Page 239 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

220 SLENDER STRUCTURES AND AXIAL FLOW

with this in mind but, because of ‘existing equipment, measuring techniques and financial
considerations’, ended up doing regular down-flow experiments with mechanically forced
excitation of the pipe (see Section 4.6).

4.4 SHORT PIPES AND REFINED FLOW MODELLING

In the foregoing (Chapter 3 and Sections 4.1 -4.3), it has been assumed that (i) the pipe
is sufficiently slender for Euler-Bernoulli beam theory to be adequate for describing the
dynamics of the pipe, and (ii) that wavelength of deformation is sufficiently long for the
plug-flow model to be acceptable, thus ignoring conditions upstream and downstream
while determining the fluid-dynamic forces at a given point. If the pipe is sufficiently
short, however, both assumptions become questionable, as will be discussed further in
the following, and the use of Timoshenko beam theory and more elaborate fluid dynamics
becomes necessary. In this section the necessary fundamentals are developed, by means
of which (a) the limits of applicability of the Euler-Bernoulli plug-flow (EBPF for short)
analytical model are determined, and (b) a theory for really short pipes conveying fluid
is established.
Since stability is of primary concern, it is noted that short thin-walled pipes lose stability
in their shell modes [n > 2; see Figure 2.7(c)] rather than in their beam modes (n = l),
as discussed in Chapter 7 (Volume 2). In what follows, however, it is presumed that the
pipe is sufficiently thick-walled for its beam-mode dynamics to be of primary interest.
Timoshenko beam theory, where shear deformation and rotatory inertia are not
neglected, was first applied to the study of dynamics of pipes conveying fluid by Paldoussis
& Laithier (1976). This theory is applicable to articulated pipes in the limit of a very
large number of articulations (Section 3.8), where the articulations permit substantial
shear deformation. It is also applicable to continuously flexible short pipes, as well
as for obtaining the dynamical behaviour of long pipes in their higher modes; in both
these cases the necessity of utilizing Timoshenko, as opposed to Euler-Bernoulli beam
theory, is well established (Meirovitch 1967). The equations of motion in Pafdoussis &
Laithier (1976) are derived by Newtonian methods, and solved by finite difference and
variational techniques. They are rederived by Laithier & Paldoussis (198 1) via Hamilton’s
principle - a nontrivial exercise. In terms of the fluid mechanics of the problem, however,
the use of the plug-flow model is retained in both cases; this theory will be referred
to as the Timoshenko plug-flow theory (TPF for short). Also, numerous finite element
schemes based on TPF-type theory have been proposed and used for stability and more
general dynamical analysis of piping conveying fluid (Sections 4.6 and 4.7), e.g. by
Chen & Fan (1987), Pramila et al. (1991), Sdlstrom & Akesson (1990) and Sallstrom
(1990, 1993).
It is nevertheless recognized that the applicability of the plug-flow model to short
pipes - or indeed to the study of the high-mode dynamical behaviour of relatively longer
pipes - is questionable, as discussed first by Niordson (1953) and also by others, e.g.
Shayo & Ellen (1974): if the wavelength of deformation is not large, as compared to the
pipe radius, the use of the plug-flow model for obtaining the fluid forces becomes invalid
[Section 4.4.3(b)]. Hence, there is need for improvement of the fluid mechanics of the
problem for studying the dynamics of this class of problem.
The dynamics and stability of short pipes conveying fluid are examined here by means
of Timoshenko beam theory for the pipe and a three-dimensional fluid-mechanical model