Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-12-01T04:39:52.913Z Has data issue: false hasContentIssue false
  • English
  • Français

Models for inviscid wakes past a normal plate

Published online by Cambridge University Press:  14 August 2012

A. Elcrat  and
L. Zannetti
A. Elcrat
Affiliation:
Wichita State University 1845 N. Fairmount, Wichita, KS 67260-0033, USA
L. Zannetti*
Affiliation:
DIMEAS, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Closed and open hollow wakes are considered as analytic models for the two-dimensional inviscid steady flow past a plate normal to the stream. It is shown that only open configurations which satisfy the Kutta condition exist. The main argument is based on considering a plate located on the edge of a step with varying height. It is shown that solutions for open wakes exist for backward-, null and forward-facing steps, while closed wakes only exist for backward-facing steps. The occurrence of secondary separation has been modelled by adding a hollow region attached to the downstream corner. Peculiar accuracy issues of the problem are pointed out which may explain other contradictory results from the literature. It is shown how the Kirchhoff wake is a limiting solution for certain values of the governing parameters.

JFM classification

Wakes/Jets: Separated flows Vortex Flows: Vortex dynamics Wakes/Jets: Wakes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 708 , 10 October 2012 , pp. 377 - 396
Copyright
Copyright © Cambridge University Press 2012

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

1. Alimov, M. M. & Mazo, A. B. 2002 On the M. A. Lavrentyev model for stationary vortex zones. Fluid Dyn. 5, 705–712.CrossRefGoogle Scholar
2. Batchelor, G. K. 1956a On steady laminar flows with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–190.CrossRefGoogle Scholar
3. Batchelor, G. K. 1956b A proposal concerning laminar wakes behind bluff bodies at large Reynolds number. J. Fluid Mech. 1, 388–398.CrossRefGoogle Scholar
4. Chernyshenko, S. I. 1998 Asymptotic theory of global separation. Appl. Mech. Rev. 51, 523–536.CrossRefGoogle Scholar
5. Clements, R. R. 1973 An inviscid model of two-dimensional vortex shedding. J. Fluid Mech. 57, 321–336.CrossRefGoogle Scholar
6. Crowdy, D. G. & Green, C. C. 2011 Analytical solution for von Kármán streets of hollow vortices. Phys. Fluids 23, 126602.CrossRefGoogle Scholar
7. Driscoll, T. 1996 A MATLAB toolbox for Schwarz–Christoffel mapping. ACM Trans. Math. Softw. 22, 168–186.CrossRefGoogle Scholar
8. Elcrat, A. 1982 Separated flow past a plate with spoiler. SIAM J. Math. Anal. 13, 632–639.CrossRefGoogle Scholar
9. Gallizio, F. 2004 Modello di Prandtl–Batchelor per il flusso normale ad una placca piana posta all’interno di un canale: studio numerico dell’esistenza e unicità della soluzione. Dissert. Tesi Laurea Ing. Aerosp. aa 2003/2004, Politecnico di Torino, Turin, Italy.Google Scholar
10. Gallizio, F., Iollo, A., Protas, B. & Zannetti, L. 2010 On continuation of inviscid vortex patches. Physica D 239, 190–201.CrossRefGoogle Scholar
11. Garrick, I. E. 1936 Potential flow about arbitrary biplane wing sections. NACA Rep. 542.Google Scholar
12. Gurevich, M. I. 1965 Theory of Jets in Ideal Fluids. Academic.Google Scholar
13. Ives, D. C. 1976 A modern look at conformal mapping, including multiply connected regions. AIAA J. 14, 1006–1011.CrossRefGoogle Scholar
14. Kirchhoff, G. 1869 Zur theorie freier flussigkeitsstrahlen. J. Reine Angew. Math. 70, 289–298.Google Scholar
15. Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
16. Lavrentiev, M. A. 1962 Variational Methods for Boundary Value Problems for Systems of Elliptic Functions. Noordhoff.Google Scholar
17. Lin, A. & Landweber, L. 1977 On a solution of the Lavrentiev wake model and its cascade. J. Fluid Mech. 79, 801–823.CrossRefGoogle Scholar
18. Llewellyn Smith, S. G. & Crowdy, D. G. 2012 Structure and stability of hollow vortex equilibria. J. Fluid Mech. 691, 170–200.CrossRefGoogle Scholar
19. Mathematica, 2008 Mathematica, version 8.0. Wolfram Research, Inc.Google Scholar
20. MATLAB, 2010 version 7.10.0 (R2010a). The MathWorks Inc.Google Scholar
21. Michell, J. H. 1890 On the theory of free stream lines. Phil. Trans. R. Soc. Lond. A 182, 394–431.Google Scholar
22. Pocklington, H. C. 1895 The configuration of a pair of equal and opposite hollow straight vortices, of finite cross-section, moving steadily through fluid. Proc. Camb. Phil. Soc. 8, 178–187.Google Scholar
23. Protas, B. 2008 Vortex dynamics models in flow control problems. Nonlinearity 21, 203–250.CrossRefGoogle Scholar
24. Routh, E. J. 1881 Some application of conjugate functions. Proc. Lond. Math. Soc. 12, 73–89.Google Scholar
25. Smith, J. H. B. & Clark, R. W. 1975 Nonexistence of stationary vortices behind a two-dimensional normal plate. AIAA J. 13 (8), 1114–1115.CrossRefGoogle Scholar
26. Sychev, V. V., Ruban, A. I., Sychev, V. V. & Korolev, G. L. 1998 Asymptotic Theory of Separated Flows. Cambridge University Press.CrossRefGoogle Scholar
27. Telib, H. & Zannetti, L. 2011 Hollow wakes past arbitrarily shaped obstacles. J. Fluid Mech. 669, 214–224.CrossRefGoogle Scholar
28. Tricomi, F. 1951 Funzioni Ellittiche. Zanichelli.Google Scholar
29. Turfus, C. 1993 Prandtl–Batchelor flow past a flat plate at normal incidence in a channel – inviscid analysis. J. Fluid Mech. 249, 59–72.CrossRefGoogle Scholar
30. Turfus, C. & Castro, I. E. 2000 Prandtl–Batchelor model of flow in the wake of a cascade of normal plates. Fluid Dyn. Res. 26, 181–202.CrossRefGoogle Scholar
31. Zannetti, L. 2006 Vortex equilibrium in the flow past bluff bodies. J. Fluid Mech. 562, 151–171.CrossRefGoogle Scholar