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Similarity solutions for unsteady stagnation point flow

Published online by Cambridge University Press:  12 September 2012

D. Kolomenskiy  and
H. K. Moffatt
D. Kolomenskiy*
Affiliation:
Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique, 42 avenue Gaspard Coriolis, 31057 Toulouse CEDEX 01, France
H. K. Moffatt
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: dkolom@gmail.com
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Abstract

A class of similarity solutions for two-dimensional unsteady flow in the neighbourhood of a front or rear stagnation point on a plane boundary is considered, and a wide range of possible behaviour is revealed, depending on whether the flow in the far field is accelerating or decelerating. The solutions, when they exist, are exact solutions of the Navier–Stokes equations, having a boundary-layer character analogous to that of the classical steady front stagnation point flow. The velocity profiles are obtained by numerical integration of a nonlinear ordinary differential equation. For the front-flow situation, the solution is unique for the accelerating case, but bifurcates for modest deceleration, while for sufficient rapid deceleration there exists a one-parameter family of solutions. For the rear-flow situation, a unique solution exists (remarkably!) for sufficiently strong acceleration, and a one-parameter family again exists for sufficient strong deceleration. Analytic results, which are consistent with the numerical results, are obtained in the limits of strong acceleration or deceleration, and for the asymptotic behaviour far from the boundary.

JFM classification

Boundary Layers: Boundary layer structure Mathematical Foundations: General fluid mechanics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 711 , 25 November 2012 , pp. 394 - 410
Copyright
Copyright © Cambridge University Press 2012

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References

1. Abramowitz, M. & Stegun, I. 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th edition. Dover.Google Scholar
2. Angot, P., Bruneau, C.-H. & Fabrie, P. 1999 A penalization method to take into account obstacles in viscous flows. Numer. Math. 81, 497–520.Google Scholar
3. Burde, G. I. 1995 The construction of special explicit solutions of the boundary-layer equations: unsteady flows. Q. J. Mech. Appl. Math. 48 (4), 611–633.CrossRefGoogle Scholar
4. Cheng, E. H. W., Özişik, M. N. & Williams, J. C. III 1971 Nonsteady three-dimensional stagnation-point flow. J. Appl. Mech. 38 (1), 282–287.Google Scholar
5. Dormand, J. R. & Prince, P. J. 1980 A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26.Google Scholar
6. Drazin, P. & Riley, N. 2006 The Navier–Stokes Equations: A Classification of Flows and Exact Solutions. Cambridge.Google Scholar
7. Edwards, R. H. & Cheng, H. K. 1982 The separation vortex in the Weis-Fogh circulation-generation mechanism. J. Fluid Mech. 120, 463–473.Google Scholar
8. Hiemenz, K. 1911 Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dingl. Polytech. J. 326, 321–410.Google Scholar
9. Kierzenka, J. & Shampine, L. F. 2001 A BVP solver based on residual control and the MATLAB PSE. ACM Trans. Math. Softw. 27 (3), 299–316.Google Scholar
10. Kolomenskiy, D., Moffatt, H. K., Farge, M. & Schneider, K. 2011 The Lighthill–Weis-Fogh clap–fling–sweep mechanism revisited. J. Fluid Mech. 676, 572–606.Google Scholar
11. Kolomenskiy, D. & Schneider, K. 2009 A Fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles. J. Comput. Phys. 228, 5687–5709.Google Scholar
12. Lighthill, M. J. 1973 On the Weis-Fogh mechanism of lift generation. J. Fluid Mech. 60 (1), 1–17.Google Scholar
13. Proudman, I. & Johnson, K. 1962 Boundary-layer growth near a rear stagnation point. J. Fluid Mech. 12 (2), 161–168.Google Scholar
14. Schlichting, H. 1960 Boundary Layer Theory, 4th edition. McGraw-Hill.Google Scholar
15. Schneider, K. 2005 Numerical simulation of the transient flow behaviour in chemical reactors using a penalization method. Comput. Fluids 34, 1223–1238.Google Scholar
16. Stoer, J. & Bulirsch, R. 1983 Introduction to Numerical Analysis. Springer.Google Scholar
17. Teipel, I. 1979 Unsteady laminar boundary layers in an incompressible three-dimensional stagnation flow. In Recent Developments in Theoretical and Experimental Fluid Mechanics: Compressible and Incompressible Flows (ed. Müller, U., Roesner, K. G. & Schmidt, B. ), pp. 439–445. Springer.Google Scholar
18. Williams, J. C. III 1968 Nonsteady stagnation-point flow. AIAA J. 6 (12), 2417–2419.Google Scholar
19. Yang, K.-T. 1958 On unsteady laminar boundary layers. Trans. ASME: J. Appl. Mech. 25, 421–427.Google Scholar