Energetic benefit and enhanced performance are considered among the most fascinating achievements of collective behaviours, e.g. fish schools and flying formations. The collective locomotion of two self-propelled flapping plates initially in a side-by-side arrangement is investigated numerically. Both in-phase and antiphase oscillations for the two plates are considered. It is found that the plates will spontaneously form some stable configurations as a result of the flow-mediated interaction, specifically, the staggered-following (SF) mode and the alternate-leading (AL) mode for the in-phase scenario and the moving abreast (MA) mode and the AL mode for the antiphase scenario. In the SF mode, the rear plate follows the front one with a staggered configuration. In the AL mode, the plates chase each other side-by-side alternately. In terms of propulsive speed and efficiency, the performance of the plates in the SF mode with small lateral spacing $H$ is found to be better than those in the tandem following case ($H=0$) and the side-by-side case (i.e. the AL mode). To achieve higher propulsive efficiency, no matter in-phase or antiphase oscillations, the two plates with moderate bending stiffness, e.g. $K\approx O(1)$, are preferred and they should be close enough in the lateral direction. For the side-by-side configuration, the performance of each plate in the antiphase and in-phase scenarios is enhanced and weakened in comparison with that of the isolated plate, respectively. Besides the pressure and vorticity contours, the normal force and thrust acting on the plates are also analysed. It is revealed that the thrust is mainly contributed by the normal force at moderate bending stiffness. The normal force and thrust are critical to the propulsive speed and efficiency. For two self-propelled plates, in view of hydrodynamics, to achieve higher performance the in-phase SF mode and antiphase flappings in the side-by-side configuration are preferred.

Scaling characteristics, essential to the mechanisms of transition in square-prism shear layers, were explored experimentally. In particular, the evolution of the dominant instability modes as a function of Reynolds number were reported in the range $1.5\times 10^{4}\lesssim Re_{D}\lesssim 7.5\times 10^{4}$. It was found that the ratio between the shear layer frequency and the shedding frequency obeys a power-law scaling relation. Adherence to the power-law relationship, which was derived from hot-wire measurements, has been supported by two additional and independent scaling considerations, namely, by particle image velocimetry measurements to observe the evolution of length and velocity scales in the shear layer during transition, and by comparison to direct numerical simulations to illuminate the properties of the front-face boundary layer. The nonlinear dependence of the shear layer instability frequency is sustained by the influence of $Re_{D}$ on the thickness of the laminar front-face boundary layer. In corroboration with the original scaling argument for the circular cylinder, the length scale of the shear layer was the only source of nonlinearity in the frequency ratio scaling, within the range of Reynolds numbers reported. The frequency ratio scaling may therefore be understood by the influence of $Re_{D}$ on the appropriate length scale of the shear layer. This length scale was observed to be the momentum thickness evaluated at a transition point, defined where the Kelvin–Helmholtz instability saturates.

Laboratory experiments with a bottom hinged surface-piercing cylinder, exposed to irregular deep water waves, are used to investigate high-frequency forcing. The focus is on the secondary load cycle, a strongly nonlinear phenomenon regarding the wave load on a vertical cylinder, first identified by Grue et al. (1993 Preprint Series. Mechanics and Applied Mathematics, pp. 1–30. University of Oslo, available at http://urn.nb.no/URN:NBN:no-52740; 1994 Ninth International Workshop on Water Waves and Floating Bodies (ed. M. Ohkusu), pp. 77–81, available at http://iwwwfb.org). For a total of 2166 single wave events, the force above $3\unicode[STIX]{x1D714}$ (where $\unicode[STIX]{x1D714}$ is the governing wave frequency) is used to identify and split the strongly nonlinear forces into two peaks: a high-frequency peak closely correlated in time with the wave crest when the total load is positive and a high-frequency peak defining the secondary load cycle which occurs close in time to the wave zero downcrossing when the total load is negative. The two peaks are studied by regression analysis as a function of either the Keulegan–Carpenter number ($KC$) or the Froude number ($Fr$). Regarding the secondary load cycle, the best correlation is found with $Fr$. The speed of the travelling edge of the undisturbed wave approximates the fluid velocity. A threshold value separating between small and large forces is found for $KC\sim 4$–5, indicating effects of flow separation. Alternatively, the threshold occurs for $Fr\sim 0.3$–0.4, indicating local wave effects at the scale of the cylinder diameter. The findings suggest that both effects are present and important.

Three-dimensional turbulent magnetoconvection at a Rayleigh number of $Ra=10^{7}$ in liquid gallium at a Prandtl number $Pr=0.025$ is studied in a closed square cell for very strong external vertical magnetic fields $B_{0}$ in direct numerical simulations which apply the quasistatic approximation. As $B_{0}$, or equivalently the Hartmann number $Ha$, are increased, the convection flow, which is highly turbulent in the absence of magnetic fields, crosses the Chandrasekhar linear stability limit for which thermal convection ceases in an infinitely extended layer and which can be assigned a critical Hartmann number $Ha_{c}$. Similar to rotating Rayleigh–Bénard convection, our simulations reveal subcritical sidewall modes that maintain a small but finite convective heat transfer for $Ha>Ha_{c}$. We report a detailed analysis of the complex two-layer structure of these wall modes, their extension into the cell interior, and a resulting sidewall boundary layer composition that is found to scale with the Shercliff layer thickness.

The phenomenon of drag reduction induced by injection of bubbles into a turbulent carrier fluid has been known for a long time; the governing control parameters and underlying physics is, however, not well understood. In this paper, we use three-dimensional numerical simulations to uncover the effect of deformability of bubbles injected in a turbulent Taylor–Couette flow on the overall drag experienced by the system. We consider two different Reynolds numbers for the carrier flow, i.e. $Re_{i}=5\times 10^{3}$ and $Re_{i}=2\times 10^{4}$; the deformability of the bubbles is controlled through the Weber number, which is varied in the range $We=0.01{-}2.0$. Our numerical simulations show that increasing the deformability of bubbles (that is, $We$) leads to an increase in drag reduction. We look at the different physical effects contributing to drag reduction and analyse their individual contributions with increasing bubble deformability. Profiles of local angular velocity flux show that, in the presence of bubbles, turbulence is enhanced near the inner cylinder while attenuated in the bulk and near the outer cylinder. We connect the increase in drag reduction to the decrease in dissipation in the wake of highly deformed bubbles near the inner cylinder.