Linear instability of a supersonic boundary layer over a rotating cone

In this paper, we conduct a systematic study of the instability of a boundary layer over a rotating cone that is inserting into a supersonic stream with zero angle of attack. The base flow is obtained by solving the compressible boundary-layer equations using a marching scheme, whose accuracy is confirmed by comparing with the full Navier-Stokes solution. Setting the oncoming Mach number and the semi-apex angle to be 3 and 7 degrees, respectively, the instability characteristics for different rotating rates ((Omega) over bar, defined as the ratio of the rotating speed of the cone to the axial velocity) and Reynolds numbers (R) are revealed. For a rather weak rotation, (Omega) over bar << 1, only the modified Mack mode (MMM) exists, which is an extension of the supersonic Mack mode in a quasi-two-dimensional boundary layer to a rotation configuration. Further increase of (Omega) over bar leads to the appearance of a cross-flow mode (CFM), coexisting with the MMM but in the quasi-zero frequency band. The unstable zones of the MMM and CFM merge together, and so they are referred to as the type-I instability. When (Omega) over bar is increased to an O(1) level, an additional unstable zone emerges, which is referred to as the type-II instability to be distinguished from the aforementioned type-I instability. The type-II instability appears as a centrifugal mode (CM) when R is less than a certain value, but appears as a new CFM for higher Reynolds numbers. The unstable zone of the type-II CM enlarges as (Omega) over bar increases. The vortex structures of these types of instability modes are compared, and their large-R behaviours are also discussed.