We study the effect of severe geometrical confinement in Rayleigh–Bénard convection with a wide range of width-to-height aspect ratio $\unicode[STIX]{x1D6E4}$, $1/128\leqslant \unicode[STIX]{x1D6E4}\leqslant 1$, and Rayleigh number $Ra$, $3\times 10^{4}\leqslant Ra\leqslant 1\times 10^{11}$, at a fixed Prandtl number of $Pr=4.38$ by means of direct numerical simulations in Cartesian geometry with no-slip walls. For convection under geometrical confinement (decreasing $\unicode[STIX]{x1D6E4}$ from 1), three regimes can be recognized (Chong et al., Phys. Rev. Lett., vol. 115, 2015, 264503) based on the global and local properties in terms of heat transport, plume morphology and flow structures. These are Regime I: classical boundary-layer-controlled regime; Regime II: plume-controlled regime; and Regime III: severely confined regime. The study reveals that the transition into Regime III leads to totally different heat and momentum transport scalings and flow topology from the classical regime. The convective heat transfer scaling, in terms of the Nusselt number $Nu$, exhibits the scaling $Nu-1\sim Ra^{0.61}$ over three decades of $Ra$ at $\unicode[STIX]{x1D6E4}=1/128$, which contrasts sharply with the classical scaling $Nu-1\sim Ra^{0.31}$ found at $\unicode[STIX]{x1D6E4}=1$. The flow in Regime III is found to be dominated by finger-like, long-lived plume columns, again in sharp contrast with the mushroom-like, fragmented thermal plumes typically observed in the classical regime. Moreover, we identify a Rayleigh number for regime transition, $Ra^{\ast }=(29.37/\unicode[STIX]{x1D6E4})^{3.23}$, such that the scaling transition in $Nu$ and $Re$ can be clearly demonstrated when plotted against $Ra/Ra^{\ast }$.

We experimentally study resonant interactions of oblique surface gravity waves in a large basin. Our results strongly extend previous experimental results performed mainly for perpendicular or collinear wave trains. We generate two oblique waves crossing at an acute angle, while we control their frequency ratio, steepnesses and directions. These mother waves mutually interact and give birth to a resonant wave whose properties (growth rate, resonant response curve and phase locking) are fully characterized. All our experimental results are found in good quantitative agreement with four-wave interaction theory with no fitting parameter. Off-resonance experiments are also reported and the relevant theoretical analysis is conducted and validated.

We consider a rapidly rotating, Boussinesq fluid stirred by buoyant anomalies. In such a system it is known that, in the absence of a magnetic field, inertial waves whose wave vectors lie normal to the rotation axis play a key role in establishing quasi-geostrophic motion. In particular, buoyant anomalies radiate low-frequency inertial wave packets which disperse along the rotation axis, leading to axially elongated columnar vortices. Here we focus on the influence of an ambient magnetic field on this process, motivated by the dynamics of planetary cores. We find that, once again, the waves responsible for establishing quasi-geostrophic structures have wave vectors normal to the rotation axis; however, these are not conventional inertial waves, but rather hybrid ‘inertial–Alfvén waves’. Their frequency equals that of an Alfvén wave but their axial group velocity is half that of the equivalent inertial wave. They have maximal kinetic, magnetic and cross-helicity, carry magnetic and kinetic energy in equal amounts, and are particularly potent in establishing columnar, helical vortices through the spontaneous emission of axially elongated wave packets. Although our hybrid inertial–Alfvén waves have been overlooked in dynamo literature to date, we speculate that they in fact play a central role in planetary dynamos.

Multi-particle dispersion is studied using direct numerical simulations of temporally evolving mixing layers and planar jets for tetrahedra consisting of four fluid particles which are seeded in the turbulent regions or in the non-turbulent regions near the turbulent/non-turbulent interface (TNTI). The modified Richardson law for decaying turbulence is observed for particle pairs. The size dependence of the mean and relative motions of the entrained tetrahedra indicates that the characteristic length scale of the entrained lumps of fluid is approximately 10 times the Kolmogorov microscale. When the tetrahedra move within the TNTI layer they are flattened and elongated by vortex stretching at a deformation rate that is characterized by the Kolmogorov time scale. The shape evolutions of the tetrahedra show that in free-shear flows, thin-slab structures of advected scalars are generated within the TNTI layers.

