Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-05-02T00:22:53.824Z Has data issue: false hasContentIssue false
  • English
  • Français

VORTEX EQUILIBRIA IN FLOW PAST A PLATE

Part of: Incompressible inviscid fluids

Published online by Cambridge University Press:  01 January 2008

N. ROBB MCDONALD
N. ROBB MCDONALD*
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK (email: robb@math.ucl.ac.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Families of vortex equilibria, with constant vorticity, in steady flow past a flat plate are computed numerically. An equilibrium configuration, which can be thought of as a desingularized point vortex, involves a single symmetric vortex patch located wholly on one side of the plate. Given that the outermost edge of the vortex is unit distance from the plate, the equilibria depend on three parameters: the length of the plate, circulation about the plate, and the distance of the innermost edge of the vortex from the plate. Families in which there is zero circulation about the plate and for which the Kutta condition at the plate ends is satisfied are both considered. Properties such as vortex area, lift and free-stream speed are computed. Time-dependent numerical simulations are used to investigate the stability of the computed steady solutions.

Keywords

vortex dynamicsvortex equilibriacontour dynamicsaerodynamics

MSC classification

Secondary: 76B47: Vortex flows 76B10: Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
Type
Research Article
Information
The ANZIAM Journal , Volume 49 , Issue 3 , January 2008 , pp. 347 - 359
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Batchelor, G. K., “Steady laminar flow with closed streamlines at large Reynolds numbers”, J. Fluid Mech. 1 (1956) 177–190.Google Scholar
[2]Deem, G. and Zabusky, N. J., “Stationary “V-states”, interactions, recurrence and breakings”, Phys. Rev. Lett. 40 (1978) 859–862.CrossRefGoogle Scholar
[3]Dritschel, D. G., “Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics”, J. Comput. Phys. 77 (1988) 240–266.CrossRefGoogle Scholar
[4]Elcrat, A., Fornberg, B., Horn, M. and Miller, K., “Some steady vortex flows past a circular cylinder”, J. Fluid Mech. 409 (2000) 13–27.Google Scholar
[5]Elcrat, A., Fornberg, B. and Miller, K., “Stability of vortices in equilibrium with a circular cylinder”, J. Fluid Mech. 544 (2005) 53–68.CrossRefGoogle Scholar
[6]Hocking, G. C., “Steady Prandtl–Batchelor flows past a circular cylinder”, ANZIAM J. 48 (2006) 165–177.CrossRefGoogle Scholar
[7]Johnson, E. R. and McDonald, N. R., “The motion of a vortex near two circular cylinders”, Proc. Roy. Soc. A 460 (2004) 939–954.Google Scholar
[8]Johnson, E. R. and McDonald, N. R., “Vortices near barriers with multiple gaps”, J. Fluid Mech. 531 (2005) 335–358.CrossRefGoogle Scholar
[9]Johnson, E. R. and McDonald, N. R., “Steady vortical flow around a finite plate”, Q. J. Mech. Appl. Math. 60 (2007) 65–72.Google Scholar
[10]Milne-Thompson, L. M., Theoretical hydrodynamics, 2nd edn (Macmillan, London, 1949).Google Scholar
[11]Pierrehumbert, R. T., “A family of steady, translating vortex pairs with distributed vorticity”, J. Fluid Mech. 99 (1980) 129–144.Google Scholar
[12]Routh, E. J., “Some applications of conjugate functions”, Proc. Lond. Math. Soc. 12 (1984) 73–89.Google Scholar
[13]Saffman, P. G., Vortex dynamics (Cambridge University Press, Cambridge, 1992).Google Scholar
[14]Saffman, P. G. and Sheffield, J. S., “Flow over a wing with an attached free vortex”, Stud. Appl. Math. 57 (1977) 107–117.CrossRefGoogle Scholar
[15]Saffman, P. G. and Tanveer, S., “Prandtl–Batchelor flow past a flat plate with a forward facing flap”, J. Fluid Mech. 143 (1984) 351–365.Google Scholar
[16]Wu, H. M., Overman, E. A. and Zabusky, N. J., “Steady state solutions of the Euler equations in two-dimensions. Rotating and translating V-states with limiting cases”, J. Comput. Phys. 53 (1984) 42–71.Google Scholar
[17]Zannetti, L., “Vortex equilibrium in flows past bluff bodies”, J. Fluid Mech. 562 (2006) 151–171.Google Scholar