Tollmien-Schlichting waves near neutral stable curve
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| Format: | Preprint |
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2025
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| _version_ | 1866913678366867456 |
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| author | Chen, Qi Wu, Di Zhang, Zhifei |
| author_facet | Chen, Qi Wu, Di Zhang, Zhifei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_02258 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tollmien-Schlichting waves near neutral stable curve Chen, Qi Wu, Di Zhang, Zhifei Analysis of PDEs 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. |
| title | Tollmien-Schlichting waves near neutral stable curve |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.02258 |