Page 227 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 209

Experimentally determined values of the dissipative constants were used in the theory,
using a mixed viscoelastic-hysteretic model, with corresponding coefficients Vd and p.

General observations

With increasing flow, externally induced beam motions become more heavily damped;
however, beyond a certain flow the trend is reversed and, at sufficiently high flow, the
stability limit is reached and flutter is precipitated.
Close to, but below, the critical flow for self-excited flutter, the system behaves as if it
has a small unstable limit cycle within a larger stable one, so that external disturbances
of a certain magnitude may precipitate flutter, yet small disturbances are damped. As
the flow gets closer to the stability limit, the inner limit cycle becomes smaller, to the
point where random, turbulence-induced disturbances are sufficient to propel the system
beyond the confines of this limit cycle, precipitating amplified oscillation (flutter). These
are clearly characteristics of a subcritical Hopf bifurcation [Figure 2.1 l(d)].
Limit cycles could generally be observed in the case of pipes hanging in air rather than
water. The amplitude involved was larger for pipes with a uniform conduit than for those
with a conical conduit. For flow velocities higher than those associated with the onset
of instability, the amplitude of the limit cycle increased further. In contrast, for pipes in
water, presumably because of buoyancy counteracting the stabilizing effect of gravity, the
oscillations continued to grow until, in 10-20 cycles, the amplitude became large enough
(i.e. about 8 pipe diameters) for the pipe to start hitting the walls of the test-section,
whereupon the experiment was discontinued for fear of damage to the apparatus; thus,
established limit-cycle motion could not actually be observed in this case.

Comparison between theory and experiment

The dimensionless critical flow velocities, uic, and the corresponding frequencies, w,., for
flutter of a cylindrical-conical pipe in air and water are shown in Figures 4.7 and 4.8,
respectively. Also shown is one experimental point for a cylindrical pipe, for comparison
purposes.
It is seen that theoretical and experimental critical flow velocities agree very
well - although the experimental values ought to have been a little lower than the
theoretical ones, this being a subcritical Hopf bifurcation. The corresponding frequencies
agree less well. However, this is not surprising, upon realizing that: (i) in the case of
pipes in air, the measured frequencies were those of limit-cycle motion, rather than those
associated with the onset of flutter; these two values could be quite different in the case of
a subcritical Hopf bifurcation, since the initial limit cycle is of non-negligible magnitude;
(ii) in the case of experiments in water, the frequency was measured during the first few
cycles of motion, before the pipe started hitting the wall, and precision of measurement
was not high.
The theoretically predicted reduction in dimensionless critical flow velocity with
increasing slenderness (and hence the even more substantial reduction in dimensional
flow velocity) is wholly supported by these experiments, as well as the theoretical finding
that the system is less stable when immersed in water than in air.
Finally, the experimental frequencies for the cylindrical-conical pipes are lower than
those of the uniform cylindrical ones, which is in agreement with theory.