Page 246 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 227

p being the fluid density. Assume now separable solutions of the form

$(r, 8, x, t) = R(r) sin 8 exp[i(Kx + Qt)], (4.54)

where the form of the 8 component has been suggested by (4.51), and the form of the
x component emerges in the course of separating the variables. Substituting into (4.50)
leads to
d2R 1 dR -++' R=O,
-+---
dr2 r dr (:2 )
admitting solutions of the form

where 11 and K1 are modified Bessel functions of the first and second kind of order 1,
and where D1 = 0 because 4 must remain finite within the pipe. C, is determined by
application of (4.51), and one finds

(4.56)

in which I', = dIl/d(Kr). Then, utilizing the relation Ik(x) = (l/x)[nI,(x) +xI,+~(x)]
(Dwight 1961) for n = 1, one obtains from (4.53)

(4.57)

From this, the force FA is found to be
2rr -M
fi = 1 pa sin Ode = (4.58)
1 + Kdz(KU)/Il (KU)
where M = pna2 has been used. Comparing (4.58) to (4.46) it is clear that M is now
replaced by M/[ 1 + ~aIz(~a)/I1 (KU)], where the denominator is generally larger than
unity. Hence, for finite wavenumbers KU (and wavelengths of motion) the effective
fluid-dynamic force is generally smaller than that given by the plug-flow approxima-
tion.
It is instructive to consider the case of KU small, i.e. motions of large wavelength.
Utilizing the series expansion I,(x) = (l/n!)($x)"[l + 6(x2)] (Dwight 1961), one
obtains
A4
lim = M,
K-a-tO 1 + KU[i(iKU)2/(!jKU)]

thus retrieving the form of FA given by equation (4.46) and proving that it only holds true
provided that the wavelength of motions is large compared to the pipe diameter.
However, for the analysis of short pipes the full form of (4.58) is retained. The pertinent
forms of 4 and Qk, - cf. equation (4.47) - are presented in the next section.