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Hypersonic boundary layer transition on a concave wall: stationary Görtler vortices

Published online by Cambridge University Press:  19 February 2019

X. Chen
Affiliation:
State Key Laboratory of Turbulence and Complex Systems, Collaborative Innovation Center for Advanced Aero-Engines, Peking University, Beijing 100871, PR China
G. L. Huang
Affiliation:
State Key Laboratory of Turbulence and Complex Systems, Collaborative Innovation Center for Advanced Aero-Engines, Peking University, Beijing 100871, PR China
C. B. Lee*
Affiliation:
State Key Laboratory of Turbulence and Complex Systems, Collaborative Innovation Center for Advanced Aero-Engines, Peking University, Beijing 100871, PR China
*
Email address for correspondence: [email protected]

Abstract

This study investigates the stability and transition of Görtler vortices in a hypersonic boundary layer using linear stability theory and direct numerical simulations. In the simulations, Görtler vortices are separately excited by wall blowing and suction with spanwise wavelengths of 3, 6 and 9 mm. In addition to primary streaks with the same wavelength as the blowing and suction, secondary streaks with half the wavelength also emerge in the 6 and 9 mm cases. The streaks develop into mushroom structures before breaking down. The breakdown processes of the three cases are dominated by a sinuous-mode instability, a varicose-mode instability and a combination of the two, respectively. Both fundamental and subharmonic instabilities are relevant in all cases. Multiple modes are identified in the secondary-instability stage, some of which originate from the primary instabilities (first and second Mack modes). We demonstrate that the first Mack mode can be destabilized to either a varicose-mode or sinuous-mode streak instability depending on its frequency and wavelength, whereas the second Mack mode undergoes a stabilizing stage before turning into a varicose mode in the 6 and 9 mm cases. An energy analysis reveals the stabilizing and destabilizing mechanisms of the primary instabilities under the influence of Görtler vortices, highlighting the role played by the spanwise production based on the spanwise gradient of the streamwise velocity in both varicose and sinuous modes. The effects introduced by the secondary streaks are examined by filtering the secondary streaks in two new simulations with nominally identical conditions to those of the 6 and 9 mm cases. Remarkably, the secondary streaks can destabilize the Görtler vortices, therefore advancing the transition. The stability theory results are in good agreement with those from direct numerical simulations.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

