Page 400 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

376 SLENDER STRUCTURES AND AXIAL FLOW

the triangles (A) at lower values of u, corresponding at the same time to a turning point
and a restabilization, for which stable periodic oscillations of small amplitude and high
frequency appear. The appearance of the second stable periodic solution explains clearly
how the smaller amplitude oscillations at u > 8.625 detected by FDM (Figure 5.46) come
into being.
(iii) The value of u of the bifurcation point leading to the appearance of stable periodic
solutions decreases dramatically with increasing p, which means that the range where
quasiperiodic oscillations can be detected decreases with p, to a point where they no
longer exist. This is confirmed by FDM calculations; for p = 0.1 there exists only a very
narrow range of quasiperiodic solutions.
(iv) The case of p = 0.1 may be considered as a limiting case for two distinct reasons:
firstly because of the previous remark, and secondly because of the small amplitude ‘jump’
observed in Figure 5.47(a) for u 2 6.1 or the corresponding small frequency ‘hump’ in
Figure 5.47(b); the evolution of the stable periodic solutions emerging from the Hopf
bifurcations is no longer smooth for p > 0.1, and new phenomena start to occur.
A bifurcation diagram for p = 0.15 is shown in Figure 5.47(c). It is recalled that for
such ‘high’ p, there is a very clear frequency jump across the secondary bifurcation in the
experiments, succeeded by chaos. The results shown indicate excellent agreement between
FDM and IHB - although the former cannot compute unstable solutions, while the latter
cannot compute nonperiodic (chaotic) ones. Nevertheless, the two in synergism are a
potent tool. Thus, the IHB results reveal that the amplitude jump at u 2 6 for p = 0.10
in Figure 5.47(a) is associated with a loop, as shown for p = 0.15 in Figure 5.47(c).
Unstable solutions emerging at u = 7.54 in Figure 5.47(c) exist up to u = 11.69, which
is beyond the scale of the figure. Furthermore, although the system is always unstable
in this big loop, the number of Floquet multipliers inside the unit circle varies several
times (4 + 5 + 3 + 5 + 6). These bifurcations are of no great importance because the
system is unstable in any case. Of more interest is the bifurcation occurring in the small-
amplitude stable periodic solution at u = 8.76: increasing u further, the stable solution
becomes unstable, again because two complex conjugate multipliers cross the unit circle,
but the solution thereafter is not quasiperiodic but chaotic, as shown in Figure 5.47(d).
Consequently, from a physical viewpoint, four distinct types of solution may be
observed for p = 0.15: (i) solutions converging to the stable equilibrium for u 5 4.66;
(ii) periodic solutions whose frequency increases with u for 4.66 -= u < 7.54; (iii) periodic
solutions of higher frequency and smaller amplitude for 7.26 < u < 8.76 (implying a
jump in the response); and (iv) chaotic oscillations for u > 8.76. This is exactly what
is observed in the experiment. As shown in Table 5.5, agreement between theory and
experiment is relatively good in terms of the critical flow velocities and the frequency
before the ‘jump’ but not after.
If the value of p is increased further, the results obtained numerically are qualitatively
similar to those for p = 0.15, except that the number of ‘humps’ increases, which means
that the number of bifurcations in the system increases as well. Alas, the quantitative agree-
ment between theory and experiment deteriorates, since the values of u for the second
bifurcation (followed almost immediately by chaotic oscillations) increase in the experi-
ment, while they decrease in theory (Table 5.6). Before giving reasons for this, the effects
of the nonlinear inertial terms on the dynamics are investigated; these terms have been
included in the analysis so far. The idea here is to compare the results with and without
the nonlinear inertial terms. If it is found that these terms do not significantly affect the