Hostname: page-component-7479d7b7d-fwgfc Total loading time: 0 Render date: 2024-07-12T06:53:39.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition to turbulence over convex surfaces

Published online by Cambridge University Press:  25 September 2018

Michael Karp Open the ORCID record for Michael Karp [Opens in a new window]  and
M. J. Philipp Hack Open the ORCID record for M. J. Philipp Hack [Opens in a new window]
Michael Karp*
Affiliation:
Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA
M. J. Philipp Hack
Affiliation:
Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: mkarp@stanford.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Although boundary-layer flows over convex surfaces are exponentially stable, non-modal mechanisms may enable significant disturbance growth which can make the flow susceptible to secondary instabilities. A parametric investigation of the transient growth and secondary instabilities in flows over convex surfaces is performed. The optimal disturbance in the steady case corresponds to alternating streaks and streamwise vortices of opposite sign that reinforce one another due to lift-up and centrifugal forces, respectively. The process repeats with a constant (naturally appearing) streamwise wavelength which is proportional to the square root of the radius. Unsteady disturbances achieve a higher optimal gain, compared to the steady case, as a result of the opposing effects of the lift-up and centrifugal mechanisms. Linear analysis shows that the curvature has a negligible effect on secondary instabilities. Direct numerical simulations of transient growth with and without secondary instabilities confirm the predictions obtained by the local stability theory. It is found that the presence of a secondary instability is not sufficient, on its own, to ensure transition to turbulence. Only sufficiently long and energetic streaks trigger the breakdown to turbulence.

JFM classification

Boundary Layers: Boundary Layers Boundary Layers: Boundary layer stability Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 855 , 25 November 2018 , pp. 1208 - 1237
Check for updates
Copyright
© 2018 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

Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134–150.Google Scholar
Andersson, P., Brandt, L., Bottaro, A. & Henningson, D. S. 2001 On the breakdown of boundary layer streaks. J. Fluid Mech. 428, 29–60.Google Scholar
Benmalek, A. & Saric, W. S. 1994 Effects of curvature variations on the nonlinear evolution of Goertler vortices. Phys. Fluids 6, 3353–3367.Google Scholar
Bottaro, A. & Luchini, P. 1999 Görtler vortices: are they amenable to local eigenvalue analysis? Eur. J. Mech. (B/Fluids) 18, 47–65.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.Google Scholar
Cossu, C. & Brandt, L. 2004 On Tollmien–Schlichting-like waves in streaky boundary layers. Eur. J. Mech. (B/Fluids) 23, 815–833.Google Scholar
Floryan, J. M. 1986 Görtler instability of boundary layers over concave and convex walls. Phys. Fluids 29, 2380–2387.Google Scholar
Floryan, J. M. 1991 On the Görtler instability of boundary layers. Prog. Aerosp. Sci 28, 235–271.Google Scholar
Floryan, J. M. & Saric, W. S. 1982 Stability of Görtler vortices in boundary layers. AIAA J. 20, 316–324.Google Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222–224.Google Scholar
Görtler, H. 1941 Instabilität laminarer Grenzschichten an konkaven Wänden gegenüber gewissen dreidimensionalen Störungen. Z. Angew. Math. Mech. 21, 250–252.Google Scholar
Goulpié, P., Klingmann, B. G. B. & Bottaro, A. 1996 Görtler vortices in boundary layers with streamwise pressure gradient: linear theory. Phys. Fluids 8, 451–459.Google Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.Google Scholar
Hack, M. J. P. & Zaki, T. A. 2014 Streak instabilities in boundary layers beneath free-stream turbulence. J. Fluid Mech. 741, 280–315.Google Scholar
Hack, M. J. P. & Zaki, T. A. 2015 Modal and non-modal stability of boundary layers forced by spanwise wall oscillations. J. Fluid Mech. 778, 389–427.Google Scholar
Hack, M. J. P. & Zaki, T. A. 2016 Data-enabled prediction of streak breakdown in pressure-gradient boundary layers. J. Fluid Mech. 801, 43–64.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.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, 151–189.Google Scholar
Hanifi, A., Schmid, P. J. & Henningson, D. S. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8, 826–837.Google Scholar
Jacobs, R. G. & Durbin, P. A. 2001 Simulations of bypass transition. J. Fluid Mech. 428, 185–212.Google Scholar
Kalburgi, V., Mangalam, S. M. & Dagenhart, J. R. 1988 Görtler instability on an aerofoil: comparison of marching solution with experimental observations. AGARD CP 438, 8.Google Scholar
Karp, M. & Cohen, J. 2014 Tracking stages of transition in Couette flow analytically. J. Fluid Mech. 748, 896–931.Google Scholar
Karp, M. & Cohen, J. 2017 On the secondary instabilities of transient growth in Couette flow. J. Fluid Mech. 813, 528–557.Google Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323.Google Scholar
Le Cunff, C. & Zebib, A. 1996 Nonlinear spatially developing Görtler vortices in curved wall jet flow. Phys. Fluids 8, 2375–2384.Google Scholar
Lee, K. & Liu, J. T. C. 1992 On the growth of mushroomlike structures in nonlinear spatially developing Goertler vortex flow. Phys. Fluids 4, 95–103.Google Scholar
Liu, W. & Domaradzki, J. A. 1993 Direct numerical simulation of transition to turbulence in Görtler flow. J. Fluid Mech. 246, 267–299.Google Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309.Google Scholar
Mahesh, K. 2013 The interaction of jets with crossflow. Annu. Rev. Fluid Mech. 45, 379–407.Google Scholar
Mangalam, S. M., Dagenhart, J. R., Hepner, T. E. & Meyers, J. F. 1985 The Görtler instability on an airfoil. In Proceedings of 23rd Aerospace Sciences Meeting, AIAA Paper 85-0491, American Institute of Aeronautics and Astronautics.Google Scholar
Mangalam, S. M., Dagenhart, J. R. & Kalburgi, V. 1987 Influence of suction and curvature on the growth of Görtler vortices on an airfoil. In Proceedings of 25th Aerospace Sciences Meeting, AIAA Paper 87-0481, American Institute of Aeronautics and Astronautics.Google Scholar
Matsson, O. J. E. 1995 On the curved wall jet influenced by system rotation and self-similar suction or blowing. Phys. Fluids 7, 3048–3059.Google Scholar
Matsson, O. J. E. 2008 Görtler vortices in Falkner–Skan flows with suction and blowing. Intl J. Numer. Meth. Fluids 56, 257–277.Google Scholar
Matsubara, M. & Alfredsson, P. H. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.Google Scholar
Pearcey, H. H. 1961 Shock-induced separation and its prevention. In Boundary Layer and Flow Control, Its Principle and Applications (ed. Lachmann, G. V.). Pergamon.Google Scholar
Reddy, S. C., Schmid, P. J., Baggett, J. S. & Henningson, D. S. 1998 On the stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech. 365, 269–303.Google Scholar
Roach, P. E. & Brierley, D. H. 1990 The influence of a turbulent free-stream on zero pressure gradient transitional boundary layer development. Part I. Test cases T3A and T3B. In ERCOFTAC Workshop: Numerical Simulation of Unsteady Flows and Transition to Turbulence, Lausanne, Switzerland, pp. 319–347. Cambridge University Press.Google 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, 124107.Google Scholar
Rosenfeld, M., Kwak, D. & Vinokur, M. 1991 A fractional step solution method for the unsteady incompressible Navier–Stokes equations in generalized coordinate systems. J. Comput. Phys. 94, 102–137.Google Scholar
Saric, W. S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26, 379–409.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.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.Google Scholar
Vaughan, N. J. & Zaki, T. A. 2011 Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks. J. Fluid Mech. 681, 116–153.Google Scholar
Westin, K. J. A., Boiko, A. V., Klingmann, B. G. B., Kozlov, V. V. & Alfredsson, P. H. 1994 Experiments in a boundary layer subjected to free-stream turbulence. Part 1. Boundary layer structure and receptivity. J. Fluid Mech. 281, 193–218.Google Scholar

Karp et al. supplementary movie

Streamwise streaks for case C in table 3, visualized by isosurfaces of _±0.1 streamwise disturbance velocity (positive, red; negative, blue). Flow from left to right.

Download Karp et al. supplementary movie(Video)
Video 7 MB