Nonlinear dynamics involving multiple modes in high-speed transitional boundary layer

Abstract

Extensive studies have investigated the transition mechanism of boundary layers initiated by a single primary instability. In a real-world scenario, however, multiple primary instabilities of different physical nature would coexist and generate more complicated stages of mode--mode interactions. For this scenario, conventional secondary stability analysis may not be applicable. In this work, a general framework is established to decompose the input--output system and to quantify the transfer of energy involving various modes. The linearized governing equation with nonlinear forcings is applied in a Mach 6 boundary layer, where two different types of primary instabilities are added simultaneously. As the primary-wave amplitudes increase to certain threshold, the nonlinear effect causes the saturation of the second mode and secondary growth of the first mode. In the generation stage of each higher-order mode, a specific leading triadic forcing term can be identified. These higher-order waves manifest solely in response to the identified dominant forcing during their generation. At the moderate and late transitional stages, the forcings are, however, not equally transferred to the response via the resolvent operator. In other words, the base-flow-associated resolvent operator exerts different levels of `leverage' to transfer different forcings to responses. The nonlinear energy transfer via triadic forcings also drives the higher-order instability to inherit physical signature from the associated lower-order instability. Finally, the interplay between secondary/tertiary waves and primary waves occurs notably earlier then one may expect, namely before transition onset or in the early transitional region. This differs from the traditional secondary instability analysis that a large-amplitude primary wave is developed first to perform the bi-global analysis in the distorted base flow.

Figures (21)

• Figure 1: Contours of optimal gain (normalized by the maximum) in the parameter space of the angular frequency and the spanwise wavenumber.
• Figure 2: Triadic resonance states involving the primary modes and several representative higher-order modes. Here (+) denotes the sum interaction and (-) denotes the difference interaction.
• Figure 3: Three-dimensional visualization of the flow field based on the DNS results.
• Figure 4: Spatial distributions of primary waves visualized by the real parts of the streamwise velocity, temperature and pressure components. The respective solid, dashed and dotted lines in the plots represents the boundary layer edge, the generalized inflection point and the sonic line. Contour bar ($\blacksquare$$\blacksquare$$\blacksquare$) ranges between $\pm0.8\times{\rm{max}}(|\hat{q}|)$ of each field.
• Figure 5: Development of the Chu's energy of the two primary waves.