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Footnotes

Present address: Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China

References

Auvity, B., Etz, M. R. & Smits, A. J. 2001 Effects of transverse helium injection on hypersonic boundary layers. Phys. Fluids 13, 30253032.10.1063/1.1401813Google Scholar
Berridge, D. C., Chou, A., Ward, A. C., Steen, L. E., Gilbert, P. L., Juliano, T. J., Schneider, S. P. & Gronvall, J. E.2010 Hypersonic boundary-layer transition experiments in a Mach 6 quiet tunnel. AIAA Paper 2010-1061.Google Scholar
Bertolotti, F. P.1993 Vortex generation and wave-vortex interaction over a concave plate with roughness and suction. Tech. Rep. 93. ICASE Report.Google Scholar
Boiko, A. V., Ivanov, A. V., Kachanov, Y. S. & Mischenko, D. A. 2010 Steady and unsteady Görtler boundary-layer instability on concave wall. Eur. J. Mech. (B/Fluids) 29, 6183.10.1016/j.euromechflu.2009.11.001Google Scholar
Borodulin, V. I., Ivanov, A. V., Kachanov, Y. S. & Mischenko, D. A. 2018 Systematic study of distributed excitation of unsteady Görtler modes by freestream vortices. Eur. J. Mech. (B/Fluids) 68, 167183.10.1016/j.euromechflu.2017.11.008Google Scholar
Bottaro, A. & Luchini, P. 1999 Görtler vortices: are they amenable to local eigenvalue analysis? Eur. J. Mech. (B/Fluids) 18, 4765.10.1016/S0997-7546(99)80005-3Google Scholar
Chen, X., Zhu, Y. & Lee, C. 2017 Interactions between second mode and low-frequency waves in a hypersonic boundary layer. J. Fluid Mech. 820, 693735.10.1017/jfm.2017.233Google Scholar
Chevalerie, D. A., De La Fonteneau, A., Luca, L. D. & Cardone, G. 1997 Görtler-type vortices in hypersonic flows: the ramp problem. Exp. Therm. Fluid Sci. 15, 6981.10.1016/S0894-1777(97)00051-4Google Scholar
Chou, A., Ward, C. A. C., Letterman, L. E., Luersen, R. P. K., Borg, M. P. & Schneider, S. P.2011 Transition research with temperature-sensitive paints in the Boeing/AFOSR Mach-6 quiet tunnel. AIAA Paper 2010-3872.Google Scholar
Day, H. P., Herbert, T. & Saric, W. S. 1990 Comparing local and marching analyses of Görtler instability. AIAA J. 28, 10101015.10.2514/3.25158Google Scholar
Dempsey, L. J., Hall, P. & Deguchi, K. 2017 The excitation of Görtler vortices by free stream coherent structures. J. Fluid Mech. 826, 6096.10.1017/jfm.2017.380Google Scholar
Denier, J., Hall, P. & Seddougui, S. O. 1991 On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness. Phil. Trans. R. Soc. Lond. A 335, 5185.Google Scholar
Floryan, J. M. 1991 On the Görtler instability of boundary layers. Prog. Aerosp. Sci. 28, 235271.10.1016/0376-0421(91)90006-PGoogle Scholar
Franko, K. J. & Lele, S. K. 2013 Breakdown mechanisms and heat transfer overshoot in hypersonic zero pressure gradient boundary layers. J. Fluid Mech. 730, 491532.10.1017/jfm.2013.350Google Scholar
Fu, Y., Hall, P. & Blackaby, N. 1993 On the Görtler instability in hypersonic flows: Sutherland flow fluids and real gas effects. Phil. Trans. R. Soc. Lond. A 342, 325377.Google Scholar
Gaster, M. 1962 A note on the relation between temporally increasing and spatially increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222224.10.1017/S0022112062001184Google Scholar
Girgis, I. G. & Liu, J. T. C. 2006 Nonlinear mechanics of wavy instability of steady longitudinal vortices and its effect on skin friction rise in boundary layer flow. Phys. Fluids 18, 33073309.10.1063/1.2158430Google Scholar
Guo, Y. & Finlay, W. H. 1991 Splitting, merging and wavelength selection of vortices in curved and/or rotating channel flow due to Eckhaus instability. J. Fluid Mech. 228, 661691.Google Scholar
Hall, P. 1982 Taylor–Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475494.10.1017/S0022112082002596Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 4158.10.1017/S0022112083000968Google Scholar
Hall, P. 1990 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 27, 151189.10.1112/S0025579300012894Google Scholar
