zbMATH — the first resource for mathematics

The excitation of Görtler vortices by free stream coherent structures. (English) Zbl 1430.76138
Summary: The effect of free stream coherent structures in the asymptotic suction boundary layer on the initiation of Görtler vortices is considered from both the “imperfect” bifurcation and receptivity viewpoints. Firstly a weakly nonlinear and a full numerical approach are used to describe Görtler vortices in the asymptotic suction boundary layer in the absence of forcing from the free stream. It is found that interactions between different spanwise harmonics occur and lead to multiple secondary bifurcations in the fully nonlinear regime. Furthermore it is shown that centrifugal instabilities of the asymptotic suction boundary layer behave quite differently than their counterparts in either fully developed flows such as Couette flow or growing boundary layers. A significant result is that the most dangerous disturbance is found to bifurcate subcritically from the unperturbed state. Within the weakly nonlinear regime the receptivity of Görtler vortices to the free stream exact coherent structures discovered by the second author and the third author [ibid. 752, 602–625 (2014; Zbl 1431.76046); ibid. 778, 451–484 (2015; Zbl 1382.76107)] is considered. The presence of free stream structures results in a resonant excitation of Görtler vortices in the main boundary layer. This leads to imperfect bifurcations reminiscent of those found by P. G. Daniels [Proc. R. Soc. Lond., Ser. A 358, 173–197 (1977; Zbl 0366.76069)] and the second author and I. C. Walton [Proc. R. Soc. Lond., Ser. A 358, 199–221 (1977; Zbl 0366.76072); J. Fluid Mech. 90, 377–395 (1979; Zbl 0411.76061)] in the context of transition to finite amplitude Bénard convection in a bounded region. In order to understand the receptivity problem for the given flow the spatial initial value problem for this interaction is also considered when the free stream structure begins at a fixed position along the wall. Remarkably, it will be shown that free stream structures are incredibly efficient generators of Görtler vortices; indeed the induced vortices are found to be larger than the free stream structure which provokes them! The relationship between the imperfect bifurcation approach and receptivity theory is described.
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76E30 Nonlinear effects in hydrodynamic stability
76F06 Transition to turbulence
76F40 Turbulent boundary layers
Full Text: DOI
[1] Boyd, J. P.1982The optimization of convergence for Chebyshev polynomial methods in an unbounded domain. J. Comput. Phys.45, 43-79. · Zbl 0488.65035
[2] Chomaz, J. M. & Perrier, M.1991Nature of the Görtler instability: a forced experiment in The Global Geometry of Turbulence (ed. J.Jimenez), Plenum Press.
[3] Collins, D. A. & Maslowe, S. A.1988Vortex pairing and resonant wave interactions in a stratified shear layer. J. Fluid Mech.191, 465-480.
[4] Daniels, P. G.1977The effect of distant sidewalls on the transition to finite amplitude Bénard convection. Proc. R. Soc. Lond. A358, 173-197. · Zbl 0366.76069
[5] Deguchi, K. & Hall, P.2014Free-stream coherent structures in parallel boundary-layer flows. J. Fluid Mech.752, 602-625.
[6] Deguchi, K. & Hall, P.2015Free-stream coherent structures in growing boundary-layers: a link to near wall streaks. J. Fluid Mech.778, 451-484. · Zbl 1382.76107
[7] Dempsey, L. J.2015 Nonlinear exact coherent structures in high Reynolds number shear flows. Imperial College thesis, http://spiral.imperial.ac.uk:8443/handle/10044/1/33319.
[8] Denier, J. P., Hall, P. & Seddougui, S. O.1991On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness. Phil. Trans. R. Soc. Lond A335, 51-85. · Zbl 0850.76211
[9] Fransson, J. H. M. & Alfredsson, P. H.2003On the disturbance growth in an asymptotic suction boundary-layer. J. Fluid Mech.482, 51-90. · Zbl 1049.76508
[10] Goldstein, M. E.1985Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variations in geometry. J. Fluid Mech.154, 509-529. · Zbl 0576.76064
[11] Gulyaev, A. N., Kozlov, V. E., Kuznetzov, V. R., Mineev, B. I. & Sekundov, A. N.1989Interaction of a laminar boundary layer with external turbulence. Izv. Akad. Nauk SSSR Mekh. Zhidkh. Gaza6, 700-710.
