Page 59 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

42 SLENDER STRUCTURES AND AXIAL FLOW

(b) The added mass from the kinetic energy
The classical way of introducing the added (or ‘virtual’) mass concept is via energy
considerations (Milne-Thomson 1949; Duncan et al. 1970). As this gives new insights, it
is presented here, parenthetically, following the treatment of Duncan et af . (1970).
Consider a rigid body moving rectilinearly with velocity U at the instant considered
in unconfined fluid, otherwise at rest. The velocity of the fluid thereby generated, at any
point, is proportional to U, and hence the velocity components may be written as u = Uu’,
v = Uv’, = Uw’. Hence, the total kinetic energy of the fluid (over the whole region
w
occupied by it) is
T = $pU2 ///(uf2 + vr2 + wf2)drdydz = {pU2~, (2.129)

where K is a constant, for motion in any given direction. Next, suppose that the velocity
of the body is variable, and let F be the force exerted by the body on the fluid. Then, by
elementary energy considerations, the change in kinetic energy is equal to the work done
by F, say in the z-direction, i.e.
dU
F dz = dT = KPU - dt,
dt
which gives
dU
F = KP-, (2.130)
dt
and the force on the body is the negative of that. In (2.130), dU/dt is the body acceleration
and, hence, by definition, PK is the added mass.
For 2-D oscillations of a circular cylinder in unbounded inviscid fluid, v =
(Ua2/r2) sin 20, and w = (Ua2/r2) cos 20, and v2 + w2 = Ua2/r2; hence, in this case
1 2n 00
K = 3 A A (v2+w2)rd0dr=xa2,

per unit length, and the added mass, also per unit length, is

Mf
m =-=pna. 2 (2.131)
I
L
Thus, the well-known result is obtained that the added mass of a long cylinder oscillating
in unconfined fluid is equal to the displaced mass of fluid. This corresponds exactly to
the result in equation (2.127) for Ro +. co, as it should.
It is worthwhile taking this one step further, to the case where there is an obstacle or
boundary in the fluid; K is then not a constant but a function of position, Le. K(z). In this
case, by following the same procedure one finds

dU 1 dK 2.
F = KP- + - - pU , (2.132)
dt 2 dz
i.e. there is now a quadratic velocity-dependent component, which for small-amplitude
motion is of second order, as already remarked in the first footnote of subsection 2.2.2(a).