Properties of single- and two-point moment generating functions (MGFs) are examined in the inertial region of wall-bounded flows. Empirical evidence for power-law scaling of the single-point MGF $\langle \text{exp}(qu^{+})\rangle$ (where $u^{+}$ is the normalized streamwise velocity fluctuation and $q$ a real parameter) with respect to the wall-normal distance is presented, based on hot-wire data from a $Re_{{\it\tau}}=13\,000$ boundary-layer experiment. The parameter $q$ serves as a ‘dial’ to emphasize different parts of the signal such as high- and low-speed regions, for positive and negative values of $q$ , respectively. Power-law scaling $\langle \text{exp}(qu^{+})\rangle \sim (z/{\it\delta})^{-{\it\tau}(q)}$ can be related to the generalized logarithmic laws previously observed in higher-order moments, such as in $\langle u^{+2p}\rangle ^{1/p}$ , but provide additional information not available through traditional moments if considering $q$ values away from the origin. For two-point MGFs, the scalings in $\langle \text{exp}[qu^{+}(x)+q^{\prime }u^{+}(x+r)]\rangle$ with respect to $z$ and streamwise displacement $r$ in the logarithmic region are investigated. The special case $q^{\prime }=-q$ is of particular interest, since this choice emphasizes rare events with high and low speeds at a distance $r$ . Applying simple scaling arguments motivated by the attached eddy model, a ‘scaling transition’ is predicted to occur for $q=q_{cr}$ such that ${\it\tau}(q_{cr})+{\it\tau}(-q_{cr})=1$ , where ${\it\tau}(q)$ is the set of scaling exponents for single-point MGFs. This scaling transition is not visible to traditional central moments, but is indeed observed based on the experimental data, illustrating the capabilities of MGFs to provide new and statistically robust insights into turbulence structure and confirming essential ingredients of the attached eddy model.

A numerical study of the onset of convection in rotating internally heated self-gravitating fluid spheres is presented. The exploration of the stability of the conduction state versus the Taylor and Prandtl numbers supplies a detailed idea of the laws that fulfil the four types of solutions obtained at low Prandtl numbers. The main result found is that axisymmetric (torsional) modes of convection are preferred at high Taylor numbers in the zero-Prandtl-number limit. This instability appears at low Rayleigh numbers and gives rise to an oscillating single vortex of very high frequency.

By means of direct numerical simulations (DNS) we investigate the effect of a tilt angle ${\it\beta}$ , $0\leqslant {\it\beta}\leqslant {\rm\pi}/2$ , of a Rayleigh–Bénard convection (RBC) cell of aspect ratio 1, on the Nusselt number $\mathit{Nu}$ and Reynolds number $\mathit{Re}$ . The considered Rayleigh numbers $\mathit{Ra}$ range from $10^{6}$ to $10^{8}$ , the Prandtl numbers range from 0.1 to 100 and the total number of the studied cases is 108. We show that the $\mathit{Nu}\,({\it\beta})/\mathit{Nu}(0)$ dependence is not universal and is strongly influenced by a combination of $\mathit{Ra}$ and $\mathit{Pr}$ . Thus, with a small inclination ${\it\beta}$ of the RBC cell, the Nusselt number can decrease or increase, compared to that in the RBC case, for large and small $\mathit{Pr}$ , respectively. A slight cell tilt may not only stabilize the plane of the large-scale circulation (LSC) but can also enforce an LSC for cases when the preferred state in the perfect RBC case is not an LSC but a more complicated multiple-roll state. Close to ${\it\beta}={\rm\pi}/2$ , $\mathit{Nu}$ and $\mathit{Re}$ decrease with increasing ${\it\beta}$ in all considered cases. Generally, the $\mathit{Nu}({\it\beta})/\mathit{Nu}(0)$ dependence is a complicated, non-monotonic function of ${\it\beta}$ .

