Contrasting the Cartesian and polar forms of the shedding-induced force vector in response to 12 subharmonic and superharmonic mechanical excitations

Abstract

We mechanically excited the near-field shedding of the von Kármán vortex street of a stationary cylinder at Reynolds number 300 by moving it sinusoidally with 12 different frequencies. We first solved the incompressible Navier–Stokes equations using the finite difference method and obtained unexcited shedding, which occurs at the Strouhal frequency. Then, we applied six subharmonic excitations with rational frequency ratios (defined as the excitation frequency to the Strouhal frequency) in the form 1/n and six superharmonic excitations with frequency ratios n, where n is an integer taking values between 2 and 7. We compared the behavior of the shedding-induced force vector acting on the cylinder in each excitation case to their counterparts in the other excitation cases and to the case of the stationary cylinder. This force vector has two representations: lift and drag (Cartesian) or magnitude and direction (polar). Whereas the first representation is commonly used in theoretical fluid mechanics and engineering applications, we found that the second representation is favored in terms of revealing how the force vector responds to the applied excitations. For example, the mean and standard deviation of the drag do not show clear trends with the excitation frequency, whereas the force magnitude changes quadratically with it. We showed that the theoretical analysis of a simplified nonlinear wake model does in fact support this behavior. We found three modes of shedding-induced force in response to mechanical excitation, depending on the ratio of excitation frequency to Strouhal frequency. Near-field shedding is not entrained by excitation for all subharmonic excitations, but it is entrained for all superharmonic excitations. For entrained shedding, we found two behaviors of the shedding-induced force vector, which is periodic only at odd frequency ratios.