Page 247 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

228 SLENDER STRUCTURES AND AXIAL FLOW

4.4.4 The fluid-dynamic force by the integral Fourier-transform
method

It is noted that K in equation (4.58) is not known a priori. Hence, there no longer exists a
‘point relationship’ between fi and x as in most of the analyses of Chapter 3: fi at any
given x depends on the deformation all along the pipe. A powerful method for the solution
of problems such as this was proposed by Dowell & Widnall (1966) - see also Widnall
& Dowell (1967) and Dowell (1975) - the essence of which will become evident with
its application in what follows.
We start by adapting what has just been obtained in Section 4.4.3(b) to a suitable form.
We first redefine

and define the Fourier transforms of +(r, x) and W(x) by

00
+*(r, a) = S_, ~(r, x)eiax dx, -* w (a) = (4.60)

and similarly for Ti* [see, e.g. Meirovitch (1967)l; the asterisk denotes the Fourier trans-
form and a is the transform variable. The inverse transforms are
’
+(r, XI = - /0° +* (r, a)epicux da, - @*(x)e-iax da, (4.61)
w(x) =
2n -‘-&
and similarly for p(r, 8, x). Furthermore, we define

(4.62)

where k is the so-called reduced frequency, F(Z) is clearly the first part of (4.56) in the
Fourier domain and E = L/2a, as already defined.
Proceeding with the analysis exactly as in Section 4.4.3(b) but in the Fourier domain,
one finds for the perturbation pressure

-* pu2a
p (a, e, x) = -(E - /c)~F(cY)z* sin 8, (4.63)
L3
which inverted gives

(4.64)

in terms of ( = x/L. The inviscid fluid-dynamic force FA is then found to be

fi =MU2(&)eini~~(a!-k)2F(a!){/~ epizt da!. (4.65)

The physical domain of the problem is [0, L]; in terms of (, it is [0, 13. However, this
domain will be expanded, by taking in some additional space beyond the downstream