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Three-dimensional boundary layers with short spanwise scales

Published online by Cambridge University Press:  02 September 2014

Richard E. Hewitt  and
Peter W. Duck
Richard E. Hewitt*
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
Peter W. Duck
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Email address for correspondence: [email protected]
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Abstract

We investigate three-dimensional (laminar) boundary layers that include a spanwise scale comparable to the boundary-layer thickness. A forcing of short spanwise scales requires viscous dissipation to be retained in the two-dimensional cross-section, perpendicular to the external flow direction, and in this respect the flows are related to previous work on corner boundary layers. We use two examples to highlight the main features of this category of boundary layer: (i) a flat plate of narrow (spanwise) width, and (ii) a narrow (spanwise) gap cut into an otherwise infinite flat plate; in both cases the plate is aligned with a uniform oncoming stream. We find that a novel feature arises in connection with the external flow; the presence of a narrow gap/plate (or indeed any comparable short-scale feature of long streamwise extent) necessarily modifies the streamwise mass flux in that vicinity, which in turn induces an associated boundary-layer transpiration on the same short spanwise length scale. This (short-scale) transpiration region leads to a half-line-source/sink correction to the outer inviscid, irrotational flow. Crucially, the volumetric flux associated with this line-source/sink must be explicitly included as part of the computational procedure for the leading-order boundary layer, and as such there is a weak interaction between the outer (inviscid) flow and the boundary layer. This is a generic feature of boundary layers that are forced through the presence of short-scale spanwise variations.

JFM classification

Boundary Layers: Boundary Layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 756 , 10 October 2014 , pp. 452 - 469
Copyright
© 2014 Cambridge University Press 

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References

Alizard, F., Robinet, J.-C. & Rist, U. 2010 Sensitivity analysis of a streamwise corner flow. Phys. Fluids 22, 014103.Google Scholar
Balachandar, S. & Malik, M. R. 1995 Inviscid instability of streamwise corner flow. J. Fluid Mech. 282, 187–202.Google Scholar
Barclay, W. H. & El-Gamal, H. A. 1983 Streamwise corner flow with wall suction. AIAA J. 21 (1), 31–37.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
Carrier, G. F. 1947 The boundary layer in a corner. Q. Appl. Maths 4 (4), 367–370.Google Scholar
Demmel, J. W., Eisenstat, S. C., Gilbert, J. R., Li, X. S. & Liu, J. W. H. 1999 A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Applics. 20 (3), 720–755.Google Scholar
Desai, S. S. & Mangler, K. W.1974 Incompressible laminar boundary layer flow along a corner formed by two intersecting planes. RAE Technical Report 74062.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
Duck, P. W. & Hewitt, R. E. 2012 A resolution of Stewartson’s quarter-infinite plate problem. Theor. Comput. Fluid Dyn. 26 (1–4), 117–140.Google Scholar
Duck, P. W., Stow, S. R. & Dhanak, M. R. 1999 Non-similarity solutions to the corner boundary-layer equations (and the effects of wall transpiration). J. Fluid Mech. 400 (1), 125–162.Google Scholar
Duck, P. W., Stow, S. R. & Dhanak, M. R. 2000 Boundary-layer flow along a ridge: alternatives to the Falkner–Skan solutions. Phil. Trans. R. Soc. Lond. A 358 (1777), 3075–3090.Google Scholar
Galionis, I. & Hall, P. 2005 Spatial stability of the incompressible corner flow. Theor. Comput. Fluid Dyn. 19 (2), 77–113.Google Scholar
Ghia, K. N. 1975 Incompressible streamwise flow along an unbounded corner. AIAA J. 13 (7), 902–907.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
Kemp, N. H.1951 The laminar three-dimensional boundary layer and a study of the flow past a side edge. MAeS thesis, Cornell University.Google Scholar
Lakin, W. D. & Hussaini, M. Y. 1984 Stability of the laminar boundary layer in a streamwise corner. Proc. R. Soc. Lond. A 393 (1804), 101–116.Google Scholar
Levy, R.1959 The boundary layer in a corner. PhD thesis, Princeton University.Google Scholar
Loitsianskii, L. G. & Bolshakov, V. P.1951 On the motion of fluid in the boundary layer near the line of intersection of two planes. NACA Tech. Mem. 1308 (Translation).Google Scholar
Luchini, P. 1996 Reynolds-number-independent instability of the boundary layer over a flat surface. J. Fluid Mech. 327, 101–116.Google Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404 (1), 289–309.Google Scholar
Neyland, 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
Park, D. H., Park, S. O., Kwon, K. J. & Shim, H. J. 2012 Particle image velocimetry measurement of laminar boundary layer in a streamwise corner. AIAA J. 50 (4), 811–817.Google Scholar
Parker, S. J. & Balachandar, S. 1999 Viscous and inviscid instabilities of flow along a streamwise corner. Theor. Comput. Fluid Dyn. 13 (4), 231–270.Google Scholar
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. CRC Press.Google Scholar
Patankar, S. V. & Spalding, D. B. 1972 A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Intl J. Heat Mass Transfer 15 (10), 1787–1806.Google Scholar
Ridha, A. 1992 On the dual solutions assoclated with boundary-layer equations in a corner. J. Engng Maths 26 (4), 525–537.Google Scholar
Rubin, S. G. 1966 Incompressible flow along a corner. J. Fluid Mech. 26 (pt 1), 97–110.Google Scholar
Rubin, S. G. & Grossman, B.1969 Viscous flow along a corner: numerical solution of corner layer equations. Polytechnic Institute of Brooklyn, Department of Aerospace Engineering and Applied Mechanics.Google Scholar
Stewartson, K. 1961 Viscous flow past a quarter-infinite plate. J. Aero. Sci. 28, 1–10.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of non-parallel and three-dimensional flows. Prog. Aerosp. Sci. 39 (4), 249–315.CrossRefGoogle Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics, vol. 964. Academic Press.Google Scholar
Zamir, M. 1973 Further solution of the corner boundary-layer equations. Aeronaut. Q. 24, 219–226.Google Scholar
Zamir, M. 1981 Similarity and stability of the laminar boundary layer in a streamwise corner. Proc. R. Soc. Lond. A 377 (1770), 269–288.Google Scholar
Zamir, M. & Young, A. D. 1970 Experimental investigation of the boundary layer in a streamwise corner. Aeronaut. Q. 21, 313–339.Google Scholar
Zamir, M. & Young, A. D. 1979 Pressure gradient and leading edge effects on the corner boundary layer. Aeronaut. Q. 30, 471–484.Google Scholar
Zuccher, S., Luchini, P. & Bottaro, A. 2004 Algebraic growth in a Blasius boundary layer: optimal and robust control by mean suction in the nonlinear regime. J. Fluid Mech. 513, 135–160.CrossRefGoogle Scholar