On vortex rings and Vogel exponents*

In heavy winds, plants bend and align themselves with the flow. These passive behaviors allow plants to resist being uprooted by reducing their drag. Here, we study the drag reduction of “leaves”, approximated by flexible disks with radial cuts. The disks are translated vertically upwards through water and bend axisymmetrically in response to the motion. We measure the drag on the deforming disks using a high precision load cell. Existing models that predict the drag on flexible disks assume a uniform pressure distribution in their wake. Spatially resolved velocity field measurements show that the pressure distribution is not uniform, as an axisymmetric vortex ring forms beneath the deforming disks. The vortex size, circulation, and non-dimensional energy, which is a proxy for its stability, are modified by the disk’s deformation during the motion and directly influence the drag. Using these time-varying quantities, we derive an improved model for the drag reduction experienced by flexible disks. We further extend our model to disks with varying numbers of radial cuts, which fold and bend differently during the same motion. The vortex-structure interactions highlighted in our work can guide the design of robust, flexible components for drones and other aerial vehicles.