We present an experimental and theoretical study of the dynamics of laterally confined marine ice sheets in the natural limit in which the long, narrow channel into which they flow is wider than the depth of the ice. A marine ice sheet comprises a grounded ice sheet in contact with bedrock that floats away from the bedrock at a‘grounding line’ to form a floating ice shelf. We model the grounded ice sheet as a viscous gravity current resisted dominantly by vertical shear stresses owing to the no-slip boundary condition applied at the bedrock. We model the ice shelf as a floating viscous current resisted dominantly by horizontal shear stresses owing to no-slip boundary conditions applied at the sidewalls of the channel. The two shear-dominated regions are coupled by jump conditions relating force and fluid flux across a short transition region downstream of the grounding line. We find that the influence of the stresses within the transition region becomes negligible at long times and we model the transition region as a singular interface across which the ice thickness and mass flux can be discontinuous. The confined shelf buttresses the sheet, causing the grounding line to advance more than it would otherwise. In the case that the sheet flows on a base of uniform slope, we find asymptotically that the grounding line advances indefinitely as $t^{1/3}$ , where $t$ is time. This contrasts with the two-dimensional counterpart, for which the shelf provides no buttressing and the grounding line reaches a steady state (Robison, J. Fluid Mech., vol. 648, 2010, pp. 363–380).

The anisotropy of the slip length and its effect on the skin-friction drag are numerically investigated for a turbulent channel flow with an idealized superhydrophobic surface having an air layer, where the idealized air–water interface is flat and does not contain the surface-tension effect. Inside the air layer, both the shear-driven flow and recirculating flow with zero net mass flow rate are considered. With increasing air-layer thickness, the slip length, slip velocity and percentage of drag reduction increase. It is shown that the slip length is independent of the water flow and depends only on the air-layer geometry. The amount of drag reduction obtained is in between those by the empirical formulae from the streamwise slip only and isotropic slip, indicating that the present air–water interface generates an anisotropic slip, and the streamwise slip length ( $b_{x}$ ) is larger than the spanwise one ( $b_{z}$ ). From the joint probability density function of the slip velocities and velocity gradients at the interface, we confirm the anisotropy of the slip lengths and obtain their relative magnitude ( $b_{x}/b_{z}=4$ ) for the present idealized superhydrophobic surface. It is also shown that the Navier slip model is valid only in the mean sense, and it is generally not applicable to fluctuating quantities.

We revisit, both numerically and analytically, the finite-time blowup of the infinite-energy solution of 3D Euler equations of stagnation-point type introduced by Gibbon et al. (Physica D, vol. 132, 1999, pp. 497–510). By employing the method of mapping to regular systems, presented by Bustamante (Physica D, vol. 240 (13), 2011, pp. 1092–1099) and extended to the symmetry-plane case by Mulungye et al. (J. Fluid Mech., vol. 771, 2015, pp. 468–502), we establish a curious property of this solution that was not observed in early studies: before but near singularity time, the blowup goes from a fast transient to a slower regime that is well resolved spectrally, even at mid-resolutions of $512^{2}.$ This late-time regime has an atypical spectrum: it is Gaussian rather than exponential in the wavenumbers. The analyticity-strip width decays to zero in a finite time, albeit so slowly that it remains well above the collocation-point scale for all simulation times $t<T^{\ast }-10^{-9000}$ , where $T^{\ast }$ is the singularity time. Reaching such a proximity to singularity time is not possible in the original temporal variable, because floating-point double precision ( ${\approx}10^{-16}$ ) creates a ‘machine-epsilon’ barrier. Due to this limitation on the original independent variable, the mapped variables now provide an improved assessment of the relevant blowup quantities, crucially with acceptable accuracy at an unprecedented closeness to the singularity time: $T^{\ast }-t\approx 10^{-140}$ .

