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Görtler vortices and streaks in boundary layer subject to pressure gradient: excitation by free stream vortical disturbances, nonlinear evolution and secondary instability

Published online by Cambridge University Press:  06 August 2020

Dongdong Xu Open the ORCID record for Dongdong Xu [Opens in a new window] ,
Jianxin Liu Open the ORCID record for Jianxin Liu [Opens in a new window]  and
Xuesong Wu Open the ORCID record for Xuesong Wu [Opens in a new window]
Dongdong Xu
Affiliation:
Department of Mechanics, Tianjin University, Tianjin300072, PR China
Jianxin Liu
Affiliation:
Department of Mechanics, Tianjin University, Tianjin300072, PR China
Xuesong Wu*
Affiliation:
Department of Mechanics, Tianjin University, Tianjin300072, PR China Department of Mathematics, Imperial College London, 180 Queen’s Gate, LondonSW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

This paper investigates streaks and Görtler vortices in a boundary layer over a flat or concave wall in a contracting or expanding stream, which provides a favourable or adverse pressure gradient, respectively. We consider first the excitation of streaks and Görtler vortices by free stream vortical disturbances (FSVD), and their nonlinear evolution. The focus is on FSVD with sufficiently long wavelength, to which the boundary layer is most receptive. The formulation is directed at the general case where the Görtler number $G_{\Lambda }$ (based on the spanwise length scale $\Lambda$ of FSVD) is of order one, and the FSVD is strong enough that the induced vortices acquire an $O(1)$ streamwise velocity in the region where the boundary layer thickness becomes comparable with $\Lambda,$ and the vortices are governed by the nonlinear boundary region equations (NBRE). An important effect of a pressure gradient is that the oncoming FSVD are distorted by the non-uniform inviscid flow outside the boundary layer through convection and stretching. This process is accounted for by using the rapid distortion theory. The impact of the distorting FSVD is analysed to provide the appropriate initial and boundary conditions, which form, along with the NBRE, the appropriate initial boundary value problem describing the excitation and nonlinear evolution of the vortices. Numerical results show that an adverse/favourable pressure gradient cause the Görtler vortices to saturate earlier/later, but at a lower/higher amplitude than that in the case of zero-pressure-gradient. On the other hand, for the same pressure gradient and at low levels of FSVD, the vortices saturate earlier and at a higher amplitude as $G_{\Lambda }$ increases. Raising FSVD intensity reduces the effects of the pressure gradient and curvature. At a high FSVD level of 14 %, the curvature has no impact on the vortices, while the pressure gradient only influences the saturation intensity. The unsteadiness of FSVD is found to reduce the boundary layer response significantly at FSVD levels, but that effect weakens as the turbulence level increases. A secondary instability analysis of the vortices is performed for moderate adverse and favourable pressure gradients. Three families of unstable modes have been identified, which may become dominant depending on the frequency and streamwise location. In the presence of an adverse pressure gradient, the secondary instability occurs earlier, but the unstable modes appear in a smaller band of frequency, and have smaller growth rates. The opposite is true for a favourable pressure gradient. The present theoretical framework, which accounts for the influence of the curvature, turbulence level and pressure gradient, allows for a detailed and integrated description of the key transition processes, and represents a useful step towards predicting the pretransitional flow and transition itself of the boundary layer over a blade in turbomachinery.

