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Laminar streak growth above a spanwise oscillating wall

Published online by Cambridge University Press:  06 March 2015

Peter D. Hicks  and
Pierre Ricco
Peter D. Hicks*
Affiliation:
School of Engineering, Fraser Noble Building, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK
Pierre Ricco
Affiliation:
Department of Mechanical Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK
*
†Email address for correspondence: [email protected]
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Abstract

The use of spanwise wall oscillations to attenuate the growth of laminar streaks within the incompressible Blasius boundary layer is investigated. As in the case of the flow above a stationary flat plate, studied by Leib et al. (J. Fluid Mech., vol. 380, 1999, pp. 169–203), free-stream convected gusts interact with the boundary layer to drive the streak growth. Spanwise wall oscillations can either reduce or increase the total energy of the laminar streaks, depending upon the wall oscillation amplitude and frequency, as well as the free-stream gust properties. Reductions in streak energies of up to 90 % are obtained, indicating that spanwise wall oscillations are an effective technique for attenuating the laminar streak growth. Therefore they may suppress secondary boundary-layer instabilities and delay transition. The laminar boundary-layer base flow matches the Blasius profile in the streamwise and wall-normal directions, while in the spanwise direction a generalized version of the classical Stokes layer profile (generated by a wall oscillating beneath a quiescent fluid) occurs, which evolves downstream due to non-parallel flow effects. Via a Wentzel–Kramers–Brillouin–Jeffreys analysis, this generalized Stokes layer is shown to approach the classical Stokes layer in the limit of large downstream distances or high-frequency plate oscillations. The laminar streaks forced by the generalized and the classical Stokes flows differ significantly, which implies that the choice of the spanwise base flow may affect the secondary instability and transition in this flow. The analysis also proves that the use of the classical Stokes layer as spanwise base flow, as employed by Hack & Zaki (Phys. Fluids. vol. 24 (3), 2012, 034101), is inappropriate.

JFM classification

Aerodynamics: Aerodynamics Boundary Layers: Boundary Layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 768 , 10 April 2015 , pp. 348 - 374
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Copyright
© 2015 Cambridge University Press 

