Page 53 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

36 SLENDER STRUCTURES AND AXIAL FLOW

cases the approximation is made that the added mass in quiescent (stagnant) and flowing
fluid is the same, although this is not rigorous. Such an approximation is definitely shaky
if the flow is grossly unsteady or accelerating. Thus, in the extreme case of oscillatory
flow, C,,, = 2 instead of 1 [see, e.g. Sarpkaya & Isaacson (1981)], as a result of induced
buoyancy - i.e. because of the presence of a pressure gradient.'
The added damping may similarly be expressed in terms of a damping coefficient cd,
which may be defined in different ways, e.g.

for oscillations in quiescent or flowing fluid; other definitions are possible.
Added stiffness may arise due to buoyancy, asymmetry$ or proximity to other solid
boundaries. For example, if a body lies close to a wall or a free surface and it is subjected
to flow, there will be a fluid force acting on it, because the flow field is nonuniform. If the
body is displaced towards or away from the aforementioned boundary by Az, this force
will change by AF. The quantity AF/Az is the so-called added stiffness, and it depends
purely on displacement and not on velocity or acceleration. Hence, one may similarly
define a stiffness force coefficient by

(2.1 12)

In equation (2.109), m, c and k are devoid of fluid effects; i.e. in an experimental system
they should ideally be measured in vacuum. Also, unless there exists a mathematical model
the linearization of which yields (2.108), m', c' and k' must be determined experimentally,
e.g. by conducting experiments first in vacuum (practically in still air) and then in fluid
(say, in water) or fluid flow; it is noted that although the c' coefficient of the fluid force
determined thereby is easily separable from the rest, since the velocity-dependent compo-
nent is in quadrature (900 out of phase) with displacement, more than one experiment
would be necessary, and in some cases it is virtually impossible, to separate rn' and k'
since they are 180" out of phase with each other (hence, they differ only in sign).
The rest of Section 2.2 is devoted to the presentation of two simple but representative
analyses - in abridged form - which illustrate the use of the foregoing and also intro-
duce some useful nomenclature for the chapters that follow. In both cases, the mean flow
is zero. Problems involving a mean axial flow, the prime concern of this book, are dealt
with in the other chapters.

2.2.2 Loading on coaxial shells filled with quiescent fluid

Consider two long, thin coaxial shells, with the annular space between them filled with
quiescent, inviscid, dense fluid (e.g. water), while within the inner shell and outside the
outer one the fluid is of much smaller density (e.g. air) or a vacuum; see Figure 2.7. The

+One way of looking at the difference between a cylinder oscillating in quiescent fluid (C,, = 1) and a
cylinder in oscillatory Row (C," = 2) is that in the former case the flow velocity at infinity tends to zero,
whereas in the latter it has the full amplitude of the oscillation: clearly two very different flow fields.
*For example, in the case of an iced conductor in uniform wind, rotation of the noncircular-section conductor
clearly results in a change in the static forces experienced by it; see, e.g. Den Hartog (1956).