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Modal and non-modal stability of boundary layers forced by spanwise wall oscillations

Published online by Cambridge University Press:  03 August 2015

M. J. Philipp Hack  and
Tamer A. Zaki
M. J. Philipp Hack
Affiliation:
Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK
Tamer A. Zaki*
Affiliation:
Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
*
†Email address for correspondence: [email protected]
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Abstract

Modal and non-modal perturbation growth in boundary layers subjected to time-harmonic spanwise wall motion are examined. The superposition of the streamwise Blasius flow and the spanwise Stokes layer can lead to strong modal amplification during intervals of the base-flow period. Linear stability analysis of frozen phases of the base state demonstrates that this growth is due to an inviscid instability, which is related to the inflection points of the spanwise Stokes layer. The generation of new inflection points at the wall and their propagation towards the free stream leads to mode crossing when tracing the most unstable mode as a function of phase. The fundamental mode computed in Floquet analysis has a considerably lower growth rate than the instantaneous eigenfunctions. Furthermore, the algebraic lift-up mechanism that causes the formation of Klebanoff streaks is examined in transient growth analyses. The wall forcing significantly weakens the wall-normal velocity perturbations associated with lift-up. This effect is attributed to the formation of a pressure field which redistributes energy from the wall-normal to the spanwise velocity perturbations. The results from linear theory explain observations from direct numerical simulations of breakdown to turbulence in the same flow configuration by Hack & Zaki (J. Fluid Mech., vol. 760, 2014a, pp. 63–94). When bypass mechanisms are dominant, the flow is stabilized due to the weaker non-modal growth. However, at high amplitudes of wall oscillation, transition is promoted due to fast growth of the modal instability.

JFM classification

Boundary Layers: Boundary layer stability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 778 , 10 September 2015 , pp. 389 - 427
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Copyright
© 2015 Cambridge University Press 

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Footnotes

‡

Present address: Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA.

