Page 444 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

416 SLENDER STRUCTURES AND AXIAL FLOW

configurations, Their equations of motion have a significant difference from those of
most of the previous studies: they include the effect of the initial forces arising from the
centrifugal effect and the pressure of the fluid. They obtained the unexpected result that, if
the initial forces are taken into account, then pipes with both ends supported do not lose
stability, no matter how high the flow velocity may be! Similar observations had been
made by Svetlitsky (Svetlitskii) in an earlier paper published in Russian (Svetlitskii 1969)
and subsequently in English (Svetlitsky 1977); see also Svetlitskii (1982). Svetlitsky
considered cantilevered pipes as well and noted that this system loses stability by flutter
at sufficiently high flow velocities, even when initial forces are taken into account.
Doll & Mote (1974, 1976) studied a more general case, where the fluid-conveying
pipe is both curved and twisted. The equations of motion were derived via Hamilton’s
principle and solutions were obtained using the finite element method. They considered
two cases: (i) the ‘constant curvature’ case, in which the curvature does not change
with flow velocity, and (ii) the ‘variable curvature’ case in which variations in curvature
with changes in flow velocity are accounted for. The first case corresponds to the analyses
of Unny et al. and Chen, while the second is similar to Hill & Davis. Both Doll & Mote
and Hill & Davis take into account the extensibility of the centreline of the curved
pipe. An important difference between the two studies is that in the latter the equilibrium
configuration and forces are calculated via a linearized set of equations, on the assumption
that the initial and the flow-deformed equilibrium configurations are close; in the Doll
& Mote study, on the other hand, a cumulative application of linearization is utilized
for small flow velocity increments, which is more general. The main conclusions of
both studies, however, are the same: the eigenfrequencies of pipes supported at both
ends are not sensitive to flow velocity, and hence no instabilities should arise. Doll
& Mote also compared their curved-pipe theory to Liu & Mote’s (1974) experimental
data for nominally straight but actually slightly curved pipes, supported at both ends
(see Section 3.4.4). This comparison is discussed in Sections 6.4.5 and 6.6.2.
More recently, Dupuis & Rousselet (1985) have carried out a study on the dynamics
of fluid-conveying planar curved pipes modelled as Timoshenko beams. The extension of
the centreline was taken into account. This study used the transfer matrix method (Pestel
& Leche 1963), in preference to either analytical or finite element methods. Once
more, flutter instabilities are predicted for cantilevered curved pipes, but results for pipes
supported at both ends are not presented.
Misra et al. (1988a,b,c) re-examined the dynamics and stability of fluid-conveying
curved pipes ab initio. The main objective was to shed light onto the underlying reasons
for the fundamentally different dynamical behaviour for pipes with supported ends as
predicted by the extensible theories of Hill & Davis, Doll & Mote and Svetlitsky, on the
one hand, and the inextensible theories of Chen and Unny et a1 ., on the other: the former
predicting no loss of stability, while the latter predict divergence at high enough flow.
For this purpose, three theories were formulated and their results were compared:
(i) the conventional inextensible theory, in which the centreline of the pipe is
assumed to be unstretched, and the steady (initial) flow-induced forces introduced
by the pressure and centrifugal forces are entirely neglected;
(ii) the extensible theory, in which the shape of the pipe changes with flow velocity
under the action of the steady flow-induced forces;
(iii) the mod$ed inextensible theory, in which the assumption of inextensibility of the
centreline is retained, but the steady flow-related forces are taken into account.