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

doi:10.1007/s10483-020-2622-6

Abstract

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 largeReynolds-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.

Key words: boundary layer, scattering, instability, Tollmien-Schlichting (T-S) wave, triple deck Chinese Library

2010 MSC Number:

Ming DONG. Scattering of Tollmien-Schlichting waves by localized roughness in transonic boundary layers. Applied Mathematics and Mechanics (English Edition), 2020, 41(7): 1105-1124.

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References

[1] MORKOVIN, M. V. Transition in open flow systems a reassessment. Bulletin of the American Physical Society, 39, 1882-1994(1994)
[2] REED, H. L., SARIC, W. S., and ARNAL, D. Linear stability theory applied to boundary layers. Annual Review of Fluid Mechanics, 28, 389-428(1996)
[3] MACK, L. Review of linear compressible stability theory. Stability of Time Dependent and Spatially Varying Flows, Springer-Verlag, New York, 164-187(1987)
[4] KACHANOV, Y. S. Physical mechanisms of laminar-boundary-layer transition. Annual Review of Fluid Mechanics, 26, 411-482(1994)
[5] ZHONG, X. L. and WANG, X. W. Direct numerical simulation on the receptivity, instability and transition of hypersonic boundary layers. Annual Review of Fluid Mechanics, 44, 527-561(2012)
[6] WU, X. S. Nonlinear theories for shear flow instabilities:physical insights and practical implications. Annual Review of Fluid Mechanics, 51, 451-485(2019)
[7] RUBAN, A. I. On Tollimien-Schlichting wave generation by sound (in Russian). Izvestija Akademii Nauk SSSR, 5, 44-52(1984)
[8] GOLDSTEIN, M. E. Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variations in surface geometry. Journal of Fluid Mechanics, 154, 509-530(1985)
[9] GOLDSTEIN, M. E. The evolution of Tollmien-Schlichting waves near a leading edge. Journal of Fluid Mechanics, 127, 59-81(1983)
[10] 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(2016)
[11] WÖRNER, A., RIST, U., and WAGNER, S. Humps/steps influence on stability characteristics of two-dimensional laminar boundary layer. AIAA Journal, 41, 192-197(2003)
[12] RIZZETTA, D. P. and VISBAL, M. R. Numerical simulation of excrescence generated transition. AIAA Journal, 52, 385-397(2014)
[13] MARXEN, O., IACCARINO, G., and SHAQFEH, E. Disturbance evolution in a Mach 4.8 boundary layer with two-dimensional roughness-induced separation and shock. Journal of Fluid Mechanics, 648, 435-469(2010)
[14] FONG, K. D., WANG, X. W., and ZHONG, X. L. Numerical simulation of roughness effect on the stability of a hypersonic boundary layer. Computers and Fluids, 96, 350-367(2014)
[15] FONG, K. D., WANG, X. W., and ZHONG, X. L. Parametric study on stabilization of hypersonic boundray-layer waves using 2-D surface roughness. 53rd AIAA Aerospace Sciences Meeting, AIAA Paper 2015-0837, Florida (2015)
[16] STEWARTSON, K. and WILLIAMS, P. G. Self-induced separation. Proceedings of the Royal Society A:Mathematical Physical and Engineering Sciences, 312, 181-206(1969)
[17] MESSITER, A. F. Boundary-layer flow near the trailing edge of a flat plate. SIAM Journal on Applied Mathematics, 18, 241-257(1970)
[18] SMITH, F. T. Laminar flow over a small hump on flat plate. Journal of Fluid Mechanics, 57, 803-824(1973)
[19] SMITH, F. T. On the non-parallel flow stability of the Blasius boundary layer. Proceedings of the Royal Society London A, 366, 91-109(1979)
[20] DONG, M. and WU, X. S. Local scattering theory and the role of an abrupt change in boundarylayer instability and acoustic radiation. 46th AIAA Fluid Dynamics Conference, AIAA Paper 2016-3194, Washington, D. C. (2016)
