Nonaxisymmetric Viscous Lower Branch Modes in Axisymmetric Supersonic Flows

In a previous paper, we have considered the weakly nonlinear interaction of a pair of axisymmetric lower branch Tollmien-Schlichting instabilities in cylindrical supersonic flows. Here the possibility that nonaxisymmetric modes might also exist is investigated. In fact it is found that such modes do exist and, on the basis of linear theory, it appears that these modes are the most important. The nonaxisymmetric modes are found to exist for flows around cylinders with nondimensional radius a less than some critical value ac• This critical value ac is found to increase monotonically with the azimuthal wavenumber n of the disturbance and it is found that unstable modes always occur in pairs. We show that in general, instability in the form of lower branch Tollmien-Schlichting waves will occur first for nonaxisymmetric modes and that in the unstable regime, the largest growth rates correspond to the latter modes. This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NASl-18107 while the authors were in residence at the Institute for Computer Applications in Science and Engineering (!CASE), NASA Langley Research Center, Hampton, VA 23665.