Direct numerical simulation of complete transition to turbulence via first- and second-mode oblique breakdown at a high-speed boundary layer

Abstract

Complete transition to turbulence via first- and second-mode oblique breakdown in a high-speed boundary layer at Mach 4.5 is studied by direct numerical simulations (DNS) and linear stability theory (LST). The initial frequency and spanwise wavenumbers for both types of oblique breakdown are determined from LST. Then, DNS is employed to study the main features of the two oblique breakdown types in detail, which has rarely been discussed in previous studies. This includes the main flow structures and evolution of various modes during the linear, nonlinear, and breakdown stages, and both different and similar features for the two oblique breakdown types are summarized. Compared with only one type of low-speed streak existing for first-mode oblique breakdown, two types occur in the second-mode oblique breakdown, and the generation mechanism, evolution process, and role of the low-speed streaks are studied. Subsequently, the generation mechanism of both the heat transfer and skin-friction overshoot during both oblique breakdowns is illustrated with emphasis on the heat transfer overshoot for the second mode, which occurs at the laminar stage. Finally, both types of oblique breakdown are the likely path to a fully developed turbulent flow, although the unstable region for the second-mode oblique waves is short and for the first-mode oblique waves is amplified slowly.

Interactions between oblique second mode and oblique waves in a high-speed boundary layer

“…In LST, the mean quantities are solved for using the compressible Blasius equations (White 2006). This approach has been shown to give results in good agreement with those obtained through DNS as reported in Zhou et al (2022b). The nonlinear term is not considered in the LST.…”

“…An additional overshoot in heat transfer is also observed in the laminar region for cases S2 and S1, with case S1 exhibiting a greater overshoot due to the higher initial amplitude of the second mode. The additional heat transfer overshoot is caused by the dilatation dissipation generated by the saturated oblique second mode, as reported in works such as Zhu et al (2018) and Zhou et al (2022b). It should be noted that the additional heat transfer overshoot in the laminar region vanishes in case S3 due to the low initial amplitudes of the second modes (18,2) and (16,2), which cannot lead to strong dilatation dissipation.…”

“…Periodic boundary conditions are utilized on both sides of the domain. At the inlet, the profiles are solved using the compressible Blasius equations, validated in our previous study (Zhou et al 2022b).…”

“…Case S1 w is the second mode oblique breakdown with a weaker initial amplitude of 0.0016 than that of case S1. The second mode oblique breakdown has been previously studied in Franko & Lele (2013) and Zhou et al (2022b). The nonlinear stage of the second mode oblique breakdown results in the generation of streamwise streaks through the nonlinear interaction of oblique instability waves.…”

“…Nevertheless, it is important to note that the amplification rate of oblique second modes with small wave angles is comparable to that of the planar second mode (Husmeier & Fasel 2007; Zhou et al. 2022 b ). Additionally, the breakdown of a boundary layer is a three-dimensional process, and the oblique second mode can contribute to the three-dimensionality of the flow field, which cannot be achieved by the planar second mode.…”