Weakly-nonlinear interactions of broadband Tollmien–Schlichting waves in a non-self-similar boundary layer on an airfoil

Abstract

Abstract The present investigation is devoted to initial nonlinear stages of laminar–turbulent transition initiated by broadband disturbances in an essentially non-self-similar boundary layer, developing on a two-dimensional airfoil. The measurements are performed at fully controlled disturbance conditions with respect to the distribution of the initial spectra of excited 3D (in general) Tollmien–Schlichting (TS) waves. The main attention is devoted to the investigation of resonant (and non-resonant) interactions and amplification of broadband low-amplitude 3D instability waves in the presence (and in absence) of one or two primary harmonic 2D TS-waves. The broadband, “noise-like” TS-waves are composed of controlled perturbations, pseudo-random in time and space, with a continuous frequency–wavenumber spectrum. These perturbations simulate typical background disturbances usually observed at so-called “natural” disturbance conditions. The harmonic primary TS-wave simulates a primary instability mode, which has been formed by the linear instability mechanism. In fact, this paper represents the third step in a set of detailed investigations (Wurz et al., 2012 [1] ) and Wurz et al., 2012 [2] ) of weakly-nonlinear stages of transition performed in the same base flow (on an airfoil) but at increased complexity of initial disturbance conditions. In contrast to the studies of periodic resonant regimes described in refs. Wurz et al. (2012) [1] and Wurz et al. (2012) [2] , no specific (preinstalled) disturbance frequencies or/and spanwise wavenumbers were excited in the boundary layer (except for the primary wave). So, the flow was free to amplify almost any mode of the frequency–wavenumber spectrum by means of any available mechanisms of TS-wave interaction. Such disturbance conditions are the closest to the so-called “natural” ones, i.e. to those observed typically for uncontrolled external perturbations. Four main groups of regimes of disturbance excitation were studied: (i) two harmonic fundamental waves and a pair of frequency detuned subharmonics, (ii) the low-amplitude broadband disturbances only and (iii) the low-amplitude broadband disturbances together with harmonic 2D primary TS-waves, and (iv) the high-amplitude broadband disturbances only. It is found that in case (i) the superposition of the resonances takes place in a nonlinear way and joint efforts of the two fundamental waves are similar to a fundamental-mode amplitude enhancement, which always leads to the increase of the subharmonic amplification rates. In case (ii) the perturbations develop downstream in agreement with the linear stability mechanism. In case (iii) strong resonant interactions between the broadband perturbations and the harmonic primary TS-wave is observed in a wide frequency range. In case (iv) very similar strong resonant interactions are observed between 2D and 3D spectral modes starting already at significantly lower amplitudes of the 2D modes in the frequency range of the primary fundamental modes. In cases (i), (iii) and (iv) the amplification is double-exponential (exponent in exponent), similar to that found in isolated tuned and detuned resonances of deterministic modes in papers Wurz et al. (2012) [1] and Wurz et al. (2012) [2] . For all resonantly interacting disturbances the phase synchronism conditions are found to be satisfied. Based on the cumulative results obtained in the framework of this intensive investigation, the most important observation is the predominant role of resonant TS-wave interactions in the non-self-similar boundary layer developing on a natural laminar flow airfoil section.