Page 261 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

242 SLENDER STRUCTURES AND AXIAL FLOW

of u,d versus E,+ while keeping A constant and infinite - what might be termed a
Euler-Bernoulli refined-flow theory, which of course is physically impossible since the
refined-flow effects come into play for small E, when Euler-Bernoulli theory is not
applicable. These results for incompressible flow (Mach number = 0) show that, as E
is decreased, u,d is raised, because of the reduced effective virtual mass - contrary to
the results in Figure 4.19, where both A and E are varied together.
The above discussion is essential in understanding the results presented by Johnson
etal., the most notable of which are the following: decreasing the slenderness E while
keeping A constant (i) raises u,d for subsonic flows, and (ii) lowers it for supersonic
flows: also, reducing the sonic speed always diminishes u,d.
Whereas the results obtained for low Mach numbers are probably sound, this is ques-
tionable in the case of near-sonic and supersonic, indeed hypersonic, flows because, as
admitted by the authors, there are fundamental weaknesses in the model used, which
supposes the fluid flow to be wholly isentropic. In the case of compressible flow, there
are ‘secondary effects’ of fluid friction which generally cannot be ignored, e.g. causing
choking: also, for the oscillating pipe, shock waves are generally inevitable. These real
effects, which are difficult to model in a simple way, are not accounted for and their
influence on the dynamics is unknown.

4.5 PIPES WITH HARMONICALLY PERTURBED FLOW

In all of the foregoing, except in the derivation of the equations of motion in Section 3.3,
the mean flow has been taken to be steady (zi = 0). Here, the case of a harmonically
perturbed flow is considered; i.e. it is supposed that a time-dependent harmonic component
is superposed on the steady flow, such that

u = uo(1 + /1. cos wt), (4.69)

where p is generally small. This form of u may induce another type of instability,
namely oscillations due to parametric resonances. These are akin to the parametric reso-
nances experienced by, say, a pinned-pinned column subjected to a compressive end-load,
F = Fo(1 + p cos wt). Especially in the case of W/WI = 2, where w1 is the first-mode
natural frequency of the column, it is easy to appreciate physically that F pushes down
most when the column end moves in the same direction ‘naturally’, at the two extremes
of the oscillation cycle; hence, work is done on the column, resulting in amplification of
the motion, i.e. in a parametric resonance (Bolotin 1964: Evan-Iwanowski 1976; Schmidt
& Tondl 1986). Clearly, w/w1 = 1 also leads to parametric resonance, although, as will
be seen, other frequency ratios can also give rise to resonances. What renders the pipe
conveying fluid particularly interesting and worth studying are the differences in dynam-
ical behaviour vis-&vis the column problem, because of the presence of gyroscopic terms
and the fact that, in the case of a cantilevered pipe, the system is nonconservative.
The first to consider the problem, in terms of pressure pulsations arising from a pump,
for example, was Roth (1964) and, in terms of a pulsating flow velocity, Chen (1971b)
and Chen & Rosenberg (1971). However, as discussed in Section 3.3.2 in conjunction
with equation (3.38), one of the terms in Chen’s equations of motion is in error.

+In fact the results are presented in terms of the ‘aspect ratio’ a/L, the inverse of the slenderness E.