LINEAR-STABILITY OF SHEAR PROFILES AND RELATION TO THE SECONDARY INSTABILITY OF THE DEAN FLOW

Solutions of the Navier-Stokes equations for flow in a curved channel have previously been computed [Bottaro, J. Fluid Mech. 251, 627 (1993) ] and the spatial development of the Dean flow is available at differe nt supercritical Reynolds number. In this work the viscous instability of local longitudinal vortex structures (obtained from the nonlinear simulations) is investigated, with a focus toward the secondary instab ility of Dean vortices. Such a secondary instability takes the form of streamwise traveling waves. High-frequency waves are termed twisting waves, low frequency are defined undulating waves. Instead of performi ng analyses in which the basic flow in the cross section is two dimens ional, significant shear profiles along y and z are considered as base flows at constant x before the establishment of a fully developed sta te. Thus one is able to discover that the twist instability is of shea r type and is caused by inflectional spanwise profiles of the streamwi se velocity component. Sinuous waves are always preferred to varicose waves, and the latter mode of instability is destabilized only at larg e Reynolds numbers. Undulating waves are related to normal profiles of the streamwise velocity; this type of secondary instability is of cen trifugal origin. Results of the analyses for both types of waves are i n good agreement with experiments.