Page 52 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS. DEFINITIONS AND METHODS 35

downstream station, and a selected group of these integral equations is solved, often by
a relatively simple numerical method (White 1974; Schetz 1993).]
Presuming now that v, has been determined, substitution of (2.99) into (2.85) gives
au,
--+u au, 1 ap, +(v,,+v,)-+- a2u, av, (av, -+- 2) (2.106)
,
at ax, axi ax; ax, ax,

where pr = P + $pK is the turbulent ‘total pressure’. Equation (2.106) may be written in
the usual, but perhaps less convenient, form
av 1
- +(V*V)V= --V~~+(V,,+V~)V~V+(VLJ,.V)V+(W).VV~, (2.107)
at P

where W is the so-called dyad, a vector (Wills 1958; Tai 1992). Examples of the use
of these equations andor the ideas summarized in this subsection are presented in Chap-
ters 7- 10.

(g) Empirical formulations
As intimated in the foregoing, mixed analytical-empirical formulations of the fluid-
dynamic forces may be the only convenient way to analyse some fluid-structure interaction
problems (e.g. provided that there is no large-scale flow separation, by analysing the flow
as if it were inviscid, thereby obtaining the pressure-related forces, and adding empir-
ical expressions for the viscous stresses acting on the body surface). Indeed, in many
cases involving complex flows, e.g. cross-flow of heat-exchanger tube arrays, the very
foundation of the theoretical model may be empirical or quasi-empirical.
In analysing the empirical (experimental) data, it is convenient to express the unsteady
fluid loading, F(t), acting on an oscillating structure in terms of components in phase
with acceleration, velocity and displacement of the structure, locally linearized; thus, for
a one-degree-of-freedom system,

F(t) = -m’z - c’i - k’z. (2.108)

When this is substituted in the equation of motion of the structure, mz + cz + kz = F(t),
one obtains
(rn + m’)z + (c + c’)z + (k + k’)z = 0, (2.109)
hence the appellation of m’,c’ and k’ as the added mass, added damping and added
stiffness [e.g. Naudascher & Rockwell (1994, Chapter 3)].
For example, for a long cylinder of cross-sectional area A and length L, oscillating in
unconfined dense fluid of density p, the added mass per unit length is m* = m’/L = PA,
if end effects are negligible. If the cylinder is in a conduit of complex geometry, m’ may
be determined analytically, numerically or experimentally, and the added mass per unit
length expressed by
m’
m* = - = C,pA. (2.110)
L
In general, C, will be a function of geometry, viscosity and frequency (hence of the
oscillatory Reynolds number), amplitude, and other factors as discussed in Sections 2.2.2
and 2.2.3 and by others (Chen 1987; Gibert 1988; Naudascher & Rockwell 1994). In many