Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-15T01:55:36.257Z Has data issue: false hasContentIssue false
  • English
  • Français

Transient growth of perturbations in Stokes oscillatory flows

Published online by Cambridge University Press:  05 April 2016

Damien Biau Open the ORCID record for Damien Biau [Opens in a new window]
Damien Biau*
Affiliation:
DynFluid Laboratory, Arts et Métiers ParisTech, 151 Boulevard de l’Hôpital, 75013 Paris, France
*
†Email address for correspondence: damien.biau@ensam.eu
Get access Rights & Permissions [Opens in a new window]

Abstract

Oscillatory Stokes flows, with zero mean, are subjected to subcritical transition to turbulence. The maximal energy growth of perturbations is computed in the subcritical regime through an optimisation method. The results show strong amplifications during half a period. The energy transfer from the base flow involves an Orr mechanism with two-dimensional vorticity waves, and the maximum energy scales exponentially with the Reynolds number. Nonlinear simulations show that low-energy perturbations are sufficient to trigger turbulent flow.

JFM classification

Boundary Layers: Boundary layer stability Instability: Instability Instability: Transition to turbulence
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 794 , 10 May 2016 , R4
Check for updates
Copyright
© 2016 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

Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991 An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 2. Numerical simulations. J. Fluid Mech. 225, 423–444.CrossRefGoogle Scholar
Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134–150.Google Scholar
Ascher, U. M., Ruuth, S. J. & Wetton, B. T. R. 1995 Implicit–explicit methods for time- dependent partial differential equations. SIAM J. Numer. Anal. 32, 797–823.CrossRefGoogle Scholar
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, 271–304.Google Scholar
Blennerhassett, P. J. & Bassom, A. P. 2002 The linear stability of flat Stokes layers. J. Fluid. Mech. 464, 393–410.CrossRefGoogle Scholar
Blennerhassett, P. J. & Bassom, A. P. 2008 On the linear stability of Stokes layers. Phil. Trans. R. Soc. Lond. A 366, 2685–2697.Google Scholar
Blondeaux, P. & Vittori, G. 1994 Wall imperfections as a triggering mechanism for Stokes-layer transition. J. Fluid Mech. 264, 107–135.CrossRefGoogle Scholar
Botella, O. 1997 On the solution of the Navier–Stokes equations using Chebyshev projection schemes with third-order accuracy in time. Comput. Fluids 26, 107–116.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Cantwell, C. D., Barkley, D. & Blackburn, H. M. 2010 Transient growth analysis of flow through a sudden expansion in a circular pipe. Phys. Fluids 22, 034101.CrossRefGoogle Scholar
Conrad, P. W. & Criminale, W. O. 1965 The stability of time-dependent laminar flow: parallel flows. Z. Angew. Math. Phys. 16, 233–254.Google Scholar
Corbett, P. & Bottaro, A. 2000 Optimal perturbations for boundary layers subject to stream-wise pressure gradient. Phys. Fluids 12, 120–130.Google Scholar
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31, 2093–2102.CrossRefGoogle Scholar
Hall, P. 1978 The linear stability of flat Stokes layers. Proc. R. Soc. Lond. A 359, 151–166.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.Google Scholar
Jimènez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 221–240.CrossRefGoogle 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
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493–517.Google Scholar
Orr, W. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: a viscous liquid. Proc. R. Irish Acad. A 27, 1907–1909.Google Scholar
Ozdemir, C., Hsu, T.-J. & Balachandar, S. 2014 Direct numerical simulations of transition and turbulence in smooth-walled Stokes boundary layer. Phys. Fluids 26, 045108.Google Scholar
Schlichting, H. 1979 Boundary-layer Theory. McGraw-Hill.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. & Henningson, D. S. 1999 Stability and Transition of Shear Flows. Springer.Google Scholar
Spalart, P. R. 1989 Theoretical and numerical study of a three-dimensional turbulent boundary layer. J. Fluid Mech. 205, 319–340.CrossRefGoogle Scholar
Thomas, C., Blennerhassett, P. J., Bassom, A. P. & Davies, C. 2015 The linear stability of a Stokes layer subjected to high-frequency perturbations. J. Fluid Mech. 764, 193–218.Google Scholar
Thomas, C. T., Bassom, A. P., Blennerhassett, P. J. & Davies, C. 2010 Direct numerical simulations of small disturbances in the classical Stokes layer. J. Engng Maths 68, 327–338.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar
Vittori, G. & Verzicco, R. 1998 Direct simulation of transition in an oscillatory boundary layer. J. Fluid Mech. 371, 207–232.Google Scholar
Von Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62 (4), 753–773.CrossRefGoogle Scholar
Zhao, M., Ghidaoui, M. S. & Kolyshkin, A. A. 2007 Perturbation dynamics in unsteady pipe flows. J. Fluid Mech. 570, 129–154.Google Scholar