Skip to main content Accessibility help
×
Home

Transient growth in the flow past a three-dimensional smooth roughness element

  • S. Cherubini (a1) (a2), M. D. De Tullio (a1), P. De Palma (a1) and G. Pascazio (a1)

Abstract

This work provides a global optimization analysis, looking for perturbations inducing the largest energy growth at a finite time in a boundary-layer flow in the presence of smooth three-dimensional roughness elements. Amplification mechanisms are described which can bypass the asymptotical growth of Tollmien–Schlichting waves. Smooth axisymmetric roughness elements of different height have been studied, at different Reynolds numbers. The results show that even very small roughness elements, inducing only a weak deformation of the base flow, can localize the optimal disturbance characterizing the Blasius boundary-layer flow. Moreover, for large enough bump heights and Reynolds numbers, a strong amplification mechanism has been recovered, inducing an increase of several orders of magnitude of the energy gain with respect to the Blasius case. In particular, the highest value of the energy gain is obtained for an initial varicose perturbation, differently to what found for a streaky parallel flow. Optimal varicose perturbations grow very rapidly by transporting the strong wall-normal shear of the base flow, which is localized in the wake of the bump. Such optimal disturbances are found to lead to transition for initial energies and amplitudes considerably smaller than sinuous optimal ones, inducing hairpin vortices downstream of the roughness element.

