Interactions between second mode and low-frequency waves in a hypersonic boundary layer

Abstract

The stability of a hypersonic boundary layer on a flared cone was analysed for the same flow conditions as in earlier experiments (Zhang et al., Acta Mech. Sinica, vol. 29, 2013, pp. 48–53; Zhu et al., AIAA J., vol. 54, 2016, pp. 3039–3049). Three instabilities in the flared region, i.e. the first mode, the second mode and the Görtler mode, were identified using linear stability theory (LST). The nonlinear-parabolized stability equations (NPSE) were used in an extensive parametric study of the interactions between the second mode and the single low-frequency mode (the Görtler mode or the first mode). The analysis shows that waves with frequencies below 30 kHz are heavily amplified. These low-frequency disturbances evolve linearly at first and then abruptly transition to parametric resonance. The parametric resonance, which is well described by Floquet theory, can be either a combination resonance (for non-zero frequencies) or a fundamental resonance (for steady waves) of the secondary instability. Moreover, the resonance depends only on the saturated state of the second mode and is insensitive to the initial low-frequency mode profiles and the streamwise curvature, so this resonance is probably observable in boundary layers over straight cones. Analysis of the kinetic energy transfer further shows that the rapid growth of the low-frequency mode is due to the action of the Reynolds stresses. The same mechanism also describes the interactions between a second-mode wave and a pair of low-frequency waves. The only difference is that the fundamental and combination resonances can coexist. Qualitative agreement with the experimental results is achieved.

Newly identified principle for aerodynamic heating in hypersonic flows

“…An RSFV picture of flow structures and the DNS-calculated temperature field near HS are both presented in figure 7. Such a comparison is reasonable because the RSFV actually reflects the temperature distribution in the field of view (Chen et al 2017). As shown, the boundary layer is initially laminar in the upstream region, but then regular rope-like structures appear at x/L = 0.55, which is typical of second-mode instability waves because their wavelength is twice the boundary layer thickness.…”

“…As shown, DNS and PCB agree very well with each other whereas the PSE data, especially for low-frequency waves, are higher than the other two. As discussed by Chen et al (2017), one possible reason is that PSE purely considers the two fastest-growing modes whereas both the experiment and DNS involve a considerable level of random disturbance in the oncoming stream. The latter two growth ratios are integrative results of a wide spectrum.…”

“…The value of the latter increases to nearly 75 % that of w θ1 at HS for PIV but only 52 % for DNS because transition is not completed in DNS. Note that low-frequency waves continuously grow until the end of the model because it belongs to a vortical instability, as discussed by Chen et al (2017). It benefits from w θ 2 = µ θ 2 (green); w ω = µω 2 (magenta).…”

“…Streamwise developments of PCB frequency spectra at (a) Re unit = 5.4 × 6 m −1 , (b) Re unit = 7.6 × 10 6 m −1 , (c) Re unit = 9.7 × 10 6 m −1 and (d) Re unit = 11.7 × 6 m −1 . 2013;Zhu et al 2016;Chen, Zhu & Lee 2017)…”

From primary instabilities to secondary instabilities in Görtler vortex flows

“…Rather, it appears to be a common process during the boundary layer transition. Previous studies [19][20][21] indicate that the primary-secondary instability transfer could have at least two routes. The first route is through continuous modulation where the mode identity remains unchanged in the sense of local stability analysis during the transformation.…”