Cross-flow linear instability in compressible boundary layers over a flat plate

Abstract

The linear instability of compressible boundary layers over a flat plate in the presence of parameterized crossflow has been investigated by means of linear stability theory. A family of boundary layer with crossflow is obtained as the base flow from the compressible Falkner-Skan-Cooke (FSC) flow model. Two factors, i.e., the local swept angle and the pressure-gradient parameter, are designed to create the crossflow with different directions and magnitudes, which further results in the variation of the cross-flow instability. Modal properties related to the cross-flow instability are emphasized. The association between the cross-flow mode and the Mack's mode is clarified by extending the base flow from two dimension to three dimension. The cross-flow instability is discovered to be integrated with the slow-mode instability, i.e., the instability related to the Tollmien-Schlichting (T-S) mode or the first mode, and it can hardly be distinguished as an individual mode in most cases. The effects of Mach number, pressure gradient, local sweep and wall temperature are studied. The behaviors of the cross-flow instability under such effects resemble those of the slow-mode instability in the two-dimensional boundary layers. It is found that the unstable modes in the three-dimensional boundary layers are mainly affected by the streamwise pressure gradient and the crossflow per se. Specifically, the first mode is much more sensitive to the crossflow than the second mode. As a result, more marked variations are commonly observed for the first mode in the boundary layers with crossflow.

Boundary-layer instability on a highly swept fin on a cone at Mach 6

“…It is difficult to declare an exact wavelength where the two groups could be considered without overlap, given the smooth variation between wavelengths. This is similar to the study presented by Liu (2022) where, even in simplistic Falkner–Skan–Cooke flow, cases arose where no clear distinction could be made between the first mode and cross-flow instabilities. It is also possible that both instabilities, the cross-flow instability and the currently unidentified ‘large-wavelength’ instability, could coexist at all wavelengths, and one instability is simply dominant over the other for different wavelengths.…”

“…This is an extremely large cross-flow velocity relative to other examples in the literature that investigate cross-flow. While not often directly reported, examples include Liu (2022), who studies a series of Falkner-Skan-Cooke boundary layers with a maximum cross-flow velocity of 8 % in Mach 4.5 flow, and Chen et al (2022b), who report the maximum cross-flow velocity as ∼7.5 % in their case studying high-enthalpy boundary layers in Mach 16 flow over a swept parabola. As an extreme case, Mack (1984) gives an incompressible example where cross-flow instability occurs when the cross-flow velocity is as small as 0.7 % of U ∞ .…”

“…Several studies demonstrate how the influence of Mach number exacerbates this issue. For example, Liu (2022) showed that the unstable frequency and wavenumber regions representing the cross-flow and first-mode instabilities in Falkner–Skan–Cooke flow are distinct at incompressible speeds, but can no longer be distinguished at Mach 4.5. Similarly, Özgen & Kırcalı (2008) calculate neutral stability curves for flat-plate boundary layers up to Mach 10, and show how the unstable regions belonging to the first and second modes merge into one indistinguishable neutral curve at Mach 10.…”