Hot streak development in hypersonic boundary layer transition on a blunt cone with cooled walls: shock tunnel experiments and numerical simulations

The uncertainty of the boundary layer transition location on a hypersonic vehicle and the corresponding uncertainty in the surface heat flux and skin friction present major challenges for sustained hypersonic flight. One source of uncertainty is the mechanism governing breakdown of instabilities in boundary layers on geometries with highly-cooled walls (low $T_w / T_e$) relevant to flight conditions. An experimental investigation was carried out using a 7$^{\circ}$ half-angle, straight cone with a $2.5~mm$ nose radius and a Reynolds number based on nose radius of $Re_n \approx 5100$ at $0^{\circ}$ angle of attack. The tests were conducted at Mach 7.4 in the High Enthalpy Shock Tunnel Göttingen (HEG) DLR (2018) of the German Aerospace Center (DLR). Dominant primary instabilities, identified as second-mode waves in previous experiments Laurence (2016), were observed as peaks in the power spectral density plot of figure 1 for surface-mounted fast pressure transducers. Broadening of these peaks at the downstream locations on the cone was attributed to nonlinear interactions of instabilities within the boundary layer. Streaks were observed in the nonlinear interaction region, by means of temperature sensitive paint, as shown in figure 2. This suggests nonlinear interaction of the second-mode waves with secondary instabilities and is a unique result from a shock tunnel testing environment. Accompanying numerical investigations of the linear and nonlinear transition regime were carried out for the wind tunnel conditions and the circular cone geometry of the HEG experiments, in order to understand the role of the secondary instability in the transition process. The primary instability investigations using Linear Stability Theory (LST) and low-amplitude wave packet calculations confirmed that axisymmetric second-mode waves are indeed the dominant primary instability. The dominant frequency range obtained from experiments (\cref{PSD_map}) and numerical calculations are in good agreement. The strongly amplified axisymmetric second-mode waves suggest the possibility of a so-called fundamental resonance, where a large amplitude axisymmetric (primary) disturbance wave interacts (resonates) nonlinearly with a pair of lower amplitude (secondary) oblique disturbance waves of the same frequency. Therefore, fundamental resonance calculations were carried out for a wide range of azimuthal wavenumbers ($k_c$). The N-factors of the secondary disturbances reveal for which azimuthal wavenumbers the fundamental resonance is "strongest" (see \cref{fig:nfac_secondary}). A strong fundamental resonance gives rise to the nonlinear generation of steady streamwise modes, which were found to be responsible for the generation of "hot" streaks on the surface of a flared cone in the "cold-flow" experiments carried out at the Boeing/AFOSR Mach 6 Quiet Tunnel (BAM6QT) at Purdue University (\cite{Chynoweth2019}). Comparable streaks of high skin-friction have been observed in numerical simulations for the HEG conditions (\cref{fig:streaks_dns}). With the "controlled" simulations a total of 250 streaks around the circumference was obtained, which is in good agreement with the number of streaks observed in the HEG experiments ($220-230$). For the final version of this paper a detailed comparison between experimental measurements and numerical investigations will be provided for the various transition stages (linear and nonlinear).