Linear and non-linear stability analysis of boundary layers over a hump by using PSE

Abstract

Linear and non-linear stability analysis of incompressible and supersonic/hypersonic boundary layers over a two-dimensional smooth hump is carried out by using parabolized stability equations (PSE). A hump with a height smaller than the boundary layer thickness was considered. For incompressible boundary layer, overall destabilization influence of the hump is confirmed for both linear and non-linear evolution of instabil-ity waves. For linear stability, effects of several parameters including frequency, hump height, and wave obliqueness are examined. Comparisons of the linear PSE results with those from linear stability theory (LST) are also given. The discrepancy between the LST and linear PSE results mainly observed in the vicinity of the hump is found to be caused by both the flow non-parallelism and the transient behavior of PSE approach. The transient behavior sometimes accompanying wiggling in the growth rate curve is found to vary depending on the flow configurations such as frequency and hump height. Non-linear PSE analysis is carried out to examine non-linear evolution of Tollmien-Schlichting wave and subharmonic breakdown. Influence of frequency, hump height, and initial amplitude is investigated. The results demonstrate that disturbances can be amplified very much by the non-linear interaction due to the presence of hump in spite of very low level of disturbance amplitude, which suggests that the hump can bring forth earlier breakdown even for the case of rather small initial amplitude disturbances. Through analyses on subharmonic breakdown, it is found that the amplitude of primary wave affects most the growth of the secondary wave. Linear evolution of oblique first mode wave and oblique breakdown in Mach 1.6 boundary layer are analyzed for supersonic case. An overall effect of the hump on destabilization was similar to the case of subsonic boundary layer. For the case of linear stability, major difference was in the location where the largest dest...