Characterization of bifurcated dual vortex streets in the wake of an oscillating foil

Abstract

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.