Sensitivity analysis using adjoint parabolized stability equations for compressible flows

An input/output framework is used to analyze the sensitivity of two- and three-dimensional disturbances in a compressible boundary layer for changes in wall and momentum forcing. The sensitivity is defined as the gradient of the kinetic disturbance energy at a given downstream position with respect to the forcing. The gradients are derived using the parabolized stability equations (PSE) and their adjoint (APSE). The adjoint equations are derived in a consistent way for a quasi-two-dimensional compressible flow in an orthogonal curvilinear coordinate system. The input/output framework provides a basis for optimal control studies. Analysis of two-dimensional boundary layers for Mach numbers between 0 and 1.2 show that wall and momentum forcing close to branch I of the neutral stability curve give the maximum magnitude of the gradient. Forcing at the wall gives the largest magnitude using the wall normal velocity component. In case of incompressible flow, the two-dimensional disturbances are the most sensitive ones to wall inhomogeneity. For compressible flow, the three-dimensional disturbances are the most sensitive ones. Further, it is shown that momentum forcing is most effectively done in the vicinity of the critical layer.