Page 260 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 24 1

4.4.10 Long pipes and refined flow theory
Despite what is said in the previous section regarding the superfluity of using Timoshenko
or refined-flow theories except for really short pipes, there is no reason why they should
not be used for longer pipes as well. This is particularly true in the case of general
computational codes applicable to long and short pipes alike. An example is the work of
Sallstrom & Akesson (1990) and Sallstrom (1990, 1993), discussed in Section 4.7, which
is based on Timoshenko beam theory.
Another example is a study by Langthjem (1995) on the dynamics of not necessarily
short cantilevered pipes, partially or totally immersed in stagnant fluid, analysed by
Timoshenko refined-flow (TRF) theory. Both the internal and external fluid dynamics
are analysed by potential flow theory. Furthermore, it is argued that if the internal and
external fluids are the same, e.g. liquid flow discharging into stagnant liquid, a turbu-
lent jet develops and the flow is subjected to a velocity gradient at the free end; hence,
yet another type of ‘outflow model’ is developed. It is found that, as a result of flow
velocity reduction for x > L, the critical flutter speed (u,f) may be diminished by 5- 10%.
Similar conclusions to those summarized in Section 4.4.9 are reached regarding the appli-
cability of simpler theory down to very short pipes, and those in Section 4.2.4 and in
Sugiyama et al. (1996a) regarding immersion effects. In particular, the destabilization
when immersion is shallow, as compared to no immersion, is explained by noting that
this enhances the ‘dragging’ form of the motion and hence optimizes energy transfer
(Section 3.2.2).
Experiments with long elastomer pipes (E = 38-60) conveying water support the theo-
retical findings, and agreement with theory in ucf is within 10- 15% - but not sufficiently
close to validate the outflow model. It is of interest that, in one case, flutter was found
to switch between planar and rotary motions in an unpredictable manner, suggesting that
the oscillation may be chaotic. This should be compared to the physically similar case of
a pipe with an additional end-mass (rather than immersion-related added mass) analysed
by Copeland & Moon (1992) - see Section 5.8.3(b).

4.4.1 1 Pipes conveying compressible fluid

The dynamics of pipes conveying compressible fluid has been considered by Johnson
et al. (1987), developing the theory initially formulated by Niordson (1953). Timoshenko
beam theory is used for the pipe and a compressible potential flow for the fluid - in
which V2$ = c-2[(a2$/at2) + 2U(a2$/ax at) + U2(a2$/ax2)] is used instead of (4.50),
c being the sonic speed. Results are given for the critical velocity for divergence, ucd,
in the first mode of a pinned-pinned pipe, obtained by Euler’s method of equilibrium
(Section 3.4.1).
The results are presented in a different and less physical manner than in
Sections 4.4.1-4.4.9: the parameters E and A, which are physically linked by
equation (4.37), are varied independently; this, despite the fact that the only way of
varying E while keeping A constant is by changing the material constants and wall-
thickness - which in practice cannot be varied widely. On the other hand, this allows
the convenient separation of fluid-mechanical effects from structural ones (i.e. whether
Timoshenko or Euler-Bernoulli theory is used). For example, some results are presented