When subject to sufficiently strong electric fields, particles and drops suspended in a weakly conducting liquid exhibit spontaneous rotary motion. This so-called Quincke rotation is a fascinating example of nonlinear symmetry-breaking phenomena. To illuminate the rotation of liquid drops we here analyse the asymptotic limit of large electric Reynolds numbers, $\mathit{Re}\gg 1$ , within the framework of a two-dimensional Taylor–Melcher electrohydrodynamic model. A non-trivial dominant balance in this singular limit results in both the fluid velocity and surface-charge density scaling as $\mathit{Re}^{-1/2}$ . The flow is governed by a self-contained nonlinear boundary-value problem that does not admit a continuous fore–aft symmetric solution, thus necessitating drop rotation. Furthermore, thermodynamic arguments reveal that a fore–aft asymmetric solution exists only when charge relaxation within the suspending liquid is faster than that in the drop. The flow problem possesses both mirror-image (with respect to the direction of the external field) and flow-reversal symmetries; it is transformed into a universal one, independent of the ratios of electric conductivities and dielectric permittivities in the respective drop phase and suspending liquid phase. The rescaled angular velocity is found to depend weakly upon the viscosity ratio. The corresponding numerical solutions of the exact equations indeed collapse at large $\mathit{Re}$ upon the asymptotically calculated universal solution.

The propagation of wave disturbances in water of uniform depth is discussed, accounting for both gravity and compressibility effects. In the linear theory, free-surface (gravity) waves are virtually decoupled from acoustic (compression) waves, because the speed of sound in water far exceeds the maximum phase speed of gravity waves. However, these two types of wave motion could exchange energy via resonant triad nonlinear interactions. This scenario is analysed for triads comprising a long-crested acoustic mode and two oppositely propagating subharmonic gravity waves. Owing to the disparity of the gravity and acoustic length scales, the interaction time scale is longer than that of a standard resonant triad, and the appropriate amplitude evolution equations, apart from the usual quadratic interaction terms, also involve certain cubic terms. Nevertheless, it is still possible for monochromatic wavetrains to form finely tuned triads, such that nearly all the energy initially in the gravity waves is transferred to the acoustic mode. This coupling mechanism, however, is far less effective for locally confined wavepackets.

In this paper, we report the results of an experimental sensitivity analysis on a thermoacoustic system – an electrically heated Rijke tube. We measure the change of the linear stability characteristics of the system, quantified as shifts in the growth rate and oscillation frequency, that is caused by the introduction of a passive control device. The control device is a mesh, which causes drag in the system. The rate of growth is slow, so the growth rate and frequency can be measured very accurately over many hundreds of cycles in the linear regime with and without control. These measurements agree qualitatively well with the theoretical predictions from adjoint-based methods of Magri & Juniper (J. Fluid Mech., vol. 719, 2013, pp. 183–202). This agreement supports the use of adjoint methods for the development and implementation of control strategies for more complex thermoacoustic systems.

We investigate how compressibility affects the turbulent statistics from a Lagrangian point of view, particularly in the parameter range where the flow transits from the incompressible type to a state dominated by shocklets. A series of three-dimensional simulations were conducted for different types of driving and several Mach numbers. For purely solenoidal driving, as the Mach number increases a new self-similar region first emerges in the Lagrangian structure functions at sub-Kolmogorov time scale and gradually extends to larger time scale. In this region the relative scaling exponent saturates and the saturated value decreases as the compressibility becomes stronger, which can be attributed to the shocklets. The scaling exponent for the inertial range is still very close to that of incompressible turbulence for small Mach number, and discrepancy becomes visible when the Mach number is high enough. When the driving force is dominated by the compressive component the shocklet-induced self-similar region occupies a much wider range of time scales than that in the purely solenoidal driving case. Regardless of the type of driving force, the probability density functions of the velocity increment collapse onto one another for the time scales in the new self-similar region after proper normalization.

A simple geometry that exhibits near motion trapping is tested experimentally, along with perturbed versions of the structure. The motion of the freely floating structure and the surrounding wave field is tracked and the near-motion-trapped mode is found, characterised by a slowly decaying heave motion with very small linear radiation of energy. It is found that the latter property is a better discriminator of the perturbed geometries as viscous damping masks fine differences in radiation damping as far as the motion of the structure is concerned. The magnitude of this viscous damping is reasonably well predicted by a simple Stokes oscillatory boundary layer analysis.

