Page 248 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 229

end of the pipe, to [0, 11, where I > 1. This is necessary, particularly in the case of
cantilevered pipes, as flow perturbations persist beyond the free end of the pipe, as
discussed in Sections 3.5.8 and 4.4.5.' Accordingly, instead of the first of equations (4.43),
the following form of the Galerkin expansion is adopted:

00
W(0 = An Yn (t), 0 5 6 5 1
n=l
IX
AnGn(t), 1 < 5 1 (4.66)
n=l
= 0, 6 < 0 and < > 1;
Y,(c) are the comparison functions associated with q(4) in equations (4.43), and G,(C)
are the so-called 'outflow-model' functions which are associated with deflections of the
fluid jet beyond the free end of a cantilevered pipe.
In the modal-analysis solution of the problem, the main interest is in the generalized
fluid-dynamic force q, rather than fi, as defined by equations (4.47) and (4.48) or (4.45).
In this case, QLL), Qkn (2) and Qt: of (4.48) are given by

with F(E) as given in (4.62)

4.4.5 Refined and plug-flow fluid-dynamic forces and specification
of the outflow model

The lateral inviscid fluid-dynamic force derived by means of refined fluid mechanics
and the integral-transform technique is intended to be used for short pipes. Nevertheless,
in the limit of sufficiently long ones, it should give identical results to those obtained
with the simpler, plug-flow model - for the reasons discussed already. In this section, a
comparison is made of the generalized fluid-dynamic force components, Qt:, i = 1,2, 3,
obtained by the refined fluid-mechanics model [equations (4.67)] and by the plug-flow
model [equations (4.49)] - for clamped-clamped and cantilevered long pipes.
(a) Clamped- clamped pipes

For a pipe clamped at both ends, there is no need for an outflow model, insofar as
the generalized fluid-dynamic force obtained by the integral (Fourier) transform method

+It is clear from (4.65), nevertheless, that the very nature of the Fourier-transform solution requires the
specification of %(e) beyond [O, 11, even if this means stating that W(6) = 0 for -cm 5 6 < 0 and 1 < 5 +w.