Tollmien-Schlichting waves near neutral stable curve
Qi Chen, Di Wu, Zhifei Zhang
TL;DR
This work resolves the existence of a neutral Tollmien–Schlichting curve for high-Re boundary-layer flows by combining a triple-deck viewpoint with a refined Rayleigh–Airy iteration for the Orr–Sommerfeld equation. The authors introduce a modified Rayleigh operator $Ray_{\hat c}$ with $\hat c=c+i c_0$ to regularize the singularity on the neutral curve and develop coupled slow/fast modes $\phi_s,\phi_f$ via a Rayleigh–Airy scheme, supported by a comprehensive Airy theory including refined Langer transformation and Green-function constructions. A dispersion relation $\dfrac{\phi_s(0)}{\partial_Y\phi_s(0)}=\dfrac{\phi_f(0)}{\partial_Y\phi_f(0)}$ links the modes and yields the neutral curve, with detailed asymptotics showing $\alpha_{\text{low}}\sim A\nu^{1/8}$ and $\alpha_{\text{up}}\sim B\nu^{1/12}$, and scaling $\mathrm{Im}(c)$ near zero. The results illuminate the transition mechanism by revealing how diffusion and sublayer dynamics compete near the neutral curve and provide a rigorous framework for TS-wave stability analysis in boundary-layer flows.
Abstract
In this paper, we study the linear stability of boundary layer flows over a flat plate. Tollmien, Schlichting, Lin et al. found that there exists a neutral curve, which consists of two branches: lower branch $α_{low}(Re)$ and upper branch $α_{up}(Re)$. Here, $α$ is the wave number and $Re$ is the Reynolds number. For any $α\in(α_{low},α_{up})$, there exist unstable modes known as Tollmien-Schlichting (T-S) waves to the linearized Navier-Stokes system. These waves play a key role during the early stage of boundary layer transition. In a breakthrough work (Duke math Jour, 165(2016)), Grenier, Guo, and Nguyen provided a rigorous construction of the unstable T-S waves. In this paper, we confirm the existence of the neutral stable curve. To achieve this, we develop a more delicate method for solving the Orr-Sommerfeld equation by borrowing some ideas from the triple-deck theory. This approach allows us to construct the T-S waves in a neighborhood of the neutral curve.