Page 90 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 13

with the surrounding fluid, expressed in linear form as c(&/at).' The subscript f in
equations (3.18) and (3.19) identifies the acceleration of thefluid and subscript p in (3.21)
that of the pipe. Terms of second order of magnitude, for example the pipe acceleration in
the x-direction, have been neglected, as well as transverse shear deformation and rotatory
inertia in accordance with the Euler-Bernoulli beam approximation.
The acceleration of the fluid may be determined in several ways. The simplest is
utilized here, while other derivations will be employed when considering variants of the
basic system. The basic assumption is that the fluid flow may be approximated as a plug
flow, i.e. as if it were an infinitely flexible rod travelling through the pipe, all points of
the fluid having a velocity U relative to the pipe; this is a reasonable approximation for
a fully developed turbulent flow profile. As it has been assumed that pipe deflections are
of long wavelength compared to the diameter, D, and that the pipe is slender, i.e. LID
is large, unsteady secondary-flow effects may be neglected. Hence, the equivalent of a
slender-body approximation to the flow is being made. The velocity of the pipe is
ar
V - -=xi+ik (3.23)
'' - at
in terms of the unit vectors in the x- and z-directions, defined in Figure 3.5(a), where r
is the position vector to a point measured from the origin; and the velocity of the centre
of the fluid element of Figure 3.6(a) is
v, = v, + ut, (3.24)

where t is the unit vector tangential to the pipe,

ax az
t=-ii-k. (3.25)
as as

(3.26)

where D( )/Dt is the material derivative for the fluid element. Recalling that z = w
and that ax/& 21 1 and ax/& - 6(c2) 2: 0 in accordance with the assumptions made, this
gives
(3.27)

In a similar manner, the acceleration is found to be
2
dU
af = D2r = -i+ [: + UL] wk,
Dt dt (3.28)
in which the bracketed quantity squared represents the successive, double application of
the differential operator, and hence
a% a2 + a2w du aw
-.
w
dt as
asat
[$ + u 4 = + 2u - u2- + - (3.29)
as2
'The surrounding fluid is supposed to be sufficiently light (e.g. air) for added-mass effects to be negligible.