Page 480 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CURVED PIPES CONVEYING FLUID 45 1

Finally, KO & Bert (1984, 1986) derived a nonlinear equation, under a set of reasonable
assumptions, for in-plane motion of a circular-arc pipe conveying fluid,+ which they solved
for the case of clamped ends by the method of multiple scales. They use the inextensibility
assumption but take into account the steady fluid forces - similarly to the Misra et al.
modified inextensible theory. In a sample calculation, KO & Bert (1986) find that the
frequency of the first asymmetric mode [Figure 6.7(a)] increases with the flow velocity.
Furthermore, the frequency displays a strong softening behaviour (Le. it decreases with
increasing amplitude).

6.4.5 Concluding remarks
As shown by the results of Figures 6.11 and 6.12 for in-plane motions and Figures 6.13
and 6.16 for out-of-plane motions, differences in the dynamical behaviour as predicted
by the modified inextensible and extensible theories are either small or virtually zero,
whereas this behaviour is dramatically different from that predicted by the conventional
inextensible theory.
It is clear that the main difference between the extensible theories and the ‘traditional
inextensible’ theory is not the extensibility of the centreline at all, but rather whether the
combined steady axial force no is taken into account or not. This resolves the apparent
paradox that, although it is physically obvious that the actual extension of the centre-
line cannot be very large, the differences in predicted behaviour between (conventional)
inextensible and extensible theory are so profound: the first predicts loss of stability by
divergence and pronounced eigenfrequency-flow effects, whereas the second predicts no
loss of stability and weak frequency-flow effects. It has now been clarified that the use of
the ‘inextensible’ and ‘extensible’ labels is rather misleading, as are those of ‘constant’
and ‘variable curvature’ utilized by Doll & Mote; the real source of the discrepancy lies
in the fact that conventional inextensible theory also neglects all steady stress effects (Le.
all steady flow-induced forces).
Unfortunately, there are no experimental data for curved pipes, apart from those of Liu
& Mote (1974) already discussed in Section 3.5.6. In these experiments, however, the
curvature was relatively small and inadvertent. The variation of the fundamental eigen-
frequency with flow was nevertheless compared with various versions of their theory
by Doll & Mote (1976); it was found that, if anything, the experimental results up
to a certain maximum U* agreed better with those of Doll & Mote’s ‘constant curva-
ture’ analysis (which corresponds to inextensible theory) than with extensible theory.
As seen in Figure 3.26, the frequency varies with U essentially as predicted by the
conventional inextensible theory! This paradox, which has ever since cast doubt on
the validity of the extensible, and hence also the modified inextensible, theory is resolved
at the end of Section 6.6. However, proper experiments with curved pipes remain to be
done - recognizing, nevertheless, that this is not a simple task.
Until then, since there is no reason why the effect of steady fluid forces on the dynamics
of the system should be neglected, and as convincingly argued by Dupuis & Rousselet

‘According to Dupuis & Rousselet (1992), their equations of motion ‘are free of the Coriolis force and
with some linear terms that are neither accounted for in their analysis, nor found in any other analysis’. The
Conolis terms were in fact omitted intentionally (Bert 1996), presumably because the theory was to be applied
exclusively to conservative pipe systems (pipes with clamped ends).