JFM classification

Boundary Layers: Boundary layer receptivity Boundary Layers: Boundary layer stability Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A15
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Abu-Ghannam, B. J. & Shaw, R. 1980 Natural transition of boundary layers-the effects of turbulence, pressure gradient, and flow history. J. Mech. Engng Sci. 22 (5), 213–228.CrossRefGoogle Scholar
Aihara, Y. & Sonoda, T. 1981 Effects of pressure gradient on the secondary instability of Görtler vortices. AIAA Paper 81-197. American Institute of Aeronautics and Astronautics.Google Scholar
Bassom, A. P. & Hall, P. 1994 The receptivity problem for O(1) wavelength Görtler vortices. Proc. R. Soc. Lond. A 446 (1928), 499–516.Google Scholar
Batchelor, G. K. & Proudman, I. 1954 The effect of rapid distortion of a fluid in turbulent motion. Q. Appl. Maths 7 (1), 83–103.CrossRefGoogle Scholar
Benmalek, A. & Saric, W. S. 1994 Effects of curvature variations on the nonlinear evolution of Görtler vortices. Phys. Fluids 6 (10), 3353–3367.CrossRefGoogle Scholar
Blair, M. F. 1982 Influence of free-stream turbulence on boundary layer transition in favorable pressure gradients. J. Engng Power 104 (4), 743–750.CrossRefGoogle Scholar
Blair, M. F. 1992 Boundary-layer transition in accelerating flows with intense freestream turbulence. Part 1. Disturbances upstream of transition onset. Trans. ASME: J. Fluids Engng 114 (3), 313–321.Google 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. Euro. J. Mech. B/Fluids 68, 167–183.CrossRefGoogle Scholar
Bottaro, A. & Luchini, P. 1999 Görtler vortices: are they amenable to local eigenvalue analysis. Euro. J. Mech. B/Fluids 18 (1), 47–65.CrossRefGoogle Scholar
Brinkerhoff, J. R. & Yaras, M. I. 2015 Numerical investigation of transition in a boundary layer subjected to favourable and adverse streamwise pressure gradients and elevated free stream turbulence. J. Fluid Mech. 781, 52–86.CrossRefGoogle Scholar
Crane, R. I. & Umur, H. 1990 Concave-wall laminar heat transfer and Görtler vortex structure: effects of pre-curvature boundary layer and favourable pressure gradients. In ASME 1990 International Gas Turbine and Aeroengine Congress and Exposition, pp. 90-GT-94. American Society of Mechanical Engineers.CrossRefGoogle 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
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.CrossRefGoogle Scholar
Finnis, M. V. & Brown, A. 1997 The linear growth of Görtler vortices. J. Heat Fluid Flow 18 (4), 389–399.CrossRefGoogle Scholar
Floryan, J. M. & Saric, W. S. 1982 Stability of Görtler vortices in boundary layers. AIAA J. 20 (3), 316–324.CrossRefGoogle Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14 (02), 222–224.CrossRefGoogle Scholar
Goldstein, M. E. 1978 Unsteady vortical and entropic distortions of potential flows around arbitrary obstacles. J. Fluid Mech. 89 (03), 433–468.CrossRefGoogle Scholar
Goldstein, M. E. 1979 Turbulence generated by the interaction of entropy fluctuations with non-uniform mean flows. J. Fluid Mech. 93 (02), 209–224.CrossRefGoogle Scholar
Goldstein, M. E. 2014 Effect of free-stream turbulence on boundary layer transition. Phil. Trans R. Soc. Lond. A 372 (2020), 1–18.CrossRefGoogle ScholarPubMed
Goldstein, M. E. & Durbin, P. A. 1980 The effect of finite turbulence spatial scale on the amplification of turbulence by a contracting stream. J. Fluid Mech. 98 (03), 473–508.CrossRefGoogle Scholar
Goldstein, M. E. & Wundrow, D. W. 1998 On the environmental realizability of algebraically growing disturbances and their relation to Klebanoff modes. Theor. Comput. Fluid Dyn. 10 (1-4), 171–186.CrossRefGoogle Scholar
Görtler, H. 1941 Instabilität laminarer Grenzschichten an konkaven Wänden gegenüber gewissen dreidimensionalen Störungen. Z. Angew. Math. Mech. 21 (4), 250–252.CrossRefGoogle Scholar
Gostelow, J. P., Blunden, A. R. & Walker, G. J. 1994 Effects of free-stream turbulence and adverse pressure gradients on boundary layer transition. Trans. ASME J. Turbomach. 116 (3), 392–404.CrossRefGoogle Scholar
Goulpié, P., Klingmann, B. G. B. & Bottaro, A. 1996 Görtler vortices in boundary layers with streamwise pressure gradient: linear theory. Phys. Fluids 8 (2), 451–459.CrossRefGoogle 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.CrossRefGoogle Scholar
Hall, P. 1988 The nonlinear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 193, 243–266.CrossRefGoogle 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
Horseman, N. J., Bassom, A. P. & Blennerhassett, P. J. 1996 Strongly nonlinear vortices in the Stokes layer on an oscillating cylinder. Proc. R. Soc. Lond. A 452, 1087–1111.Google Scholar
Hunt, J. C. R. 1973 A theory of turbulent flow around two-dimensional bluff bodies. J. Fluid Mech. 61 (04), 625–706.CrossRefGoogle Scholar
Jahanmiri, M. 2011 Boundary layer transitional flow in gas turbines. Tech. Rep. 01. Chalmers University of Technology.Google Scholar
Johnson, M. W. & Pinarbasi, A. 2014 The effect of pressure gradient on boundary layer receptivity. Flow Turbul. Combust. 93 (1), 1–24.CrossRefGoogle 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.CrossRefGoogle Scholar