In their bipolar-coordinate analysis of circular-cylinder electrophoresis near a dielectric wall, Keh et al. (J. Fluid Mech., vol. 231, 1991, pp. 211–228) found that, when an electric field is applied parallel to the wall, the translational and rotational electrophoretic mobilities increase monotonically as the ratio $\unicode[STIX]{x1D6FF}$ of the cylinder–wall separation to the cylinder radius decreases, eventually diverging as $\unicode[STIX]{x1D6FF}^{-1/2}$ when $\unicode[STIX]{x1D6FF}\rightarrow 0$. Considering the singular limit $\unicode[STIX]{x1D6FF}\ll 1$ from the outset, we conduct here an asymptotic analysis of that electrokinetic problem, providing insight to the manner by which the intense electric field in the narrow gap is transformed into $O(\unicode[STIX]{x1D6FF}^{-3/2})$ shear stresses; these stresses, in turn, overcome the large Stokes resistance so as to provide the large electrophoretic mobilities. In a companion problem, where the cylinder is exposed to a uniform current emanating from a nearby reactive electrode, the intense gap-scale electric field results in an $O(\unicode[STIX]{x1D6FF}^{-2})$ pressure, giving rise in turn to a large repulsive force. In that problem we find that the cylinder velocity perpendicular to the wall approaches a finite limit as $\unicode[STIX]{x1D6FF}\rightarrow 0$. We also discuss the role of ‘dielectrophoretic’ forces which are inevitable in the above semi-bounded configurations.

A novel method to estimate unsteady aerodynamic force coefficients from pointwise velocity measurements is presented. As opposed to other existing methodologies, time-resolved full velocity fields are not required. The methodology is based on a resolvent-based reduced-order model which requires the mean flow to obtain physical flow structures and pointwise measurement to calibrate their amplitudes. A computationally affordable time-stepping methodology to obtain resolvent modes in non-trivial flow domains is introduced and compared with previous existing matrix-free and matrix-forming strategies. The technique is applied to the unsteady flow around an inclined square cylinder at low Reynolds number. The potential of the methodology is demonstrated through good agreement between the fluctuating pressure distribution on the cylinder and the temporal evolution of the unsteady lift and drag coefficients predicted by the model and those computed by direct numerical simulation.

Since their discovery by Yves Couder and Emmanuel Fort, droplets walking on a vibrating liquid bath have attracted considerable attention because they unexpectedly exhibit certain features reminiscent of quantum particles. While the behaviour of walking droplets in unbounded geometries has to a large extent been rationalized theoretically, no such rationale exists for their behaviour in the presence of boundaries, as arises in a number of key quantum analogue systems. We here present the results of a combined experimental and theoretical study of the interaction of walking droplets with a submerged planar barrier. Droplets exhibit non-specular reflection, with a small range of reflection angles that is only weakly dependent on the system parameters, including the angle of incidence. The observed behaviour is captured by simulations based on a theoretical model that treats the boundaries as regions of reduced wave speed, and rationalized in terms of momentum considerations.

Direct numerical simulations (DNS) are performed to study the behaviour of flow past a sphere in the regime of high stratification (low Froude number $Fr$). In contrast to previous results at lower Reynolds numbers, which suggest monotone suppression of turbulence with increasing stratification in flow past a sphere, it is found that, below a critical $Fr$, increasing the stratification induces unsteady vortical motion and turbulent fluctuations in the near wake. The near wake is quantified by computing the energy spectra, the turbulence energy equation, the partition of energy into horizontal and vertical components, and the buoyancy Reynolds number. These diagnostics show that the stabilizing effect of buoyancy changes flow over the sphere to flow around the sphere. This qualitative change in the flow leads to a new regime of unsteady vortex shedding in the horizontal planes and intensified horizontal shear which result in turbulence regeneration.

