Page 224 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

206 SLENDER STRUCTURES AND AXIAL FLOW

one (uic 2: 5.0). By reconsidering the arguments originally made by Benjamin (1961a)
and discussed in Section 3.2.2, flutter arises when the work done by centrifugal force
MU2($’w/ax2) piAiU;(a2w/ax2) overcomes that done by the Coriolis force. In the
s
case of the nonuniform pipe, however, this term is equal to [piAi(x)Ui(x)Ui(L)](d2w/ax2),
where piAi(x)Ui(x) const. Hence, since Ui(L) Ui(O), the destabilizing force is higher
=
>
at all points x > 0 in the cylindrical-conical system vis-&vis the uniform one. In this case
the ratio of critical flow velocities is 2.25/5.0 = 0.45, which is close to the diameter ratio
=
(1 - ai)/l = Di(L)/Di(O) 0.5. Similar calculations confirm that uic indeed decreases
almost linearly with increasing ‘truncation factor’ ai. Thus, the destabilizing effect of
conicity of the flow passage is similar to that of mounting a convergent nozzle at the
end of an otherwise uniform pipe [Sections 3.3.5 and 3.5.6 and Gregory & PaYdoussis
(1966b)l.
Figure 4.4(b) shows the effect of density of the surrounding fluid on the dynamics
of the cylindrical-conical pipe. The dimensionless frequency is defined, in terms of the
dimensional circular frequency f2, by

(4.25)

Intuitively one would have supposed that when the surrounding fluid is water, the system
would be more stable than when it is air. Yet, the opposite is found to be true. The
increase in the surrounding fluid density acts in two ways: (i) to increase the effective
inertia of the pipe through the added-mass effect and (ii) to decrease the gravity effect
through buoyancy. Both have a destabilizing effect with increasing density of the external
fluid, pe. The latter is physically obvious. The former may be accepted by analogy to the
case of uniform pipes where it was found that, as the mass ratio piAi/(piAi + m) becomes
smaller, the system is less stable (Section 3.5); the external stagnant fluid effectively adds
peA, to m, producing the same effect.
In Figure 4.4(b) the real parts of the dimensionless frequencies %e(wj), j being the
mode number, are lower for the pipe immersed in liquid than in air, which is reason-
able in view of the added-mass effect; this is even more pronounced in dimensional
terms - refer to equation (4.25). However, the .9rn(wj) are also lower in liquid than
in air, which is contrary to physical intuition, as the added damping in liquid should
be higher than in air. Nevertheless, it is recalled that the true measure of damping is
Cj = .9m(wj)/%e(wj), and this does show the expected behaviour. It may be shown by
a perturbation analysis for small ui that (a) 4m(wj) = 2ui[(S2 + yi)/(y, + yi)]*/* for all
j, in the absence of dissipative forces, here taken to be zero for simplicity, and (b) the
%e(wj) are approximately equal to their values at ui = 0. These may be used to obtain
estimates of for small enough ui.
Figure 4.5 shows the eigenfrequencies of some of the lowest modes of a conical-
conical pipe in still water; internal dissipation has been taken into accountt in one case.
It is seen that the behaviour of the system in both cases is considerably different from
that of the previous systems. First, the critical flow velocities are much lower, reflecting

+A modified viscoelastic dissipation model is utilized in this case to approximate the expenmentally observed
behaviour of silicone rubber, which exhibits hysteretical behaviour at high frequencies but is viscoelastic at low
frequencies. This is achieved by replacing Vd by Ud[l + (Ud/&)lWl]-’, where fid is the hysteretic damping
coefficient as w + 00.