Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-09T05:46:22.642Z Has data issue: false hasContentIssue false
  • English
  • Français

Onset of absolutely unstable behaviour in the Stokes layer: a Floquet approach to the Briggs method

Published online by Cambridge University Press:  06 October 2021

Alexander Pretty Open the ORCID record for Alexander Pretty [Opens in a new window] ,
Christopher Davies Open the ORCID record for Christopher Davies [Opens in a new window]  and
Christian Thomas Open the ORCID record for Christian Thomas [Opens in a new window]
Alexander Pretty
Affiliation:
School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK
Christopher Davies
Affiliation:
School of Engineering, University of Leicester, University Road, Leicester LE1 7RH, UK
Christian Thomas*
Affiliation:
Department of Mathematics and Statistics, Macquarie University, New South Wales 2109, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

For steady flows, the Briggs (Electron-Stream Interaction with Plasmas. MIT Press, 1964) method is a well-established approach for classifying disturbances as either convectively or absolutely unstable. Here, the framework of the Briggs method is adapted to temporally periodic flows, with Floquet theory utilised to account for the time periodicity of the Stokes layer. As a consequence of the antiperiodicity of the flow, symmetry constraints are established that are used to describe the pointwise evolution of the disturbance, with the behaviour governed by harmonic and subharmonics modes. On coupling the symmetry constraints with a cusp-map analysis, multiple harmonic and subharmonic cusps are found for each Reynolds number of the flow. Therefore, linear disturbances experience subharmonic growth about fixed spatial locations. Moreover, the growth rate associated with the pointwise development of the disturbance matches the growth rate of the disturbance maximum. Thus, the onset of the Floquet instability (Blennerhassett & Bassom, J. Fluid Mech., vol. 464, 2002, pp. 393–410) coincides with the onset of absolutely unstable behaviour. Stability characteristics are consistent with the spatio-temporal disturbance development of the family-tree structure that has hitherto only been observed numerically via simulations of the linearised Navier–Stokes equations (Thomas et al., J. Fluid Mech., vol. 752, 2014, pp. 543–571; Ramage et al., Phys. Rev. Fluids, vol. 5, 2020, 103901).

JFM classification

Boundary Layers: Boundary layer stability Instability: Absolute/convective instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 928 , 10 December 2021 , A23
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Akhaven, R., Kamm, R.D. & Shapiro, A.H. 1991 An investigation of transition to turbulence in bounded oscillatory flows. Part 1. Experiments. J. Fluid Mech. 225, 395–422.CrossRefGoogle 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. 2006 The linear stability of high-frequency oscillatory flow in a channel. J. Fluid Mech. 556, 1–25.CrossRefGoogle Scholar
Brevdo, L. & Bridges, T.J. 1996 Absolute and convective instabilities of spatially periodic flows. Phil. Trans. R. Soc. A 354, 1027–1064.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
Briggs, R.J. 1964 Electron-Stream Interaction with Plasmas. MIT Press.CrossRefGoogle Scholar
Clamen, M. & Minton, P. 1977 An experimental investigation of flow in an oscillating pipe. J. Fluid Mech. 81, 421–431.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.CrossRefGoogle 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
Gaster, M. 1968 Growth of disturbances in both space and time. Phys. Fluids 11, 723–727.CrossRefGoogle Scholar
Hall, P. 1978 The linear stability of flat Stokes layers. Proc. R. Soc. 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.CrossRefGoogle Scholar
Huerre, P. 2002 Open shear flow instabilities. In Perspectives in Fluid Dynamics (ed. M.G. Worster, G.K. Batchelor, & H.K. Moffatt), chap. 4, pp. 159–229. Cambridge University Press.Google Scholar
Hwang, Y., Kim, J. & Choi, H. 2013 Stabilization of absolute instability in spanwise wavy two-dimensional wakes. J. Fluid Mech. 727, 346–378.CrossRefGoogle Scholar
Kupfer, K., Bers, A. & Ram, A.K. 1987 The cusp map in the complex-frequency plane for absolute instabilities. Phys. Fluids 30, 3075–3082.CrossRefGoogle Scholar
Lingwood, R.J. 1995 Absolute instability of the boundary layer on a rotating disk. J. Fluid Mech. 299, 17–33.CrossRefGoogle Scholar
Lingwood, R.J. 1997 a On the applications of the Briggs’ and steepest-descent methods to a boundary-layer flow. Stud. Appl. Maths 98, 213–254.CrossRefGoogle Scholar
Lingwood, R.J. 1997 b On the effects of suction and injection on the absolute instability of the rotating-disk boundary layer. Phys. Fluids 9, 1317–1328.CrossRefGoogle Scholar
Pier, B. 2003 Finite-amplitude crossflow vortices, secondary instability and transition in the rotating-disk boundary layer. J. Fluid Mech. 487, 315–343.CrossRefGoogle Scholar
Ramage, A. 2017 Linear disturbance evolution in the semi-infinite Stokes layer and related flows. PhD thesis, School of Mathematics, Cardiff University.Google Scholar
Ramage, A., Davies, C., Thomas, C. & Togneri, M. 2020 Numerical simulation of the spatio-temporal development of linear disturbances in Stokes layers: absolute instability and the effects of high frequency harmonics. Phys. Rev. Fluids 5, 103901.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Thomas, C. 2020 The linear stability of an acceleration-skewed oscillatory Stokes layer. J. Fluid Mech. 895, A27.CrossRefGoogle Scholar
Thomas, C. 2021 Effects of velocity skewness on the linear stability of the oscillatory Stokes layer. Phys. Fluids 33, 034104.CrossRefGoogle Scholar
Thomas, C., Bassom, A.P. & Blennerhassett, P.J. 2012 The linear stability of oscillating pipe flow. Phys. Fluids 24, 014106.CrossRefGoogle 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. A 467, 2643–2662.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.CrossRefGoogle 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.CrossRefGoogle Scholar
Trefethen, L.N. 2000 Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Von Kerczek, C. & Davis, S.H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753–773.CrossRefGoogle Scholar