Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T06:44:24.007Z Has data issue: false hasContentIssue false
  • English
  • Français

The linear stability of a Stokes layer subjected to high-frequency perturbations

Published online by Cambridge University Press:  23 December 2014

Christian Thomas ,
P. J. Blennerhassett ,
Andrew P. Bassom  and
Christopher Davies
Christian Thomas
Affiliation:
Department of Mathematics, South Kensington Campus, Imperial College London, London SW7 2AZ, UK
P. J. Blennerhassett
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
Andrew P. Bassom*
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia
Christopher Davies
Affiliation:
Department of Mathematics, Cardiff University, Cardiff CF24 4AG, UK
*
†Email address for correspondence: Andrew.Bassom@uwa.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Quantitative results for the linear stability of planar Stokes layers subject to small, high-frequency perturbations are obtained for both a narrow channel and a flow approximating the classical semi-infinite Stokes layer. Previous theoretical and experimental predictions of the critical Reynolds number for the classical flat Stokes layer have differed widely with the former exceeding the latter by a factor of two or three. Here it is demonstrated that only a 1 % perturbation, at an appropriate frequency, to the nominal sinusoidal wall motion is enough to result in a reduction of the theoretical critical Reynolds number of as much as 60 %, bringing the theoretical conditions much more in line with the experimentally reported values. Furthermore, within the various experimental observations there is a wide variation in reported critical conditions and the results presented here may provide a new explanation for this behaviour.

JFM classification

Boundary Layers: Boundary Layers Boundary Layers: Boundary layer stability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 764 , 10 February 2015 , pp. 193 - 218
Check for updates
Copyright
© 2014 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 1. Experiments. J. Fluid Mech. 225, 395–422.Google Scholar
Alizard, F., Cherubini, S. & Robinet, J.-C. 2009 Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids 21, 064108.Google Scholar
Blennerhassett, P. J. & Bassom, A. P. 2002 The linear stability of flat Stokes layers. J. Fluid Mech. 464, 393–410.Google Scholar
Blennerhassett, P. J. & Bassom, A. P. 2006 The linear stability of high-frequency oscillatory flow in a channel. J. Fluid Mech. 556, 1–25.CrossRefGoogle Scholar
Blennerhassett, P. J. & Bassom, A. P. 2007 The linear stability of high-frequency flow in a torsionally oscillating cylinder. J. Fluid Mech. 576, 491–505.Google Scholar
Blondeaux, P. & Vittori, G. 1994 Wall imperfections as a triggering mechanism for Stokes-layer transition. J. Fluid Mech. 67, 107–135.CrossRefGoogle Scholar
Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transistion in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517, 167–198.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.Google Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Clamen, M. & Minton, P. 1977 An experimental investigation of flow in an oscillatory pipe. J. Fluid Mech. 77, 421–431.Google Scholar
Conrad, P. W. & Criminale, W. O. 1965 The stability of time-dependent laminar flow: parallel flows. Z. Angew. Math. Phys. 16, 233–254.CrossRefGoogle Scholar
Davies, C. & Carpenter, P. W. 2001 A novel velocity–vorticity formulation of the Navier–Stokes equations with applications to boundary layer disturbance evolution. J. Comput. Phys. 172, 119–165.Google Scholar
Eckmann, D. M. & Grotberg, J. B. 1991 Experiments on transition to turbulence in oscillatory pipe flow. J. Fluid Mech. 222, 329–350.CrossRefGoogle Scholar
Fornberg, B. 1996 A Practical Guide to Pseudospectral Methods. Cambridge University Press.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.Google Scholar
Grotberg, J. 1994 Pulmonary flow and transport phenomena. Annu. Rev. Fluid Mech. 26, 529–571.Google Scholar
Hall, P. 1978 The linear stability of flat Stokes layers. Proc. R. Soc. Lond. A 359, 151–166.Google Scholar
Hino, M., Sawamoto, M. & Takasu, S. 1976 Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech. 75, 193–207.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Jensen, B., Sumer, B. & Fredsøe, J. 1989 Turbulent oscillatory boundary layers at high Reynolds numbers. J. Fluid Mech. 206, 265–297.CrossRefGoogle Scholar
von Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753–773.Google Scholar
Lodahl, C. R., Sumer, B. M. & Fredsøe, J. 1998 Turbulent combined oscillatory flow and current in a pipe. J. Fluid Mech. 373, 313–348.Google Scholar
Luchini, P., Giannetti, F. & Pralits, J. 2008 Structural sensitivity of linear and nonlinear global modes. In Proceedings of the Fifth AIAA Theoretical Fluid Mechanics Conference (AIAA), 5th AIAA Theoretical Fluids Conference, 23–26 June 2008, Seattle. AIAA Paper 2008–4227. 19 pages.Google Scholar
Merkli, P. & Thomann, H. 1975 Transition to turbulence in oscillating pipe flow. J. Fluid Mech. 68, 567–575.Google Scholar
Pralits, J., Brandt, L. & Giannetti, F. 2010 Instability and sensitivity of the flow around a rotating circular cylinder. J. Fluid Mech. 650, 513–536.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Thomas, C., Bassom, A. P. & Blennerhassett, P. J. 2012 The linear stability of oscillating pipe flow. Phys. Fluids 24, 014105.CrossRefGoogle Scholar
Thomas, C., Bassom, A. P., Blennerhassett, P. J. & Davies, C. 2010 Direct numerical simulations of small disturbances in the classical Stokes layer. J. Engng Maths 67, 327–338.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
Thomas, C., Davies, C., Bassom, A. P. & Blennerhassett, P. J. 2014 Evolution of disturbance wavepackets in an oscillatory Stokes layer. J. Fluid Mech. 752, 543–571.Google Scholar
Trefethen, L. N. 1997 Pseudospectra of linear operators. SIAM Rev. 39, 383–406.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. SIAM.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
Verzicco, R. & Vittori, G. 1996 Direct simulation of transition in Stokes boundary layers. Phys. Fluids 8, 1341–1343.Google Scholar
Vittori, G. & Verzicco, R. 1998 Direct simulation of transition in an oscillatory boundary layer. J. Fluid Mech. 371, 207–232.Google Scholar
Wright, T. G.2002 Eigtool. http://www.comlab.ox.ac.uk/pseudospectra/eigtool.Google Scholar