Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-30T14:45:40.109Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear evolution and secondary instability of steady and unsteady Görtler vortices induced by free-stream vortical disturbances

Published online by Cambridge University Press:  25 September 2017

Dongdong Xu ,
Yongming Zhang  and
Xuesong Wu Open the ORCID record for Xuesong Wu [Opens in a new window]
Dongdong Xu
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, China
Yongming Zhang
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, China
Xuesong Wu*
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, China Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
*
†Email address for correspondence: x.wu@ic.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the nonlinear development and secondary instability of steady and unsteady Görtler vortices which are excited by free-stream vortical disturbances (FSVD) in a boundary layer over a concave wall. The focus is on low-frequency (long-wavelength) components of FSVD, to which the boundary layer is most receptive. For simplification, FSVD are modelled by a pair of oblique modes with opposite spanwise wavenumbers $\pm k_{3}$, and their intensity is strong enough (but still of low level) that the excitation and evolution of Görtler vortices are nonlinear. For the general case that the Görtler number $G_{\unicode[STIX]{x1D6EC}}$ (based on the spanwise wavelength $\unicode[STIX]{x1D6EC}$ of the disturbances) is $O(1)$, the formation and evolution of Görtler vortices are governed by the nonlinear unsteady boundary-region equations, supplemented by appropriate upstream and far-field boundary conditions, which characterize the impact of FSVD on the boundary layer. This initial-boundary-value problem is solved numerically. FSVD excite steady and unsteady Görtler vortices, which undergo non-modal growth, modal growth and nonlinear saturation for FSVD of moderate intensity. However, for sufficiently strong FSVD the modal stage is bypassed. Nonlinear interactions cause Görtler vortices to saturate, with the saturated amplitude being independent of FSVD intensity when $G_{\unicode[STIX]{x1D6EC}}\neq 0$. The predicted modified mean-flow profiles and structure of Görtler vortices are in excellent agreement with several steady experimental measurements. As the frequency increases, the nonlinearly generated harmonic component $(0,2)$ (which has zero frequency and wavenumber $2k_{3}$) becomes larger, and as a result the Görtler vortices appear almost steady. The secondary instability analysis indicates that Görtler vortices become inviscidly unstable in the presence of FSVD with a high enough intensity. Three types of inviscid unstable modes, referred to as sinuous (odd) modes I, II and varicose (even) modes I, are identified, and their relevance is delineated. The characteristics of dominant unstable modes, including their frequency ranges and eigenfunctions, are in good agreement with experiments. The secondary instability is intermittent when FSVD are unsteady and of low frequency. However, the intermittence diminishes as the frequency increases. The present theoretical framework, which allows for a detailed and integrated description of the key transition processes, from generation, through linear and nonlinear evolution, to the onset of secondary instability, represents a useful step towards predicting the pre-transitional flow and transition itself of the boundary layer over a blade in turbomachinery.