The maximum spreading of drops impacting on smooth and rough surfaces is measured from low to high impact velocity for liquids with different surface tensions and viscosities. We demonstrate that dynamic wetting plays an important role in the spreading at low velocity, characterized by the dynamic contact angle at maximum spreading. In the energy balance, we account for the dynamic wettability by introducing the capillary energy at zero impact velocity, which relates to the spreading ratio at zero impact velocity. Correcting the measured spreading ratio by the spreading ratio at zero velocity, we find a correct scaling behaviour for low and high impact velocity and, by interpolation between the two, we find a universal scaling curve. The influence of the liquid as well as the nature and roughness of the surface are taken into account properly by rescaling with the spreading ratio at zero velocity, which, as demonstrated, is equivalent to accounting for the dynamic contact angle.

Suspended particles migrate towards inertial focusing positions close to walls and align into trains in finite inertia conduit flow. The relative contribution of inertial and viscous forces at the particle length scale, defined by the particle Reynolds number ( $\mathit{Re}_{p}$ ), is a key parameter, where $\mathit{Re}_{p}=\langle \dot{\unicode[STIX]{x1D6FE}}\rangle D^{2}/\unicode[STIX]{x1D708}$ depends on the mean shear rate $\langle \dot{\unicode[STIX]{x1D6FE}}\rangle$ , particle diameter $D$ and fluid kinematic viscosity  $\unicode[STIX]{x1D708}$ . Controlling the location of inertial focusing positions and the interparticle distance is critical in applications such as flow cytometry, imaging and cell entrapment in droplets. By using experimental observations in rectangular microchannels and lattice Boltzmann numerical simulations of dilute suspension flow, the spacing between particles aligned in trains is measured. From the modes of the probability density function of interparticle spacing, preferred spacings at $5D$ and $2.5D$ are observed. At lower $\mathit{Re}_{p}$ , the preferred spacing forms around $5D$ , and with increasing $\mathit{Re}_{p}$ the spacing at $2.5D$ becomes more pronounced. With increasing concentration of the suspension the spacing is influenced by particle crowding effects until stable trains are no longer observed.

The onset of monami – the synchronous waving of seagrass beds driven by a steady flow – is modelled as a linear instability of the flow. Unlike previous works, our model considers the drag exerted by the grass in establishing the steady flow profile, and in damping out perturbations to it. We find two distinct modes of instability, which we label modes 1 and 2. Mode 1 is closely related to Kelvin–Helmholtz instability modified by vegetation drag, whereas mode 2 is unrelated to Kelvin–Helmholtz instability and arises from an interaction between the flow in the vegetated and unvegetated layers. The vegetation damping, according to our model, leads to a finite threshold flow for both of these modes. Experimental observations for the onset and frequency of waving compare well with model predictions for the instability onset criteria and the imaginary part of the complex growth rate respectively, but experiments lie in a parameter regime where the two modes can not be distinguished.

Prior modal stability analysis (Kojima et al., Phys. Fluids, vol. 27, 1984, pp. 19–32) predicted that a rising or sedimenting droplet in a viscous fluid is stable in the presence of surface tension no matter how small, in contrast to experimental and numerical results. By performing a non-modal stability analysis, we demonstrate the potential for transient growth of the interfacial energy of a rising droplet in the limit of inertialess Stokes equations. The predicted critical capillary numbers for transient growth agree well with those for unstable shape evolution of droplets found in the direct numerical simulations of Koh & Leal (Phys. Fluids, vol. 1, 1989, pp. 1309–1313). Boundary integral simulations are used to delineate the critical amplitude of the most destabilizing perturbations. The critical amplitude is negatively correlated with the linear optimal energy growth, implying that the transient growth is responsible for reducing the necessary perturbation amplitude required to escape the basin of attraction of the spherical solution.

