Page 250 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 23 1

(b) Cantilevered pipes and outflow models

Unlike pipes with fixed ends, a cantilevered pipe discharges the fluid freely from
its downstream end. The emerging jet continues its sinuous path in the ambient air,
[Figure 4.16(b)], as briefly discussed in Section 3.5.8. The motion of the cantilever is
therefore coupled with that of the downstream jet (at least in this kind of formulation) - as
first discussed by Shayo & Ellen (1978). Thus, in a study of the flow-induced instability
of a cantilevered pipe, it becomes necessary to construct an artificial ‘outflow model’
which describes the manner in which W(6) and hence the perturbations in the fluid are
attenuated beyond the free end of the pipe.
For long pipes conveying fluid, the plug-flow model is fully expected to give reasonable
approximations to the fluid-dynamic forces, and hence to predict reasonably well the
dynamical behaviour of the system. Moreover, the results have been found to be in
good agreement with experiments, and the plug-flow model may be considered to be
quite adequate for long cantilevers conveying fluid. Therefore, the following approach
is adopted: different outflow-model functions Gn(6) and various values of Z(1 > 1) are
tried and adjusted, so that the generalized fluid-dynamic forces (4.67) obtained by refined
fluid mechanics agree with those obtained by simple fluid mechanics [plug-flow model
with equations (4.49)] for a long cantihered pipe. It is then assumed that the same
outflow-model functions G,(e) and value of 1 would apply for short pipes - indeed to
the very short cantilevered pipes which are the subject of this section. The validity of this
assumption is tested, partially at least, by comparison with experimental measurements
(Section 4.4.8). Following the mathematical formulation suggested by Shayo & Ellen
(1978) for both the beam- and shell-mode dynamical behaviour of a cantilevered shell
conveying fluid, three different downstream flow models are tried for the cantilevered
tubular beam; their characteristics are summarized in Table 4.2, together with the ‘no
model’ situation, in which the deflection of the perturbation in the fluid is supposed to
vanish abruptly at 6 = I = 1; for the ‘first’, ‘second’ and ‘third’ models, the motion of
the fluid beyond the free end is described by progressively higher-order polynomials
of the fluid-jet deflection. The ‘first model’ is Shayo & Ellen’s ‘collector pipe model’
(Section 3.5.8). The ‘second model’ is described mathematically by
-1 -1
Gn(<) = Y,(1) [I - + YA(1) [(c - 1) - forl<C(/,
(I - 1)* (1 - 1)
(4.68)

The ‘third model’, which involves a cubic polynomial in 6, is given in detail in Luu (1983);
this model transcends physical reality by unjustifiably specifying a zero slope for the free
jet far downstream (Table 4.2). Calculations done for a very long cantilever (A = lo’*)
according to the various models of Table 4.2* show that the second model, equation (4.68),
with 1 = 2.8 gives optimum results, as may be seen in Table 4.3. The second and third modes
were also tested, and the second model with I = 2.8 again gives the best results. Hence, it

+In these calculations, the ?((I used in (4.67) are the eigenfunctions of clamped-free Timoshenko beam
without flow; the integration range for ?i is [-150, 1501 and the integration step fi = 2. These give convergent
results and have been used thruughout in calculations for cantilevered pipes.