Adjoint-Based Approach for Prediction of Non-Modal Growth in Spatially Evolving Boundary-Layer Flows

A novel method for approximating transient growth in spatially evolving flows within the framework of local stability theory by using bi-orthogonality is presented. Although the local linear stability analysis is based on the parallel-flow assumption, it is able to accurately predict modal growth of discrete waves in boundary-layer flows using the e^n-method. However, the classical local approach fails for the concept of non-modal disturbances, where the growth is substantially under-predicted. In this study, it is shown that the prediction can be significantly improved by employing local adjoint stability solutions for a sequential downstream projection of the disturbances. While still using local theory with the parallel-flow assumption, this can be physically explained by the redistribution of energy between different scales, which appears to be one of the aspects of non-local disturbance growth. The results for the lift-up effect and for the Orr mechanism in a Blasius flow are presented and discussed. By comparing the results from parabolized stability equations and direct numerical simulations, conclusions are drawn about the physical effects involved in non-modal perturbation growth in spatially evolving boundary-layer flows.