The wake of an oscillating teardrop hydrofoil with combined heaving and pitching motion was studied numerically at Reynolds number of 8000 and Strouhal numbers of $St=0.21{-}0.94$ . The lower Strouhal number exhibited high efficiency propulsion with small thrust generation. However, larger thrust generation at high $St$ required more power, which lowered the propulsive efficiency. Quantitative assessment of vortex evolution, along with qualitative investigation of the formation and interaction of primary structures, revealed the association with elliptic instability characteristics for both co-rotating and counter-rotating vortex structures in both wakes. With respect to advection of the leading-edge vortex, the pressure distribution further depicted evidence of spanwise instability with distinct temporal evolution along the suction and pressure surfaces of the oscillating foil. Three-dimensional assessment of wake structures located downstream of the trailing edge depicted the existence of dislocations associated with primary vortex ‘rollers’. At low $St$ , these were limited to fine spanwise corrugations (valleys and bulges) on weaker leading edge rollers, which enlarged as the rollers advected downstream. In contrast, at high $St$ , the wake exhibited conjoint hairpin-horseshoe vortex structures that led to stronger deformations on the coupled vortex rollers. The statistical characteristics of secondary structures resembled the long wavelength mode and mode A identified previously for purely pitching and heaving foils, respectively. They also mimicked mode B for stationary cylinders. Novel wake models are introduced based on a complete vivid three-dimensional depiction of coherent wake structures.

Wake evolution of an oscillating foil with combined heaving and pitching motion is evaluated numerically for a range of phase offsets ( $\phi$ ), chord-based Strouhal numbers ( $St_c$ ) and Reynolds numbers ( $Re$ ). The increase in $\phi$ from $90^\circ$ to $180^\circ$ at a given $St_c$ and $Re$ coincides with a transition of pitch- to heave-dominated kinematics that further reveals novel transitions in wake topology characterized by bifurcated vortex streets. At $Re= 1000$ , each of the dual streets constitutes a dipole-like paired configuration of counter-rotating coherent structures that resemble qualitatively the formation of $2P$ mode. A new mathematical relation between the relative circulation of coherent dipole-like paired structures and kinematic parameters is proposed, including heave-based ( $St_h$ ), pitch-based ( $St_{\theta }$ ) and combined motion ( $St_A$ ) Strouhal numbers, as well as $\phi$ . This model can predict accurately the wake transition towards $2P$ mode characterized by a bifurcation, at low $Re= 1000$ . At $Re= 4000$ , however, the relationship was inaccurate in predicting the wake transition. A shear splitting process is observed at $Re= 4000$ , which leads to the formation of reverse Bénard–von Kármán mode in conjunction with $2P$ mode. Increasing $\phi$ further depicts a consistent prolongation of the splitting process, which coincides with a unique transition in terms of absence and reappearance of bifurcated dipole-like pairs at $\phi = 120^\circ$ and $180^\circ$ , respectively. Changes in the spatial arrangement of $2P$ pairs observed consistently for oscillating foils with the combined motion constitute a novel wake transition that becomes more dominant at higher Reynolds numbers.

The wake dynamics of sharp-edge rigid panels is examined using Overset Grid Assembly (OGA) utilized in OpenFOAM, an open-source platform. The OGA method is an efficient solution technique based on overlap of a single or multiple moving grids on a stationary background grid. Five test cases for a stationary panel at different angle of attack are compared with available computational data, which show a good agreement in predicting global flow variables, such as mean drag. The models also provided accurate results in predicting the main flow features and structures. The flow past a pitching square panel is also investigated at two Reynolds numbers. The study of surface pressure distribution and shear forces acting on the panel suggests that a higher streamwise pressure gradient exists for the high Reynolds number case, which leads to an increase in lift, whereas the highly viscous effects at low Reynolds number lead to an increased drag production. The wake visualizations for the stationary and pitching motion cases show that the vortex shedding and wake characteristics are captured accurately using the OGA method.

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