The evolution of a finite-amplitude localized disturbance in an irrotational unbounded plane stagnation flow

Abstract

The evolution of a finite-amplitude three-dimensional localized disturbance, having an initial dipole Gaussian vorticity distribution, embedded in an external unbounded irrotational plane stagnation flow (U = (Ay,Ax,0)) is investigated. Using the fluid impulse integral as a characteristic of such a disturbance, the viscous vorticity equation is integrated analytically. Accordingly, the fluid impulse associated with such disturbances decays and grows exponentially along the principal axes x=y and x=-y, respectively. Numerical simulations, carried out for both linear and nonlinear disturbances at a Reynolds number of 40, confirm the above predictions. The simulations have also been compared with the solution of the linear viscous vorticity disturbance equation. While the solution predicts the vorticity distribution for the linear case, it fails to predict the essential characteristics of a nonlinear disturbance associated with its self-induced movement. Finally, it is shown that the fluid impulse and the disturbance kinetic energy follow the same trend, i.e. when the fluid impulse increases with time so does the kinetic energy and vice versa. The correspondence between them suggests the use of the fluid impulse to predict the stability of a localized disturbance. © 2006 Cambridge University Press.