Page 60 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS, DEFINITIONS AND METHODS 43

It should also be noted that in cases of symmetric confinement of the fluid, this term may
entirely vanish.

IC) Magnitude considerations: wide and narrow annuli
By re-writing expressions (2.120a,b) in terms of R;/R,, and taking the limit (R;/R,) -+ 0,
i.e. as the outer cylinder radius becomes essentially infinite, one obtains

(2.133a)

and
F,,,,; = 0, as (R;/R,) -+ 0. (2.133b)
Equation (2.133a) for n = 1 yields the result just obtained in (2.131) in another way,
that the added mass for R, +. 00 is equal to the ‘displaced’ mass of fluid. Also, the
physically reasonable result is obtained in equation (2.133b) that, for an infinitely distant
outer cylinder, the effect of accelerations of the inner one is infinitely faint.
In the other limit, writing R, = R; + h and R; Y R, 2: R, and taking h to be small,

(2.134)

This expression shows that for thin shells, and also for light, hollow cylinders in narrow
annuli, the added mass can easily exceed and be several times larger than the struc-
tural mass; i.e. M’ >> M in (2.126), for instance. Expression (2.134) would suggest that
the added mass becomes infinitely large as h -+ 0. This is not so, however, because
the Stokes number becomes small before that limit is reached, signalling that the limit
of applicability of inviscid theory has been surpassed; for oscillations of the shell or
cylinder of amplitude Eh and frequency w, where E is a small number, the Stokes (or
oscillatory Reynolds) number is B = cwhR/u ew(h/R)R2/v. An alternative, and more
general, pertinent Stokes number is B = wlt2/u. In either case, it is clear that as h + 0, or
h/R -+ 0 and E < 1, B becomes sufficiently small for viscous effects not to be negligible
(see Section 2.2.3). Furthermore, in addition to the added damping, the forces associ-
ated with shell motions become extremely large, as seen from (2.134), due to the very
large accelerations in the narrow fluid annulus; hence, sustained oscillation under the
circumstances does not occur.
It is finally noted in (2.120a,b), (2.133a) and (2.134) that the added mass becomes
smaller as n is increased, which is reasonable in physical terms: the hills and valleys
associated with deformation of the shell are half a circumference apart for n = 1, while
they are much closer for large n; hence the fluid accelerations are correspondingly smaller
for the larger n, and so is the added mass.

(dl Fluid coupling and the added mass matrix
If both shells are flexible, the only thing that changes in the formulation is that boundary
condition (2.1 15b) needs to take a form similar to (2.1 15a). Recalling the meaning of
influence coefficients in solid mechanics, by analogy (and as already done in the foregoing)
one can think of a force on the inner shell due to nth mode motion of the inner shell,
F;,n;, or of a force on the inner shell due to motion of the outer one, F;,,,, and so on.