A critical-layer analysis of the resonant triad in boundary-layer transition: nonlinear interactions

Abstract

A systematic theory is developed to study the nonlinear spatial evolution of the resonant triad in Blasius boundary layers. This triad consists of a plane wave at the fundamental frequency and a pair of symmetrical, oblique waves at the subharmonic frequency. A low-frequency asymptotic scaling leads to a distinct critical layer wherein nonlinearity first becomes important, and the critical layer's nonlinear, viscous dynamics determine the development of the triad.The plane wave initially causes double-exponential growth of the oblique waves. The plane wave, however, continues to follow the linear theory, even when the oblique waves’ amplitude attains the same order of magnitude as that of the plane wave. However, when the amplitude of the oblique waves exceeds that of the plane wave by a certain level, a nonlinear stage comes into effect in which the self-interaction of the oblique waves becomes important. The self-interaction causes rapid growth of the phase of the oblique waves, which causes a change of the sign of the parametric-resonance term in the oblique-waves amplitude equation. Ultimately this effect causes the growth rate of the oblique waves to oscillate around their linear growth rate. Since the latter is usually small in the nonlinear regime, the net outcome is that the self-interaction of oblique waves causes the parametric resonance stage to be followed by an ‘oscillatory’ saturation stage.

Weakly nonlinear stages of boundary-layer transition initiated by modulated Tollmien–Schlichting waves

“…At a fully nonlinear stage of development the disturbances exhibited rapid explosive growth, which was, again, qualitatively similar to that found by other theoretical approaches (see Spalart & Yang 1987;Zelman & Maslennikova 1993). Approximately the same theoretical approach as that of Goldstein & Lee (1992) was applied by Mankbadi et al (1993) to the Blasius boundary layer. In addition, they took into account viscous effects and the self-interaction of three-dimensional subharmonic waves.…”

“…For nonlinear stages of boundary-layer transition, the first theories of resonant interactions of TS waves were developed by Maseev (1968) and Craik (1971) and later by Herbert (1975), Volodin & Zelman (1978), Smith & Stewart (1987), Goldstein & Lee (1992), Mankbadi, Wu & Lee (1993) and Wu (1995) and in a great number of other subsequent studies; for a review see Borodulin, Kachanov & Koptsev (2002b), for example. The classical Craik's triad consists of a two-dimensional (2D) primary (fundamental) instability wave, with frequency f 1 , having relatively large amplitude and two weak oblique subharmonic waves, with frequency f s = f 1 /2, propagating at angles to the mean flow direction with the same values but opposite signs.…”

Effect of a Synthetic Jet Actuator on Airfoil Trailing Edge Noise

“…At high Reynolds number, the transition process becomes 3D (see for instance, Ref. [25]). However, at low-to-moderate Reynolds numbers, for which the current results are conducted, experimental and computational results in Ref.…”

Non-equilibrium, nonlinear critical layers in laminar-turbulent transition

“…Unfortunately, they assumed that the critical layer was passive. A correct asymptotic theory for resonanttriad of T-S waves was given by Mankbadi et al [29] and Wu [3~ who showed that contrary to what was assumed in Ref. [28], the critical layer is in fact the most active region, contributing the dominant nonlinear term.…”