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Scattering of Tollmien-Schlichting waves by localized roughness in transonic boundary layers

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Abstract

The laminar-turbulent transition in boundary-layer flows is often affected by wall imperfections, because the latter may interact with either the freestream perturbations or the oncoming boundary-layer instability modes, leading to a modification of the accumulation of the normal modes. The present paper particularly focuses on the latter mechanism in a transonic boundary layer, namely, the effect of a two-dimensional (2D) roughness element on the oncoming Tollmien-Schlichting (T-S) modes when they propagate through the region of the rapid mean-flow distortion induced by the roughness. The wave scattering is analyzed by adapting the local scattering theory developed for subsonic boundary layers (WU, X. S. and DONG, M. A local scattering theory for the effects of isolated roughness on boundary-layer instability and transition: transmission coefficient as an eigenvalue. Journal of Fluid Mechanics, 794, 68–108 (2006)) to the transonic regime, and a transmission coefficient is introduced to characterize the effect of the roughness. In the sub-transonic regime, in which the Mach number is close to, but less than, 1, the scattering system reduces to an eigenvalue problem with the transmission coefficient being the eigenvalue; while in the super-transonic regime, in which the Mach number is slightly greater than 1, the scattering system becomes a high-dimensional group of linear equations with the transmission coefficient being solved afterward. In the large-Reynolds-number asymptotic theory, the Kármán-Guderley parameter is introduced to quantify the effect of the Mach number. A systematical parametric study is carried out, and the dependence of the transmission coefficient on the roughness shape, the frequency of the oncoming mode, and the Kármán-Guderley parameter is provided.

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Acknowledgements

The author would like to acknowledge Dr.M.KRAVTSOVA of Imperial College London for valuable discussion on coding the steady triple-deck equations. The author is also grateful to one of the referees for providing detailed suggestions to improve the manuscript.

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Authors and Affiliations

  1. Department of Mechanics, Tianjin University, Tianjin, 300072, China

    Ming Dong

  2. Laboratory for High-Speed Aerodynamics, Tianjin University, Tianjin, 300072, China

    Ming Dong

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  1. Ming Dong
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Correspondence to Ming Dong.

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Project supported by the National Natural Science Foundation of China (No. 11772224)

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Dong, M. Scattering of Tollmien-Schlichting waves by localized roughness in transonic boundary layers. Appl. Math. Mech.-Engl. Ed. 41, 1105–1124 (2020). https://doi.org/10.1007/s10483-020-2622-6

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  • DOI: https://doi.org/10.1007/s10483-020-2622-6

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