Bispectral analysis of nonlinear dynamics and its connection to skin friction in heated/cooled transonic boundary layer transition

We study the impacts of wall heating/cooling on the nonlinear dynamics in the boundary layer transition and its relation to skin friction. Direct numerical simulations (DNS) of the transition over a 10%-heated/cooled flat plate are conducted. The transition is initiated by a tiny planer wave, a counterpart of the Tollmien-Schlichting (TS) wave in an incompressible flow, and a subharmonic oblique wave (OW), which can cause the H-type transition. The DNS results show a distinct difference in the streamwise growth of skin friction (Cf) between the adiabatic/heated and cooled cases. For the former cases, there is a monotonically steep rise in Cf in the nonlinear (transitional) region, whereas the cooled case shows a two-stage rise: gradual at first, then steep. We identify that this difference results from suppressing the TS mode's linear growth with wall cooling; its amplitude is up to about ten times smaller than the heated case. We quantify the degree of nonlinear interaction of modes using the bispectrum computed by bispectral mode decomposition (BMD). The bispectrum reveals that the TS mode plays a marginal role in the nonlinear dynamics for the cooled case in contrast to the adiabatic/heated cases, while the OW mode and steady stream (ST) mode nonlinearly generated by OW mode play an essential role. Moreover, the BMD suggests that the ST mode's self-interaction is relevant to the early stage with the gradual rise in Cf while the OW mode's self-interaction to the late stage with the rapid rise in Cf.