Page 62 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS, DEFINITIONS AND METHODS 45

where C,,, is given by equation (2.127), e = (smallest gap between the cylinders)/R;,
r = R,/R;. For larger values of r, the results are given in figure form. Results for a variety
of other systems may be found in the compilations of Chen (1987) and Blevins (1979).

(ii) Aspect ratio effects. As two-dimensionality of the flow is violated, the validity
of the foregoing deteriorates. A particularly simple example illustrating this is a simply-
supported cylindrical beam oscillating in a narrow annulus [so that the approximations
leading to (2.134) are valid], the ends of which are open to large cavities (Gibert 1988).
It is found that
(2.139)

where C,,, is given by expression (2.134) for n = 1. Clearly, the shorter the beam, the
smaller is C:am, as compared to an infinitely long one. The physical reason is that, near the
ends, the fluid takes the easy way around the beam, partly in the third (axial) direction;
hence, less than the total force that would be obtained by 2-D analysis is realized. A
more general analysis (PaYdoussis et al. 1984; Chen 1987), not making the assumption of
a narrow annulus, gives

(2.140)

where I1 and K1 are, respectively, the first-order modified Bessel functions of the first
and second kind; the primes denote derivatives with respect to the argument. The effect
of R;/L is strong for 1 < R,/R; < 2, but relatively weak for wider annuli.
(iii) Effects of compressibility and two-phase flow. If the flow is compressible,
the wave equation, V2+ + k2+ = 0, k2 = o/c, needs to be solved instead of the Laplace
equation, V2@ = 0. Hence, the results are found to depend also on an oscillatory Mach
number, hfk = wR;/c, where c is the speed of sound. The effect of compressibility for
Mk 5 0.2 is rather weak (Chen 1987).
It has been found (Carlucci 1980; Carlucci & Brown 1983) that in gas-liquid two-phase
flows the measured added mass is generally considerably lower than that predicted by
homogeneous mixture theory [in which average quantities are assumed for the mixture;
e.g. if the void fraction is cr and the densities of the liquid and gaseous phases are p/ and
pg, the mixture density is p = (1 - cr)p, + upg]. Since the two-phase flow may be consid-
ered as a flow with the density of the liquid phase and the compressibility of the gaseous
one, it was supposed that the discrepancy may have been due to the neglected effects of
compressibility (PaYdoussis & Ostoja-Starzewski 198 I). Also, the effect of random varia-
tions in the surrounding fluid density, inherent in two-phase flows was investigated (Klein
1981). These effects, although qualitatively working in the right direction (Chapter 8),
proved incapable of accounting fully for the discrepancy quantitatively, and the search
for more elaborate models continues.
(iv) Amplitude effects. All of the foregoing apply to cases where the amplitude of
oscillation is small enough for separation in the cross-flow not to occur. This brings into
play another dimensionless number, the Keulegan-Carpenter number, KC = 2nV,/(oD),
where V, is the amplitude in velocity fluctuations. For a harmonically oscillating cylinder
in quiescent flow, this reduces to KC = 2n(A/D), where A is the amplitude of motion