## Journal of Fluid Mechanics

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

a1 Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan

a2 Department of Mechanical Engineering, National Taiwan University, Taipei 10617, Taiwan

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.

(Received February 28 2011)

(Reviewed October 28 2011)

(Accepted January 21 2012)

(Online publication March 16 2012)

Key Words:

• flow–structure interactions;
• general fluid mechanics;
• vortex interactions

Correspondence:

c1 Email addresses for correspondence: fulingyang@ntu.edu.tw, dlyoung@ntu.edu.tw

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