Page 193 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 175

quantities a unit length could be used for L, or an appropriate length scale associated with
the initial disturbance under consideration. Considering solutions of the form

q(<, 5) = AeiK~eiU'e-*', (3.139)

it is found that equation (3.138) is satisfied if it is found that equation (3.138) is satisfied
if
+
k
w2 - h2 + ~wBI/~uK [K~ - (u' - f + fl)~~ - ho] = 0,
-
(3.140)
2h(w + B'12UK) - WCJ = 0,
which leads to
1
w1 .- = -B1''UK p, h1.2 = 20 f q, (3.141 a)
7

I
X = K4 - [U'(l - p) - f + n]K2 - k - 7CJ (3.14 1 b)
(Roth 1964). The similarity in the structure of w1.2 in (3.141a) when CJ = 0 to w1.2 = 211.2~
from (3.133) should be noted. Remarking that the form of equation (3.139) with K replaced
by -K and w by -w is also a solution, as easily seen from (3.140). one obtains for a
general waveform the general solution
co
q(<, t) = ~e-illlTIA, cos(K,,C + writ) +B,, sin(K,,t + wnr)I
11 =o

n=o
The arbitrary constants A,, to D, are determined from the initial conditions. Thus, if
q(<, 0) = a(<). li(t, 0) = b(<) are periodic functions with K,, = nr, the constants may be
determined by the use of Fourier series, while a solution for a nonperiodic and spatially
more general disturbance may be obtained with the aid of Fourier integrals (Roth 1964).
In solution (3.142) it is noted that the frequencies wln and wzn are each associated with
the phase velocities u1 = --w1,?/~,~ and 212 = -w ?,*/K,, , for downstream- and upstream-
travelling waves. For an observer travelling downstream with velocity p1/2~. these waves
propagate with wave speeds *P~/K,~. where pn is as in (3.141a).
For stability of the pipe, hl and h2 in solution (3.139) must be positive. This requires that

&2 - (U2 - r + mK2 + k] > 0, (3.143)

which is true for all damping values, CJ. The minimum of the function in square brackets
occurs at K = [i(u2 - r + n)]'I2 and is equal to A(K) = k - i(u2 - r + n)'. Hence.
condition (3.143) is satisfied if A(K) > 0, or
2 = 2 - r + n < 2fi. (3.144a)

This result could be obtained also by the work leading to equation (3.101a). It is of interest
that if k = 0, then 'u = 0, Le. the pipe is unstable for all u in the case of f = = 0. This