References

Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991 An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 1. Experiments. J. Fluid Mech. 225, 395–422.Google Scholar
Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11 (1), 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.CrossRefGoogle Scholar
Bippes, H. 1999 Basic experiments on transition in three-dimensional boundary layers dominated by crossflow instability. Prog. Aerosp. Sci. 35, 363–412.Google Scholar
Blennerhassett, P. J. & Bassom, A. P. 2002 The linear stability of flat Stokes layers. J. Fluid Mech. 464, 393–410.Google Scholar
Blesbois, O., Chernyshenko, S. I., Touber, E. & Leschziner, M. A. 2013 Pattern prediction by linear analysis of turbulent flow with drag reduction by wall oscillation. J. Fluid Mech. 724, 607–641.Google Scholar
Brandt, L. 2014 The lift-up effect: The linear mechanism behind transition and turbulence in shear flows. Eur. J. Mech. (B/Fluids) 47, 80–96.Google Scholar
Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517, 167–198.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 1637–1650.Google Scholar
Clarion, C. & Pelissier, R. 1975 A theoretical and experimental study of the velocity distribution and transition to turbulence in free oscillatory flow. J. Fluid Mech. 70, 59–79.Google Scholar
Corbett, P. & Bottaro, A. 2001 Optimal linear growth in swept boundary layers. J. Fluid Mech. 435, 1–23.CrossRefGoogle Scholar
Cowley, S. J. 1987 High frequency Rayleigh instability of Stokes layers. In Stability of Time Dependent and Spatially Varying Flows (ed. Dwoyer, D. L. & Hussaini, M. Y.), pp. 244–260. Springer.Google Scholar
Davis, S. H. 1976 The stability of time-periodic flows. Annu. Rev. Fluid Mech. 8, 57–74.Google Scholar
Deyhle, H. & Bippes, H. 1996 Disturbance growth in an unstable three-dimensional boundary layer and its dependence on environmental conditions. J. Fluid Mech. 316, 73–113.Google Scholar
Deyhle, H., Hoehler, G. & Bippes, H. 1993 Experimental investigation of instability wave propagation in a three-dimensional boundary-layer flow. AIAA J. 31 (4), 637–645.CrossRefGoogle Scholar
Duque-Daza, C. A., Baig, M. F., Lockerby, D. A., Chernyshenko, S. I. & Davies, C. 2012 Modelling turbulent skin-friction control using linearized Navier–Stokes equations. J. Fluid Mech. 702, 403–414.Google Scholar
Golub, G. H. & Van Loan, C. F. 1996 Matrix Computations, 3rd edn. The Johns Hopkins University Press.Google Scholar
Hack, M. J. P. & Zaki, T. A. 2012 The continuous spectrum of time-harmonic shear layers. Phys. Fluids 24 (3), 034101.Google Scholar
Hack, M. J. P. & Zaki, T. A. 2014a The influence of harmonic wall motion on transitional boundary layers. J. Fluid Mech. 760, 63–94.Google Scholar
Hack, M. J. P. & Zaki, T. A. 2014b Streak instabilities in boundary layers beneath free-stream turbulence. J. Fluid Mech. 741, 280–315.Google Scholar
Hall, P. 1978 The linear stability of flat Stokes layers. Proc. R. Soc. Lond. A 359, 151–166.Google Scholar
Hall, P. 2003 On the instability of Stokes layers at high Reynolds numbers. J. Fluid Mech. 482, 1–15.CrossRefGoogle Scholar
Hino, M., Sawamoto, M. & Takasu, S. 1976 Experiments on transition to turbulence in an oscillating pipe flow. J. Fluid Mech. 75, 193–207.Google Scholar
Hunt, J. C. R. & Carruthers, D. J. 1990 Rapid distortion theory and the ‘problems’ of turbulence. J. Fluid Mech. 212, 497–532.Google Scholar
Jacobs, R. G. & Durbin, P. A. 1998 Shear sheltering and the continuous spectrum of the Orr–Sommerfeld equation. Phys. Fluids 10 (8), 2006–2011.Google Scholar
Jacobs, R. G. & Durbin, P. A. 2001 Simulations of bypass transition. J. Fluid Mech. 428, 185–212.Google Scholar
Koch, W., Bertolotti, F. P., Stolte, A. & Hein, S. 2000 Nonlinear equilibrium solutions in a three-dimensional boundary layer and their secondary instability. J. Fluid Mech. 406, 131–174.Google Scholar
Landahl, M. T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Maths 28 (4), 735–756.CrossRefGoogle Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.Google Scholar
Luo, J. & Wu, X. 2010 On the linear instability of a finite Stokes layer: instantaneous versus Floquet modes. Phys. Fluids 22, 054106.CrossRefGoogle Scholar
Mandal, A. C., Venkatakrishnan, L. & Dey, J. 2010 A study on boundary-layer transition induced by free-stream turbulence. J. Fluid Mech. 660, 114–146.Google Scholar
Matsubara, M. & Alfredsson, P. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.Google Scholar
Merkli, P. & Thomann, H. 1975 Transition to turbulence in oscillating pipe flow. J. Fluid Mech. 68, 567–575.CrossRefGoogle Scholar
Monkewitz, P. A. & Bunster, A. 1987 The stability of the Stokes layer: visual observations and some theoretical considerations. In Stability of Time Dependent and Spatially Varying Flows (ed. Dwoyer, D. L. & Hussaini, M. Y.), pp. 244–260. Springer.Google Scholar
Morkovin, M. V. & Obremski, H. J. 1969 Application of a quasi-steady stability model to periodic boundary-layer flows. AIAA J. 7 (7), 1298–1301.Google Scholar
Nagarajan, S., Lele, S. K. & Ferziger, J. H. 2007 Leading-edge effects in bypass transition. J. Fluid Mech. 572, 471–504.Google Scholar
Nolan, K. P. & Zaki, T. A. 2013 Conditional sampling of transitional boundary layers in pressure gradients. J. Fluid Mech. 728, 306–339.CrossRefGoogle Scholar
Radeztsky, R. H., Reibert, M. S. & Saric, W. S. 1999 Effect of isolated micron-sized roughness on transition in swept-wing flows. AIAA J. 37 (11), 1370–1377.Google Scholar
Ricco, P. 2011 Laminar streaks with spanwise wall forcing. Phys. Fluids 23, 064103.Google Scholar
Saric, W. S., Reed, H. L. & White, E. B. 2003 Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35, 413–440.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schrader, L.-U., Brandt, L. & Henningson, D. S. 2009 Receptivity mechanisms in three-dimensional boundary-layer flows. J. Fluid Mech. 618, 209–241.CrossRefGoogle Scholar
Schrader, L.-U., Subir, A. & Brandt, L. 2010 Transition to turbulence in the boundary layer over a smooth and rough swept plate exposed to free-stream turbulence. J. Fluid Mech. 646, 297–325.Google Scholar
Tempelmann, D., Hanifi, A. & Henningson, D. S. 2010 Spatial optimal growth in three-dimensional boundary layers. J. Fluid Mech. 646, 5–37.Google Scholar
Thomas, C., Bassom, A. P., Blennerhassett, P. J. & Davies, C. 2011 The linear stability of oscillatory poiseuille flow in channels and pipes. Proc. R. Soc. Lond. A 467, 2643–2662.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
Von Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753–773.CrossRefGoogle Scholar
Wassermann, P. & Kloker, M. 2002 Mechanisms and passive control of crossflow-vortex-induced transition in a three-dimensional boundary layer. J. Fluid Mech. 456, 49–84.CrossRefGoogle Scholar
White, E. B. & Saric, W. S. 2005 Secondary instability of crossflow vortices. J. Fluid Mech. 525, 275–308.Google Scholar
Zaki, T. A. & Saha, S. 2009 On shear sheltering and the structure of vortical modes in single- and two-fluid boundary layers. J. Fluid Mech. 626, 111–147.CrossRefGoogle Scholar