Page 170 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

152 SLENDER STRUCTURES AND AXIAL FLOW

a ‘point relationship’ between the fluid-dynamic force acting on the pipe at a particular
point and the deflection at that point, resulting in equations (3.32) and (3.28) for instance.
This makes the analysis much easier than if an ‘integral relationship’ were necessary,
requiring, for instance, knowledge of the relationship between the unsteady pressure and
streamwise position all along and beyond the end of the pipe for the force at any point x
to be specified.
However, there is no guarantee in all of this that for sufficiently short pipes the jet
behaviour beyond x = L will not influence the dynamics of the pipe, or that this will
be so in the case of shell motions. Furthermore, in some analyses, notably for short
pipes and shell motions [Section 4.4 and Chapter 71, (i) three-dimensional potential flow
theory is used for the formulation of the fluid-dynamic forces, which means that they are
determined via integration of the unsteady pressure around the pipe circumference, and
(ii) the generalized-fluid-dynamic-force Fourier-transform technique is employed which
does require knowledge of the jet behaviour sufficiently far downstream of the free
end - sufficiently far for the perturbation pressure to vanish; this, in effect, amounts to
the specification of a downstream boundary condition for the fluid. As a result, a number
of so-called outflow models have been proposed, starting with Shayo & Ellen (1978).
In most of these models (Shayo & Ellen 1978; Paidoussis et al. 1986, 1991b; Nguyen
et al. 1993) the manner in which jet oscillations decay to zero is prescribed, based on
more or less reasonable assumptions. A more physical approach, in which the dynamics
of the free jet issuing from a vertical pipe with a terminal nozzle are coupled into the
overall analysis, is adopted by Ilgamov et al. (1994).
Here some results obtained by Shayo & Ellen (1978) are presented, while other outflow
models are discussed in Section 4.4 and Chapter 7. Shayo & Ellen proposed two such
models. In the first, the so-called ‘collector pipe model’, it is supposed that there exists
a collector pipe which is actuated by a sensor, so that its deflection matches that of the
pipe outlet without touching it; the collector swallows up the fluid and discharges it at its
other end, which is anchored on the undeformed x-axis, some distance downstream. In the
analysis, the following extension to the cantilevered beam eigenfunctions @; (0, utilized
as comparison functions, is introduced to describe the behaviour of the fluid for 6 > 1:

(3.1 13)

where 2 is chosen sufficiently large in the numerical calculations such that changes in its
value have no effect on the fluid forces calculated. Thus, in this model it is presumed
that the deflection dies out linearly to zero in a dimensional distance (1 - 1)L, L being
the length of the pipe. In the second, so-called ‘free-flow model’, it is supposed that the
sinuous deflections persist in the fluid beyond = 1, such that

(3.114)
Shayo & Ellen were concerned mostly with shell oscillations, but they also conducted
calculations for beam-mode instabilities, albeit via the more complex three-dimensional
potential flow theory (see Section 4.4.3 and Chapter 7) and shell theory for the pipe,
instead of the simpler plug-flow Euler-Bernoulli beam theory. However, as shown by
Pafdoussis (1975) and discussed in Chapter 7, the results of the two theories converge
for thin-walled slender pipes. Here, some of Shayo & Ellen’s results are presented in
Table 3.6, in terms of u = U/(E/p,(l - v2)}’/* for the given h/a and p = pa/p,h, where