Page 125 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 107

area is generally a function of internal pressure; (iii) the pipe has an initial curvature as a
result of being coiled during manufacture while still warm and of plastic set during storage.
Of these, item (i) plays no role in the determination of ucd, (ii) is not too important if the
fluid discharges at x = L so that the pressure is not too high at any point upstream, and
(iii) was solved, according to the authors, by hanging the pipes vertically and pouring hot
water through them.
In the clamped-pinned arrangement, the downstream support was provided quite simply
by a greased steel rod in contact with the downstream end of the pipe. As the flow velocity
was increased, the pipe began to bow slightly. At a certain critical speed the pipe was
observed to statically diverge rapidly and to slide completely off the steel rod. This
means that the measured ucd was slightly higher than the real one. The experimental
ucd = 4.70 nevertheless compares favourably with the theoretical ucd = 4.49 given by
equation (3.90~).
A more recent, successful experiment for a clamped-pinned pipe, again with a sliding
downstream end, was conducted by Yoshizawa et al. (1985, 1986) and is discussed in
Section 5.5.3.
The main purpose of these studies was to validate items (i)-(iii) of the first paragraph
of this section and it was partly achieved. It was also shown, by the way, that large flow
velocities are necessary to induce divergence; hence, it is unlikely to arise in practice,
except in specialized applications. Nevertheless, there is a high degree of idealization in
the systems studied so far; certainly, systems of the type of Figure 3.21 are unlikely to be
found in engineering applications. In more practical systems, the pipe would not discharge
to atmosphere but would be connected to another component at a pressure higher than
atmospheric [Figure 3.16(a)] - except after an accidental break (Section 4.7); moreover,
axial sliding, if any, would not occur freely and destabilizing pressurization effects would
come into play. In the next set of such studies, the dynamics under these more realistic
conditions was considered.
A careful study of the effects of pressurization and tensioning was made by
Naguleswaran & Williams (1968). Unfortunately, in the paper they do not give any of
the dimensions and properties of their apparatus, nor any of their results in dimensional
form. Nevertheless, Naguleswaran ( 1996) was kind enough to provide the approximate
principal dimensions of the neoprene pipes used: Do = 15 mm, h = 2mm, and variable
length, up to 880mm. The pipe was attached on either side to rigid copper pipes, one of
which was connected to the water mains and the other, after a certain length, discharged
to atmosphere. The mean pressure in the whole system could be regulated, presumably
by valves on the downstream end, so that pressurization was possible. Furthermore, axial
tension could be applied by loading one of the copper pipe connections statically and
then fixing it; thereafter, sliding was prevented (8 = 1). The flow rate was determined by
collecting and weighing the discharged water over a known time interval. Motions of the
pipe were sensed at two locations along the span via capacitance transducers. The Poisson
ratio, v, of the pipe was determined in special tests by measuring the change in volume
resulting from axial extension, and EZ was determined from the natural frequency of a
short cantilevered length of the pipe.
It was found that pressurization affected appreciably the first-mode natural frequency, to
the extent that the pipe could be made to buckle quite readily without flow. For this reason,
preliminary tests were made without flow. The variation of %e(Q1) with l7/r = FA/T
is shown in Figure 3.24; %e(Ql)~ is the value for the pipe under T but for p = 0. Since