Page 159 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 141

his experiments with articulated pipes. It is of interest that, if the tubular cantilever
is initially (i.e. at zero flow) not substantially straight, flow can produce large lateral
movements which are much larger than the initial departures from straightness. This can
be observed by conducting an experiment using as the cantilever a piece of commercial
rubber tubing, which normally has a set bow in it. Flow exaggerates the original bow, the
shape of the tube continually changing with increasing flow velocity. Clearly, this could
be misinterpreted as buckling of a straight pipe.
The second point of interest is that, in some of these experiments, it was possible to
demonstrate the nonlinear dynamical behaviour displayed in Figures 2.12 and 2.13 about
the origin. Over a very small range of flow velocities, it was found that: (i) weak taps to the
pipe caused it to oscillate, but the oscillation decayed and the pipe returned to its equilib-
rium state; (ii) stronger taps induced the system to develop limit-cycle oscillation - thus
demonstrating the existence of a small unstable limit cycle and a larger stable one.
Several experiments were conducted with different lengths (different y) of a number of
pipes with varying #?. The pipes were all with Do = 15.5 mm and h = 2.79-9.14 mm; the
initial length was - 480 mm and experiments were conducted with L = 230-480 mm.
Two different materials were used, Silastic A and Silastic B (Appendix D), the latter
having a larger E and higher damping. In comparing with theory, the dissipation was
modelled as a hysteretic effect, and average values were used: p = 0.02 for Silastic A
and p = 0.10 for Silastic B.
Typical results for the experimental uL.f and wL.f for spontaneous flutter of hanging
cantilevers (y > 0) are shown in Figures 3.49 and 3.50 for water flow and Table 3.4
for air flow, where they are compared with theory. It is clear that agreement between
theory and experiment is reasonably good, especially when dissipation is taken into
account. It is interesting that in some cases the measurements provide indirect experi-
mental support to the theoretical prediction that damping may destabilize the system (e.g.
for #? = 0.241, y 2: 16 and for B = 0.645, y 2: 8.6).
In assessing agreement between theory and experiment, greater weight should be placed
on the critical flow velocity than on the critical frequency, as the latter is measured after the
limit cycle has been established, when nonlinear forces not taken into account in the theory
have already come into play. Accordingly, the fact that taking into account dissipation
seems to worsen agreement in the frequency between theory and experiment- in nearly
all cases, cannot be interpreted as a weakness of the theory; rather, it should be viewed
as being symptomatic of the limitations in the experimental procedure (in identifying the
limit-cycle frequency with w,f).
As already remarked in Section 3.5.2, the impetus for these experiments was partly
provided by Benjamin’s (1 96 1 a,b) findings in connection with dynamical behaviour of
articulated pipes conveying fluid. Benjamin found that divergence is sometimes possible
in cases of vertically hanging articulated cantilevers conveying water; yet it does not occur
if the conveyed fluid is air, the only form of instability possible in that case being flutter.
However, in the case of continuous (hanging) cantilevers, it was found that divergence
is not possible at all whatever the fluid conveyed, only flutter. This matter is clarified in
Section 3.8.
We next consider the experiments with standing cantilevers conveying air only, for
obvious reasons. The dynamical behaviour of the system was of three distinct types,
which for ease of description will be categorized as applying to long, intermediate and
short cantilevers.