Inviscid and Viscous Models of the Vortex Breakdown Phenomenon

摘要

In the case of high swirl and large Reynolds number, the Navier-Stokes equations for rotationally symmetric incompressible flow are shown to reduce to (1) a viscous parabolic system for slender (quasicylindrical) flows and (2) an inviscid elliptic system for expanding (or contracting) flows. The inviscid system is solved for the case of flow with initial rigid rotation in a cylindrical stream surface. Assuming different downstream boundary conditions, Fourier-Bessel series solutions are computed for the supercritical (nonoscillatory) case and plotted. For very high swirl values, closed and open bubbles of recirculating fluid are obtained for certain cases, where the closed bubbles resemble those observed in vortex tube experiments. The viscous slender problem is formulated in an integral method, using approximating functions for axial velocity and circulation which satisfy boundary and asymptotic requirements. Weighting functions are used to generate a linearly independent set of equations of sufficient number to determine the coefficients in the velocity and circulation approximations as functions of one (the axial) coordinate. The formulation is for any order of approximation. Computational difficulties appear near the suspected breakdown point. Flows resembling breakdown flows are obtained. (Author)