The wake behind a circular cylinder in Mach 4 flow is examined experimentally in the Reynolds number range $2\times 10^{4}$ to $5\times 10^{5}$ . Periodic oscillations of the sliplines in the wake are observed. The Strouhal number of the oscillations based on the diameter of the cylinder is found to increase monotonically from 0.30 to 0.50 with increasing Reynolds number. If the Strouhal number is formed using the length of the sliplines, however, it has a constant value of approximately 0.48 for all Reynolds numbers studied. This scaling indicates that the oscillations in supersonic flow are likely driven by acoustic signals propagating back and forth through the subsonic region between the separation points on the cylinder and the neck where the sliplines converge, unlike in subsonic flow where oscillations are caused by vortices shed from the cylinder surface.

When a drop impacts onto a solid surface, the lubrication pressure in the air deforms its bottom into a dimple. This makes the initial contact with the substrate occur not at a point but along a ring, thereby entrapping a central disc of air. We use ultra-high-speed imaging, with 200 ns time resolution, to observe the structure of this first contact between the liquid and a smooth solid surface. For a water drop impacting onto regular glass we observe a ring of microbubbles, due to multiple initial contacts just before the formation of the fully wetted outer section. These contacts are spaced by a few microns and quickly grow in size until they meet, thereby leaving behind a ring of microbubbles marking the original air-disc diameter. On the other hand, no microbubbles are left behind when the drop impacts onto molecularly smooth mica sheets. We thereby conclude that the localized contacts are due to nanometric roughness of the glass surface, and the presence of the microbubbles can therefore distinguish between glass with 10 nm roughness and perfectly smooth glass. We contrast this entrapment topology with the initial contact of a drop impacting onto a film of extremely viscous immiscible liquid, where the initial contact appears to be continuous along the ring. Here, an azimuthal instability occurs during the rapid contraction at the triple line, also leaving behind microbubbles. For low impact velocities the nature of the initial contact changes to one initiated by ruptures of a thin lubricating air film.

We present a general theory for determining the force (and torque) exerted on a boundary (or body) in active matter. The theory extends the description of passive Brownian colloids to self-propelled active particles and applies for all ratios of the thermal energy $k_{B}T$ to the swimmer’s activity $k_{s}T_{s}={\it\zeta}U_{0}^{2}{\it\tau}_{R}/6$ , where ${\it\zeta}$ is the Stokes drag coefficient, $U_{0}$ is the swim speed and ${\it\tau}_{R}$ is the reorientation time of the active particles. The theory, which is valid on all length and time scales, has a natural microscopic length scale over which concentration and orientation distributions are confined near boundaries, but the microscopic length does not appear in the force. The swim pressure emerges naturally and dominates the behaviour when the boundary size is large compared to the swimmer’s run length $\ell =U_{0}{\it\tau}_{R}$ . The theory is used to predict the motion of bodies of all sizes immersed in active matter.

Laboratory experiments involving downward air–helium fountains are presented. The large density differences between these releases and the ambient allow us to investigate how non-Boussinesq effects modify fountain heights and fountain fluctuations in comparison with the Boussinesq case (i.e. marginal density differences). In these experiments, the source Froude number is varied over a wide range covering (i) the very weak, (ii) the weak and (iii) the forced fountain regimes. It is shown that the classical Boussinesq correlations can be extended to the non-Boussinesq case provided that the Froude number is multiplied by the square root of the ratio between the released fluid density and that of the ambient. In the range investigated, no influence of the source Reynolds number is observed.

Enhancement of heat transport across a fluid layer is of fundamental interest as well as great technological importance. For decades, Rayleigh–Bénard convection has been a paradigm for the study of convective heat transport, and how to improve its overall heat-transfer efficiency is still an open question. Here, we report an experimental and numerical study that reveals a novel mechanism that leads to much enhanced heat transport. When vertical partitions are inserted into a convection cell with thin gaps left open between the partition walls and the cooling/heating plates, it is found that the convective flow becomes self-organized and more coherent, leading to an unprecedented heat-transport enhancement. In particular, our experiments show that with six partition walls inserted, the heat flux can be increased by approximately 30 %. Numerical simulations show a remarkable heat-flux enhancement of up to 2.3 times (with 28 partition walls) that without any partitions.