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Injection into boundary layers: solutions beyond the classical form

Published online by Cambridge University Press:  07 June 2017

R. E. Hewitt Open the ORCID record for R. E. Hewitt [Opens in a new window] ,
P. W. Duck  and
A. J. Williams
R. E. Hewitt*
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
P. W. Duck
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
A. J. Williams
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
*
†Email address for correspondence: [email protected]
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Abstract

This theoretical and numerical study presents three-dimensional boundary-layer solutions for laminar incompressible flow adjacent to a semi-infinite flat plate, subject to a uniform free-stream speed and injection through the plate surface. The novelty in this case arises from a fully three-dimensional formulation, which also allows for slot injection over a spanwise length scale comparable to the boundary-layer thickness. This approach retains viscous effects in both the spanwise and transverse directions, and effectively results in a parabolised Navier–Stokes system (sometimes referred to as the ‘boundary-region equations’). Any injection profile can be described in this approach, but we restrict attention to three-dimensional states driven by a finite-width slot aligned with the flow direction and self-similar in their downstream development. The classical two-dimensional states are known to only exist up to a critical (‘blow off’) injection amplitude, but the three-dimensional solutions here appear possible for any injection velocity. These new states take the form of low-speed streamwise-aligned streaks whose geometry depends on the amplitude of injection and the spanwise width of the injection slot; intriguingly, although very low wall shear is typically obtained, streamwise flow reversal is not observed, however hard the blowing. Asymptotic descriptions are provided in the limit of increasing slot width and fixed injection velocity, which allow for classification of the solutions according to two bounding injection rates.

JFM classification

Boundary Layers: Boundary layer structure Aerodynamics: High-speed flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , pp. 617 - 639
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Copyright
© 2017 Cambridge University Press 

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