CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 2. Number 2, Spring 1994

RESONANT FLOW OF A ROTATING FLUID PAST AN OBSTACLE:

THE RADIALLY UNBOUNDED CASE

R. GRIMSHAW AND Y. ZHU

ABSTRACT. We consider the resonant interaction of a swirling flow past an axisymmetric obstacle on the axis of the flow. Here we consider the case when the swirling flow is radially unbounded, thus extending the work of Grimshaw 121 who considered the analogous problem for a radially con- fined flow. We show that in the weakly nonlinear long-wave regime the governing evolution equation for the amplitude of the dominant resonant mode is similar in structure to the forced Korteweg-de Vries equation derived by Grimshaw [2] in the radially confined case, and is a forced version of an equa- tion derived by Leibovich [B] for freely propagating weakly nonlinear waves on a radially unbounded swirling flow.

1. Introduction. The interaction of a swirling flow with an ax- isymmetric obstruction to that flow has been invoked by Randall and Leibovich [9] as a model of vortex breakdown (see also the review by Leibovich [7]). Further, Leibovich and Randall [8] and more recently Grimshaw [2] showed that, for weakly nonlinear long waves, the gov- erning equation is the forced Korteweg-de Vries (f KdV) equation,

Here A(X,T) is the amplitude of the dominant resonant linear long- wave mode, T and X are scaled time and axial distance, respectively, A is a detuning parameter for the resonance where A = 0 defines the critical condition with respect to the linear long-wave mode, and F(X) is a representation of the obstacle. The fKdV equation (1.1) arises in many different physical contexts and is a generic model for the interaction of a resonant flow with topography (see, for instance, the review by Grimshaw [3]). A comprehensive set of numerical solutions with theoretical interpretations has been given by Grimshaw and Smyth [4]. For the resonant case (i.e., when [A1 is sufficiently small) a typical

Accepted by the editors for publication on January 20, 1994 Copyright 01994 Rocky Mountain Mathematics Consortium

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R. GRIMSHAW AND Y. ZHU

FIGURE 1. The coordinate system.

solution of (1.1) consists of a family of solitary-like waves propagating upstream from the obstacle, a depression in the lee of the obstacle and a downstream oscillatory wavetrain.

The derivation of the f KdV equation (1.1) by Leibovich and Randall [a] and Grimshaw [2] was for flows which are radially confined to a circular tube of finite radius. The purpose of this paper is to consider the case when the basic swirling flow is radially unbounded and, outside the central core, consists of a uniform axial flow with uniform circulation. An axisymmetric obstacle lies on the axis of the flow, inside the core region. Leibovich [6] showed that in the absence of any forcing by the obstacle, weakly nonlinear long waves were governed by an evolution equation similar to the KdV equation, although he did not describe any solutions of that equation. Here we show that in the presence of the obstacle, the evolution equation is a forced version of the equation derived by Leibovich [6]. In Section 2 we present the governing equations for the axisymmetric flow of an inviscid, incompressible and homogeneous fluid, and discuss the linear long-wave theory arising from these equations. Then in Section 3 we derive the nonlinear evolution equation for the resonant, or critical, case. In Section 4 we discuss this equation and present some preliminary numerical solutions. In general, we conclude that this present case of radially unbounded flow is similar to the previous case of radially confined flow.

2. Formulation. We consider the axisymmetric flow of an inviscid, incompressible and homogeneous fluid, using nondimensional coordi- nates based on a length scale a (the radius of the inner rotating core)