Page 493 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

Appendix A

First-principles Derivation of the

Equation of Motion of a Pipe

Conveying Fluid

Consider the system of Figure A.l(a), free to oscillate in the horizontal (X, ZJ-plane, so
that gravity is inoperative. Externally imposed tension and pressurization effects are not
present and, for simplicity, dissipative effects are neglected. Elements of the fluid and the
pipe of length 6x are shown in Figure A. 1 (c,d), with the forces and moments at the ends
apportioned slightly differently from Figure 3.6.
The acceleration of the fluid element (still making the plug-flow approximation) is
derived by the standard dynamics approach, following Ginsberg (1973). An inertial refer-
ence frame (X, Y, 2) with Y into the plane of the paper and unit vectors I, J, K,
and an (x, y, :) frame embedded in the pipe element with unit axes i, j, k are utilized
[Figure A. 1 (b)], together with the expression

af = a0 + I, x r + 20 x v,,1 + o x (o x r) + arel. (A. 1)
which may be found in any book on dynamics [e.g. Meriam (1980)l; o is the angular
velocity of the pipe (and of the (x, y, 2) frame) with respect to the inertial frame, and the
subscript ‘rel’ denotes quantities relative to the (x, y, z} frame. The various components
of (A. 1 ) may be expressed and then approximated according to the assumptions made in
Section 3.3.1 as follows:
a’u azw a’ U’ aw
a0 = -I+ -K 2: -K, v,~ = Ui = U cos @I+ U sin @Kz UI+ U -K,
at’ at2 at’ as
(A.3)
a@
dU
dU
U2
_- Jz __ arel = -i + - 2 -I+ dU aw K , a2w K,
k
as :t ($) J, dt vn dt - - + U- ~ 8s’
dt as
assuming a positive (counterclockwise) rotation, a@/& it is also noted that r, the distance
from the origin of {x, y, z) to other points within the element is of second order smallness,
so that the second and fourth terms of (A. 1) are negligible. Hence, (A. 1) may be written as

correct to O(E) - which is the same as equation (3.28).

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