Low-Order Modeling of Bistable Side Forces on a Sphere Measured for a Transient Inflow in a Wind Tunnel

The unsteady forces acting on a sphere laterally mounted in a wind tunnel with transient sinusoidal inflow are investigated. The study shows that the low-frequency dynamic is characterized by a spontaneous switching between two partially stable states, also known as bistability. Additionally, oscillatory forces exist, which, on the one hand, are induced by the sinusoidal transient inflow and, on the other hand, originate from natural vortex shedding. By separating the bistability and the oscillatory forces using proper orthogonal decomposition, a low-order model based on the Duffing equation is developed both for the bistability and the oscillatory multi-scale induced forces. The bistability is represented by a chaotic bistable Duffing equation, whose parameters are determined by manual adjustment. In contrast, the parameters of the oscillatory multi-scale forcing are determined by expanding the forcing of the Duffing equation to three terms and preserving the phase angles between the three characteristic frequencies, the flapping frequency, its multiples and the natural vortex shedding.