Page 65 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

48 SLENDER STRUCTURES AND AXIAL FLOW

(2.152)

and I, and K, are modified Bessel functions of the first and second kind, respectively.
It is noted that, by virtue of the presence of 4 in the argument (in A), H is complex.
and
To evaluate H, therefore, one can either (i) evaluate J,, (??A) Y,, (Zd), the ordinary
Bessel functions, utilizing the expressions and tables in Jahnke & Emde (1945), for
instance, for the real and imaginary parts of each of them, and then convert to I,, and K,, ,
or (ii) utilize the ber, bei, ker and kei functions,

i”1, (xd) = ber, x + i bei, x, i-”K,, (xd) = kern x + i kei, x

and the expressions given by Dwight ( 1961).i
Expressing the force F of equation (2.150) in terms of added mass and added damping
as in equations (2.1 10) and (2.1 1 l), one can write

d2t dz
F = +C,pA- - CdJ2pA-i (2.153)
dt2 dt

hence
C, = %e(H) and Cd = -Sini(H). (2.154)

The results for %e(H) and .$nr(H) for various Stokes numbers S = DZ??/v are given in
Figure 2.9. Several observations may be made, as follows:

(i) both C,,, and Cd increase dramatically as R,/Ri is reduced towards unity, but
rises more rapidly;
(ii) for sufficiently high S, the values of C,, approach those obtained by inviscid
theory (S = oo), but increasingly diverge from inviscid theory as S is diminished;
(iii) for sufficiently narrow annuli, the results for C,,, sensibly collapse onto a single
curve - in the scale of the figure.

Chen et al. (1976), Yeh & Chen (1978) and Chen (1981, 1987) give a number of useful
approximations for H. These have been rechecked, corrected in some cases, and rewritten
into a congruous set, in terms of the parameters

as follows:

+It may be of interest that difficulties are encountered in trying to obtain solutions by standard software pack-
ages, including some symbolic mnnipulation systems. Thus, neither 1z4qde nor Marlab could do it; Marbemarica
could. but it was painfully slow.