Page 195 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 177
velocity, up = W/K, yielding

(3.148)

where u, is obtained from (3.144b) when it is transformed into an equality, while setting
1
r = 0: uz = [2k1/2 - n]/( - B); K, = k'l4 is a critical wavenumber, which corresponds
to the value of K for which all positive roots of (3.148, whether K > K, or < K,, have
phase velocities greater than that for K,. However, it is possible to obtain some positively
travelling waves with up < ~JK,,) from the negative roots of (3.148), namely for u/u, >
(1 - p)'/'. The dependence of up on u may be assessed from (3.148). For any K, for the
positively travelling wave, wp increases with u; up to u > u,,(K,), whereafter increasing u
causes up to increase for some wavenumbers and to decrease for others. For the negatively
travelling waves, wp diminishes continuously with u, for all u < u,.
One may retrieve from equation (3.148) Roth's result that for an observer travel-
ling with a velocity B1/2~,t upstream- and downstream-travelling waves would appear
to have equal velocities; this would imply that the distribution of waves would always
be symmetric about this translating axis for a symmetric disturbance about the origin.
However, Stein & Tobriner show that this is true only asymptotically (in time), because
the solution does not satisfy the boundary conditions in the limit as 6 -+ 00.
Some typical numerical results are shown in Figures 3.72-3.74 for a steel pipe
conveying water, with zero dissipation (c = 0); the larger foundation modulus, k =
6.54, is typical of crushed gravel. The initial disturbance is taken to be q(6,O) =
=
~0 e~p[-~(x/L)~] 70 exp(-;c2), with L = 12.5ft (3.81 m); this same L is used in
1
obtaining u, I7 and k from the corresponding dimensional quantities.
Figure 3.72 shows the time evolution of the disturbance at 6 = 0 fork = 0 and k = 6.3,
when u = 0.160 (U = 30.48ds or lOOft/s) and I7 = 0. In (a) it is seen that the system
is unstable, as discussed, and the oscillations are amplified with time. In (b), condition
(3.14413) is satisfied and hence the oscillation is stable (u < u,); the amplitude of the
oscillation at 6 = 0 is diminished with time as the disturbance energy is shared with
progressively larger parts of the pipe, 161 > 0, as shown in Figure 3.73. Stein & Tobriner
(1970) also show a case with li' = 0.0256 and k = 2.43 x lop4, where u = u, and a
neutrally stable oscillation at 6 = 0 is obtained.
In Figure 3.73(a) is shown the development of the initial disturbance when 14 = 0. It is
seen that up- and downstream propagating waves are symmetric about the origin. It is also
seen that the amplitudes for the lower k values are more severely attenuated than for the
largest k. When u > 0, as in Figure 3.73(b), the symmetry about the origin is destroyed,
and the waveform becomes symmetric with respect to an axis travelling at #?3/2u. In the
figure this is visible only for large k; for the smaller k, this symmetry which occurs for
large enough t has not yet developed for the range of t shown in the figure.
In Figure 3.74 we look at a particular point along the pipe, 6 = 8, versus time. It is seen
that for a stiff enough foundation (k = 6.54), the wave retains its cohesion and propagates
downstream as a 'wave packet', roughly at B'I2u; the upstream-propagating component

.'It is noted that v,/u = p-'/2(Vp/U), where the capital letters are for the dimensional quantities, because
of the different nondimensionalizing factors for vp and u; thus, in dimensional terms, the observer travels with
velocity pU.