[12] Hall, P.1980Centrifugal instabilities in finite containers: a periodic model. J. Fluid Mech.99, 575-596. · Zbl 0433.76086
[13] Hall, P.1982Taylor-Görtler vortices in fully developed or boundary layer flows. J. Fluid Mech. B23, 715-735.
[14] Hall, P.1983The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech.130, 41-58. · Zbl 0515.76040
[15] Hall, P.1988The nonlinear development of Görtler vortices in growing boundary layers. J. Fluid Mech.193, 243-266. · Zbl 0643.76041
[16] Hall, P.1990Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and nonlinear breakdown stage. Mathematika37, 151-189. · Zbl 0708.76053
[17] Hall, P. & Horseman, N. J.1991The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech.232, 357-375. · Zbl 0738.76029
[18] Hall, P. & Lakin, W. D.1988The fully nonlinear development of Gp̈ortlier vertices in growing boundary layers. Proc. R. Soc. Lond. A415, 421-444. · Zbl 0669.76063
[19] Hall, P. & Sherwin, S.2010Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech.661, 178-205. · Zbl 1205.76085
[20] Hall, P. & Smith, F. T.1988The nonlinear interaction of Görtler vortices and Tollmien-Schlichting waves in curved channel flows. Proc. R. Soc. Lond. A417, 255-282. · Zbl 0657.76047
[21] Hall, P. & Smith, F. T.1989Tollmien-Schlichting/vortex interaction in boundary layers. Eur. J. Mech. (B/Fluids)8, 179-205. · Zbl 0676.76045
[22] Hall, P. & Smith, F. T.1991On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech.227, 641-666. · Zbl 0721.76027
[23] Hall, P. & Walton, I. C.1977The smooth transition to a convective regime in a two-dimensional box. Proc. R. Soc. Lond. A358, 199-221. · Zbl 0366.76072
[24] Hall, P. & Walton, I. C.1979Bénard convection in a finite box: secondary and imperfect bifurcations. J. Fluid Mech.90, 377-395. · Zbl 0411.76061
[25] Hocking, L. M.1975Non-linear instability of the asymptotic suction velocity profile. J. Fluid Mech.90, 377-395.
[26] Hughes, T. H. & Reid, W. H.1965On the stability of the asymptotic suction boundary-layer profile. J. Fluid Mech.23, 715-735.
[27] Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckhardt, B. & Henningson, D. S.2012Localized edge states in the asymptotic suction boundary layer. J. Fluid Mech.717, R6. · Zbl 1284.76106
[28] Maslowe, S. A.1977Weakly nonlinear stability theory for stratified shear flows. Q. J. R. Meteorol. Soc.103, 769-783.
[29] Milinazzo, F. A. & Saffman, P. G.1985Finite-amplitude steady waves in plane viscous shear flows. J. Fluid Mech.160, 281-295. · Zbl 0594.76028
[30] Nagata, M.1990Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech.217, 519-527.
[31] Nayfeh, A. H. & Saric, W. S.1972Nonlinear Kelvin-Helmholz instability. J. Fluid Mech.46, 209-231. · Zbl 0229.76040
[32] Park, D. & Huerre, P.1988On the convective nature of the Görtler instability. Bull. Am. Phys. Soc.33, 2552.
[33] Park, D. & Huerre, P.1995Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate. J. Fluid Mech.283, 249-272. · Zbl 0835.76028
[34] Ruban, A. I.1984On Tolmien-Schlichting wave generation by sound. Izv. Akad. Nauk. SSSR Mekh. Zhidk. Gaza5, 44-52.
[35] Saffman, P. G.1983Vortices, stability and turbulence. Ann. N.Y. Acad. Sci.404, 12-24.
[36] Waleffe, F.1997On a self-sustaining process in shear flows. Phys. Fluids9, 883-900.
[37] Waleffe, F.2001Exact coherent structures in channel flow. J. Fluid Mech.435, 93-102. · Zbl 0987.76034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.