Page 351 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 33 1

results are shown in Figure 5.19(b). It is interesting that the zones where the system loses
stability out-of-plane (zones I, I11 and V), correspond nearly exactly to the regions where
Rousselet & Herrmann (1981) predict that the system is stabilized by ‘nonlinearities’ !
Suppressing the small region around B = 0.295 in Figure 5.18 (where dissipation is known
to have a destabilizing effect - Section 3.5.3), the comparison is made in Table 5.1.
Evidently, the nozzle affects stability in a similar way as ‘nonlinearities’: when the nozzle
has a stabilizing effect in its own plane, the pipe loses stability out-of-plane; and when
the effect is destabilizing, then in-plane flutter is obtained.
The results were tested experimentally for varying values of B, and accord between
theory and experiment, as to whether in-plane or out-of-plane limit cycles arise, is quite
good, considering especially that the experiments were with vertical pipes, whereas in the
theory gravity effects are neglected.
The definitive study into the nature of the Hopf bifurcations leading to flutter, for
the same system as that examined by Rousselet & Herrmann, is due to Bajaj et al.
(1980), who conducted a sophisticated analysis, utilizing the tools of modern dynamics
theory. The equation of motion analysed involves a constant upstream pressure and a
deformation-dependent flow velocity. The partial differential equation is transformed to
vector form, and then the linear problem and its adjoint are solved. The solution procedure
is nonstandard, but is based on the ideas of centre manifold theory and averaging [refer
also to the discussion of Bajaj’s (1987b) work in Section 5.9.21. A periodic solution with
undetermined amplitude and phase is assumed, and the equations are eventually reduced
to the following normal form:

r = H(/L + ar2) + S(&, lj = w, + E[/LC + br2] + 0(t2). (5.113)

Depending on the sign of a, the emerging limit cycle is stable (a < 0, supercritical Hopf)
or unstable (a > 0, subcritical Hopf). The results are shown in Figure 5.20, and depend
on the parameter a! = f L/&, f being the friction factor and A = the flow area;
thus, a! c( L/Di, represents the slenderness of the pipe. It is seen that for sufficiently
long (slender) pipes, hence large a!, the bifurcation is always supercritical, whereas for
short enough pipes’ it can be subcritical. Significantly, for any given a!, the regions of
sub- and supercritical Hopf bifurcations in Figure 5.20 do not correspond to those of
destabilization and stabilization due to ‘nonlinearities’ in Table 5.1, and hence contradict
the results of Rousselet & Herrmann (1981); the reason for this is likely the fact that
‘artificial damping’ is included among the ‘nonlinearities’ in Rousselet & Herrmann’s

Table 5.1 Effect of ‘nonlinearities’ (Rousselet & Herrmann 1981)
and plane of flutter for a pipe fitted with an inclined end-nozzle
(Lundgren et al. 1979) for different ranges of /?; in-plane means in
the plane of the nozzle, and out-of-plane perpendicular to the plane
of the nozzle.
_____ ~ ~
B Effect of B Plane of flutter
‘nonlinearities’
0.02-0.21 Stabilizing 0.00-0.23 Out-of-plane
0.21 -0.42 Destabilizing 0.23-0.42 In-plane
0.42-0.66 Stabilizing 0.42-0.63 Out-of-plane

+Still presuming that they are long enough for Euler-Bernoulli theory to hold.