Page 88 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

(1+e)2+(E)
PIPES CONVEYING FLUID: LINEAR DYNAMICS I 71
=l.
2
(3.15)

In both (3.14) and (3.13, xo may be replaced by s.
If the pipe cannot be considered to be inextensible, e.g. in Figure 3.l(a), 6x0 and 6s
are no longer equal; they must be related through (3.13) which, with the aid of (3.12),
leads to
-I 12
ax,= [(l+-g)2+(e)2] (3.16)
as

The final preliminary point that needs be examined is related to the orders of magni-
tude of the displacements, which define the degree of approximation and simplification
that is admissible in the derivations to follow. First, it is reasonable to assume, partic-
ularly in linear analysis, that the lateral displacement w is small compared to the pipe
length, i.e.
w/L - f%E), (3.17a)

where E << 1. By expanding (3.15) and neglecting as compared to 2(au/axo),
and also replacing xo by s, it is clear that

I aw
u 2: - 1’ (%) ds, u/L - O(E*); (3.17b)
-
02
i.e. longitudinal displacements are one order smaller than the lateral ones. It is also
well known that, in the Newtonian approach, if all terms are correct to order E, so is
the equation of motion. In the Hamiltonian approach, however, since the energies are
generally quadratic expressions of displacements and velocities, the various terms should
be correct to order c2. Hence, in the Newtonian derivation of Section 3.3.2 one may
take x = xo = s and consider only the lateral deflection of the pipe, w = w(x, t). In the
Hamiltonian derivation of Section 3.3.3, however, one has to take account of u(x, t) as
well, and to take care to differentiate xo or s from x, since then generally x $ s for
inextensible pipes and also xo # s for extensible ones.

3.3.2 Newtonian derivation

Consider the system of Figure 3.1 (a-c), a uniform pipe of length L, internal perimeter S,
flow-area A, mass per unit length m, and flexural rigidity EZ, conveying fluid of mass per
unit length M, with mean axial flow velocity U. The flow in the pipe is fully developed
turbulent. Consider the undisturbed axis of the pipe to be vertical, along the x-axis, and
the effect of gravity to be generally non-negligible. The flow velocity may be subject to
small perturbations, imposed externally, so that dU/dt # 0 generally.
The pipe is considered to be slender, and its lateral motions, w(x, t), to be small and
of long wavelength compared to the diameter; thus, in accordance with the discussion
in Section 3.3.1, the curvilinear coordinate s along the centreline of the pipe and the
coordinate x may be used interchangeably. Consider then elements 6s of the fluid and the
pipe, as shown in Figure 3.6.