Page 82 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 65

The energy transfer mechanism was also demonstrated in terms of rudimentary repre-
sentations of the operation of a pump and a radial-flow turbine by Benjamin (1961a), as
follows.
Suppose first that in the course of some free motion the pipe rotates about A without
bending elsewhere, as shown in Figure 3.2(a), This motion requires transfer of energy
from the pipe to the fluid, since the Coriolis forces on the fluid have reactions on the
pipe in a direction always opposing motion. [For the motion to continue (with the pipe
remaining straight between A and C), work from an external source would have to be
done on the pipe, over and above that for bending it at A.] Thus, this energy transfer
mechanism causes the fluid to gain kinetic energy in passing through the pipe, and the
centripetal acceleration of the fluid results in a suction developing at the inlet, A; on
reflection, this is essentially the action of a centrificgal pump.

A \ B \ c

Figure 3.2 Rudimentary representation of (a) a pump and (b) a radial-flow turbine, illustrating
the mechanisms of energy transfer in a cantilevered pipe conveying fluid, as proposed by Benjamin
(1961a). From Paldoussis (1973a).

Consider next the pipe momentarily ‘frozen’ in the shape shown in Figure 3.2(b); the
change in direction of the momentum of the fluid stream about B gives rise to a reaction
on the pipe, resulting in a clockwise couple. In this case, energy is transferred from the
fluid to the pipe, causing it to accelerate to a speed at which the rate of energy gain just
balances the work done in bending the pipe at B. The energy-transfer mechanism in this
case corresponds to that of a radial-$ow turbine. (It is noted, however, that if the rotation
about A becomes sufficiently rapid, pumping action will again prevail.)
In general, in the course of free motions of the system both mechanisms will be
operative. If the first predominates, oscillatory motions will be damped; but if the second
prevails, they will be amplified continuously, i.e. an oscillatory instability will develop.