Hunt, J. C. R., Wray, A. A. & Moin, P. 1988 Eddies, streams, and convergence zones in turbulent flows. In Proceedings of the 1988 Summer Research Program, Center for Turbulence Research. pp. 193208. Stanford University.Google Scholar
Kravchenko, A. G., Choi, H. & Moin, P. 1993 On the relation of near-wall streamwise vortices to wall skin friction in turbulent boundary layers. Phys. Fluids 5, 33073309.10.1063/1.858692Google Scholar
Li, F., Choudhari, M., Chang, C.-L., Wu, M. & Greene, P.2010a Development and breakdown of Görtler vortices in high speed boundary layers. AIAA Paper 2010-0705.Google Scholar
Li, F. & Malik, M. R. 1995 Fundamental and subharmonic secondary instabilities of Görtler vortices. J. Fluid Mech. 297, 77100.10.1017/S0022112095003016Google Scholar
Li, X., Fu, D. & Ma, Y. 2008 Direct numerical simulation of hypersonic boundary-layer transition over a blunt cone. AIAA J. 46, 28992913.10.2514/1.37305Google Scholar
Li, X., Fu, D. & Ma, Y. 2010b Direct numerical simulation of hypersonic boundary layer transition over a blunt cone with a small angle of attack. Phys. Fluids 22, 025105.10.1063/1.3313933Google Scholar
Liang, X., Li, X., Fu, D. & Ma, Y. 2010 Effects of wall temperature on boundary layer stability over a blunt cone at Mach 7.99. Comput. Fluids 39, 359371.10.1016/j.compfluid.2009.09.015Google Scholar
Liu, J. T. C. 2008 Nonlinear instability of developing streamwise vortices with applications to boundary layer heat transfer intensification through an extended Reynolds analogy. Phil. Trans. R. Soc. Lond. A 366, 26992716.10.1098/rsta.2008.0057Google Scholar
Liu, J. T. C. & Sabry, A. S. 1991 Concentration and heat transfer in nonlinear Gortler vortex flow and the analogy with longitudinal momentum transfer. Phil. Trans. R. Soc. Lond. A 432, 112.Google Scholar
Liu, W. & Domaradzki, J. A. 1993 Direct numerical simulation of transition to turbulence in Görtler flow. J. Fluid Mech. 246, 267299.10.1017/S0022112093000126Google Scholar
de Luca, L., Cardone, G., de la Chevalerie, D. A. & Fonteneau, A. 1993 Goertler instability of a hypersonic boundary layer. Exp. Fluids 16, 1016.10.1007/BF00188500Google Scholar
Malatesta, V., Souza, L. F., Liu, J. T. C. & Kloker, M. J. 2015 Heat transfer analysis in a flow over concave wall with primary and secondary instabilities. Proc. IUTAM 14, 487495.10.1016/j.piutam.2015.03.077Google Scholar
Malik, M. R. 1990 Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86, 376413.10.1016/0021-9991(90)90106-BGoogle Scholar
Mitsudharmadi, H., Winoto, S. H. & Shah, D. A. 2005 Splitting and merging of Görtler vortices. Phys. Fluids 17, 124102.10.1063/1.2151227Google Scholar
Mitsudharmadi, H., Winoto, S. H. & Shah, D. A. 2006 Development of most amplified wavelength Görtler vortices. Phys. Fluids 18, 014101.10.1063/1.2160523Google Scholar
Paredes, P., Choudhari, M. M. & Li, F. 2017a Instability wave–streak interactions in a supersonic boundary layer. J. Fluid Mech. 831, 524553.10.1017/jfm.2017.630Google Scholar
Paredes, P., Choudhari, M. M. & Li, F.2017b Stabilization of hypersonic boundary layers by linear and nonlinear optimal perturbations. AIAA Paper 2017-3634.Google Scholar
Ren, J. & Fu, S. 2015a Secondary instabilities of Görtler vortices in high-speed boundary layer flows. J. Fluid Mech. 781, 388421.10.1017/jfm.2015.490Google Scholar
Ren, J. & Fu, S. 2015b Study of the discrete spectrum in a Mach 4.5 Görtler flow. Flow Turbul. Combust. 94, 339357.10.1007/s10494-014-9575-zGoogle Scholar
Ren, J., Fu, S. & Hanifi, A. 2016 Stabilization of the hypersonic boundary layer by finite-amplitude streaks. Phys. Fluids 28, 024110.10.1063/1.4941989Google Scholar
Rogenski, J. K., Souza, L. F. & Floryan, J. M. 2016 Non-linear aspects of Görtler instability in boundary layers with pressure gradient. Phys. Fluids 28, 124107.10.1063/1.4972241Google Scholar
