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Injection into boundary layers: solutions beyond the classical form

  • R. E. Hewitt (a1), P. W. Duck (a1) and A. J. Williams (a1)
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.

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Corresponding author
Email address for correspondence: richard.e.hewitt@manchester.ac.uk
References
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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), 501520.
Brown S. N. & Stewartson K. 1965 On similarity solutions of the boundary-layer equations with algebraic decay. J. Fluid Mech. 23 (04), 673687.
Catherall D., Stewartson K. & Williams P. G. 1965 Viscous flow past a flat plate with uniform injection. Proc. R. Soc. Lond. A 284 (1398), 370396.
Cole J. D. & Aroesty J. 1968 The blowhard problem inviscid flows with surface injection. Intl J. Heat Mass Transfer 11 (7), 11671183.
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), 17931815.
van Dommelen L. L. & Yapalparvi R. 2014 Laminar boundary-layer separation control by Görtler-scale blowing. Eur. J. Mech. (B/Fluids) 46, 116.
Elliott L. 1968 Two-dimensional boundary layer theory with strong blowing. Q. J. Mech. Appl. Maths 21 (1), 7791.
Emmons H. W. & Leigh D. C.1954 Tabulation of the Blasius function with blowing and suction ARC CP no. 157.
Fernandez E., Kumar R. & Alvi F. 2013 Separation control on a low-pressure turbine blade using microjets. J. Propul. Power 29 (4), 867881.
Goldstein M. E., Sescu A., Duck P. W. & Choudhari M. 2016 Nonlinear wakes behind a row of elongated roughness elements. J. Fluid Mech. 796, 516557.
Goldstein R. J. 1971 Film cooling. Adv. Heat Transfer 7, 321379.
Goldstein S. 1948 On laminar boundary-layer flow near a position of separation. Q. J. Mech. Appl. Maths 1 (1), 4369.
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), 198221.
Haidari A. H. & Smith C. R. 1994 The generation and regeneration of single hairpin vortices. J. Fluid Mech. 277, 135162.
Hewitt R. E. & Duck P. W. 2014 Three-dimensional boundary layers with short spanwise scales. J. Fluid Mech. 756, 452469.
Hewitt R. E., Duck P. W. & Stow S. R. 2002 Continua of states in boundary-layer flows. J. Fluid Mech. 468, 121152.
Kassoy D. R. 1970 On laminar boundary layer blowoff. SIAM J. Appl. Maths 18 (1), 2940.
Kassoy D. R. 1971 On laminar boundary-layer blow-off. Part 2. J. Fluid Mech. 48, 209228.
Kassoy D. R. 1974 A resolution of the blow-off singularity for similarity flow on a flat plate. J. Fluid Mech. 62, 145161.
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.
Klemp J. B. & Acrivos A. 1972 High Reynolds number flow past a flat plate with strong blowing. J. Fluid Mech. 51, 337356.
Liu T. M. & Libby P. A. 1971 Flame sheet model for stagnation point flows. Combust. Sci. Technol. 2 (5–6), 377388.
Neiland V. Ya., Bogolepov V. V., Dudin G. N. & Lipatov I. I. 2008 Asymptotic Theory of Supersonic Viscous Gas Flows. Butterworth-Heinemann.
Pal A. & Rubin S. G. 1971 Asymptotic features of viscous flow along a corner. Q. Appl. Maths 29, 91108.
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.
Rosenhead L. 1963 Laminar Boundary Layers. Clarendon.
Schlichting H. & Gersten K. 2003 Boundary-layer Theory. Springer.
Stewartson K. 1954 Further solutions of the Falkner–Skan equation. Math. Proc. Camb. Phil. Soc. 50 (03), 454465.
Watson E. J. 1966 The equation of similar profiles in boundary-layer theory with strong blowing. Proc. R. Soc. Lond. A 294 (1437), 208234.
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, 229262.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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