Page 245 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

226 SLENDER STRUCTURES AND AXIAL FLOW

reversed acceleration as given by equation (3.29), here with dUldt = 0; hence,

(4.46)
ax at

Expressed nondimensionally and in the form required by the modal analysis method
(Section 4.4.2), the generalized fluid-dynamic force, q, may be written as follows:

(4.47)

where

(b) The inviscid fluid-dynamic force for 3-0 potential flow

The fluid is assumed to be inviscid and the flow irrotational, consisting of the mean flow
Ui along the pipe and a small perturbation v(r, 8, x, t) associated with small motions of
the pipe, which may be expressed in terms of a perturbation potential via v = V@. This
potential must satisfy equation (2.73a), V2@ = 0, which for this system is

a2@ 1 a@ 1 a2@ a2@
-+--+-- +--0, (4.50)
ar2 r ar r2 ao2 ax2
as well as the compatibility and boundary conditions
$I,.=, a@ = ( at aw sin@, OixzL, Oi0<2n, (4.5 1 )
aw
-+U-
ax)
= 0, x < 0,

where motions are assumed to take place in the 0 = in plane and a is the internal pipe
radius, and
lim @ = 0, lim (a@/ax> = 0. (4.52)
X'iZ.60 X'lk.60
The force on the pipe is determined by integrating the pressure p p(a, 0, x, t) on
the inner pipe boundary, which may be determined by substituting v = V@ and v = 0 in
equation (2.67a), leading to

(4.53)

+Although these are equal to lg), = 1,2,3, respectively, defined in conjunction with equation (4.44), they
j
are denoted differently to indicate that they are related to the right-hand side of (4.45) or (4.47).