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A class of exact Navier–Stokes solutions for homogeneous flat-plate boundary layers and their linear stability

Published online by Cambridge University Press:  07 July 2014

Michael O. John ,
Dominik Obrist  and
Leonhard Kleiser
Michael O. John*
Affiliation:
Institute of Fluid Dynamics, ETH Zurich, CH-8092 Zurich, Switzerland
Dominik Obrist
Affiliation:
Institute of Fluid Dynamics, ETH Zurich, CH-8092 Zurich, Switzerland
Leonhard Kleiser
Affiliation:
Institute of Fluid Dynamics, ETH Zurich, CH-8092 Zurich, Switzerland
*
†Email address for correspondence: john@ifd.mavt.ethz.ch
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Abstract

We introduce a new boundary layer formalism on the basis of which a class of exact solutions to the Navier–Stokes equations is derived. These solutions describe laminar boundary layer flows past a flat plate under the assumption of one homogeneous direction, such as the classical swept Hiemenz boundary layer (SHBL), the asymptotic suction boundary layer (ASBL) and the oblique impingement boundary layer. The linear stability of these new solutions is investigated, uncovering new results for the SHBL and the ASBL. Previously, each of these flows had been described with its own formalism and coordinate system, such that the solutions could not be transformed into each other. Using a new compound formalism, we are able to show that the ASBL is the physical limit of the SHBL with wall suction when the chordwise velocity component vanishes while the homogeneous sweep velocity is maintained. A corresponding non-dimensionalization is proposed, which allows conversion of the new Reynolds number definition to the classical ones. Linear stability analysis for the new class of solutions reveals a compound neutral surface which contains the classical neutral curves of the SHBL and the ASBL. It is shown that the linearly most unstable Görtler–Hämmerlin modes of the SHBL smoothly transform into Tollmien–Schlichting modes as the chordwise velocity vanishes. These results are useful for transition prediction of the attachment-line instability, especially concerning the use of suction to stabilize boundary layers of swept-wing aircraft.

JFM classification

Boundary Layers: Boundary layer stability Boundary Layers: Boundary Layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 752 , 10 August 2014 , pp. 462 - 484
Copyright
© 2014 Cambridge University Press 

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