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Prandtl-Batchelor flow on a circular cylinder and on aerofoil sections

Published online by Cambridge University Press:  04 July 2016

G. Giannakidis
G. Giannakidis*
Affiliation:
Department of AeronauticsImperial College of Science, Technology and MedicineLondon, UK
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Abstract

Steady, inviscid, incompressible, two dimensional flows containing vortex patches bounded by vortex sheets are used to model finite separated regions on certain closed bodies as an approach to modelling separation bubbles. First of all the problem of a vortex patch on a circle is considered. Calculations were done for symmetric flows covering a wide range of values of the separation position, the strength of the vortex sheet and the circulation at infinity and for non-symmetric flows at various angles of incidence. Smooth separation was detected in some cases. This model is used as an intermediate step for the calculation of the flow around an ellipse and a Joukowski aerofoil with the use of the Joukowski transformation.

Type
Research Article
Information
The Aeronautical Journal , Volume 100 , Issue 991 , January 1996 , pp. 15 - 26
Copyright
Copyright © Royal Aeronautical Society 1996 

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References

1. Prandtl, L. Motion of fluids with very little viscosity, NACA TM 452, 1928. Translation from Prandtl, L. Uber Flussigkeitsbewegung bei sehr kleiner Reibung, Verh. III, Int. Math-Kongr, Heidelberg, 1904, pp 484–491, Leipzig: Teubner, Reprinted in Gesammelte Abhandlungen 2:575-584.Google Scholar
2. Batchelor, G. A proposal concerning laminar wakes behind bluff bodies at large Reynolds number, J Fluid Mech, 1956, 1, pp 388–398.Google Scholar
3. Smith, F. A structure of laminar flow past a bluff body at high Reynolds number, J Fluid Mech, 1985, 55, pp 175–191.Google Scholar
4. Smith, F. Concerning inviscid solutions for large-scale separated flows, J Eng Maths, 1986, 20, pp 271–292.Google Scholar
5. Smith, J. The representation of planar separated flow by regions of uniform vorticity, in Vortex Motion, pp 157–172, Vieweg Braunschveig, 1982.Google Scholar
6. Saffman, P. and Tanveer, S. The touching pair of equal and opposite uniform vortices, Phys Fluid, 1982, 25, (11), pp 1929–1930.Google Scholar
7. Childress, S. Solution of Euler’s equations containing finite eddies, Phys Fluid, 1960, 9, (5), pp 860–872.Google Scholar
8. Riley, N. Inviscid separated flows of finite extent, J Eng Maths, 1987, 21, pp 349–361.Google Scholar
9. Sadovskii, V. Vortex regions in a potential stream with a jump of Bernoulli constant at the boundary, J Appl Maths Mech, 1971, 35, pp 729–735.Google Scholar
10. Sadovskii, V. and Sinitsyna, N. Mixed flow of an ideal fluid on a plane containing a recess, Izv Akad Nauk SSSR, Mekh Zhidk. Gaza, 1983, pp 306–309, 1983. Translation in Fluid Dynamics, 18.Google Scholar
11. Saffman, P. and Tanveer, S. Prandtl-Batchelor flow past a flat plate with a forward-facing flap, J Fluid Mech, 1984, 143, pp 351–365.Google Scholar
12. Chernyshenko, S. The calculation of separated flows of low viscosity liquids using the Batchelor model, Library Translation 2133, Royal Aircraft Establishment, April 1985.Google Scholar
13. Moore, D. Saffman, P. and Tanveer, S. The calculation of some Batchelor flows: The Sadovskii vortex and rotational corner flow, Phys Fluid, 1988, 31, pp 978–990.Google Scholar
14. Fornberg, B. Steady viscous flow past a circular cylinder up to Re 600, J Comp Phys, 61, pp 297–320, 1985.Google Scholar
15. MacLachlan, R. A steady separated viscous corner flow, J Fluid Mech, 1991, 231, pp 1–34.Google Scholar
16. Giannakidis, G. Prandtl-Batchelor flow in a channel, Phys Fluids A, 1993, 5, (4), pp 863–867.Google Scholar
17. Fornberg, B. Steady incompressible flow past a row of circular cylinders, J Fluid Mech, 1991, 225, pp 655–671.Google Scholar
18. Turfus, C. Prandtl-Batchelor flow past a flat plate at normal incidence in a channel — Inviscid analysis, J Fluid Mech, 1991, 249, pp 59–72.Google Scholar
19. Chernyshenko, S. Stratified Sadovskii flow in a channel, J Fluid Mech, 1993, 250, pp 423–431.Google Scholar
20. Pullin, D. The non-linear behaviour of a constant vorticity layer at a wall, J Fluid Mech, 1981, 108, pp 401–421.Google Scholar
21. Pullin, D. A constant-vorticity Riabouchinsky free-streamline flow, Q J Mech Appl Math, 1984, 37, pp 619–631.Google Scholar
22. Pullin, D. Contour dynamics methods, Ann Rev Fluid Dyn, 1992, 24, pp 89–115.Google Scholar
23. Giannakidis, G. Problems of Vortex Sheets and Vortex Patches, PhD Thesis, Imperial College, University of London, 1993.Google Scholar
24. Smith, J.H.B. Behaviour of a vortex sheet separating from a smooth surface, Technical Report 77058, RAE, April 1977.Google Scholar
25. Doedel, E. Auto: Software for continuation and bifurcation problems in ordinary differential equations, Pasadena: CalTech, 1986.Google Scholar
26. Saffman, P. and Szeto, R. Structure of a linear array of uniform vortices, Stud Appl Maths, 1981, 65, pp 223–248.Google Scholar
27. Broadbent, E. and Moore, D. Waves of extreme form on a layer of uniform vorticity, Phys Fluids, May 1985, 28, pp 1561–1563.Google Scholar
28. Chen, B. and Saffman, P. Numerical evidence for the existence of new types of gravity waves of permanent form on deep water, Stud Appl Maths, 1980, 62, pp 1–21.Google Scholar