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References

Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11 (1), 134–150.Google Scholar
Balakumar, P. & Hall, P. 1999 On the control of the fastest growing Görtler vortex. Theor. Comput. Fluid Dyn. 13, 21–31.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bodonyi, R. J. & Duck, P. W.1990 Boundary-layer receptivity due to a wall suction and control of Tollmien–Schlichting waves. Tech. Rep., NASA Contractor Rep. 182103, ICASE Rep. No. 90-62.Google Scholar
Bodonyi, R. J. & Duck, P. W. 1992 Boundary-layer receptivity due to a wall suction and control of Tollmien–Schlichting waves. Phys. Fluids 4 (6), 1206–1214.CrossRefGoogle Scholar
Byström, M. G., Levin, O. & Henningson, D. S. 2007 Optimal disturbances in suction boundary layers. Eur. J. Mech. (B/Fluids) 26 (3), 330–343.Google Scholar
Cebeci, T. 2002 Convective Heat Transfer. Springer.Google Scholar
Davidsson, E. N. & Gustavsson, L. H. 2008 Elementary solutions for streaky structures in boundary layers with and without suction. Fluid Dyn. Res. 40 (3), 212–231.CrossRefGoogle Scholar
Dhanak, M. R. & Si, C. 1999 On reduction of turbulent wall friction through spanwise wall oscillations. J. Fluid Mech. 383, 175–195.CrossRefGoogle Scholar
Dong, M. & Wu, X. 2013a On continuous spectra of the Orr–Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances. J. Fluid Mech. 732, 616–659.CrossRefGoogle Scholar
Dong, M. & Wu, X.2013b On continuous spectra of the Orr–Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances in the Blasius boundary layer. In Proceedings of the 43rd AIAA Fluid Dynamics Conference, San Diego, CA, 24–27 June. American Institute of Aeronautics and Astronautics.Google Scholar
Dryden, H. L.1936 Air flow in the boundary layer near a plate. NACA Tech. Rep. 562. NACA.Google Scholar
El-Hady, N. M. 1992 Secondary instability of high-speed flows and the influence of wall cooling and suction. Phys. Fluids 4 (4), 727–743.Google Scholar
Fang, T. & Lee, C.-F. 2009 Three-dimensional wall-bounded laminar boundary layer with span-wise cross free stream and moving boundary. Acta Mechanica 204, 235–248.Google Scholar
Floryan, J. M. & Saric, W. S. 1983 Effects of suction on the Görtler instability of boundary layers. AIAA J. 21, 1635–1639.Google Scholar
Fransson, J. H. M. & Alfredsson, P. H. 2003 On the disturbance growth in an asymptotic suction boundary layer. J. Fluid Mech. 482, 51–90.Google Scholar
Galionis, I. & Hall, P. 2005 On the stabilization of the most amplified Görtler vortex on a concave surface by spanwise oscillations. J. Fluid Mech. 527, 265–283.CrossRefGoogle Scholar
Glauert, M. B. 1956 The laminar boundary layer on oscillating plates and cylinders. J. Fluid Mech. 1 (1), 97–110.Google Scholar
Goldstein, M. E. 1983 The evolution of Tollmien–Schlichting waves near a leading edge. J. Fluid Mech. 127, 59–81.Google Scholar
Gulyaev, A. N., Kozlov, V. E., Kuznetsov, V. R., Mineev, B. I. & Sekundov, A. N. 1989 Interaction of a laminar boundary layer with external turbulence. Fluid Dyn. 24, 700–710.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
Hall, P. 1978 The linear stability of flat Stokes layers. Proc. R. Soc. Lond. A 359 (1697), 151–166.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.Google Scholar
Hubbard, S. & Riley, N. 1995 Boundary-layer control by heat and mass transfer. Intl J. Heat Mass Transfer 38 (17), 3209–3217.CrossRefGoogle Scholar
Jones, R. T.1947 Effects of sweepback on boundary-layer and separation. NACA Tech. Note 1402. NACA.Google Scholar
Jung, W. J., Mangiavacchi, N. & Akhavan, R. 1992 Suppression of turbulence in wall-bounded flows by high-frequency spanwise oscillations. Phys. Fluids A 4 (8), 1605–1607.Google Scholar
Klebanoff, P. S. 1971 Effect of free-stream turbulence on a laminar boundary layer. Bull. Am. Phys. Soc. 16, 1323.Google Scholar
Laadhari, F., Skandaji, L. & Morel, R. 1994 Turbulence reduction in a boundary layer by a local spanwise oscillating surface. Phys. Fluids 6 (10), 3218–3220.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.Google Scholar
Liepmann, H. W. & Fila, G. H.1947 Investigation of effects of surface temperature and single roughness elements on boundary-layer transition. NACA Tech. Note 1196. NACA.Google Scholar
Lighthill, M. J. 1954 The response of laminar skin friction and heat transfer to fluctuations in the stream velocity. Proc. R. Soc. Lond. A 224 (1156), 1–23.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
Luo, J. & Wu, X. 2010 On the linear instability of a finite Stokes layer: instantaneous versus Floquet modes. Phys. Fluids 22 (5), 054106.Google Scholar
Mans, J., Kadijk, E. C., de Lange, H. C. & van Steenhoven, A. A. 2005 Breakdown in a boundary layer exposed to free-stream turbulence. Exp. Fluids 39 (6), 1071–1083.Google Scholar
Matsubara, M. & Alfredsson, P. H. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.CrossRefGoogle Scholar
Quadrio, M. & Ricco, P. 2003 Initial response of a turbulent channel flow to spanwise oscillation of the walls. J. Turbul. 4, N7.Google Scholar
Rabin, S. M. E., Caulfield, C. P. & Kerswell, R. R. 2014 Designing a more nonlinearly stable laminar flow via boundary manipulation. J. Fluid Mech. 738, R1.Google Scholar
Ricco, P. 2009 The pre-transitional Klebanoff modes and other boundary-layer disturbances induced by small-wavelength free-stream vorticity. J. Fluid Mech. 638, 267–303.Google Scholar
Ricco, P. 2011 Laminar streaks with spanwise wall forcing. Phys. Fluids 23 (6), 064103.Google Scholar
Ricco, P. & Dilib, F. 2010 The influence of wall suction and blowing on boundary-layer laminar streaks generated by free-stream vortical disturbances. Phys. Fluids 22 (4), 044101.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
Ricco, P., Tran, D.-L. & Ye, G. 2009 Wall heat transfer effects on Klebanoff modes and Tollmien–Schlichting waves in a compressible boundary layer. Phys. Fluids 21 (2), 024106.Google Scholar
Ricco, P. & Wu, X. 2007 Response of a compressible laminar boundary layer to free-stream vortical disturbances. J. Fluid Mech. 587, 97–138.Google Scholar
Riley, N. 1965 Oscillating viscous flows. Mathematika 12, 161–175.CrossRefGoogle Scholar
Riley, N. 1967 Oscillatory viscous flows. Review and extension. IMA J. Appl. Maths 3 (4), 419–434.CrossRefGoogle Scholar
Riley, N. 1991 Oscillating viscous flows: II. Superposed oscillations. Mathematika 38, 203–216.Google Scholar
Ruban, A. I., Bernots, T. & Pryce, D. 2013 Receptivity of the boundary layer to vibrations of the wing surface. J. Fluid Mech. 723, 480–528.CrossRefGoogle Scholar
Saric, W. S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26 (1), 379–409.Google Scholar
Schlichting, H. & Gersten, K. 2001 Boundary Layer Theory, 8th edn. Springer.Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9 (2), 8–106.Google Scholar
Taylor, G. I. 1939 Some recent developments in the study of turbulence. In Proceedings of the Fifth International Congress for Applied Mechanics (ed. Hartog, J. P. D. & Peters, H.), pp. 294–310. Wiley.Google Scholar
Wu, X. 2001 Receptivity of boundary layers with distributed roughness to vortical and acoustic disturbances: a second-order asymptotic theory and comparison with experiments. J. Fluid Mech. 431, 91–133.Google Scholar
Wu, X. & Choudhari, M.2001 Effects of long-wavelength Klebanoff modes on boundary-layer instability. Tech. Rep., Annual Research Briefs, Center for Turbulence Research.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., 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
Wuest, W. 1952 Grenzschichten an zylindrischen Körpern mit nichtstationärer Querbewegung. Z. Angew. Math. Mech. 32 (6), 172–178.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.Google Scholar