This study presents a numerical investigation of wall-mounted tandem flexible plates with unequal lengths in a laminar boundary layer flow, examining both two-dimensional (2-D) and three-dimensional (3-D) configurations. Key parameters influencing the system include the plate’s bending stiffness ($K$), Reynolds number (${Re}$) and length ratio ($L^*$). Five motion modes are identified: dual collapse (DC), flapping collapse (FC), dual flapping (DF), static flapping (SF) and dual static (DS). A phase diagram in the ($K,L^*$) space is constructed to illustrate their regimes. We focus on DF and SF modes, which significantly amplify oscillations in the downstream plate – critical for energy harvesting. These amplification mechanisms are classified into externally driven and self-induced modes, with the self-induced mechanism, which maximises the downstream plate’s amplitude, being the main focus of our study. A rigid–flexible (RF) configuration is introduced by setting the upstream plate as rigid, showing enhanced performance at high ${Re}$, with oscillation amplitudes up to 100 % larger than the isolated flexible (IF) plate configuration. A relation is developed to explain these results, relating oscillation amplitude to trailing-edge velocity, oscillation frequency and chord length. Force analysis reveals that the RF configuration outperforms both IF and flexible–flexible (FF) configurations. Unlike frequency lock-in, the RF configuration exhibits frequency unlocking, following a $-2/3$ scaling law between the Strouhal number ($St$) and ${Re}$. Results from the 3-D RF configuration confirm that the 2-D model remains applicable, with the self-induced amplification mechanism persisting in 3-D scenarios. These findings enhance understanding of fluid–structure interactions, and offer valuable insights for designing efficient energy harvesting systems.

Natural fliers and marine swimmers twist and turn their lifting or control surfaces to manipulate the unsteady forces experienced in air and water. The passive deformation of such surfaces has been investigated by several researchers, but the aspect of controlled deformation has received comparatively less attention. In this paper, we experimentally measure the forces and the flow fields of a flat-plate wing (aspect ratio (AR) = 3), translating at a constant Reynolds number (Re) of 10 000, with a dynamically twisting span. We show that the unsteady forces can be dependably estimated by a three-dimensional discrete vortex model. In this model, we account for the leading-edge separation with the help of the leading-edge-suction parameter. Experiments are conducted for two angles of attack (AoAs), $5^\circ$ and $15^\circ$. In addition, two rates of twisting are implemented where part of the leading edge, closer to the tip region, is twisted away from the incoming flow, increasing the effective AoA. The results show that twisting away from the flow augments the lift forces in all cases, although the rate of increase of lift is higher for the highest twist rate. The act of twisting causes an increase in effective AoA beyond the static stall angle in the AoA $=15^\circ$ case. This is highlighted by a distinct dip in the force data following the initial rise after twisting is activated. The increase in effective AoA from the reference case (without twisting) causes separation of the flow below the mid-span. This, in turn, creates higher levels of vorticity in those regions and results in a leading-edge vortex with increased cross-section and strength when compared with the reference case without twisting. Finally, we apply force partitioning and reveal that dynamic twisting leads to a localised increase in vorticity-induced forces along the twisted part of the span, which is approximately twice that of the untwisted case.

We experimentally investigate the rotational dynamics of neutrally buoyant flat bodies of revolution (spheroids, disks and rings with different cross-sectional shapes) in shear flows. In the Stokes regime, the axis of revolution of these rigid particles moves in one of a family of closed periodic Jeffery orbits. Inertia is able to lift the orbit degeneracy and induces drift among several rotations towards limiting stable orbits. Furthermore, permanent alignment can be achieved for disks and rings with triangular cross-sectional shapes, provided the inertia is sufficiently high. The bifurcations between the different dynamics are compared with those predicted by small-inertia asymptotic theories and numerical simulations.

This paper considers the propagation, arrest and recession of a planar hydraulic fracture in a porous elastic medium whose footprint is constrained to a growing or shrinking rectangular region with a constant height. Hydraulic fractures with large aspect ratio rectangular footprints are frequently referred to as PKN fractures in recognition of the original researchers (Perkins & Kern 1961 J. Petrol. Tech. 13, 937–949) and (Nordgren 1972 J. Petrol Technol. 1972, 306–314) who first analyzed models of such fracture geometries. We investigate the one-dimensional non-local PKN approximation to a fully planar rectangular hydraulic fracture model in a three-dimensional elastic medium. By analysing the tip behaviour of the non-local PKN model, a transformation procedure is established to render the asymptotic equations for the dynamics of the steady semi-infinite PKN and plane strain models formally identical, which implies that all the existing multiscale plane strain asymptotes can be converted directly to the PKN case by making use of this transformation. Using this transformation, it is shown that the appropriate PKN asymptotes for the average aperture $\bar {w}$ with distance $\hat {x}$ to the fracture front are $\bar {w}\sim \hat {x}^{1/2},\ \hat {x}^{5/8}\ {\textrm{and}}\, \ \hat {x}^{2/3}$ in the toughness, leak-off and viscous modes of propagation, respectively; as well as the linear elastic fracture mechanics tip asymptote $\bar {w}\sim \hat {x}^{1/2}$ for arrest, which transitions to the linear asymptote tip $\bar {w}\sim \hat {x}$ for a fracture driven to recede due to fluid leak-off. Both the arrest and recession tip asymptotes share the intermediate leak-off asymptote $\bar {w}\sim \hat {x}^{3/4}$. A scaling analysis yields the arrest time, length and aperture as functions of a dimensionless injection-cessation time $\omega$. An asymptotic analysis of the non-local PKN model is used to establish the fundamental decoupling between dynamics and kinematics, which leads to the emergence of a similarity solution – termed the sunset solution – close to the time of collapse of the fracture. The multiscale PKN numerical solutions agree well with those for a fully planar multiscale rectangular hydraulic fracture model in a three-dimensional elastic medium. The scaling laws and the emergence of the sunset solution are confirmed by the PKN numerical model. The sunset solution also emerges in the fully planar numerical model and persists beyond the collapse time of the PKN model, by which time its footprints have separated from the upper and lower constraining sedimentary layer boundaries and have assumed self-similar elliptic shapes that shrink as they approach collapse.

This study employs a direct numerical simulation method to investigate the wake pattern evolutions of flows past an insulated spheroid and provides expressions of force and torque coefficients influenced by a streamwise magnetic field in an incompressible, conducting, viscous fluid. A total of 1150 cases are examined covering a parameter range of Reynolds number $50 \leqslant \textit{Re} \leqslant 250$, aspect ratio $1.5 \leqslant \beta \leqslant 6$, inclination angle $0^\circ \leqslant \theta \leqslant 90^\circ$, and interaction parameter $0 \leqslant N \leqslant 10$, where $\beta$ and $N$, respectively, reflect the anisotropy of the spheroid and the strength of magnetic field. Nine wake patterns are classified based on wake structure features and summarised in three maps of regimes according to the inclination angle. The transition mechanisms among these wake patterns are also investigated under the influence of a streamwise magnetic field. Furthermore, expressions for drag, lift and torque coefficients are derived with the help of three fundamental physical criteria. Results indicate that the force and torque expressions give a good prediction within the present parameter space $\{\textit{Re}, \beta , \theta , N\}$.