JFM classification

Boundary Layers: Boundary layer receptivity Instability: Instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 829 , 25 October 2017 , pp. 681 - 730
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aihara, Y. & Koyama, H. 1981 Secondary instability of Görtler vortices–formation of periodic three-dimensional coherent structure. Trans. Japan Soc. Aero. Astron. 24, 78–94.Google Scholar
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57 (9), 1435–1458.Google Scholar
Bassom, A. P. & Seddougui, S. O. 1995 Receptivity mechanisms for Görtler vortex modes. Theor. Comput. Fluid Dyn. 7 (5), 317–339.Google Scholar
Bippes, H.1972 Experimental study of the laminar-turbulent transition of a concave wall in a parallel flow. Tech. Rep. 75243. NASA Tech. Mem.Google Scholar
Bippes, H. & Deyhle, H. 1992 The receptivity problem in boundary layers with streamwise vortex disturbances. Z. Flugwiss. Weltraumforsch. 16, 34–41.Google Scholar
Bippes, H. & Görtler, H. 1972 Dreidimensionale Störungen in der grenzschicht an einer konkaven wand. Acta Mechanica 14 (4), 251–267.Google Scholar
Bottaro, A. & Luchini, P. 1999 Görtler vortices: are they amenable to local eigenvalue analysis? Eur. J. Mech. (B/Fluids) 18 (1), 47–65.Google Scholar
Day, H. P., Herbert, T. & Saric, W. S. 1990 Comparing local and marching analyses of Görtler instability. AIAA J. 28 (6), 1010–1015.CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2017 The relationship between free-stream coherent structures and near-wall streaks at high Reynolds numbers. Phil. Trans. R. Soc. Lond. A 375 (2089), 20160078.Google Scholar
Denier, J. P., 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 (1636), 51–85.Google Scholar
Dong, M. & Wu, X. 2013 On continuous spectra of the Orr-Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances. J. Fluid Mech. 732, 616–659.Google Scholar
Floryan, J. M. 1991 On the Görtler instability of boundary layers. Prog. Aerosp. Sci. 28 (3), 235–271.Google Scholar
Floryan, J. M. & Saric, W. S. 1982 Stability of Görtler vortices in boundary layers. AIAA J. 20 (3), 316–324.CrossRefGoogle 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 (2), 024102.Google Scholar
Goldstein, M. E. & Leib, S. J. 1993 Three-dimensional boundary-layer instability and separation induced by small-amplitude streamwise vorticity in the upstream flow. J. Fluid Mech. 246, 21–41.Google Scholar
Goldstein, M. E., Leib, S. J. & Cowley, S. J. 1992 Distortion of a flat-plate boundary layer by free-stream vorticity normal to the plate. J. Fluid Mech. 237, 231–260.CrossRefGoogle Scholar
Görtler, H. 1941 Instabilität laminarer Grenzschichten an konkaven Wänden gegenüber gewissen dreidimensionalen Störungen. Z. Angew. Math. Mech. 21 (4), 250–252.Google Scholar
Hall, P. 1982 Taylor Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475–494.CrossRefGoogle Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.Google Scholar
Hall, P. 1988 The nonlinear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 193, 243–266.Google Scholar
Hall, P. 1990 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37 (74), 151–189.Google Scholar
Hall, P. & Horseman, N. J. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375.CrossRefGoogle Scholar
Hall, P. & Lakin, W. D. 1988 The fully nonlinear development of Görtler vortices in growing boundary layers. Phil. Trans. R. Soc. Lond. A 415 (1849), 421–444.Google Scholar
Herbert, T. 1976 On the stability of the boundary layer along a concave wall. Arch. Mech. 28, 1039–1055.Google Scholar
Ito, A. 1985 Breakdown structure of longitudinal vortices along a concave wall. Trans. Jpn. Soc. Aeronaut. Space Sci. 33 (374), 166–173.Google Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.Google Scholar
Kottke, V. 1988 On the instability of laminar boundary layers along concave walls towards Görtler vortices. In Propagation and Nonequilibrium Systems (ed. Westfield, J. E. & Brand, H.), pp. 390–398. Springer.Google Scholar
Kovasznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aero. Sci. 20, 657–682.Google Scholar
Lee, K. & Liu, J. T. C. 1992 On the growth of mushroomlike structures in nonlinear spatially developing Görtler vortex flow. Phys. Fluids A 4 (1), 95–103.Google Scholar
