Understanding the vortex interactions and wake transitions for flapping flexible foils is important because of their increased usage in bioinspired aquatic and aerial robotic propulsors. Although wake transitions have been studied for rigid foils, we experimentally investigate how flexibility alters the transitions and vortex interactions for flexible foils, which are closer to the natural flapping foils in fish, birds and insects. We conduct the experiments in a flowing soap film on a pitching airfoil with a flexible filament at its trailing edge (TE). We find that, apart from the Strouhal number (${\textit{St}}$), flexural rigidity (${\textit{EI}}$) is important to determine the transitions. We vary ${\textit{EI}}$ of the flexible filament by three orders of magnitude and also investigate an extreme case of ${\textit{EI}} \rightarrow \infty$. Flexibility triggers the shedding of multiple small ‘secondary vortices’ (SVs) along with big ‘primary vortices’ (PVs), unlike only PVs for the rigid foil. Continuous deformations of the flexible filament play crucial roles in determining the interaction of boundary layer vortices and trailing edge vortices and, ultimately, the generation and evolution of PVs and SVs. We identify five vortex interaction mechanisms (VIMs). Depending on how SVs interact with PVs, the wake assumes different patterns. We construct the ${\textit{St}}$–${\textit{EI}}$ phase maps for wake transitions and newly identified VIMs. We devise a non-dimensional parameter $\varUpsilon$, referred to as ‘Yashavant number’. One order increase in $\varUpsilon$ reduces the number of VIMs by one. Instead of following the usual transition route, the flexible foil reveals counterintuitive transition trends that strongly depend on the filament ${\textit{EI}}$.

Steady tip streaming in the limit of vanishing flow rate has been experimentally and numerically documented, yet theoretical solutions describing local conical Stokes flows have remained elusive. Here, we derive approximate analytical solutions for local conical flows in liquid–liquid flow focusing scenarios, addressing the limit of negligible emitted flow rate. Our analysis demonstrates the existence of a universal relationship between the inner-to-outer liquid viscosity ratio and the cone angle, establishing the theoretical underpinning for precise control of microscopic jet formation. A posteriori comparison with previously published experiments reveals that digitised cusp-like meniscus profiles collapse quantitatively onto the predicted slender-body similarity solution. These findings pave the way for technologies that require exact manipulation of fluid flows at nearly molecular dimensions.

Energy transfer across scales is fundamental in fluid dynamics, linking large-scale flow motions to small-scale turbulent structures in engineering and natural environments. Triadic interactions among three wave components form complex networks across scales, challenging understanding and model reduction. We introduce triadic orthogonal decomposition (TOD), a method that identifies coherent flow structures optimally capturing spectral momentum transfer, quantifies their coupling and energy exchange in an energy-budget bispectrum and reveals the regions where they interact. Triadic orthogonal decomposition distinguishes three components – a momentum recipient, donor and catalyst – and recovers laws governing pairwise, six-triad and global triad conservation. We apply TOD to three examples: the classical cylinder wake, experimental wind turbine wake data and a direct numerical simulation of isotropic turbulence. Energy transfer can be spatially distributed but vanish upon integration or spatially localised but facilitate net interscale exchange, so a complete characterisation of nonlinearity requires examination of both integral and local transfers. In the cylinder wake, we link backscatter of energy from high to low frequencies to a compact attenuation region downstream of the cylinder. In the turbine wake, we confirm the known association between energy amplification and decay and vortex tilting, but observe more complex secondary mechanisms in suboptimal modes. For isotropic turbulence, we derive and confirm inertial-range frequency scaling for convective–recipient covariances, then demonstrate self-similar energy transfer at each rank.

Direct numerical simulations of turbulent channel flows with thermally unstable stratification laden with finite-size particles are performed using the fictitious domain method. The effects of particle concentration, size and specific heat capacity on turbulence and heat transfer are investigated at the friction Reynolds number 180, the Prandtl number 0.7, the Richardson number 20 and both density ratio and thermal conductivity ratio being unity. The natural convection circulations in the unstable stratification case cause the occurrence of streamwise streaks of vortex, velocity and temperature. Compared with the neutral case, the particle-induced flow drag enhancement is more significant for the unstable stratification case, mainly because the particle-induced reduction in the fluid Reynolds shear stress is much less significant for the unstable stratification case, which may be caused by the weaker suppression effects of the particles on the Rayleigh–Bénard circulations and the streamwise vortex packages. In contrast to the neutral case where the particles attenuate the fluid turbulent heat flux and thereby the Nusselt number, the particles enhance the fluid turbulent heat flux and thereby the Nusselt number for the unstable stratification case. The above particle effects are stronger for higher particle volume fractions or smaller particle sizes, when the other parameters are fixed. As the specific heat capacity ratio increases, the Nusselt number increases as a result of the increase in the solid turbulent heat flux contribution.