Bounded and unbounded solutions of the von Kármán vortex trail

Abstract

The general initial value problem for the linearized von Karman vortex trail is solved. In particular, the fundamental differences between von Karman’s normal mode solutions and aperiodic solutions with compact support are discussed. Within the natural space ℓ 2 {\ell _2} , it is shown that the von Karman trail at its special aspect ratio (neutrally stable case) supports unbounded solutions. Two invariants of the equations determine whether a solution is bounded or unbounded. The asymptotic behaviour of aperiodic solutions is discussed. It is found that unbounded solutions grow at the rate O ( l n τ ) O\left ( {ln \tau } \right ) in ℓ ∞ {\ell _\infty } while bounded solutions decay as O ( 1 / τ ) O\left ( {1/\sqrt \tau } \right ) . In addition, a Loschmidt’s demon is constructed for the reversible dispersive equations which demonstrates the focussing effect in the von Karman trail.