Roghelia, A., Olivier, H., Egorov, I. & Chuvakhov, P. 2017 Experimental investigation of Görtler vortices in hypersonic ramp flows. Exp. Fluids 58, 139.Google Scholar
Saric, W. S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26, 379409.10.1146/annurev.fl.26.010194.002115Google Scholar
Schrader, L., Brandt, L. & Zaki, T. A. 2011 Receptivity, instability and breakdown of Görtler flow. J. Fluid Mech. 682, 362396.10.1017/jfm.2011.229Google Scholar
Schultz, M. P. & Volino, R. J. 2003 Effects of concave curvature on boundary layer transition under high freestream turbulence conditions. Trans. ASME J. Fluids Engng 125, 1827.10.1115/1.1522410Google Scholar
Sescu, A., Sassanis, V., Haywood, J. S. & Visbal, M.2015 Study of the impact of localized roughness elements on Görtler instabilities. AIAA Paper 2015-0275.Google Scholar
Sivasubramanian, J. & Fasel, H. F. 2015 Direct numerical simulation of transition in a sharp cone boundary layer at Mach 6: fundamental breakdown. J. Fluid Mech. 768, 175218.10.1017/jfm.2014.678Google Scholar
Souza, L. F. 2017 On the odd and even secondary instabilities of Görtler vortices. Theor. Comput. Fluid Dyn. 31, 405425.10.1007/s00162-017-0431-3Google Scholar
Souza, L. F., Mendonca, M. T., Medeiros, M. A. F. & Kloker, M. 2004 Seeding of Görtler vortices through a suction and blowing strip. J. Braz. Soc. Mech. Sci. Engng 26, 269279.Google Scholar
Spall, R. E. & Malik, M. R. 1989 Goertler vortices in supersonic and hypersonic boundary layers. Phys. Fluids 1, 18221835.10.1063/1.857508Google Scholar
Swearingen, J. D. & Blackwelder, R. F. 1987 The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255290.10.1017/S0022112087002337Google Scholar
Tandiono, Winoto, S. H. & Shah, D. A. 2008 On the linear and nonlinear development of Görtler vortices. Phys. Fluids 20, 094103.Google Scholar
Tandiono, Winoto, S. H. & Shah, D. A. 2009 Wall shear stress in Görtler vortex boundary layer flow. Phys. Fluids 21, 084106.Google Scholar
Wang, C. W. & Zhong, X.2001 Secondary Görtler instability in hypersonic boundary layers. AIAA Paper 2001-0273.Google Scholar
Wang, Q., Wang, Z. & Zhao, Y. 2018 Visualization of Görtler vortices in supersonic concave boundary layer. J. Vis. 21, 5762.10.1007/s12650-017-0443-5Google Scholar
White, F. M. 2006 Viscous Fluid Flow. McGraw-Hill.Google Scholar
Winoto, S. H., Tandiono, Shah, D. A. & Mitsudharmadi, H. 2008 Flows over concave surfaces: development of pre-set wavelength Görtler vortices. Intl J. Fluid Mech. Syst. 1 (1), 1023.10.5293/IJFMS.2008.1.1.010Google Scholar
Wu, X., Zhao, D. & Luo, J. 2011 Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances. J. Fluid Mech. 682, 66100.10.1017/jfm.2011.224Google Scholar
Xu, D., Zhang, Y. & Wu, X. 2017 Nonlinear evolution and secondary instability of steady and unsteady Görtler vortices induced by free-stream vortical disturbances. J. Fluid Mech. 829, 681730.10.1017/jfm.2017.572Google Scholar
Yu, X. & Liu, J. T. C. 1994 On the mechanism of sinuous and varicose modes in three-dimensional viscous secondary instability of nonlinear Görtler rolls. Phys. Fluids 6, 736750.10.1063/1.868312Google Scholar
Zhang, C. H., Tang, Q. & Lee, C. B. 2013 Hypersonic boundary-layer transition on a flared cone. Acta Mech. Sinica 29, 4853.10.1007/s10409-013-0009-2Google Scholar
Zhu, Y., Chen, X., Wu, J., Chen, S., Lee, C. & Gadelhak, M. 2018 Aerodynamic heating in transitional hypersonic boundary layers: role of second-mode instability. Phys. Fluids 30, 011701.10.1063/1.5005529Google Scholar

Chen et al. supplementary movie 1

Normalized temperature contour for case L3

Download Chen et al. supplementary movie 1(Video)
Video 8.7 MB

Chen et al. supplementary movie 2

Normalized temperature contour for case L6

Download Chen et al. supplementary movie 2(Video)
Video 8.2 MB

Chen et al. supplementary movie 3

Normalized temperature contour for case L9

Download Chen et al. supplementary movie 3(Video)
Video 9 MB