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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
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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

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