Nonlinear modelling of vortex shedding control in cylinder wakes

Abstract Karman vortex shedding behind a cylinder placed at right angles to a uniform flow is known to be a limit cycle oscillation that results from the saturation of a global instability of the wake flow. In this paper we study the feedback control of Karman vortex shedding for Reynolds numbers (based on cylinder diameter) close to the critical value of Re c ≈ 47 using “single input - single output” (SISO) proportional control. A model is presented that combines the linear streamwise global mode amplitude equation and the nonlinear spanwise Ginzburg-Landau equation and correctly models the three-dimensional effects observed in the controlled wake of finite length cyclinders. In particular it is demonstrated that for long cylinders vortex shedding can only be suppressed at the spanwise location of the sensor even though the actuation occurs uniformly over the entire span. At a fixed streamwise position the spanwise variation of the shedding angle is thereby given by the “hole solution” of Nozaki and Bekki, J. Phys. Soc. Jpn. 53 (1984) 1581.