nematode’s tail. These findings are compared with recent theoretical and numerical results.

We investigate numerically transient linear growth of three-dimensional perturbations in a stratified shear layer to determine which perturbations optimize the growth of the total kinetic and potential energy over a range of finite target time intervals. The stratified shear layer has an initial parallel hyperbolic tangent velocity distribution with Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=U_0 h/\nu =1000$ and Prandtl number $\nu /\kappa =1$, where $\nu $ is the kinematic viscosity of the fluid and $\kappa $ is the diffusivity of the density. The initial stable buoyancy distribution has constant buoyancy frequency $N_0$, and we consider a range of flows with different bulk Richardson number ${\mathit{Ri}}_b=N_0^2h^2/U_0^2$, which also corresponds to the minimum gradient Richardson number ${\mathit{Ri}}_g(z)=N_0^2/(\mathrm{d}U/\mathrm{d} z)^2$ at the midpoint of the shear layer. For short target times, the optimal perturbations are inherently three-dimensional, while for sufficiently long target times and small ${\mathit{Ri}}_b$ the optimal perturbations are closely related to the normal-mode ‘Kelvin–Helmholtz’ (KH) instability, consistent with analogous calculations in an unstratified mixing layer recently reported by Arratia et al. (J. Fluid Mech., vol. 717, 2013, pp. 90–133). However, we demonstrate that non-trivial transient growth occurs even when the Richardson number is sufficiently high to stabilize all normal-mode instabilities, with the optimal perturbation exciting internal waves at some distance from the midpoint of the shear layer.