Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer

Abstract

This article deals with the linear dynamics of a transitional boundary layer subject to two-dimensional Tollmien–Schlichting instabilities. We consider a flat plate including the leading edge, characterized by a Reynolds number based on the length of the plate equal to Re = 6 × 105, inducing a displacement thickness-based Reynolds number of 1,332 at the end of the plate. The global linearized Navier–Stokes equations only display stable eigenvalues, and the associated eigen-vectors are known to poorly represent the dynamics of such open flows. Therefore, we resort to an input–output approach by considering the singular value decomposition of the global resolvent. We then obtain a series of singular values, an associated orthonormal basis representing the forcing (the so-called optimal forcings) as well as an orthonormal basis representing the response (the so-called optimal responses). The objective of this paper is to analyze these spatial structures and to closely relate their spatial downstream evolution to the Orr and Tollmien–Schlichting mechanisms. Analysis of the spatio-frequential diagrams shows that the optimal forcings and responses are respectively localized, for all frequencies, near the upstream neutral point (branch I) and the downstream neutral point (branch II). It is also shown that the spatial growth of the dominant optimal response favorably compares with the eN method in regions where the dominant optimal forcing is small. Moreover, thanks to an energetic input–output approach, it is shown that this spatial growth is solely due to intrinsic amplifying mechanisms related to the Orr and Tollmien–Schlichting mechanisms, while the spatial growth due to the externally supplied power by the dominant optimal forcing is negligible even in regions where the dominant optimal forcing is strong. The energetic input–output approach also yields a general method to assess the strength of the instability in amplifier flows. It is based on a ratio comparing two quantities of same physical dimension, the mean-fluctuating kinetic energy flux of the dominant optimal response across some boundary and the supplied mean external power by the dominant optimal forcing. For the present boundary-layer flow, we have computed this amplification parameter for each frequency and discussed the results with respect to the Orr and Tollmien–Schlichting mechanisms.