Page 63 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

46 SLENDER STRUCTURES AND AXIAL FLOW

of the cylinder. If KC < 4, separation generally does not occur (Sarpkaya & Isaacson
1981; Naudascher & Rockwell 1994). If KC > 8 approximately, the flow field is entirely
different, with the cylinder now oscillating in the remnants of vortices shed from previous
cycles of oscillation; this type of flow, arising also in wave-induced oscillatory flows, has
been studied extensively in conjunction with offshore mechanics applications (Sarpkaya
& Isaacson 1981).
(f) Numerical calculations of added mass

Some early attempts to calculate the added mass by numerical (CFD) methods are due
to Levy & Wilkinson (1975), PaTdoussis et af. (1977) and Yang & Moran (1979), for
instance. Nowadays, any CFD package capable of heat transfer calculations, hence of
solving the Laplace equation, would be suitable - based on finite element, finite differ-
ence or other methods. A few examples of finite-element (FEM) based packages are
FIDAP from Fluid Dynamics International, U.S.A., and CASTEM 2000 from Commis-
sariat B 1’Energie Atomique, France; and finite-volume (FVM) based packages FLOW3D
from Hanvell Laboratories, U.K., and PHEONICS from Cham Ltd, U.K.
Other numerical methods also exist, e.g. based on spectral methods (Mateescu,
Paidoussis & Sim 1994a,b), finite difference methods (Mateescu, Paidoussis & BClanger
1994a,b), or the boundary integral equation method (BIEM) (Groh 1992).

2.2.3 Loading on coaxial shells filled with quiescent viscous fluid

Consider the same system as in Figure 2.7(a), but with only the inner cylinder free to
oscillate, and then only as a beam (n = 1) or as a rigid body in the plane of the paper,
while the outer one is rigid and immobile. The annular space is filled with a quiescent
viscous fluid. Again, the task is to determine the fluid forces generated by harmonic
motion of the inner cylinder.
If the cylinders are sufficiently long, the flow is essentially two-dimensional in cross-
flow. Writing equation (2.63) in Cartesian coordinates and eliminating the pressure
between the two equations, or simply taking the curl of (2.63), one obtains a single
equation

(2.141)

in terms of the vorticity,

(2.142)

uz and uy are the flow velocity components in the z and y directions, which may be
expressed in terms of the stream function: u, = a+/ay, u,, = -a+/az. The continuity
equation (2.62), is satisfied automatically. Moreover, since o = -V2@, equation (2.141)
leads to (Schlichting 1960, chapter IV)

(2.143)