Hostname: page-component-84b7d79bbc-dwq4g Total loading time: 0 Render date: 2024-07-26T22:25:23.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized two-dimensional Lagally theorem with free vortices and its application to fluid–body interaction problems

Published online by Cambridge University Press:  16 March 2012

C. T. Wu ,
F.-L. Yang  and
D. L. Young
C. T. Wu
Affiliation:
Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan
F.-L. Yang*
Affiliation:
Department of Mechanical Engineering, National Taiwan University, Taipei 10617, Taiwan
D. L. Young*
Affiliation:
Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan
*
†Email addresses for correspondence: fulingyang@ntu.edu.tw, dlyoung@ntu.edu.tw
†Email addresses for correspondence: fulingyang@ntu.edu.tw, dlyoung@ntu.edu.tw
Get access Rights & Permissions [Opens in a new window]

Abstract

The Lagally theorem describes the unsteady hydrodynamic force on a rigid body exhibiting arbitrary motion in an inviscid and incompressible fluid by the properties of the singularities employed to generate the flow and the body motion and to meet the boundary condition. So far, only sources and dipoles have been considered, and the present work extends the theorem to include free vortices in a two-dimensional flow. The present extension is validated by reproducing the system dynamics or the force evolution of three literature problems: (i) a free cylinder interacting with a free vortex; (ii) the moving Föppl problem; and (iii) a cylinder in constant normal approach to a fixed identical cylinder. This work further extends the bifurcation analysis on the moving Föppl problem by including the solid-to-liquid density ratio as a new parameter, in addition to the system total impulse and the vortex strength. We then apply the theorem to the problem where a moving Föppl system is made to approach a fixed or a free neutrally buoyant target cylinder of identical size from far away. The force developed on each cylinder is examined with respect to the vortex pair configuration and the target mobility. When approaching a fixed target, a greater force is developed if the vortex pair has stronger circulation and larger structure. The mobility of the target cylinder, however, can modify the hydrodynamic force by reducing its magnitude and reversing the force ordering with respect to the vortex pair configuration found for the case with fixed target. Possible mechanisms for such a change of force nature are given based on the currently derived equation of motion.

JFM classification

Mathematical Foundations: General fluid mechanics Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 698 , 10 May 2012 , pp. 73 - 92
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. Bampalas, N. & Graham, J. M. R. 2008 Flow-induced forces arising during the impact of two circular cylinders. J. Fluid Mech. 616, 205–234.CrossRefGoogle Scholar
2. Biesheuvel, A. 1985 A note on the generalized Lagally theorem. J. Engng Maths 19, 69–77.CrossRefGoogle Scholar
3. Birkhoff, G. 1950 Hydrodynamics. Princeton University Press.Google Scholar
4. Borisov, A. V., Mamaev, I. S. & Ramodanov, S. M. 2003 Motion of a circular cylinder and n point vortices in a perfect fluid. Regular Chaotic Dyn. 8, 449–462.CrossRefGoogle Scholar
5. Crowdy, D. G. 2008 Explicit solution for the potential flow due to an assembly of stirrers in an inviscid fluid. J. Engng Maths 62, 333–344.CrossRefGoogle Scholar
6. Crowdy, D. G. 2010 A new calculus for two-dimensional vortex dynamics. Theor. Comput. Fluid Dyn. 24, 9–24.CrossRefGoogle Scholar
7. Crowdy, D. G. & Marshall, J. S. 2005a Analytical formulae for the Kirchhof–Routh path function in multiply connected domains. Proc. R. Soc. Lond. A 461, 2477–2501.Google Scholar
8. Crowdy, D. G. & Marshall, J. S. 2005b The motion of a point vortex around multiple circular islands. Phys. Fluids 17, 056602.CrossRefGoogle Scholar
9. Cummins, W. E. 1953 The forces and moments acting on a body moving in an arbitrary potential stream. David Taylor Model Basin, Rep. no. 708.Google Scholar
10. Eames, I., Landeryou, M. & Flór, J. B. 2007 Inviscid coupling between point symmetric bodies and singular distributions of vorticity. J. Fluid Mech. 589, 33–56.CrossRefGoogle Scholar
11. Föppl, L. 1913 Wirbelbewegung hinter einem Kreiszylinder. Sitz. K. Bäyr. Akad. Wiss. 1, 7–18.Google Scholar
12. Kanso, E. & Oskouei, B. G. 2008 Stability of a coupled body–vortex system. J. Fluid Mech. 600, 77–94.CrossRefGoogle Scholar
13. Lagally, M. 1922 Berechnung der Kräfte und Momente, die strömende Flüssigkeiten auf ihre Begrenzung ausüben. Z. Angew. Math. Mech. 2, 409–422.CrossRefGoogle Scholar
14. Lamb, H. 1932 Hydrodynamics. Dover.Google Scholar
15. Landweber, L. 1956 On a generalization of Taylor’s virtual mass relation for Rankine bodies. Q. Appl. Maths 14, 51–56.CrossRefGoogle Scholar
16. Landweber, L. & Chwang, A. T. 1989 Generalization of Taylor’s added-mass formula for two bodies. J. Ship Res. 33, 1–9.CrossRefGoogle Scholar
17. Landweber, L. & Miloh, T. 1980 Unsteady Lagally theorem for multipoles and deformable bodies. J. Fluid Mech. 96, 33–46.CrossRefGoogle Scholar
18. Landweber, L. & Shahshahan, A. 1991 Added masses and forces on two bodies approaching central impact in an inviscid fluid. Tech. Rep. 346. Iowa Institute of Hydraulic Research.CrossRefGoogle Scholar
19. Landweber, L. & Yih, C.-S. 1956 Forces, moments, and added masses for Rankine bodies. J. Fluid Mech. 1, 319–336.CrossRefGoogle Scholar
20. Marshall, J. S. 2001 Inviscid Incompressible Flow. Wiley-Interscience.Google Scholar
21. Milne-Thompson, L. M. 1968 Theoretical Hydrodynamics. Dover.CrossRefGoogle Scholar
22. Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
23. Sedov, L. I. 1965 Two-Dimensional Problems in Hydrodynamics and Aerodynamics. John Wiley & Sons.CrossRefGoogle Scholar
24. Shashikanth, B. N. 2006 Symmetric pairs of point vortices interacting with a neutrally buoyant two-dimensional circular cylinder. Phys. Fluids 18, 127103.CrossRefGoogle Scholar
25. Shashikanth, B. N., Marsden, J. E., Burdick, J. W. & Kelly, S. D. 2002 The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with N point vortices. Phys. Fluids 14, 1214–1227.CrossRefGoogle Scholar
26. Taylor, G. I. 1928 The energy of a body moving in an infinite fluid, with an application to airships. Proc. R. Soc. Lond. A 120, 13–21.Google Scholar
27. Tchieu, A. A., Crowdy, D. & Leonard, A. 2010 Fluid–structure interaction of two bodies in an inviscid fluid. Phys. Fluids 22, 107101.CrossRefGoogle Scholar
28. Wang, Q. X. 2004 Interaction of two circular cylinders in inviscid fluid. Phys. Fluids 16, 4412–4425.CrossRefGoogle Scholar
29. Yang, F.-L. 2010 A formula for the wall-amplified added mass coefficient for a solid sphere in normal approach to a wall and its application for such motion at low Reynolds number. Phys. Fluids 22, 123303.CrossRefGoogle Scholar