[21] DONG, M. and WU, X. S. Generation of convective instability modes in the near-wake flow of the trailing edge of a flat plate. 8th AIAA Theoretical Fluid Mechanics Conference, AIAA Paper 2017-4022, Colorado (2017)
[22] DONG, M. and ZHANG, A. Scattering of Tollmien-Schlichting waves as they pass over forward-/backward-facing steps. Applied Mathematics and Mechanics (English Edition), 39(10), 1411-1424(2018) https://doi.org/10.1007/s10483-018-2381-8
[23] HUANG, Z. F. and WU, X. S. A local scattering approach for the effects of abrupt changes on boundary-layer instability and transition:a finite-Reynolds-number formulation for isolated distortions. Journal of Fluid Mechanics, 822, 444-483(2017)
[24] ZHAO, L., DONG, M., and YANG, Y. G. Harmonic linearized Navier-Stokes equatioin on describing the effect of surface roughness on hypersonic boundary-layer transition. Physics of Fluids, 31, 034108(2019)
[25] TIMOSHIN, S. N. Asymptotic form of the lower branch of the neutral curve in transonic boundary layer. Tsagi Uchenye Zapiski, 21, 50-57(1990)
[26] BOWLES, R. I. and SMITH, F. T. On boundary-layer transition in transonic flow. Journal of Engineering Mathematics, 27, 309-342(1993)
[27] BOGDANOV, A. N., DIESPEROV, V. N., ZHUK, V. I., and CHERNYSHEV, A. V. Triple-deck theory in transonic flows and boundary layer stability. Computational Mathematics and Mathematical Physics, 50, 2095-2108(2010)
[28] RUBAN, A. I., BERNOTS, T., and KRAVTSOVA, M. A. Linear and nonlinear receptivity of the boundary layer in transonic flows. Journal of Fluid Mechanics, 786, 154-189(2016)
[29] SAVENKOV, I. V. Effect of surface elasticity on the transformation of acoustic disturbances into Tollmien-Schlichting waves in a boundary layer at transonic free-stream velocities. Computational Mathematics and Mathematical Physics, 46, 907-913(2006)
[30] PERRAUD, J., ARNAL, D., and KUEHN, W. Laminar-turbulent transition prediction in the presence of surface imperfections. International Journal of Engineering Systems Modelling and Simulation, 6, 162-170(2014)
[31] BEGUET, S., PERRAUD, J., FORTE, M., and BRAZIER, J. P. Modeling of transverse gaps effects on boundary-layer transition. Journal of Aircraft, 54, 794-801(2017)
[32] EDELMANN, C. A. and RIST, U. Impact of forward-facing steps on laminar-turbulent transition in transonic flows. AIAA Journal, 53, 2504-2511(2015)
[33] ZAHN, J. and RIST, U. Impact of deep gaps on laminar-trubulent transition in compressible boundary-layer flow. AIAA Journal, 54, 66-76(2016)
[34] ZAHN, J. and RIST, U. Active and natural suction at forward-facing steps for delaying laminarturbulent transition. AIAA Journal, 55, 1333-1344(2017)
[35] THOMAS, C., MUGHAL, S. M., ROLAND, H., ASHWORTH, R., and MARTINEZ-CAVA, A. Effect of small surface deformations on the stability of Tollmien-Schlichting disturbances. AIAA Journal, 56, 2157-2165(2018)
[36] THOMAS, C., MUGHAL, S., and ASHWORTH, R. Development of Tollmien-Schlichting disturbances in the presence of laminar separation bubbles on an unswept infinite wavy wing. Physical Review Fluids, 2, 043903(2017)
[37] LIN, C. C. On the stability of two-dimensional parallel flows, part III:stability in a viscous fluid. Quarterly of Applied Mathematics, 3, 277-301(1946)
[38] KRAVTSOVA, M. A., ZAMETAEV, V. B., and RUBAN, A. I. An effective numerical method for solving viscous-inviscid interaction problems. Philosophical Transaction of the Royal Society A, 363, 1157-1167(2005)
[39] WU, X. S. On generation of sound in wall-bounded shear flows:back action of sound and global acoustic coupling. Journal of Fluid Mechanics, 689, 279-316(2011)
[40] WU, X. S. and HOGG, L. Acoustic radiation of Tollmien-Schlichting waves as they undergo rapid distortion. Journal of Fluid Mechanics, 550, 307-347(2006)