Analysis and Computation of Extremum Points with Application to Boundary-Layer Stability

Abstract

A method for analysis and computation of derivatives and extremum points of variable-coefficients differential eigenvalue problems is presented. The method utilizes the orthogonality of the adjoint eigenfunctions to the homogenous part of the once or more differentiated problem to derive an analytical expression for the rate of change of eigenvalue with respect to a free parameter. The extremum point can be analyzed and computed by setting and driving, respectively, the first rate of change of the eigenvalue with respect to the free parameters to zero. Higher order derivatives can be computed by solving, sequentially, sets of inhomogeneous two-point boundary value problems. The method is applied to analyze and compute the most amplified inviscid instability wave in two-dimensional compressible boundary layers and the most amplified viscous instability wave in three-dimensional incompressible boundary layers. It is shown analytically that while the most-amplified spatial instability wave in two-dimensional incompressible boundary layer is two dimensional, the corresponding most amplified wave in three-dimensional boundary layer is generally oblique. It is also shown analytically that the most-amplified disturbance in three-dimensional boundary layer is generally a traveling disturbance. Furthermore, it is shown analytically that the inviscid growth rate is an extremum point.