FUNDAMENTAL AND SUBHARMONIC SECONDARY INSTABILITIES OF GORTLER VORTICES

The nonlinear development of stationary Gortler vortices leads to a hi ghly distorted mean flow field where the streamwise velocity depends s trongly not only on the wall-normal but also on the spanwise coordinat es. In this paper, the inviscid instability of this flow field is anal ysed by solving the two-dimensional eigenvalue :problem associated wit h the governing partial differential equation. It is found that the fl ow field is subject to the fundamental odd and even (with respect to t he Gortler vortex) unstable modes. The odd mode, which was also found by Hall and Horseman (1991), is initially more unstable. However, ther e exists an even mode which has higher growth rate further downstream. It is shown that the relative significance of these two modes depends upon the Gortler vortex wavelength such that the even mode is stronge r for large wavelengths while the odd mode is stronger for short wavel engths. Our analysis also shows the existence of new subharmonic (both odd and even) modes of secondary instability. The nonlinear developme nt of the fundamental secondary instability modes is studied by solvin g the (viscous) partial differential equations under a parabolizing ap proximation. The odd mode leads to the well-known sinuous mode of brea k down while the even mode leads to the horseshoe-type vortex structur e. This helps explain experimental observations that Gortler vortices break down sometimes by sinuous motion and sometimes by developing a h orseshoe vortex structure. The details of these break down mechanisms are presented.