Skip to main content
    • Aa
    • Aa

Wall imperfections as a triggering mechanism for Stokes-layer transition

  • P. Blondeaux (a1) and G. Vittori (a1)

The boundary layer generated by the harmonic oscillations of a wavy wall in a fluid otherwise at rest is studied. First the wall waviness is assumed to be of small amplitude and large values of the Reynolds number are considered. The results obtained by means of a linear analysis, where the time variable appears only as a parameter, show that resonance may occur. Indeed it is found that when the Reynolds number is larger than a critical value, an instant within the decelerating part of the cycle exists such that a waviness of infinitesimal amplitude induces unbounded perturbations of the flow in the Stokes layer. The passage through resonance is then studied by means of a multiple-timescale approach, taking into account the damping effect of local acceleration within a small time range around resonance. The asymptotic approach fails beyond a threshold value of the Reynolds number, because the damping effect of the local acceleration terms spreads over the whole cycle. The problem is then tackled by means of an approach that takes into account the above damping effect throughout the whole cycle. Finally, a numerical procedure is used that also allows the inclusion of nonlinear terms and the study of the interactions among forced and free modes. The numerical approach reveals that, even for relatively large values of the amplitude of the wall waviness, nonlinear effects are negligible and the damping of resonance is mainly due to local acceleration effects. The relevance of the results to the understanding of transition to turbulence in Stokes layers is discussed.

Hide All
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, 423444.

Bayly, B. J.1986Three-dimensional instability of elliptical flow. Phys. Rev. Lett.57, 21602163.

Bayly, B. J., Orszag, S. A. & Herbert, T.1988Instability mechanisms in shear-flow transition. Ann. Rev. Fluid Mech.20, 359391.

Blondeaux, P. 1990 Sand ripples under sea waves. Part 1. Ripple formation. J. Fluid Mech. 218, 117.

Blondeaux, P. & Vittori, G. 1991a Vorticity dynamics in an oscillatory flow over a rippled bed. J. Fluid Mech. 226, 257289.

Blondeaux, P. & Vittori, G.1991bA route to chaos in an oscillatory flow: Feigenbaum scenario. Phys. Fluids A 3, 24922495.

Corke, T. C., Bar Sever, A. & Morkovin, M. V.1986Experiments on transition enhancement by distributed roughness. Phys. Fluids29, 31993213.

Crouch, J. D. 1992 Non-localized receptivity of boundary layers. J. Fluid Mech. 244, 567581.

Eckmann, D. M. & Grotberg, J. B. 1991 Experiments on transition to turbulence in oscillatory pipe flow. J. Fluid Mech. 222, 329350.

Goldstein, M. E. & Hultgren, L. S.1989Boundary-layer receptivity to long-wave free-stream disturbances. Ann. Rev. Fluid Mech.21, 137166.

Hall, P.1978The linear stability of flat Stokes layers. Proc. R. Soc. Lond. A 359, 151166.

Hara, T. & Mei, C. C. 1990 Oscillating flows over periodic ripples. J. Fluid Mech. 211, 183209.

Hino, M., Sawamoto, M. & Takasu, S. 1976 Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech. 75, 193207.

Kaneko, A. & Honjii, H. 1979 Double structures of steady streaming in the oscillatory viscous flow over a wavy wall. J. Fluid Mech. 93, 727736.

Kervorkian, J.1971Passage through resonance for a one-dimensional oscillator with slowly varying frequency. SIAM J. Appl. Maths20, 364373.

Landman, M. J. & Saffman, P. G.1987The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids30, 23392342.

Lyne, W. H. 1971 Unsteady viscous flow over wavy wall. J. Fluid Mech. 50, 3348.

Merkly, P. & Thomann, H. 1975 Transition to turbulence and velocity distribution in an oscillatory pipe flow. J. Fluid Mech. 68, 567575.

Orszag, S. A. 1971 Numerical simulation of incompressible flows within simple boundaries: accuracy. J. Fluid Mech. 49, 75112.

Pierrehumbert, R. T.1986Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett.57, 21572159.

Reshotko, E.1976Boundary layer stability and transition. Ann. Rev. Fluid Mech.8, 311349.

Seminara, G. & Hall, P.1975Linear stability of slowly varying unsteady flows in a curved channel. Proc. R. Soc. Lond. A 346, 279303.

Seminara, G. & Hall, P.1976Centrifugal instability of a Stokes layer: linear theory. Proc. R. Soc. Lond. A 350, 299316.

Sergeev, S. I.1966Fluid oscillations in pipes at moderate Reynolds number. Fluid Dyn.1, 121122.

Shen, S. F.1961Some considerations on the laminar stability of time dependent basic flows. J. Aero Sci.28, 397404.

Sleath, J. F. A.1976On rolling-grain ripples. J. Hydraul. Res.14, 6981.

Vittori, G.1988Nonlinear viscous oscillatory flow over a small amplitude wavy wall. J. Hydraul. Res.27, 267280.

Vittori, G.1992Oscillatory boundary layer induced by sea waves over brick-pattern ripples. J. Hydraul. Engng118, 12411259.

Von Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753773.

Wu, X. 1992 The nonlinear evolution of high-frequency resonant triad waves in an oscillatory Stokes layer at high Reynolds number. J. Fluid Mech. 245, 553597.

Wu, X., Lee, S. S. & Cowley, S. J. 1993 On the weakly nonlinear three-dimensional instability of shear layers to pairs of oblique waves: the Stokes layer as a paradigm. J. Fluid Mech. 253, 681721.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 12 *
Loading metrics...

Abstract views

Total abstract views: 71 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th October 2017. This data will be updated every 24 hours.