Nonlinear instabilities of a non-self-similar boundary layer on an airfoil: Experiments, DNS, and theory

Abstract

Transition prediction of a laminar boundary layer developing on a two-dimensional airfoil section at pure subsonic speeds is still a challenge, because the commonly used semi-empirical prediction tools are mainly based on results derived from the linear stability theory (LST). This approach completely neglects non-linear interactions at later stages of the transition development and therefore provides limited accuracy for cases when the non-linear part has significant length in comparison to the linear stage. Nevertheless, the later one can be of significant importance for airfoil design. The onset and further development of non-linear, in general resonant wave interactions is depending on the pre-history of the boundary layer and the specific linear and non-linear stability characteristics driven by the pressure gradient, which varies in streamwise direction for typical airfoil sections. The variation of the pressure gradient leads to a non self-similar boundary layer development connected with a continuous change of the resonance conditions for wave interactions. The present paper is devoted to the detailed experimental, numerical and theoretical study of weakly nonlinear subharmonic resonant interactions of Tollmien–Schlichting waves in such a two-dimensional non self-similar boundary layer on an airfoil. The experimental approach is based on phase locked hot-wire measurements under controlled disturbance conditions in a low turbulence wind tunnel at a Reynolds number of Re=0.7×10 6. Direct numerical simulations (DNS) based on the vorticity–velocity formulation of the complete Navier–Stokes equations are utilized to provide a detailed comparison for the test cases. The results of weakly non-linear theory (WNT) enabled a profound understanding of the underlying physical mechanisms observed in the experiments and simulations. The joint study provides a complete set of data, starting from the base flow and stability characteristics up to non-linear disturbance development with different involved wave triplets. The phase synchronization mechanism is investigated in detail as well as the resonance efficiency for tuned cases with respect to the exact subharmonic frequency. In general, a good agreement between the experiment, DNS and WNT is found. The essential non-self-similarity of the airfoil boundary layer does not prevent strong resonant interactions and in accordance with investigations in self-similar flows, rapid, close to double-exponential amplification of subharmonic modes is observed. At the resonant stage, the phase-synchronization condition is shown to be satisfied, which provides equal phase speeds for all involved modes. For tuned cases, the initial phase relation between fundamental and subharmonic modes can lead, partly, to a suppression of the subharmonic wave amplification. Variation of the fundamental wave frequency shows that the integral resonance efficiency is significantly decreased with the reduction of the frequency. This effect is explained as a frequency dependence of the nonlinear coupling coefficients as well as the detuning of the subharmonic-fundamental phase speed associated mainly with the base-flow non-self-similarity. Depending on the frequency content of the initial disturbances this can lead to a direct influence on the corresponding transition position.