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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: [email protected]
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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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References

Arnal, D., Juillen, J. C., Reneaux, J. & Gasparian, G. 1997 Effect of wall suction on leading edge contamination. Aerosp. Sci. Technol. 1 (8), 505–517.CrossRefGoogle Scholar
Cooke, J. C. 1950 The boundary layer of a class of infinite yawed cylinders. Math. Proc. Camb. Phil. Soc. 46 (04), 645–648.Google Scholar
Dorrepaal, J. M. 1986 An exact solution of the Navier–Stokes equation which describes non-orthogonal stagnation-point flow in two dimensions. J. Fluid Mech. 163, 141–147.CrossRefGoogle Scholar
Drazin, P. G. & Riley, N. 2006 The Navier–Stokes Equations: A Classification of Flows and Exact Solutions. Cambridge University Press.CrossRefGoogle Scholar
Fransson, J. H. M. & Alfredsson, P. H. 2003 On the disturbance growth in an asymptotic suction boundary layer. J. Fluid Mech. 482, 51–90.Google Scholar
Gaster, M. 1967 On the flow along swept leading edges. Aeronaut. Q. 18, 165–184.Google Scholar
Görtler, H. 1955 Dreidimensionale instabilität der ebenen Staupunktströmung gegenüber wirbelartigen Störungen. In 50 Jahre Grenzschichtforschung. Eine Festschrift in Originalbeiträgen (ed. Görtler, H. & Tollmien, W.), pp. 304–314. Vieweg.Google Scholar
Hall, P., Malik, M. R. & Poll, D. I. A. 1984 On the stability of an infinite swept attachment line boundary layer. Proc. R. Soc. Lond. A 395 (1809), 229–245.Google Scholar
Hämmerlin, G. 1955 Zur Instabilitätstheorie der ebenen Staupunktströmung. In 50 Jahre Grenzschichtforschung. Eine Festschrift in Originalbeiträgen (ed. Görtler, H. & Tollmien, W.), pp. 315–327. Vieweg.Google Scholar
Herron, I. H., Von Kerczek, C. H. & Tozzi, J. 1985 Instability characteristics of the asymptotic suction profile. Trans. ASME J. Appl. Mech. 52 (2), 487.CrossRefGoogle Scholar
Hiemenz, K. 1911 Die grenzschicht an einem in den gleichförmigen flüssigkeitsstrom eingetauchten geraden kreiszylinder. Dingl. Polytech. J. 326 (21–26), 321–324; 344–348; 357–362; 372–376; 391–393; 407–410.Google Scholar
Hocking, L. M. 1975 Nonlinear instability of the asymptotic suction velocity profile. Q. J. Mech. Appl. Maths 28, 341–353.Google Scholar
Howarth, L. 1951 The boundary layer in three-dimensional flow. Part 2. The flow near a stagnation point. Phil. Mag. 42 (335), 1433–1440.Google Scholar
John, M. O., Obrist, D. & Kleiser, L. 2012 An exact Navier–Stokes solution for three-dimensional, spanwise-homogeneous boundary layers. Proc. Applied Maths Mech. 12 (1), 477–478.CrossRefGoogle Scholar
John, M. O., Obrist, D. & Kleiser, L. 2014 Stabilization of subcritical bypass transition in the leading-edge boundary layer by suction. J. Turbul. (to appear); doi: 10.1080/14685248.2014.933226.Google Scholar
Kolomenskiy, D. & Moffatt, H. K. 2012 Similarity solutions for unsteady stagnation point flow. J. Fluid Mech. 711 (1), 394–410.Google Scholar
Lew, H. G. 1956 Asymptotic suction characteristics of the boundary layer over a circular cylinder. J. Aeronaut. Sci. 23 (9), 895–897.Google Scholar
Lok, Y. Y., Amin, N. & Pop, I. 2006 Non-orthogonal stagnation point flow towards a stretching sheet. Intl J. Non-Linear Mech. 41 (4), 622–627.Google Scholar
Obrist, D., Henniger, R. & Kleiser, L. 2012 Subcritical spatial transition of swept Hiemenz flow. Intl J. Heat Fluid Flow 35, 61–67.CrossRefGoogle Scholar
Obrist, D. & Schmid, P. J. 2003 On the linear stability of swept attachment-line boundary layer flow. Part 1. Spectrum and asymptotic behaviour. J. Fluid Mech. 493, 1–29.Google Scholar
Owen, P. R. & Randall, D. G.1952 Boundary layer transition on a sweptback wing. Tech. Rep. RAE-TM-AERO 277. Royal Aircraft Establishment, Farnborough.Google Scholar
Pfenninger, W.1965 Flow phenomena at the leading edge of swept wings. Tech. Rep. AGARDograph 97. North Atlantic Treaty Organization, Advisory Group for Aeronautical Research and Development.Google Scholar
Poll, D. I. A. 1979 Transition in the infinite swept attachment line boundary layer. Aeronaut. Q. 30, 607–629.Google Scholar
Preston, J. H.1948 The boundary-layer flow over a permeable surface through which suction is applied. Tech. Rep. RM 2244. Ministry of Supply, Aeronautical Research Council, London.Google Scholar
Robitaillié-Montané, C.2005 Une approche non-locale pour l’étude des instabilités linéaires: application à l’écoulement de couche limite compressible le long d’une ligne de partage. PhD thesis, Ecole Nationale Supérieure de l’Aéronautique et de l’Espace, Toulouse, France.Google Scholar
Rosenhead, L. 1963 Laminar Boundary Layers. Clarendon Press.Google Scholar
Saric, W. S., Reed, H. L. & White, E. B. 2003 Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35, 413–440.Google Scholar
Schlichting, H. 1979 Boundary-Layer Theory, 7th edn. McGraw-Hill.Google Scholar
Stuart, J. T. 1959 The viscous flow near a stagnation point when the external flow has uniform vorticity. J. Aero. Sci. 26 (2), 124–125.Google Scholar
Tamada, K. 1979 Two-dimensional stagnation-point flow impinging obliquely on a plane wall. J. Phys. Soc. Japan 46 (1), 310–311.Google Scholar
Theofilis, V., Fedorov, A. V., Obrist, D. & Dallmann, U. C. 2003 The extended Görtler–Hämmerlin model for linear instability of three-dimensional incompressible swept attachment-line boundary layer flow. J. Fluid Mech. 487, 271–313.Google Scholar
Tooke, R. M. & Blyth, M. G. 2008 A note on oblique stagnation-point flow. Phys. Fluids 20 (3), 033101.Google Scholar
Trefethen, L. N. & Bau, D. 1997 Numerical Linear Algebra. Society for Industrial and Applied Mathematics.Google Scholar
Wang, C. Y. 1991 Exact solutions of the steady-state Navier–Stokes equations. Annu. Rev. Fluid Mech. 23, 159–177.Google Scholar
Weidman, P. D. 2012 Non-axisymmetric Homann stagnation-point flows. J. Fluid Mech. 702, 460–469.Google Scholar