Page 443 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

6

Curved Pipes Conveying Fluid

6.1 INTRODUCTION
As may be appreciated from the contents of Chapters 3-5, a great deal of work has been
done on the dynamics of straight pipes conveying fluid over the past 45 years or so.
Relatively less effort has been directed towards the investigation of the dynamics and
stability of fluid-conveying curved pipes. In general, piping may be curved and twisted
into complex spatial forms. In this book, however, reflecting the state of the art, mostly
curved pipes which initially lie within a given plane are considered. In this case, one
may distinguish motions in the plane of curvature and perpendicular to it, which will be
referred to as in-plane and out-of-plane motions for short; as will be seen, depending
on the assumptions made, these two sets of motions are sometimes uncoupled from
each other.
Among the first to study the hydroelastic vibration of curved pipes was Svetlitskii
(1966). He investigated the out-of-plane motion of a fluid-conveying perfectly flexible
hose, treating it as a string, and therefore neglecting the bending rigidity. The ends of
the hose were fixed and its initial shape was a catenary. Unny et al. (1970) considered the
in-plane divergence of initially circular tubular beams with fixed ends. The equations of
motion were derived using Hamilton’s principle, and critical flow velocities for instability
were obtained for pinned and clamped ends; the equations of motion, however, were
subsequently shown to be incorrect (Chen 1972b).
The dynamics and stability of curved pipes in the form of circular arcs were
studied extensively by Chen (1972b,c, 1973). He derived the equations governing in-
plane motions using both the Newtonian (Chen 1972b) and Hamiltonian (Chen 1972c)
formulations, and equations governing out-of-plane motions from the Hamiltonian
viewpoint (Chen 1972c, 1973). In all cases, it was assumed that the centreline of the pipe
is inextensible. It was found that in the case of clamped-clamped and pinned-pinned
boundary conditions, the pipe loses stability by divergence when the flow velocity or the
fluid pressure exceeds a certain critical value. This behaviour is qualitatively similar to
that of a straight pipe. Chen also studied the stability of cantilevered curved pipes. He
found that for in-plane motions such pipes are generally subject to both divergence and
flutter instabilities, with divergence occurring first, except in cases where the subtended
angle is very small (so that the system comes closer to a straight pipe), when only flutter
was found to arise (Chen 1972~). In the case of out-of-plane motions, only flutter was
predicted, with stability characteristics similar to those of a straight pipe (Chen 1973).
Hill & Davis (1974) studied the dynamics and stability of clamped-clamped pipes
conveying fluid, shaped as circular arcs, as well as S-shaped, L-shaped and spiral

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