Copyright

Corresponding author

Email address for correspondence: s.cherubini@gmail.com

References

Hide All
Acalar, M. & Smith, C. 1987 A study of hairpin vortices in a laminar boundary layer: part 1, hairpin vortices generated by a hemisphere protuberance. J. Fluid Mech. 175, 141.
Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19, 041301.
Andersson, P., Brandt, L., Bottaro, A. & Henningson, D. S. 2001 On the breakdown of boundary layer streaks. J. Fluid Mech. 428, 2960.
Barkley, D., Gomes, G. M. & Henderson, R. D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167190.
Barkley, D. & Henderson, R. D. 1996 Floquet stability analysis of the periodic wake of a circular cylinder. J. Fluid Mech. 322, 215241.
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271304.
Bottaro, A. 1990 Note on open boundary conditions for elliptic flows. Numer. Heat Transfer B 18, 243256.
Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transition in a boundary layers subject to free stream turbulence. J. Fluid Mech. 517, 167198.
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 16371650.
Cherubini, S., De Palma, P., Robinet, J.-Ch. & Bottaro, A. 2010a Rapid path to transition via nonlinear localized optimal perturbations. Phys. Rev. E 82, 066302.
Cherubini, S., De Palma, P., Robinet, J.-Ch. & Bottaro, A. 2011 The minimal-seed of turbulent transition in a boundary layer. J. Fluid Mech. 689, 221253.
Cherubini, S., Robinet, J.-Ch., Bottaro, A. & De Palma, P. 2010b Optimal wave packets in a boundary layer and initial phases of a turbulent spot. J. Fluid Mech. 656, 231259.
Cherubini, S., Robinet, J.-C. & De Palma, P. 2010c The effects of non-normality and nonlinearity of the Navier–Stokes operator on the dynamics of a large laminar separation bubble. Phys. Fluids 22 (1), 014102.
Choudhari, M. & Fischer, P. 2005 Roughness-induced transient growth: nonlinear effects. In 35th AIAA Fluid Dynamics Conference and Exhibit, Toronto, Ontario Canada, AIAA-2005-4765.
Corbett, P. & Bottaro, A. 2000 Optimal perturbations for boundary layers subject to stream-wise pressure gradient. Phys. Fluids 12, 120130.
Fadlun, E. A, Verzicco, R., Orlandi, P. & Mohd-Yusof, J. 2000 Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J. Comput. Phys. 161, 35.
Farrell, B. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31, 20932102.
Fransson, J. H. M., Brandt, L., Talamelli, A. & Cossu, C. 2004 Experimental and theoretical investigation of the nonmodal growth of steady streaks in a flat plate boundary layer. Phys. Fluids 16 (10), 36273638.
Fransson, J. H. M., Brandt, L., Talamelli, A. & Cossu, C. 2005 Experimental study of the stabilization of Tollmien–Schlichting waves by finite amplitude streaks. Phys. Fluids 17 (5), 054110.
Fransson, J. H. M., Talamelli, A., Brandt, L. & Cossu, C. 2006 Delaying transition to turbulence by a passive mechanism. Phys. Rev. Lett. 96 (6), 064501.
Gallaire, F., Marquillie, M. & Ehrenstein, U. 2007 Three-dimensional transverse instabilities in detached boundary-layers. J. Fluid Mech. 571, 221233.
Gaster, M., Grosch, C. E. & Jackson, T. L. 1994 The velocity field created by a shallow bump in a boundary layer. Phys. Fluids 6, 3079.
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167197.
Gregory, N. & Walker, W. S. 1956 The effect on transition of isolated surface excrescences in the boundary layer. A.R.C. Tech Rep. R. & M. No. 2779.
Hoepffner, J., Brandt, L. & Henningson, D. S. 2005 Transient growth on boundary layer streaks. J. Fluid Mech. 537, 91100.
Joslin, R. D. & Grosch, C. E. 1995 Growth characteristics downstream of a shallow bump: computation and experiment. Phys. Fluids 7 (12), 30423047.
Lipatov, I. I. & Vinogradov, I. V. 2000 Three-dimensional flow near surface distortions for the compensation regime. Phil. Trans. R. Soc. Lond. A 358, 31433153.
Luchini, P. 2000 Reynolds number indipendent instability of the Blasius boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289309.
Marquet, O., Sipp, D., Chomaz, J.-M. & Jacquin, L. 2008 Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429443.
Mohd-Yusof, J. 1997 Combined Immersed Boundaries/B-Splines Methods for Simulations of Flows in Complex Geometries. CTR annual research briefs. NASA Ames/Stanford University.
Monokrousos, A., Akervik, E., Brandt, L. & Henningson, D. S. 2010 Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time steppers. J. Fluid Mech. 650, 181214.
O’Rourke, J. 1993 Computational Geometry in C. Cambridge University Press.
Orr, W. MF. 1907 The stability or instability of the steady motions of a liquid. Part I. Proc. R. Irish Acad. A 27, 968.
Piot, E., Casalis, G. & Rist, U. 2008 Stability of the laminar boundary layer flow encountering a row of roughness elements: biglobal stability approach and dns. Eur. J. Mech. (B/Fluids) 27 (6), 684706.
Polak, E. & Ribière, G. 1969 Note sur la convergence de directions conjugées. Rev. Franc. Inform. Rech. Operat. 16, 3543.
Schmid, P. J. 2000 Linear stability theory and by pass transition in shear flows. Phys. Plasmas 7, 17881794.
Schmid, P. & Henningson, D. 2001 Stability and Transition in Shear Fows. Springer.
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57108.
Tani, I., Komoda, H., Komatsu, Y. & Iuchi, M. 1962 Boundary-layer transition by isolated roughness. Tech Rep. 375.
Theofilis, V. 2003 Advances in global linear instability of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 249315.
Theofilis, V., Hein, S. & Dallmann, U. 2000 On the origins of unsteadiness and three-dimensionality in a laminar separation bubble. Phil. Trans. R. Soc. Lond. 358 (1777), 32293246.
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for the three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123 (2), 402414.
White, E. B. 2002 Transient growth of stationary disturbances in a flat plate boundary layer. Phys. Fluids 14 (12), 4429.
White, E. B. & Ergin, F. G. 2003 Receptivity and transient growth of roughness-induced disturbances. 33rd AIAA Fluid Dynamics Conference and Exhibit, 23–26 June 2003, Orlando, FL, AIAA 2003-4243.
Zuccher, S., Luchini, P. & Bottaro, A. 2004 Algebraic growth in a blasius boundary layer: optimal and robust control by mean suction in the nonlinear regime. Eur. J. Mech. (B/Fluids) 513, 135160.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Transient growth in the flow past a three-dimensional smooth roughness element

  • S. Cherubini (a1) (a2), M. D. De Tullio (a1), P. De Palma (a1) and G. Pascazio (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed