Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-16T23:31:40.287Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2017 Cambridge University Press 

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

Amestoy, P. R., Duff, I. S. & L’Excellent, J.-Y. 2000 Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Meth. Appl. Mech. Engng 184 (2), 501–520.Google Scholar
Brown, S. N. & Stewartson, K. 1965 On similarity solutions of the boundary-layer equations with algebraic decay. J. Fluid Mech. 23 (04), 673–687.Google Scholar
Catherall, D., Stewartson, K. & Williams, P. G. 1965 Viscous flow past a flat plate with uniform injection. Proc. R. Soc. Lond. A 284 (1398), 370–396.Google Scholar
Cole, J. D. & Aroesty, J. 1968 The blowhard problem inviscid flows with surface injection. Intl J. Heat Mass Transfer 11 (7), 1167–1183.Google Scholar
Dhanak, M. R. & Duck, P. W. 1997 The effects of freestream pressure gradient on a corner boundary layer. Proc. R. Soc. Lond. A 453 (1964), 1793–1815.Google Scholar
van Dommelen, L. L. & Yapalparvi, R. 2014 Laminar boundary-layer separation control by Görtler-scale blowing. Eur. J. Mech. (B/Fluids) 46, 1–16.Google Scholar
Elliott, L. 1968 Two-dimensional boundary layer theory with strong blowing. Q. J. Mech. Appl. Maths 21 (1), 77–91.Google Scholar
Emmons, H. W. & Leigh, D. C.1954 Tabulation of the Blasius function with blowing and suction ARC CP no. 157.Google Scholar
Fernandez, E., Kumar, R. & Alvi, F. 2013 Separation control on a low-pressure turbine blade using microjets. J. Propul. Power 29 (4), 867–881.Google Scholar
Goldstein, M. E., Sescu, A., Duck, P. W. & Choudhari, M. 2016 Nonlinear wakes behind a row of elongated roughness elements. J. Fluid Mech. 796, 516–557.CrossRefGoogle ScholarPubMed
Goldstein, R. J. 1971 Film cooling. Adv. Heat Transfer 7, 321–379.Google Scholar
Goldstein, S. 1948 On laminar boundary-layer flow near a position of separation. Q. J. Mech. Appl. Maths 1 (1), 43–69.CrossRefGoogle Scholar
Gross, J. F., Hartnett, J. P., Masson, D. J. & Gazley, C. 1961 A review of binary laminar boundary layer characteristics. Intl J. Heat Mass Transfer 3 (3), 198–221.CrossRefGoogle Scholar
Haidari, A. H. & Smith, C. R. 1994 The generation and regeneration of single hairpin vortices. J. Fluid Mech. 277, 135–162.CrossRefGoogle Scholar
Hewitt, R. E. & Duck, P. W. 2014 Three-dimensional boundary layers with short spanwise scales. J. Fluid Mech. 756, 452–469.Google Scholar
Hewitt, R. E., Duck, P. W. & Stow, S. R. 2002 Continua of states in boundary-layer flows. J. Fluid Mech. 468, 121–152.Google Scholar
Kassoy, D. R. 1970 On laminar boundary layer blowoff. SIAM J. Appl. Maths 18 (1), 29–40.Google Scholar
Kassoy, D. R. 1971 On laminar boundary-layer blow-off. Part 2. J. Fluid Mech. 48, 209–228.CrossRefGoogle Scholar
Kassoy, D. R. 1974 A resolution of the blow-off singularity for similarity flow on a flat plate. J. Fluid Mech. 62, 145–161.Google Scholar
Kemp, N. H.1951 The laminar three-dimensional boundary layer and a study of the flow past a side edge. M.Ae.S. thesis, Cornell University.Google Scholar
Klemp, J. B. & Acrivos, A. 1972 High Reynolds number flow past a flat plate with strong blowing. J. Fluid Mech. 51, 337–356.CrossRefGoogle Scholar
Liu, T. M. & Libby, P. A. 1971 Flame sheet model for stagnation point flows. Combust. Sci. Technol. 2 (5–6), 377–388.Google Scholar
Neiland, V. Ya., Bogolepov, V. V., Dudin, G. N. & Lipatov, I. I. 2008 Asymptotic Theory of Supersonic Viscous Gas Flows. Butterworth-Heinemann.Google Scholar
Pal, A. & Rubin, S. G. 1971 Asymptotic features of viscous flow along a corner. Q. Appl. Maths 29, 91–108.Google Scholar
Ricco, P. & Dilib, F. 2010 The influence of wall suction and blowing on boundary-layer laminar streaks generated by free-stream vortical disturbances. Phys. Fluids 22 (4), 044101.Google Scholar
Rosenhead, L. 1963 Laminar Boundary Layers. Clarendon.Google Scholar
Schlichting, H. & Gersten, K. 2003 Boundary-layer Theory. Springer.Google Scholar
Stewartson, K. 1954 Further solutions of the Falkner–Skan equation. Math. Proc. Camb. Phil. Soc. 50 (03), 454–465.Google Scholar
Watson, E. J. 1966 The equation of similar profiles in boundary-layer theory with strong blowing. Proc. R. Soc. Lond. A 294 (1437), 208–234.Google Scholar
Wundrow, D. W. & Goldstein, M. E. 2001 Effect on a laminar boundary layer of small-amplitude streamwise vorticity in the upstream flow. J. Fluid Mech. 426, 229–262.Google Scholar