Exponential vs algebraic growth and transition prediction in boundary layer flow

For applications regarding transition prediction, wing design and control of boundary layers, the fundamental understanding of disturbance growth in the flat-plate boundary layer is an important issue. In the present work we investigate the energy growth of eigenmodes and non-modal optimal disturbances. We present a set of linear governing equations for the parabolic evolution of wavelike disturbances valid both for the exponential and algebraic growth scenario. The base flow is taken as the Falkner-Skan similarity solution with favorable, adverse and zero pressure gradients. The optimization is carried out over the initial streamwise position as well as the spanwise wave number and frequency. The exponential growth is maximized in the sense that the envelope of the most amplified eigenmode is calculated. In the case of algebraic growth, an adjoint-based optimization technique is used. We find that the optimal algebraic disturbance introduced at a certain downstream position gives rise to a larger growth than for the optimal disturbance introduced at the leading edge. The exponential and algebraic growth is compared and a unified transition-prediction method based on available experimental data is suggested.