Unsteady Lagally theorem for multipoles and deformable bodies

Abstract

The Lagally theorem for unsteady flow expresses the forces and moments acting on a rigid body moving in an inviscid and incompressible fluid in terms of the singularities of the analytically continued flow within the body. Previous generalizations of the Lagally theorem, originally given by Lagally (1922) for steady flows, are due to Cummins (1957) and Landweber & Yih (1956), who consider the effect of flow unsteadiness on the forces and moments. In these, the system of image singularities within the body was assumed to consist of isolated or continuous (surface or volume) distributions of sources and doublets. A further extension of Lagally's theorem ie due to Landweber (1967), who derived expressions for the steady forces and moments acting on a rigid body generated by isolated or a continuous distribution of multipoles. The purpose of the present paper is to generalize the Lagally theorem so as to include the effects of multipoles in unsteady flow, and deformability of the body, as well as to present a briefer derivation of the resulting formulae. Two examples, illustrating the application of the force and moment formulae, will be presented.

The motion of pairs of gas bubbles in a perfect liquid

“…However, for systems of bodies moving through the liquid, no exact general theory is available because of the complexity of the interactions. For the present problem which involves two bodies, we make use of a result obtained recently by Landweber and Miloh [7] for the force exerted on a body moving through an arbitrary flow in the case in which this flow can be represented by multipoles of given strength. Let a multipole of order q (a source has order zero, a dipole has order one, etc.)…”

“…The force F on the body with volume ~-and moving with velocity v in the arbitrary potential flow is, as obtained by Landweber and Miloh [7],…”

Cloud Cavitation Dynamics

“…Based on bubble dynamics scalings, the Froude number (characterizing the effect of buoyancy) is 9.8 × 10 −5 , which is negligible. The initial surface velocity, at t = 0, is set to zero and the bubbles [7] Cloud cavitation dynamics 205…”

“…We conclude with a simulation on irregular arrays with unequal radii, where an individual radius is chosen to vary randomly according to a i = η R m (1 + ζ (i)), and the internal pressure is chosen so as to achieve a maximum Rayleigh radius of [8] x-coordinate z-coordinate a i = η R m (1 + ζ (i)). Figure 4 demonstrates the variation in amplitude for a 20-element array with ζ (i) ∈ (−0.3, 0.3), and R m , η equal to 0.0001 m and 0.2 respectively.…”

“…In the Lagally formulation of the multipole method (see Taylor [13], Landweber and 200 M. Wilson, J. R. Blake and P. M. Haese [2] Yih [9], and Landweber and Miloh [7,8] for further details), this leads to a particularly simple expression relating the rate of change of the Kelvin impulse to the singularities of the analytically continued potential flow. If the potential solution is known or can be calculated, then this expression results in an extremely effective method of calculating a bubble's evolution.…”