Page 191 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 173

3.7 LONG PIPES AND WAVE PROPAGATION
If the pipe is very long between supports, or infinitely long, then the question of wave
propagation becomes especially important. The main interest in this is for application to
pipelines resting on the ground or on the ocean floor, or pipelines with many, periodically
spaced supports. These two topics are treated here, after some preliminary discussion on
wave propagation in simple systems.

3.7.1 Wave propagation

Some general characteristics of wave propagation will be reviewed here with the aid of
some work by Chen & Rosenberg (1971) on ‘pipe-strings’ conveying fluid.
Consider first a totally unsupported very long, straight pipe of negligible rigidity, under
tension - a very useful tutorial system. Since EZ = 0 in (3.1), the equation of motion
is rendered dimensionless by defining ii = (M/T)’/* U, r = [T/(rn + M)] ‘/2t/L, together
with q, < and B as in (3.69) and (3.71), yielding

The nondimensionalization gives c = 1 ; nevertheless, the equation is written like this to
facilitate the physical interpretation of the results. Thus, if ii = 0, equation (3.131) is the
wave equation and c is the dimensionless wave velocity.
Consider now a wave of the form q = A exp[iK(c - v,t)], where K is the wavenumber
and up the phase velocity; K = l/h, where h is the wavelength. Substituting into (3.131),
it is easy to see that the equation is of the hyperbolic type provided that U2(1 - B) < c2.
In that case, either progressive or standing waves can exist, and the general solution is of
the form (Morse 1948; Meirovitch 1967)

(3.133)

Considering the two component parts of (3.133), together with the form of (3.132), it
is easy to show that (i) if U < c, two waves propagate in the pipe-string, one in the
downstream and the other in the upstream direction, with phase velocities 111 and el?,
respectively, where q > uz; (ii) if ii > c, then both waves travel downstream; and (iii) if
-
u = e, there is one propagating and one standing wave. A disturbance, e.g. q(<,O) =
exp( -C2), leads to waves travelling upstream and downstream without alteration in form.
However. whereas for U = 0 the two waves propagate with the same phase velocity and
have the same form, for ii # 0 they do not: the wave with the larger phase velocity has
smaller amplitude - unlike the classical string (Chen & Rosenberg 1971).
The case of a pipe-string of finite length and fixed ends is examined next. In this case,
solutions of the form

+
v(<, r) = AI exp[i(KiC + or)] +A2 exp[i(~~< or)] (3.134)