Cross-flow linear instability in compressible boundary layers over a flat plate

Abstract

The linear instability of compressible boundary layers over a flat plate in the presence of parameterized crossflow has been investigated by means of linear stability theory. A family of boundary layer with crossflow is obtained as the base flow from the compressible Falkner–Skan–Cooke (FSC) flow model. Two factors, that is, the local swept angle and the pressure-gradient parameter, are designed to create the crossflow with different directions and magnitudes, which further results in the variation of the cross-flow instability. Modal properties related to the cross-flow instability are emphasized. The association between the cross-flow mode and the Mack's mode is clarified by extending the base flow from two dimension to three dimension. The cross-flow instability is discovered to be integrated with the slow-mode instability, that is, the instability related to the Tollmien–Schlichting (T–S) mode or the first mode, and it can hardly be distinguished as an individual mode in most cases. The effects of Mach number, pressure gradient, local sweep, and wall temperature are studied. The behaviors of the cross-flow instability under such effects resemble those of the slow-mode instability in the two-dimensional boundary layers. It is found that the unstable modes in the three-dimensional boundary layers are mainly affected by the streamwise pressure gradient and the crossflow per se. Specifically, the first mode is much more sensitive to the crossflow than the second mode. As a result, more marked variations are commonly observed for the first mode in the boundary layers with crossflow.