Nonlinear regimes of spanwise modulated waves in plane Poiseuille flow

Abstract

Wave modulation is an unavoidable ingredient of natural transition and wavepackets composed of a continuous range of frequencies and wavenumbers are considered as a good model for it. Conclusions regarding wavepacket nonlinear regimes are essentially based on comparison of the dominant mode in the modulated signal nonlinear bands with predictions of the most unstable mode in the corresponding unmodulated cases. The modulated signal bands are very broad and establishing the dominant mode is difficult. If the Reynolds number changes along the packet evolution, the bands also change to adapt to local conditions, which further hinders data interpretation and weakens the conclusions. In view of this, a study at a constant Reynolds number is proposed, the Poiseuille plane flow being chosen as the base flow. The flow choice also allowed an investigation of the phenomenon at different positions in the stability loop, an aspect that has never been addressed before. The work was numerical. Only spanwise modulation was considered and two different regimes were observed. Close to the first branch of the instability loop the packet splits into two patches. Oblique transition was the dominant nonlinear mechanism, which required spanwise interaction of packets. Elsewhere the dominant mechanism was fundamental instability (or$K$-type), which was fed by a previous nonlinear phenomenon and led to a subsequent one. The analysis involved comparison of growth rates, threshold amplitudes and amplitude scalings among other aspects, for modulated and corresponding unmodulated cases. Perfect agreement was found if appropriate variables were used, which enabled firm conclusions to be drawn about the phenomena investigated. This level of agreement was only possible because of the constant Reynolds number character of the flow, but the main conclusions are applicable to other wall shear flows.