Page 45 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

28 SLENDER STRUCTURES AND AXIAL FLOW

The essence of slender-body theory is to take advantage of the linearity of the problem
and to express it as the superposition of the following two problems: (i) the axisymmetric
flow past the body of revolution with flow velocity V, cos a, and (ii) the cross-flow
around the body with flow velocity V, sin a (Ward 1955; Karamcheti 1966). Thus,
defining $ = $1 + 42 and u, = u,l + u,2, equation (2.80) may be re-written as
dR dR
- (U + u,.)- 2 u -, (2.8 1 a)
dx dx
dR
= uX2 - - W sin 8, (2.81b)
dx

where
U = V, cos a, W = V, sin a. (2.82)

The solution to (2.81a) is usually obtained by representing the body through a distribution
of singularities (e.g. sources and sinks) along the centreline, while the solution to (2.81b)
may be obtained via standard potential-flow analysis (Streeter 1948; Milne-Thomson 1949;
Karamcheti 1966).
Consider next a very slender cylindrical body for which dR/dx 2: 0, or exactly 0,
except near the extremities [Figure 2.5(b)]. The body is subjected to an oscillatory lateral
displacement w(x, r) in the 8 = in plane. Then, according to slender-body theory, the
flow can be regarded as compounded of (a) the steady flow around the stretched-straight
body, which we shall ignore here [and hence (2.81a) also] since dR/dx is nearly or
exactly zero over most of the length of the body, and (b) the flow due to displacements
w(x, t) (Lighthill 1960). Hence, only the velocity component related to (2.81b) remains,
namely (a$2/ar)lr=R 2: -W. The lateral velocity of the fluid relative to the moving body
is made up of (i) the component of U normal to the inclined body, equal to -U sin a,
where a = tan-'(aw/ax), and (ii) the lateral velocity of the body, awlat, reversed, if at
that instant the body is moving upwards as in the inset of Figure 2.5(b). Therefore, for
sufficiently small a, one may write

aw aw
V(x,r) = - + U--, (2.83)
at ax
on the implicit assumption that, locally, the body shape differs little from that of a long
(infinite) cylinder C, of the same cross-section all the way along. Thus, according to the
slender-body approximation, this lateral flow near any point of the cylinder is identical
with the two-dimensional potential flow that would result from the motion of C, through
fluid at rest, with velocity V(x, r). Lighthill (1960) then goes on to obtain the rate of
change of lateral momentum of the fluid passing over the flexible body,

(2.84)

where A(x) is the slowly varying (or constant) cross-sectional area along the length of
the body. This equation is further discussed in Chapters 8 and 9, where the slender-body
approach is used extensively.