A spectral force representation and its physical implication for vortex shedding past a stationary sphere

In this study, we consider the connection of the hydrodynamic force with vortex shedding behind a stationary sphere in an unbounded fluid at several Reynolds numbers, covering the range of critical transition from plane-symmetric to non-symmetric flow. A principal spectrum analysis (PSA) is employed/proposed to obtain the spectral representation (SR) modes for each time-varying force coefficient; this enables us to identify (i) one single characteristic frequency at Re = 300, 350, and 380, i.e., the fast frequency f(VS) which represents the frequency of vortex shedding and (ii) two characteristic frequencies at Re = 390, 400, and 420: one is the first fast frequency f(1 )= f(VS), representing the frequency of vortex shedding, and the other is the second fast frequency f(2), representing the frequency of side oscillations of the vortex flow in the transverse flow direction. In the situations of the latter set of Reynolds numbers, the SR modes indicate that not only the strength but also the phase of shed vortex varies slightly at each successive vortex shedding, and most notably, we have identified rotation of the vorticity pattern in the near wake about the central axis. The PSA-SR mode analysis reveals that these three vortex activities all co-operate at the integer multiples of the greatest common-divisor of the two characteristic frequencies ( f(1), f(2)). In other words, the intricate subtlety of vortex shedding behind a stationary sphere can well be uncovered through the PSA-SR mode analysis for establishing the close connections between the force coefficients and the vorticity activities behind the sphere. The present work represents a sequel to our previous study: A spectral force representation and its physical implication for vortex shedding past a stationary or an oscillating circular cylinder at low Reynolds numbers [Lu et al., "A spectral force representation and its physical implication for vortex shedding past a stationary or an oscillating circular cylinder at low Reynolds number," Phys. Fluids 35(5), 053606 (2023)], where we initially proposed the PSA-SR analysis to establish the connections between the force coefficients and the vortex shedding of the circular cylinder under pre-, sub-, and normal synchronization.