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Evolution of disturbance wavepackets in an oscillatory Stokes layer

Published online by Cambridge University Press:  09 July 2014

Christian Thomas ,
Christopher Davies ,
Andrew P. Bassom  and
P. J. Blennerhassett
Christian Thomas
Affiliation:
Department of Mathematics, South Kensington Campus, Imperial College London, London SW7 2AZ, UK
Christopher Davies*
Affiliation:
School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK
Andrew P. Bassom
Affiliation:
School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia
P. J. Blennerhassett
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
*
†Email address for correspondence: [email protected]
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Abstract

Numerical simulation results are presented for the linear and nonlinear evolution of disturbances in a flat Stokes layer. The response to a spatially localised impulsive forcing is investigated and it is found that the spatial–temporal development of the flow displays an intriguing family-tree-like structure, which involves the birth of successive generations of distinct wavepacket components. It is shown that some features of this unexpected structure can be predicted using the results of a linear stability analysis based on Floquet theory.

JFM classification

Instability: Absolute/convective instability Boundary Layers: Boundary layer stability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 752 , 10 August 2014 , pp. 543 - 571
Copyright
© 2014 Cambridge University Press 

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References

Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991a An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 1. Experiments. J. Fluid Mech. 225, 395–422.Google Scholar
Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991b An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 2. Numerical simulations. J. Fluid Mech. 225, 423–444.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.CrossRefGoogle Scholar
Blondeaux, P. & Vittori, G. 1994 Wall imperfections as a triggering mechanism for Stokes-layer transition. J. Fluid Mech. 67, 107–135.CrossRefGoogle Scholar
Bowles, R. I., Davies, C. & Smith, F. T. 2003 On the spiking stages in deep transition and unsteady separation. J. Engng Maths 45, 227–245.Google Scholar
Brevdo, L. & Bridges, T. J. 1997 Absolute and convective instabilities of temporally oscillating flows. Z. Angew. Math. Phys. 48, 290–309.Google Scholar
Clamen, M. & Minton, P. 1977 An experimental investigation of flow in an oscillatory pipe. J. Fluid Mech. 77, 421–431.Google Scholar
Clarion, C. & Pelissier, P. 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
Cowley, S. J. 1987 High frequency Rayleigh instability analysis of Stokes layers. In Stability of Time-dependent and Spatially Varying Flows (ed. Dwoyer, D. L. & Hussaini, M. Y.), pp. 261–275. Springer.CrossRefGoogle Scholar
Davies, C. 2003 Convective and absolute instabilities of flow over compliant walls. Fluid Mech. Appl. 72, 69–93.Google 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
Davies, C. & Carpenter, P. 2003 Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. J. Fluid Mech. 486, 287–329.Google Scholar
Davis, S. H. 1976 The stability of time-periodic flows. Annu. Rev. Fluid Mech. 8, 57–74.Google Scholar
Eckmann, D. M. & Grotberg, J. B. 1991 Experiments on transition to turbulence in oscillatory pipe flow. J. Fluid Mech. 222, 329–350.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 stability of the Stokes layers at high Reynolds numbers. J. Fluid Mech. 482, 1–15.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.CrossRefGoogle Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in open flows. In Hydrodynamics and Nonlinear Instabilities (ed. Godreche, C. & Manneville, P.), pp. 81–294. Cambridge University Press.CrossRefGoogle Scholar
Kloker, M., Konzelmann, U. & Fasel, H. 1993 Outflow boundary conditions for spatial Navier–Stokes simulations of transition boundary layers. AIAA J. 31, 602–628.CrossRefGoogle Scholar
Luo, J. & Wu, X. 2010 On linear instability of a finite Stokes layer: Instantaneous versus Floquet modes. Phys. Fluids 22, 054106.CrossRefGoogle Scholar
Merkli, P. & Thomann, H. 1975 Transition to turbulence in oscillating pipe flow. J. Fluid Mech. 68, 567–575.Google 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 68, 327–338.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
von Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753–773.Google Scholar
Womersley, J. R. 1955 Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127, 553–563.CrossRefGoogle ScholarPubMed