Kim, J., Simon, T. W. & Russ, S. G. 1992 Free-stream turbulence and concave curvature effects on heated, transitional boundary layers. Trans. ASME J. Heat Transfer 114 (2), 338–347.CrossRefGoogle Scholar
Kottke, V. 1988 On the instability of laminar boundary layers along concave walls towards Görtler vortices. In Propagation in Systems Far from Equilibrium (ed. Westifield, J. E. & Brand, H.), pp.390–398. Springer.CrossRefGoogle Scholar
Lee, K. & Liu, J. T. C. 1992 On the growth of mushroomlike structures in nonlinear spatially developing Görtler vortex flow. Phys. Fluids 4 (1), 95–103.CrossRefGoogle 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.CrossRefGoogle Scholar
Li, F. & Malik, M. R. 1995 Fundamental and subharmonic secondary instabilities of Görtler vortices. J.Fluid Mech. 297, 77–100.CrossRefGoogle Scholar
Malik, M. R. 1990 Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86 (2), 376–413.CrossRefGoogle Scholar
Marensi, E., Ricco, P. & Wu, X. 2017 Nonlinear unsteady streaks engendered by the interaction of free-stream vorticity with a compressible boundary layer. J. Fluid Mech. 817, 80–121.CrossRefGoogle Scholar
Matsson, O. J. E. 2008 Görtler vortices in Falkner–Skan flows with suction and blowing. Int. J. Numer. Methods Fluids 56 (3), 257–277.CrossRefGoogle Scholar
Mayle, R. E. 1991 The role of laminar-turbulent transition in gas turbine engines. Trans. ASME J. Turbomach. 113, 509–537.CrossRefGoogle Scholar
Ragab, S. A. & Nayfeh, A. H. 1980 Effect of pressure gradients on Görtler instability. AIAA Paper 1980-1377. American Institute of Aeronautics and Astronautics.CrossRefGoogle 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.CrossRefGoogle Scholar
Ricco, P. & Wu, X. 2007 Response of a compressible laminar boundary layer to free-stream vortical disturbances. J. Fluid Mech. 587, 97–138.CrossRefGoogle Scholar
Rogenski, J. K., de Souza, L. F. & Floryan, J. M. 2016 Non-linear aspects of Görtler instability in boundary layers with pressure gradient. Phys. Fluids 28 (12), 124107.CrossRefGoogle Scholar
Schetz, J. A. 1993 Boundary Layer Analysis. Prentice Hall.Google Scholar
Schrader, L. U., Brandt, L. & Zaki, T. A. 2011 Receptivity, instability and breakdown of Görtler flow. J. Fluid Mech. 682, 362–396.CrossRefGoogle Scholar
Schreiber, H., Steinert, W. & Küsters, B. 2002 Effects of Reynolds number and free-stream turbulence on boundary layer transition in a compressor cascade. Trans. ASME J. Turbomach. 124 (1), 1–9.CrossRefGoogle Scholar
Schultz, M. P. & Volino, R. J. 2003 Effects of concave curvature on boundary layer transition under high freestream turbulence conditions. J. Fluids Engng 125 (1), 18–27.CrossRefGoogle Scholar
Sescu, A. & Afsar, M. Z. 2018 Hampering Görtler vortices via optimal control in the framework of nonlinear boundary region equations. J. Fluid Mech. 848, 5–41.CrossRefGoogle 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.CrossRefGoogle Scholar
Smith, A. M. O. 1955 On the growth of Taylor-Görtler vortices along highly concave walls. Q. Appl. Maths 13, 233–262.CrossRefGoogle Scholar
Solomon, W. J., Walker, G. J. & Gostelow, J. P. 1996 Transition length prediction for flows with rapidly changing pressure gradients. Trans. ASME J. Turbomach. 118 (4), 744–751.CrossRefGoogle 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
Talamelli, A., Fornaciari, N. & Westin, K. 2002 Experimental investigation of streaky structures in a relaminarizing boundary layer. J. Turbul. 3, 18.CrossRefGoogle Scholar
Tani, I. 1962 Production of longitudinal vortices in the boundary layer along a concave wall. J. Fluid Mech. 67 (8), 3075–3080.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43 (1), 319–352.CrossRefGoogle Scholar
Volino, R. J. & Simon, T. W. 1995 Bypass transition in boundary layers including curvature and favorable pressure gradient effects. Trans. ASME J. Turbomach. 117 (1), 166–174.CrossRefGoogle Scholar
Volino, R. J. & Simon, T. W. 1997 Boundary layer transition under high free-stream turbulence and strong acceleration conditions. Part 1. Mean flow results. Trans. ASME J. Heat Transfer 119 (3), 420–426.CrossRefGoogle Scholar
Volino, R. J. & Simon, T. W. 2000 Spectral measurements in transitional boundary layers on a concave wall under high and low free-stream turbulence conditions. Trans. ASME J. Turbomach. 122 (3), 450–457.CrossRefGoogle Scholar
Walker, G. J. 1993 The role of laminar-turbulent transition in gas turbine engines: a discussion. Trans. ASME J. Turbomach. 115 (2), 207–216.CrossRefGoogle 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.CrossRefGoogle 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
Xu, D. 2020 Receptivity, instability and separation of boundary layer over a curved surface subject to elevated free-stream vortical disturbances. PhD thesis, Tianjin University.Google 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, 681–730.CrossRefGoogle 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 (2), 736–750.CrossRefGoogle Scholar
Zaki, T. A. & Durbin, P. A. 2006 Continuous mode transition and the effects of pressure gradient. J.Fluid Mech. 563, 357–388.CrossRefGoogle 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. American Institute of Aeronautics and Astronautics.CrossRefGoogle Scholar