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Hydrodynamic forces acting on the elliptic cylinder performing high-frequency low-amplitude multi-harmonic oscillations in a viscous fluid

Abstract The paper is devoted to the study of the periodic rectilinear motion of an elliptic cylinder at an arbitrary angle of attack in a viscous incompressible fluid according to an arbitrary multi-harmonic law. The study develops the classical method of inner and outer asymptotic expansions of a solution in a small parameter, which is taken as the ratio of the thickness of the Stokes boundary layer to the semi-major axis of the elliptic cylinder. Another parameter of the problem, which is the Reynolds number calculated based on the thickness of the boundary layer, is considered to be of the order of unity. The selected range of values of parameters describes the low-amplitude high-frequency oscillations of the cylinder in a fluid. The asymptotic solution of the problem built in the study allows us to determine the first three terms of the expansion of the hydrodynamic force acting on the cylinder. For the detailed analysis of the obtained solution, several cases of the periodic motion of cylinders with different axis ratios for different oscillation parameters according to harmonic and multi-harmonic laws were considered. In these cases, the asymptotic dependencies of the hydrodynamic forces were compared with the data of direct numerical simulations and with known results of previous asymptotic and experimental research. The results of the analysis allow us to establish the limits of applicability of the asymptotic theory and to determine the structural changes of the hydrodynamic forces when transition to the zone of moderate oscillation amplitudes occurs.