Page 43 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

26 SLENDER STRUCTURES AND AXIAL FLOW

In other cases, e.g. when fluid motion is entirely caused by small-amplitude oscillatory
motion of a structure, all components of V may be small, and (2.69) is further simplified to

1
aV
-
- - -- vp+vv2v. (2.70)
at P
Because there is no mean flow velocity in this case, the Reynolds number as such does
not exist. Hence, to decide whether viscous effects are important or not, the ‘oscillatory
Reynolds number’ is used instead. For a circular cylinder of diameter D, this may be
defined as B = lAJD/v, where lAl is the amplitude of the oscillatory velocity of the body.
Further, denoting the amplitude of motion by ED, E << 1, and the oscillation frequency by
Q, one obtains [AI = QeD and hence /3 = QeD2/v, from which it is obvious that this is
a modified Stokes number. Clearly, if B is sufficiently large, then viscous effects become
unimportant, and the approximation

(2.71)

may be used (see, Section 2.2.2 and Chapter 11). This may be combined with the conti-
nuity equation to give
v2p = 0, (2.72)
the Laplace equation. In terms of the velocity potential, 4, the continuity equation and
equation (2.71) may be written as
v%#J 0 (2.73a)
=
and
(2.73b)

(e) Slender-body theory

A particular class of linearized flows pertains to slender bodies, i.e. bodies of small cross-
sectional dimensions as compared to their length [e.g. for a body of revolution of radius
R(x), if R(x) << L] and no abrupt changes of cross-section (dR/dr << l), with the flow
being irrotational and along the long axis of the body or at a small angle to that axis
[Figure 2.5(a)]. Let the body be defined by

F(r, 8, x) = r - R(x) = 0. (2.74)
The flow field may be expressed as

v = v, + v4, (2.75)
where 4 is associated with the perturbations to the flow associated with the presence of
the body and satisfies
024 = 0 (2.76)

and the boundary conditions
(V, + V4). VF = 0 on F(r, 8, x) = 0 (2.77a)