The instability of a vortex ring impinging on a free surface

Abstract

Direct numerical simulation is used to study the development of a single laminar vortex ring as it impinges on a free surface directly from below. We consider the limiting case in which the Froude number approaches zero and the surface can be modelled with a stress-free rigid and impermeable boundary. We find that as the ring expands in the radial direction close to the surface, the natural Tsai–Widnall–Moore–Saffman (TWMS) instability is superseded by the development of the Crow instability. The Crow instability is able to further amplify the residual perturbations left by the TWMS instability despite being of differing radial structure and alignment. This occurs through realignment of the instability structure and shedding of a portion of its outer vorticity profile. As a result, the dominant wavenumber of the Crow instability reflects that of the TWMS instability, and is dependent upon the initial slenderness ratio of the ring. At higher Reynolds number a short-wavelength instability develops on the long-wavelength Crow instability. The wavelength of the short waves is found to vary around the ring dependent on the local displacement of the long waves.