Leib, S. J., Wundrow, D. W. & Goldstein, M. E. 1999 Effect of free-stream turbulence and other vortical disturbances on a laminar boundary layer. J. Fluid Mech. 380, 169–203.Google Scholar
Li, F. & Malik, M. R. 1995 Fundamental and subharmonic secondary instabilities of Görtler vortices. J. Fluid Mech. 297, 77–100.Google Scholar
Malik, M. R. 1990 Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86 (2), 376–413.Google Scholar
Martin, J. A., Martel, C., Paredes, P. & Theofilis, V. 2015 Numerical studies of non-linear intrinsic streaks in the flat plate boundary layer. Procedia IUTAM. vol. 14, pp. 479–486. Elsevier.Google Scholar
Mitsudharmadi, H., Winoto, S. H. & Shah, D. A. 2004 Development of boundary-layer flow in the presence of forced wavelength Görtler vortices. Phys. Fluids 16 (11), 3983–3996.Google Scholar
Mitsudharmadi, H., Winoto, S. H. & Shah, D. A. 2005 Secondary instability in forced wavelength Görtler vortices. Phys. Fluids 17 (7), 074104.Google Scholar
Mitsudharmadi, H., Winoto, S. H. & Shah, D. A. 2006 Development of most amplified wavelength Görtler vortices. Phys. Fluids 18 (1), 014101.Google Scholar
Peerhossaini, H. & Bahri, F. 1998 On the spectral distribution of the modes in nonlinear Görtler instability. Exp. Therm. Fluid Sci. 16 (3), 195–208.Google Scholar
Ren, J. & Fu, S. 2014 Floquet analysis of fundamental, subharmonic and detuned secondary instabilities of Görtler vortices. Sci. China Phys. Mech. Astron. 57 (3), 555–561.CrossRefGoogle Scholar
Ren, J. & Fu, S. 2015 Secondary instabilities of Görtler vortices in high-speed boundary layer flows. J. Fluid Mech. 781, 388–421.Google Scholar
Ricco, P., Luo, J. & Wu, X. 2011 Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances. J. Fluid Mech. 677, 1–38.Google Scholar
Sabry, A. S. & Liu, J. T. C. 1991 Longitudinal vorticity elements in boundary layers: nonlinear development from initial Görtler vortices as a prototype problem. J. Fluid Mech. 231, 615–663.Google Scholar
Saric, W. S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26 (1), 379–409.Google Scholar
Schrader, L.-U., Brandt, L. & Zaki, T. A. 2011 Receptivity, instability and breakdown of Görtler flow. J. Fluid Mech. 682, 362–396.Google Scholar
Sescu, A. & Thompson, D. 2015 On the excitation of Görtler vortices by distributed roughness elements. Theor. Comput. Fluid Dyn. 29 (1–2), 67–92.Google 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, 255–290.CrossRefGoogle Scholar
Tandiono, S. H., Winoto, S. H. & Shah, D. A. 2008 On the linear and nonlinear development of Görtler vortices. Phys. Fluids 20 (9), 094103.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43 (1), 319–352.CrossRefGoogle Scholar
Winoto, S. H., Shah, D. A. & Mitsudharmadi, H. 2011 Concave surface boundary layer flows in the presence of streamwise vortices. Int. J. Fluid Mach. Syst. 4 (1), 33–46.Google Scholar
Wu, X. & Choudhari, M. 2003 Linear and nonlinear instabilities of a Blasius boundary layer perturbed by streamwise vortices. Part 2. Intermittent instability induced by long-wavelength Klebanoff modes. J. Fluid Mech. 483, 249–286.Google Scholar
Wu, X. & Dong, M. 2016 Entrainment of short-wavelength free-stream vortical disturbances in compressible and incompressible boundary layers. J. Fluid Mech. 797, 683–728.Google Scholar
Wu, X., Zhao, D. & Luo, J. 2011 Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances. J. Fluid Mech. 682, 66–100.Google Scholar
Wundrow, D. W. & Goldstein, M. E. 2001 Effect on a laminar boundary layer of small-amplitude streamwise vorticity in the upstream flow. J. Fluid Mech. 426, 229–262.CrossRefGoogle Scholar
Yu, X. & Liu, J. T. C. 1991 The secondary instability in Görtler flow. Phys. Fluids A 3 (8), 1845–1847.Google 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 A 6 (2), 736–750.Google Scholar
Zhang, Y. & Luo, J. 2015 Application of Arnoldi method to boundary layer instability. Chin. Phys. B 24 (12), 124701.Google Scholar
Zhang, Y., Zaki, T., Sherwin, S. & Wu, X.2011 Nonlinear response of a laminar boundary layer to isotropic and spanwise localized free-stream turbulence. AIAA Paper 2011-3292.Google Scholar
Zhao, L., Zhang, C., Liu, J. & Luo, J. 2016 Improved algorithm for solving nonlinear parabolized stability equations. Chin. Phys. B 25 (8), 084701.Google Scholar