A simple low order model of the forced Karman wake

Abstract

The present work develops a very simple mathematical model for the 2D von Karman vortex shedding wake. The case where this wake is periodically excited at the vortex shedding frequency is considered. The goal is to arrive at an approximate model that is simple enough to allow a full analysis of the underlying nonlinear dynamics. Since such a simple model cannot be expected (and is not intended) to replicate the Navier–Stokes equations the comparison criteria is the ability of the model to replicate the sequence of bifurcations as the forcing amplitude parameter is varied. Such a model can be useful for flow control applications. Equivariant bifurcation theory is employed to obtain the low order discrete model for the dynamics of the Karman wake when ‘reflection-symmetrically’ forced at the vortex shedding frequency. The discrete dynamical system (amplitude equations) modeling the mode interactions is derived in Poincaré space. Model parameters are then determined via POD from numerical simulations of the simple stationary cylinder case. A quantitative analysis of the wake dynamics based on the model above is presented. For Re=1000, dynamics of the 2D cylinder wake are shown to be closely linked, via the Bogdanov–Takens bifurcation scenario to physical and mathematical systems having SO(2) symmetry. For Re=200, a torus breakdown following the Afraimovich–Shilnikov scenario is found. The result is a complex, possibly chaotic, wake flow. Experimental results, in the form of measured POD modes, are also presented. The results suggest that the 3D wake transition does not destroy the 2D Karman wake dynamics; the latter apparently remains dominant even at moderately higher Reynolds numbers. The goal of the present work is to show that much can be learnt from a very simple model of the wake dynamics when the role of symmetry is carefully considered. This could also be viewed the other way around; that the dynamical bifurcation behavior of the apparently complex forced wake flow can, somewhat surprisingly, be described by a fairly simple model.