Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T13:21:32.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Extension of the Prandtl–Batchelor theorem to three-dimensional flows slowly varying in one direction

Published online by Cambridge University Press:  17 June 2010

M. SANDOVAL  and
S. CHERNYSHENKO
M. SANDOVAL*
Affiliation:
Department of Aeronautics, Imperial College, London SW7 2AZ, UK
S. CHERNYSHENKO
Affiliation:
Department of Aeronautics, Imperial College, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

According to the Prandtl–Batchelor theorem for a steady two-dimensional flow with closed streamlines in the inviscid limit the vorticity becomes constant in the region of closed streamlines. This is not true for three-dimensional flows. However, if the variation of the flow field along one direction is slow then it is possible to expand the solution in terms of a small parameter characterizing the rate of variation of the flow field in that direction. Then in the leading-order approximation the projections of the streamlines onto planes perpendicular to that direction can be closed. Under these circumstances the extension of the Prandtl–Batchelor theorem is obtained. The resulting equations turned out to be a three-dimensional analogue of the equations of the quasi-cylindrical approximation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 654 , 10 July 2010 , pp. 351 - 361
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Batchelor, G. K. 1956 On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–190.CrossRefGoogle Scholar
Blennerhassett, P. 1979 A three-dimensional analogue of the Prandtl–Batchelor closed streamline theory. J. Fluid Mech. 93, 319–324.Google Scholar
Buldakov, E. V., Chernyshenko, S. I. & Ruban, A. I. 2000 On the uniqueness of steady flow past a rotating cylinder with suction. J. Fluid Mech. 411, 213–232.Google Scholar
Bunyakin, A., Chernyshenko, S. & Stepanov, G. 1988 High-Reynolds-number Batchelor-model asymptotics of a flow past an aerofoil with a vortex trapped in a cavity. J. Fluid Mech. 358, 283–297.Google Scholar
Chernyshenko, S. I. 1983 a Heat transfer in toroidal pipes at high Prandtl number (in Russian). Vestn. MGU, Ser. 1, Mat. Mech. 2, 40–45.Google Scholar
Chernyshenko, S. I. 1983 b Steady flow of fluid of small viscosity in channels and tubes of periodic shape (in Russian). Dokl. Akad. Nauk SSSR 168 (2), 314–316.Google Scholar
Childress, S., Landman, M. & Strauss, H. 1989 Steady motion with helical symmetry at large Reynolds number. In Proceedings of IUTAM Symposium on Topological Fluid Mechanics (ed. Moffat, H. & Tsinober, A.), pp. 216–224. Cambridge University Press.Google Scholar
Grimshaw, B. 1968 On steady recirculating flows. J. Fluid Mech. 39, 695–703.CrossRefGoogle Scholar
Kamachi, M., Saitou, T. & Honji, H. 1985 Characteristics of stationary closed streamlines in stratified fluids. Part 1. Theory of laminar flows. Tech. Rep. 4. Yamaguchi University.Google Scholar
Mezic, I. 2002 An extension of Prandtl–Batchelor theory and consequences for chaotic advection. Phys. Fluids 14, 61–64.CrossRefGoogle Scholar
Neiland, V. Y. 1970 On the asymptotic theory of plane steady supersonic flows with separation regions (in Russian). Izv. Acad. Nauk SSSR, Mekh. Zhidk. Gaza 5, 22–32.Google Scholar
Neiland, V. Y. & Sychev, V. V. 1970 On the theory of flows in steady separation regions (in Russian). Uch. Zap. TsAGI 1 (1), 14–23.Google Scholar
Prandtl, L. 1905 Über Flüssigkeitsbewegung beu sehr kleiner Reibung. In Vehr. d. III. Intern. Math.–Kongr., Heidelberg, 1904, pp. 484–491. Leipzig: Teubner.Google Scholar
Revuelta, A., Sanchez, A. & Linan, A. 2004 The quasi-cylindrical description of submerged laminar swirling jets. Phys. Fluids 16, 848–851.CrossRefGoogle Scholar
Squire, H. B. 1956 Note on the motion inside a region of recirculation (cavity flow). J. R. Aeronaut. Soc. 60, 203–205.CrossRefGoogle Scholar
Wood, W. W. 1957 Boundary layers whose streamlines are closed. J. Fluid Mech. 2, 77–87.Google Scholar