On the catalytic role of the phase-locked interaction of Tollmien–Schlichting waves in boundary-layer transition

Abstract

This paper is concerned with the nonlinear interaction between a planar and a pair of oblique Tollmien–Schlichting (T-S) waves which are phase-locked in that they travel with (nearly) the same phase speed. The evolution of such a disturbance is described using a high-Reynolds-number asymptotic approach in the so-called ‘upper--branch’ scaling regime. It follows that there exists a well-defined common critical layer (i.e. a thin region surrounding the level at which the basic flow velocity equals the phase speed of the waves to leading order) and the dominant interactions take place there. The disturbance is shown to evolve through several distinctive stages. In the first of these, the critical layer is in equilibrium and viscosity dominated. If a small mismatching exists in the phase speeds, the interaction between the planar and oblique waves leads directly to super-exponential growth/decay of the oblique modes. However, if the modes are perfectly phase-locked, the interaction in the first instance affects only the phase of the amplitude function of the oblique modes (so causing rapid wavelength shortening), while the modulus of the amplitude still evolves exponentially until the wavelength shortening produces a back reaction on the modulus (which then induces a super-exponential growth). Whether or not there is a small mismatch or a perfect match in the phase speeds, once the growth rate of the oblique modes becomes sufficiently large, the disturbance enters a second stage, in which the critical layer becomes both non-equilibrium and viscous in nature. The oblique modes continue to experience super-exponential growth, albeit of a different form from that in the previous stages, until the self-interaction between them, as well as their back effect on the planar mode, becomes important. At that point, the disturbance enters a third, fully interactive stage, during which the development of the disturbance is governed by the amplitude equations with the same nonlinear terms as previously derived for the phase-locked interaction of Rayleigh instability waves. The solution develops a singularity, leading to the final stage where the flow is governed by fully nonlinear three-dimensional inviscid triple-deck equations. The present work indicates that seeding a planar T-S wave can enhance the amplification of all oblique modes which share approximately its phase speed.

On the disturbance evolution downstream of a cylindrical roughness element

“…It is seen that the difference mode is amplified by a linear instability at first. An order-of-magnitude analysis for the onset of the first nonlinear stage described by Wu et al (2007) is in agreement with the experimental findings. Moreover, the nonlinear interactions in the experiments occur in the critical layer for modes with approximately equal phase speeds in agreement with the phase-locked interaction mechanism described by Wu et al (2007).…”

“…An order-of-magnitude analysis for the onset of the first nonlinear stage described by Wu et al (2007) is in agreement with the experimental findings. Moreover, the nonlinear interactions in the experiments occur in the critical layer for modes with approximately equal phase speeds in agreement with the phase-locked interaction mechanism described by Wu et al (2007). Hence, the most dominant mechanism for the growth of the low-frequency difference modes in the first, weakly nonlinear stage is associated with a 'subharmonic-type frequency-detuned resonance' between the difference and the fundamental modes (de Paula et al 2013).…”

“…Corke & Mangano 1989;Kachanov 1994), this phase-locked wave interaction mechanism works under less restrictive conditions. Therefore, Wu et al (2007) suggest that the above-described mechanism might be responsible for a growth of low-frequency, broadband disturbances in the nonlinear stages of transition. In recent experimental investigations by de Paula et al…”

“…Accordingly, the low-frequency disturbances were understood as subsequent nonlinear mode interactions. A further explanation for the growth of low-frequency modes is given by the concept of phase-locked wave interactions, which was introduced for the Rayleigh instability (Wu & Stuart 1996) and more recently applied to a flat-plate laminar boundary layer (Wu, Stuart & Cowley 2007). Based on a high-Reynolds-number asymptotic theory, an interaction of a planar and an oblique TS mode is considered, which share approximately the same phase speed and result in the generation of modes at the sum and the difference frequency.…”

Nonlinear interaction of wind-driven oblique surface waves and parametric growth of lower frequency modes

“…The phase-locked interaction of Wu & Stewart (1996) can occur in any shear flow which is inviscidly unstable. A multi-stage nonlinear interaction between the phase-locked triad of a two-dimensional and a pair of oblique Tollmien-Schlichting waves in boundary-layer transition was studied by Wu, Stewart & Cowley (2007). In both phase-locked and sub-harmonic resonant-triad interactions in incompressible flows, the two-dimensional mode was the most amplified linear mode that produced the initial nonlinear interaction.…”