Nonlinear input/output analysis: application to boundary layer transition

• Published in: Journal of fluid mechanics Vol. 911
• Main Authors: Rigas, Georgios, Sipp, Denis, Colonius, Tim
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 25.03.2021; Cambridge University Press (CUP)
• Subjects: Algorithms; Analysis; Ascent; Boundary layer transition; Boundary layers; Breakdown; Chemical Sciences; Computer Science; Engineering Sciences; Fluid mechanics; Frequency domain analysis; Friction; Harmonic balance method; Harmonics; Input output analysis; JFM Papers; Mathematical analysis; Mathematics; Nonlinear analysis; Nonlinear systems; Nonlinearity; Optimization; Periodic variations; Perturbation; Physics; Plate boundaries; Three dimensional boundary layer; Wavelengths; boundary layer stability; transition to turbulence; OPTIMISATION AERODYNAMIQUE; INSTABILITE COUCHE LIMITE; TRANSITION LAMINAIRE TURBULENT
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2020.982

We extend linear input/output (resolvent) analysis to take into account nonlinear triadic interactions by considering a finite number of harmonics in the frequency domain using the harmonic balance method. Forcing mechanisms that maximise the drag are calculated using a gradient-based ascent algorithm. By including nonlinearity in the analysis, the proposed frequency-domain framework identifies the worst-case disturbances for laminar-turbulent transition. We demonstrate the framework on a flat-plate boundary layer by considering three-dimensional spanwise-periodic perturbations triggered by a few optimal forcing modes of finite amplitude. Two types of volumetric forcing are considered, one corresponding to a single frequency/spanwise wavenumber pair, and a multi-harmonic where a harmonic frequency and wavenumber are also added. Depending on the forcing strategy, we recover a range of transition scenarios associated with $K$-type and $H$-type mechanisms, including oblique and planar Tollmien–Schlichting waves, streaks and their breakdown. We show that nonlinearity plays a critical role in optimising growth by combining and redistributing energy between the linear mechanisms and the higher perturbation harmonics. With a very limited range of frequencies and wavenumbers, the calculations appear to reach the early stages of the turbulent regime through the generation and breakdown of hairpin and quasi-streamwise staggered vortices.