A laboratory investigation was performed to study distinctive features of the laminar-to-turbulent transition over distributed roughness characterized by two-dimensional (2D) and three-dimensional (3D) periodic, low-order topographies at roughness Reynolds number $Re_{k}\approx 300$. Systematic experiments were performed using high-spatial-resolution planar particle image velocimetry (PIV) in a refractive-index-matching (RIM) channel, where the roughness covered the entire length of the test section. The results show that the flow over the 2D roughness becomes turbulent much sooner than its 3D counterpart ($Re_{x}=50\,000$ versus 120 000). This is attributed to the presence of a velocity inflection point resulting from flow separation within the troughs of the 2D roughness. In the transitional region, unsteady disturbances above the two roughnesses appear upstream of near-roughness disturbances. The above-roughness disturbances are associated with the inflection point in the vertically displaced boundary layer for the 2D case, and with the mean velocity deficit resulting from the interaction of the wakes of upstream elements for the 3D case. The near-roughness fluctuations are associated with the shear layer present behind the crests of both roughnesses. The transitional region is characterized by the interaction between above- and near-roughness disturbances, which merge, leading to a rapid vertical growth of the turbulent fluctuations.

In this paper we present an exact and explicit nonlinear solution of a $\unicode[STIX]{x1D6FD}$-plane approximation to the governing equations which retains all Coriolis terms. The solution represents an equatorially trapped wave propagating in the presence of a constant underlying background current. In particular, we show that retention of the (relatively) small-scale centripetal forces in the governing equations enables us to admit currents of any physically plausible magnitude in the background flow.

We describe new experiments in which a bubble plume, produced from a point source of bubbles, rises through an ambient fluid composed of two layers of fluid of different density. In the lower layer, the speed of the plume exceeds the bubble rise speed and the motion is well described using classical theory of turbulent buoyant plumes. As the mixture enters the upper layer, it is either buoyant and rises to the top of the layer, or is negatively buoyant and forms a fountain. In our experiments, in which a fountain forms in the upper layer, the bubble rise speed exceeds the characteristic speed of this fountain, and a separated flow develops. The bubbles rise to the top of the system, while the lower layer fluid in the fountain rises a finite distance into the upper layer, entrains some of the upper layer fluid, and then collapses. This mixture of fluids then feeds a growing layer of density which is intermediate between the upper and lower layer. The height of rise of the fountain scales with the square of the Froude number of the fountain and the rate of entrainment of upper layer fluid into the fountain is directly proportional to the height of the fountain. This is analogous to the scaling for single-phase fountains with Froude numbers in the same range, $1<Fr<7$, but the constants of proportionality are smaller. We illustrate the relevance of the work for the design of mixing and aeration systems in freshwater reservoirs.

The effect of differential diffusion of two passive scalars having Schmidt numbers of unity and 0.25, respectively, is investigated using direct numerical simulation of a temporally evolving jet. The objective of the research is twofold: (i) to compare the turbulent/non-turbulent (T/NT) interface position using the scalar criterion between the unity- and low-Schmidt-number scalar; and (ii) to determine the impact of the T/NT interface on differential diffusion. For the latter, the T/NT interface is detected using the vorticity criterion. To quantify the effect of differential diffusion, a normalised differential diffusion parameter is analysed, clearly showing the dominance of differential diffusion at the T/NT interface. A transport equation for the scalar differences is then evaluated, which shows that differential diffusion originates at the interface. Further, the separation between the passive scalars, arising due to differential diffusion, is studied using conventional and conditional statistics with respect to the interface distance. Since differential diffusion is known to be present at large and small scales, the connection between them is analysed using the scalar dissipation rate. Moreover, the physical mechanism responsible for the departure of the two scalars is analysed using the scalar gradient alignment, the ratio of the diffusive fluxes and by a transport equation for the scalar gradients.

The subgrid-scale (SGS) Prandtl number ($Pr$) is an important parameter in large-eddy simulation. Prior models often assume that the ‘$-5/3$’ inertial subrange scaling applies to the wavenumber range from 0 to $k_{\unicode[STIX]{x1D6E5}}$ (the wavenumber corresponding to the filter scale $\unicode[STIX]{x1D6E5}$) and yield a $Pr$ that is stability-independent and scale-invariant, which is inconsistent with experimental data and the results of dynamic models. In this study, the SGS Prandtl number is revisited by solving the co-spectral budgets of momentum and heat fluxes in an idealized but thermally stratified atmospheric surface layer. The SGS Prandtl number from the co-spectral budget model shows a strong dependence on the atmospheric stability and increases (decreases) as the atmosphere becomes stable (unstable), which is in good agreement with recent field experimental data. The dependence of $Pr$ on the filter scale is also captured by the co-spectral budget model: as the filter scale becomes smaller, the SGS Prandtl number decreases. Finally, the value of SGS Prandtl number under neutral conditions is shown to be caused by the dissimilarity between momentum and heat in the pressure decorrelation term and the flux transfer term. When the dissimilarity exists only in the flux transfer term, the fact that under neutral conditions the SGS Prandtl number is usually smaller than the turbulent Prandtl number for Reynolds-averaged Navier–Stokes simulations is an indication of a stronger spectral transfer coefficient for heat than for momentum. The model proposed in this study can be readily implemented.

Vortex reconnection, as the topological change of vortex lines or surfaces, is a critical process in transitional flows, but is challenging to accurately characterize, particularly in shear flows. We apply the vortex-surface field (VSF), whose isosurface is the vortex surface consisting of vortex lines, to study vortex reconnection in the Klebanoff-type temporal transition in channel flow. The VSF evolution can capture the reconnection of the hairpin-like vortical structures evolving from the initial vortex sheets in opposite halves of the channel. The incipient vortex reconnection is characterized by the vanishing minimum distance between a pair of vortex surfaces and the reduction of vorticity flux through the region enclosed by the wall and the VSF isoline of the channel half-height on the spanwise symmetric plane. We find that the surge of the wall-friction coefficient begins at the identified reconnection time. From the Biot–Savart law, the rapid reconnection of vortex lines can induce a velocity opposed to the mean flow, which partially blocks the flow near the central region and generally accelerates the near-wall fluid motion in the flow with constant mass flux. Therefore, the vortex reconnection appears to play an important role in the sudden increase of wall friction in transitional channel flows.

Understanding the origin of large-scale structures in high-Reynolds-number wall turbulence has been a central issue over a number of years. Recently, Rawat et al. (J. Fluid Mech., vol. 782, 2015, pp. 515–540) have computed invariant solutions for the large-scale structures in turbulent Couette flow at $Re_{\unicode[STIX]{x1D70F}}\simeq 128$ using an overdamped large-eddy simulation with the Smagorinsky model to account for the effect of the surrounding small-scale motions. Here, we extend this approach to Reynolds numbers an order of magnitude higher in turbulent channel flow, towards the regime where the large-scale structures in the form of very-large-scale motions (long streaky motions) and large-scale motions (short vortical structures) emerge energetically. We demonstrate that a set of invariant solutions can be computed from simulations of the self-sustaining large-scale structures in the minimal unit (domain of size $L_{x}=3.0h$ streamwise and $L_{z}=1.5h$ spanwise) with midplane reflection symmetry at least up to $Re_{\unicode[STIX]{x1D70F}}\simeq 1000$. By approximating the surrounding small scales with an artificially elevated Smagorinsky constant, a set of equilibrium states are found, labelled upper- and lower-branch according to their associated drag. It is shown that the upper-branch equilibrium state is a reasonable proxy for the spatial structure and the turbulent statistics of the self-sustaining large-scale structures.

