Detuned resonances of Tollmien-Schlichting waves in an airfoil boundary layer: Experiment, theory, and direct numerical simulation

Abstract

Previous studies of the interaction between boundary layer streaks and Tollmien-Schlichting (TS) waves have shown puzzling effects. Streaks were shown to reduce the growth rate of primary TS waves and, thereby, to delay transition; however, they can also promote transition by inducing a secondary instability. The outcome of the interaction depends on the spanwise wavelength and intensity of the streaks as well as on the amplitude of the TS waves. A Floquet analysis of secondary instability is able to explain many of these features. The base state is periodic in two directions: it is an Ansatz composed of a saturated TS wave (periodic in x) and steady streaks (periodic in z). Secondary instability analysis is extended to account for the doubly periodic base flow. Growth rate computations show that, indeed, the streak can either enhance or diminish the overall stability of the boundary layer. The stabilizing effect is a reduction in the growth rate of the primary two-dimensional TS wave; the destabilizing effect is a secondary instability. Secondary instability falls into two categories, depending on the spanwise spacing of the streaks. The response of one category to perturbations is dominated by fundamental and subharmonic instability; the response of the other is a detuned instability. Previous studies of the interaction between boundary layer streaks and Tollmien-Schlichting TS waves have shown puzzling effects. Streaks were shown to reduce the growth rate of primary TS waves and, thereby, to delay transition; however, they can also promote transition by inducing a secondary instability. The outcome of the interaction depends on the spanwise wavelength and intensity of the streaks as well as on the amplitude of the TS waves. A Floquet analysis of secondary instability is able to explain many of these features. The base state is periodic in two directions: it is an Ansatz composed of a saturated TS wave periodic in x and steady streaks periodic in z. Secondary instability analysis is extended to account for the doubly periodic base flow. Growth rate computations show that, indeed, the streak can either enhance or diminish the overall stability of the boundary layer. The stabilizing effect is a reduction in the growth rate of the primary two-dimensional TS wave; the destabilizing effect is a secondary instability. Secondary instability falls into two categories, depending on the spanwise spacing of the streaks. The response of one category to perturbations is dominated by fundamental and subharmonic instability; the response of the other is a detuned instability.

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

“…The amplification factors of these modes turned out to be greater than those of the exact subharmonics. This phenomenon was also observed by Sartorius et al (2004) and recently in Würz et al (2012b), for a non-self-similar boundary layer, and an explanation was found based on theoretical analysis within the framework of linear and weakly nonlinear stability theories.…”

“…At s = 120 mm the quasi-subharmonic-mode amplitudes corresponding to the resonances (i) (figure 9b) and (ii) (figure 9d) display the appearance of very strong beats, typical of frequencydetuned resonances; see e.g. Kachanov & Levchenko (1984), Borodulin et al (2002d) and Würz et al (2012b). The physical nature of these beats (observed for sinusoidal fundamental waves) is associated with a variation of the initial phase relationship between the frequency-detuned quasi-subharmonic mode and the fundamental wave.…”

“…In particular, it was shown that for certain parameters the frequency-detuned resonance can lead to even greater amplitudes of the quasisubharmonic waves. Detailed investigations of properties of the frequency-detuned resonances have also been performed recently in experiments by Würz et al (2012b) for the case of a non-self-similar boundary layer on an aerofoil. In the latter paper the mechanism responsible for the greater amplification of the frequency-detuned modes has been explained.…”

“…,Borodulin et al (2002b) andWu et al (2007) andWürz et al (2012b) that waves under strong resonant interaction exhibit a phase synchronism. The time-traces shown in figure 9 are obtained by extracting the involved modes in Fourier space and performing a backward transformation.…”

“…The modulation of the TS waves during the transition process was simulated in several experimental investigations by means of excitation of either localized wave packets (see Gaster 1979;Cohen 1994;Breuer et al 1997;Medeiros & Gaster 1999a,b;Borodulin, Gaponenko & Kachanov 2000) or by modulated periodic TS waves usually consisting of a superposition of two or more harmonic waves (see Saric & Reynolds 1980;Kachanov et al 1985Kachanov et al , 1989Corke 1990;Würz et al 2012b). The corresponding theoretical models were studied within the framework of the weakly nonlinear approach (see e.g.…”