Effect of Phase Difference between Instability Modes on Boundary Layer Transition

Abstract

Laminar-to-turbulent transition is a crucial phenomenon in boundary layer flow. One natural transition scenario due to sub-harmonic resonance is investigated here to study effects of the phase difference between instabilities on the flow stability. Non-linear interaction between a two-dimensional fundamental Tollmien–Schlichting (TS) and three-dimensional sub-harmonic oblique waves is numerically analyzed using the concept of parametric resonance. A simple system with fundamental and sub-harmonic modes described with Mathieu's equation is studied first to understand the effect of the phase difference on the amplitude growth of the sub-harmonic mode under the influence of a fundamental-frequency mode. Then, sub-harmonic resonance in boundary layer is investigated with the parametric equations derived from the Navier-Stokes equations. The current parametric analysis provides the phase-locked condition where the growth rate of the sub-harmonic mode is the maximum under the influence of the fundamental mode. Comparison to the relevant analysis of the Parabolized Stability Equation (PSE) indicates that an arbitrary initial phase difference evolves to the phase-locked condition obtained in the current study.