The concept of edge states is investigated in the asymptotic suction boundary layer in relation to the receptivity process to noisy perturbations and the nucleation of turbulent spots. Edge tracking is first performed numerically, without imposing any discrete symmetry, in a large computational domain allowing for full spatial localisation of the perturbation velocity. The edge state is a three-dimensional localised structure recurrently characterised by a single low-speed streak that experiences erratic bursts and planar shifts. This recurrent streaky structure is then compared with predecessors of individual spot nucleation events, triggered by non-localised initial noise. The present results suggest a nonlinear picture, rooted in dynamical systems theory, of the nucleation process of turbulent spots in boundary-layer flows, in which the localised edge state plays the role of state-space mediator.

The mechanism of trailing vortex wandering has long been debated and is often attributed to either wind-tunnel effects or an instability. Using particle image velocimetry data obtained in the wake of a NACA0012 airfoil, we remove the effect of wandering from the measured velocity field and, through a triple decomposition, recover the coherent wandering motion. Based on this wandering motion, the most energetic structures are computed using the proper orthogonal decomposition (POD) and exhibit a helical mode with an azimuthal wavenumber of $|m|=1$ whose kinetic energy grows monotonically in the downstream direction. To investigate the nature of the vortex wandering, we perform a spatial stability analysis of a matched Batchelor vortex. The primary stability mode is found to be marginally stable and nearly identical in both size and structure to the leading POD mode. The strikingly similar structure, coupled with the measured energy growth, supports the proposition that the vortex wandering is the result of an instability. We conclude that the cause of the wandering is the non-zero radial velocity of the $|m|=1$ mode on the vortex centreline, which acts to transversely displace the trailing vortex, as observed in experiments. However, the marginal nature of the stability mode prevents a definitive conclusion regarding the specific type of instability.

We present results of numerical simulations of leading-edge acoustic receptivity for acoustic waves impinging on the leading edge of a finite-thickness flat plate. We use both compressible and incompressible flow solvers fitted with high-order high-accuracy numerical methods and independent methods of estimating the receptivity coefficient. The results show that the level of acoustic receptivity in the existing literature appears to be one order of magnitude too high. Our review of previous numerical simulations and experiments clearly identifies some contradictory trends. In the limit of an infinitely thin flat plate, our results are consistent with asymptotic theory and numerical simulations.

Leading-edge vortices (LEVs) can form and remain attached to a rotating wing indefinitely, but the mechanisms of stable attachment are not well understood. Taking for granted that such stable structures do form, a practical question arises: what is the trajectory of the LEV core? Noting that span-wise flow exists within the LEV core, it is apparent that a mean streamline aligned with the axis of the stable LEV must exist. The present work uses the Navier–Stokes equations along such a steady, axial streamline in order to consider the accelerations that act in the streamline-normal direction to affect its local curvature. With some simplifying assumptions, a coupled system of ordinary differential equations is derived that describes the trajectory of an axial streamline through the vortex core. The model is compared to previous work, and is found to predict the trajectory of the LEV core well at span-wise locations inboard of the midspan. This result suggests that Coriolis acceleration is responsible for limiting the span-wise extent of a stable LEV by tilting it into the wake within several chord lengths from the centre of rotation. The downwash due to the tip vortex also appears to play a role, as the only significant differences between model-predicted LEV trajectories and previous results are in the plate-normal direction.

We consider dynamo action driven by three-dimensional rotating anelastic convection in a spherical shell. Motivated by the behaviour of the solar dynamo, we examine the interaction of hydromagnetic modes with different symmetries and demonstrate how complicated interactions between convection, differential rotation and magnetic fields may lead to modulation of the basic cycle. For some parameters, type 1 modulation occurs by the transfer of energy between modes of different symmetries with little change in the overall amplitude; for other parameters, the modulation is of type 2, where the amplitude is significantly affected (leading to grand minima in activity) without significant changes in symmetry. Most importantly, we identify the presence of ‘supermodulation’ in the solutions, where the activity switches chaotically between type 1 and type 2 modulation; this is believed to be an important process in solar activity.