Page 238 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 219

Clearly the flow field is entirely different in ‘forward’ and ‘reverse’ flow through the
sprinkler. This is the key that finally led the author to the resolution of the conundrum,
for both the sprinkler and the pipe problem. Consider the stationary aspirating sprinkler,
and imagine a flared funnel, not connected to it, channelling the flow in, thus modelling
the sink flow. On reflection, the flow in the funnel is no different from that considered in
Section 4.2 for nonuniform pipes. Hence, neglecting gravity, the axial balance of forces
in the funnel is given by a form of equation (4.13),

(4.28)

where x and U; are directed as in Figure 4.12(b), and all quantities except pi are functions
of x. T is taken up by the imaginary funnel supports and may be ignored. Also, this
expression may be simplified by taking A, -Ai =A, and by writing Q = U and pi -
pe = p. and recalling that piAiUi = MU = const. Then integrating from x = 00, where
p + 0 and U + 0, to x = L, the inlet of the sprinkler, we obtain (pA>r. = -(MU2jL.
Hence, since MU2 is the same for all x < L, one can write
2
FA -MU, (4.29)
which clearly shows that at the sprinkler inlet, and hence throughout, there is a suction or
negative pressurization, 7 = -pU2 = -MU2/A. Its effect is profound, as may be seen in
Figure 4.14(d). The negative pressurization produces a lateral force FAIR = -MU2/R, R
being the radius of curvature, which totally cancels the centrifugal force M U2/R; hence,
the sprinkler remains inert!? Of course, these arguments do not hold once some rotation
of the sprinkler takes place, but may be considered to be correct to first order.
The same applies to the pipe problem. Unlike the case of discharging fluid where
the pressure at the free end (above the ambient) is zero, for the aspirating pipe there
is a suction at the free end, equal to -pUUj, and hence a negative pressurization
equal to that, throughout the pipe (cf. Section 3.3.4). Therefore, a term 7A(a2w/ax2) =
-MUUj(a2w/ax’) must be added to equation (4.26), which is incorrect as it stands. This
cancels out the centrifugal force required for flutter (Section 3.2.2)!
Still, seeing is believing. Accordingly, an experiment was performed at McGill in 1997,
in which two similar elastomer pipes were mounted as vertical cantilevers, immersed in
a transparent water tank; at the free end of each pipe there was a light plastic 90” elbow.
The clamped ends of the two pipes were interconnected via a pump. Once the pump was
started, the pipe discharging fluid deformed in reaction to the emerging jet, as expected.
The aspirating pipe, however, after a starting transient, returned to its original, no-flow
configuration and thereafter remained limply straight.* Therefore, it is now clear that
aspirating pipes cannot aspire to flutter!
Before closing this section, it ought to be mentioned that there is another engineering
application involving pipes aspirating fluid, namely the Ocean Thermal Energy Conversion
(OTEC) plants. Shilling & Lou (1980) initially intended to conduct ‘up-flow’ experiments

+An alternative demonstration of this result may be made by control volume considerations and the fact that
inlet and outlet vorticity is zero; however, some colleagues considered this less convincing.
‘The experiment was initially done with very flexible coiled Tygon tubing. In this case, there was steady-
state flow-induced deformation, with the aspirating pipe coiling itself tighter. It was discovered, however, that
this was due to the fact that, under suction, the pipe cross-section became oval, and the coiled pipe behaved
like a Bourdon pressure gauge! This shows that there is no such thing as a simple experiment.