Page 119 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 101

Hence, it is clear that for pinned ends I&.d = n, while for clamped ends = 2n, since
the Coriolis term is not involved in the divergence instability.
Gravity effects are considered next. If gravity is taken into account (i.e. if the system
is vertical), but still taking k = 0 (no elastic foundation) in equation (3.70), the critical
conditions are found to be as in Figure 3.19. Clearly, equations (3.90a,b) still apply, with
'u replacing u - with u as in the second of equations (3.100).

8

6

24

2

0
-20 -10 0 10 20 30 40 50
Y

Figure 3.19 The critical value of v,d for divergence of vertical pipes with supported ends
(f = l7 = k = 0), showing the effect of y. P-P: pinned-pinned (simply-supported) pipes; C-C:
clamped-clamped pipes; v is defined in the second of equations (3.100).

A value of y < 0 signifies that gravity is in the opposite direction to the flow
vector - i.e. upwards in Figure 3.1(b). Thus, for y -= 0 the pipe is under gravity-induced
compression, while for y > 0 it is under gravity-induced tension, which explains why
u,d for y < 0 is smaller than for y > 0; indeed, for y sufficiently large and negative,
the system diverges (buckles) under its own weight. [In the case of Figure 3.l(a), it is
implicitly presumed that the pipe is hung before the downstream end is positively fixed;
thus the pipe is subjected to the same gravity-induced tensiodcompression as in the case
of Figure 3.l(b).]
It is also noted that, as y increases, the ratio of Vcd for clamped and pinned pipes is
diminished: 2n/n = 2 for y = 0 and 7.80/5.56 = 1.4 for y = 50. Physically, one may
think of a larger y as representing a longer pipe [equations (3.71)]; in the limit, the pipe
will resemble a string rather than a beam, and hence will be less sensitive to boundary
conditions. This breaks down for y < 0, since in the case of pinned ends, as the critical y
is approached for divergence due to its own weight, v,d is diminished very fast, while this
is not yet true for clamped ends, for the range of y in Figure 3.19. The critical y values
for divergence at u = 0 are ycr = -18.55 for pinned ends and -66.34 for clamped ones.
For other aspects and/or details of the effects of pressurization, the interested reader is
referred to the work of Haringx (1952), Heinrich (1956), Hu & Tsoon (1957), Roth &
Christ (1962), Naguleswaran & Williams (1968), Stein & Tobriner (1970) and PaYdoussis