Page 296 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

5

Pipes Conveying Fluid:

Nonlinear and Chaotic Dynamics

5.1 INTRODUCTORY COMMENTS

One of the main reasons why the dynamics of pipes conveying fluid has remained
of intense interest to dynamicists well into the 1980s and 1990s is the fact that (i) it
displays interesting and sometimes perplexing nonlinear dynamical behaviour and (ii) it
has become a handy tool in developing or testing modern dynamics theory.
In Chapters 3 and 4 we have mainly dealt with the stability of systems from the linear
point of view, thus predicting loss of stability by divergence or flutter and, in some cases, a
sequence of instabilities as the flow is increased beyond the onset of the first. As discussed
in Section 2.3 with the aid of a one-degree-of-freedom model, divergence may arise via a
pitchfork bifurcation, whereby the original equilibrium point becomes statically unstable,
while two new equilibria are generated; on the other hand, flutter is often generated via a
Hopfbijkation and implies the generation of a limit cycle (Figures 2.10, 2.11 and 3.4).
Additional bifurcations, e.g. a saddle-node and a period-doubling one, will be discussed
in this chapter in due course.
Some key questions associated with these physical phenomena cannot be answered
except by nonlinear theory, among them: (i) for the static instability, where are the
new fixed points located for any value of the parameter being vaned and are they
foci (sinks) or saddles, and hence is the pitchfork bifurcation giving rise to the insta-
bility supercritical or subcritical? (ii) for the dynamic instability, is there an unstable
‘inner’ limit cycle in addition to a stable outer one (Figures 2.12 and 2.13), and hence
is the Hopf bifurcation supercritical or subcritical (Figure 2.1 1); also, what is the ampli-
tude of the limit cycle, and hence the amplitude of oscillation associated with flutter,
and how does the frequency of oscillation vary with amplitude? Many of these ques-
tions are answered via the construction of appropriate bifurcation diagrams which, in
compact form, display both (a) qualitative changes in the character of motion (bifur-
cations) and (b) the evolution in between some characteristic of the motion (typically
the amplitude) with U. Also, the existence and nature of successive instabilities can be
tackled by nonlinear theory, e.g. the question of post-divergence coupled-mode flutter,
extensively discussed in linear terms in Chapters 3 and 4 by assuming implicitly that
post-divergence oscillatory motions occur about the original equilibrium state; however,
because the original equilibrium has become unstable after divergence, motions actually
take place about the new equilibrium points, and hence stability has to be reassessed in
this light.

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