Page 355 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 335
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TW 0

’
/L/+ I I
I
k’ I I
/ I I
/ I I I I
I
fy I I : I I I 1 I I
I
I
I
I I I I I I
/I .i I I I I

Figure 5.22 Bifurcation diagram of the perturbed cantilevered pipe system with a small stiffness
asymmetry, showing the amplitude, A, of planar motions (‘standing waves’, SW), rotary motions
(‘travelling waves’, TW) and modulated waves (MW), as a function of the flow parameter h for
#I 0.25 and S = 0.1 (Bajaj & Sethna 1991).
=

bifurcation is subcritical): for small perturbations the solution converges to SW,, and for
larger ones to TW - the two coexisting for Amin < h < A/,. For h > A/,, only rotary (TW)
motions are possible.
These theoretical predictions are qualitatively supported by experiment, involving
slender, vertical cellulose acetate butyrate pipes (L = 1.83 m, D, = 6.48 mm, D,, =
7.95mm). The pipes were fitted with straight nozzles at the free end, with area
ratios of A,/A = 0.3 and 0.4, resulting in /3 = 0.173 and 0.231 [cf. Section 3.5.6 and
equations (3.74)] - the latter case being close to /3 = 0.25, as in Figure 5.22. Experiments
were conducted with a round pipe and then with flats machined on diametrally opposite
sides of its outer surface, resulting in 6 2 0.05 (cf. 6 = 0.1 in Figure 5.22). The following
remarkable set of observations are made.
(i) For the round, nearly perfectly symmetric pipe with /3 = 0.173, the initial limit-cycle
motion is confined to one plane - as determined by whatever imperfections are present.
With further increase in the flow rate, the amplitude of motion increases but it remains
planar, in the same plane. This is as predicted in Figure 5.21, where the planar motion is
stable, while the circular one is unstable.
(ii) For the same configuration but /3 = 0.231, the initial limit-cycle motion is again
planar. For a slightly larger flow rate, however, the motion becomes circular (rotary),
which is the situation predicted by theory (Figure 5.21).