Hostname: page-component-6bf8c574d5-5ws7s Total loading time: 0 Render date: 2025-03-09T13:31:08.386Z Has data issue: false hasContentIssue false
  • English
  • Français

On shear sheltering and the structure of vortical modes in single- and two-fluid boundary layers

Published online by Cambridge University Press:  10 May 2009

TAMER A. ZAKI  and
SANDEEP SAHA
TAMER A. ZAKI*
Affiliation:
Mechanical Engineering, Imperial College, London SW7 2AZ, UK
SANDEEP SAHA
Affiliation:
Mechanical Engineering, Imperial College, London SW7 2AZ, UK
*
†Email address for correspondence: t.zaki@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Studies of vortical interactions in boundary layers have often invoked the continuous spectrum of the Orr–Sommerfeld (O-S) equation. These vortical eigenmodes provide a link between free-stream disturbances and the boundary-layer shear – a link which is absent in the inviscid limit due to shear sheltering. In the presence of viscosity, however, a shift in the dominant balance in the operator determines the structure of these eigenfunctions inside the mean shear. In order to explain the mechanics of shear sheltering and the structure of the continuous modes, both numerical and asymptotic solutions of the linear perturbation equation are presented in single- and two-fluid boundary layers. The asymptotic analysis identifies three limits: a convective shear-sheltering regime, a convective–diffusive regime and a diffusive regime. In the shear-dominated limit, the vorticity eigenfunction possesses a three-layer structure, the topmost being a region of exponential decay. The role of viscosity is most pronounced in the diffusive regime, where the boundary layer becomes ‘transparent’ to the oscillatory eigenfunctions. Finally, the convective–diffusive regime demonstrates the interplay between the the accumulative effect of the shear and the role of viscosity. The analyses are complemented by a physical interpretation of shear-sheltering mechanism. The influence of a wallfilm, in particular viscosity and density stratification, and surface tension are also evaluated. It is shown that a modified wavenumber emerges across the interface and influences the penetration of vortical disturbances into the two-fluid shear flow.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 626 , 10 May 2009 , pp. 111 - 147
Copyright
Copyright © Cambridge University Press 2009

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

Belcher, S. E. & Hunt, J. C. R. 1998 Turbulent flow over hills and waves. Annual Rev. Fluid Mech. 30, 507–538.CrossRefGoogle Scholar
Berger, S. A. & Aroesty, J. 1977 ‘e 9’: stability theory and boundary layer transition. Tech Rep. R-1898-ARPA Rand Corporation.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three dimensional optimal perturbation in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Charru, F. & Hinch, E. J. 2000 ‘Phase diagram’ of interfacial instabilities in a two-layer Couette flow and mechanism of the long-wave instability. J. Fluid Mech. 414, 195–223.CrossRefGoogle Scholar
Craik, A. 1991 The continuous spectrum of the Orr–Sommerfeld equation: note on a paper of Grosch and Salwen. J. Fluid Mech. 226, 565–571.CrossRefGoogle Scholar
Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. The spectrum and the eigenfunctions. J. Fluid Mech. 87, 33–54.CrossRefGoogle Scholar
Hernon, D., Walsh, E. J. & McEligot, D. M. 2007 Experimental investigation into the routes to bypass transition and the shear-sheltering phenomenon. J. Fluid Mech. 591, 461–479.CrossRefGoogle Scholar
Hooper, A. P. & Boyd, W. G. C. 1983 Shear-flow instability at the interface between viscous fluids. J. Fluid Mech. 128, 507–528.CrossRefGoogle Scholar
Hooper, A. P. & Boyd, W. G. C. 1987 Shear-flow instability due to a wall and a viscosity discontinuity at the interface. J. Fluid Mech. 179, 201–225.CrossRefGoogle Scholar
Hunt, J. C. R. 1977 A review of rapidly distorted turbulent flows and its applications. Fluid Dyn. Res. 9 (01), 121–152.Google Scholar
Hunt, J. C. R. & Durbin, P. A. 1999 Perturbed vortical layers and shear sheltering. Fluid Dyn. Res. 24 (06), 375–404.CrossRefGoogle Scholar
Jacobs, R. G. & Durbin, P. A. 1998 Shear sheltering and the continuous spectrum of the Orr–Sommerfeld equation. Phys. fluids 10 (8), 2006–2011.CrossRefGoogle Scholar
Jacobs, R. G. & Durbin, P. A. 2001 Simulations of bypass transition. J. Fluid Mech. 428, 185–212.CrossRefGoogle Scholar
Jordinson, R. 1971 Spectrum of eigenvalues of the Orr–Sommerfeld equation for Blasius flow. Phys. fluids 14 (11), 2535–2537.CrossRefGoogle Scholar
Mack, L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497–520.CrossRefGoogle Scholar
Maslowe, S. A. & Spiteri, R. J. 2001 The continuous spectrum for a boundary layer in a streamwise pressure gradient. Phys. fluids 13, 1294–1299.CrossRefGoogle Scholar
Morkovin, M. V. 1969 Viscous Drag Reduction. Plenum.Google Scholar
Nakamura, L., Kershaw, R. & Grait, N. 1996 Prediction of near-surface gusts generated by deep convection. Meteorol. Appl. 3, 157–167.CrossRefGoogle Scholar
Nelson, J. J., Alving, A. E. & Joseph, D. D. 1995 Boundary layer flow of air over water on a flat plate. J. Fluid Mech. 284, 159–169.CrossRefGoogle Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50 (4), 689–703.CrossRefGoogle Scholar
Salwen, H. & Grosch, C. E. 1981 The continuous spectrum of the Orr–Sommerfeld equation. Part 2. Eigenfunction expansions. J. Fluid Mech. 104, 445–465.CrossRefGoogle Scholar
Schlichting, H. 1933 Berechnung der Anfachung kleiner Störungen bei der Plattenströmung. ZAMM 13, 171–174.Google Scholar
Schlichting, H. 1987 Boundary-Layer Theory, 7th ed.McGraw-Hill.Google Scholar
Tollmien, W. 1929 Über die Entstehung der Turbulenz. Nachr. Ges. Wiss. Göttingen 1, 21–44.Google Scholar
Tumin, A. 2003 Multimode decomposition of spatially growing perturbations in a two-dimensional boundary layer. Phys. Fluids 15 (9), 2525–2540.CrossRefGoogle Scholar
Yecko, P. & Zaleski, S. 2005 Transient growth in two-phase mixing layers. J. Fluid Mech. 528, 43–52.CrossRefGoogle Scholar
Yih, C. S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.CrossRefGoogle Scholar
Zaki, T. A. & Durbin, P. A. 2005 Mode interaction and the bypass route to transition. J. Fluid Mech. 531, 85–111.CrossRefGoogle Scholar
Zaki, T. A. & Durbin, P. A. 2006 Continuous mode transition and the effects of pressure gradient. J. Fluid Mech. 563, 357–388.CrossRefGoogle Scholar