Page 58 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS, DEFINITIONS AND METHODS 41

the absence of flow, result in a force; in an inviscid fluid, neither can a velocity of
the body.+
It is customary to define a virtual or added mass, by expressing the fluid loading in the
form of a d'Alembert (mass) x (acceleration) term. For ease of interpretation, consider first
the case of n = 1 [see Figure 2.7(c)], so that the shell (only the inner one for simplicity)
oscillates transversely as a whole, without deformation of its cross-section - essentially
as a beam or a rigid cylinder would. Then considering WI; cos = w, and v, = 0
(Figure 2.8), the equation of motion of the cylinder in the z-direction may be written as

(2.125)

M, and K could be the modal mass, damping and stiffness elements in a one axial-mode
C
Galerkin approximation for the structure, or one can think of a long rigid cylinder of mass
M, flexibly supported by a spring of stiffness K and a dashpot with damping coefficient
C; L is the length of the shell. The quantity in square brackets is defined as the added
mass, and may be denoted by M', so that equation (2.125) may be written as

(M +M')i + Ci+ KZ = 0, (2.126)

thus making obvious the usefulness of this concept and the appellation of 'added' mass.
Dividing this added mass by the fluid mass of the volume occupied by ('displaced' by?
the presence) of the shell, gives the so-called added mass coefficient,

(2.127)

For shell-type motions, n > 1, one cannot associate added mass or added mass coef-
ficients with motions in a particular direction as in (2.125) and (2.127), but rather with
motions associated with particular modes of deformation, e.g. the n th circumferential
mode. In any case, for the analysis of shell motions, forces due to the fluid per unit
surface area are more pertinent, as is done in Chapter 7. The added mass coefficient,
however, is defined in the same way as in the foregoing; thus, corresponding to the forces
in (2.120a,b) and (2.121), we have

(2.128a)

(2.128b)

see also Chen (1987; Chapter 4).§

+For a body in unbounded fluid this is a consequence of the d' Alembert paradox (stating that an ideal fluid
flow exerts zero net force on any body immersed in it). In the presence of solid boundaries this is generally not
so, and velocity-dependent forces may arise, but they are proportional to the square of the velocity (Duncan
et al. 1970), and so, in the present context, they are negligible.
*'Displaced', in the original sense in Archimedes' 'experiment' in Syracuse, when he immersed himself in
his bath, thus displacing an equal volume of the fluid - and evoking the famous eureku!
$Note, however, a typographical error in equation (4.39) therein.