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

Abstract

Weakly nonlinear interactions involving amplitude-modulated Tollmien-Schlichting waves in an incompressible, two-dimensional aerofoil boundary layer are investigated experimentally. Selected resonant regimes are examined with emphasis on the regimes where more than one fundamental Tollmien-Schlichting (TS) wave is present in the flow. The experiments were performed on an NLF-type aerofoil section for glider applications. Disturbances with controlled frequency-spanwise-wavenumber spectra were excited in the boundary layer and studied by phase-locked hot-wire measurements. The results show that nonlinear mechanisms connected with the steepening of the primary TS wave modulation do not play any significant role in the transition scenarios studied. It is also shown that modulations of the twodimensional fundamental waves tend to generate additional modes at modulation frequency. These low-frequency disturbances are found to be produced by a nonresonant quadratic combination of spectral components of the primary, modulated TS wave. The investigations show that the efficiency of the process is higher for three-dimensional low-frequency modes in comparison with two-dimensional modes. Thus, the emergence of three-dimensionality for the low-frequency waves does not require any resonant interactions. In a subsequent nonlinear stage, the self-generated detuned subharmonics are found to be strongly amplified due to resonant interactions with the primary TS waves. The sequence of weakly nonlinear mechanisms found and investigated here seems to be the most likely route to the laminar-turbulent transition, at least for two-dimensional boundary layers of aerofoils with a long extent of laminar flow and in a 'natural' disturbance environment.

On the disturbance evolution downstream of a cylindrical roughness element

“…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).…”

“…Further downstream the mean flow relaxes towards a uniform state and also the linear stability characteristics are recovered for the fundamental modes. Therefore, it seems appropriate to compare the present experimental results at least in the far roughness wake to the previous theoretical (Wu et al 2007) and experimental (de Paula et al 2013) findings. The experiments by de Paula et al (2013) confirm the present observations, in that the shape of the f ∆ modes compares well with linear theory for three-dimensional modes.…”

“…The loudspeakers are installed in an airtight box and balanced by a static pressure port at the streamwise slit position. A 128-channel digital signal generator with a memory of 8192 points per channel provides the signal for the loudspeakers (de Paula et al 2010(de Paula et al , 2013. A common quartz-based clock is used to trigger all 128 channels and provides the sample clock for the data acquisition in order to assure strictly phase-locked measurements (compare § 2.3).…”

“…Recently, experimental investigations by de Paula et al (2013) have highlighted the role of low-frequency disturbances in the transition development initiated by modulated TS waves. This scenario seems closely related to the present experiments (compare § 3), for which an oblique wave train evolves downstream of a medium roughness height element.…”

Nonlinear Mode Interactions in the Wake of a Medium Height Roughness Element

“…Upstream of the roughness controlled TS-wave modes are excited at x/c = 0.1 by a small slit (width 0.2 mm), which is integrated into the model surface. Beneath the slit, 128 tubes are connected to individually controllable loudspeakers, which are driven by a custom signal generator 15 . In order to ensure phase synchronism for the disturbance excitation the same clock is used for all 128 channels of the signal generator.…”