9 SLENDER-BODY THEORY 9.1 Introduction Consider a slender body with d << L, that is mostly straight. The body could be asymmetric in cross-section, or even flexible, but we require that the lateral variations are small and (Continued on next page)� � 40 9 SLENDER-BODY THEORY smooth along the length. The idea of the slender-body theory, under these assumptions, is to think of the body as a longitudinal stack of thin sections, each having an easily- computed added mass. The effects are integrated along the length to approximate lift force and moment. Slender-body theory is accurate for small ratios d/L, except near the ends of the body. As one example, if the diameter of a body of revolution is d(s), then we can compute ζma(x), where the nominal added mass value for a cylinder is β ζma = π d2ζx. (125)4 The added mass is equal to the mass of the water displaced by the cylinder. The equation above turns out to be a good approximation for a number of two-dimensional shapes, includ-ing flat plates and ellipses, if d is taken as the width dimension presented to the flow. Many formulas for added mass of two-dimensional sections, as well as for simple three-dimensional bodies, can be found in the books by Newman and Blevins. 9.2 Kinematics Following the Fluid The added mass forces and moments derive from accelerations that fluid particles experience when they encounter the body. We use the notion of a fluid derivative for this purpose: the operator d/dt indicates a derivative taken in the frame of the passing particle, not the vehicle. Hence, this usage has an indirect connection with the derivative described in our previous discussion of rigid-body dynamics. For the purposes of explaining the theory, we will consider the two-dimensional heave/surge problem only. The local geometry is described by the location of the centerline; it has vertical location (in body coordinates) of zb(x, t), and local angle ∂(x, t). The time-dependence indicates that the configuration is free to change with time, i.e., the body is flexible. Note that the curvilinear coordinate s is nearly equal to the body-reference (linear) coordinate x. The velocity of a fluid particle normal to the body at x is wn(t, x): �zb wn = cos ∂ − U sin ∂. (126)�t The first component is the time derivative in the body frame, and the second due to the deflection of the particle by the inclined body. If the body reference frame is rotated to the flow, that is, if w = 0, then �zb /�t will contain w. For small angles, sin ∂ � tan ∂ = �zb /�x,≥and we can write �zb �zb wn .� �t − U �x The fluid derivative operator in action is as follows: dzb � � wn = = zb. dt �t − U �x�µ �µ �µ �µ � � 9.3 Derivative Following the Fluid 41 9.3 Derivative Following the Fluid A more formal derivation for the fluid derivative operator is quite simple. Let µ(x, t) represent some property of a fluid particle. d 1 [µ(x, t)] = lim [µ(x + ζx, t + ζt) − µ(x, t)]dt νt�0 ζt ⎬� = . �t − U �x The second equality can be verified using a Taylor series expansion of µ(x + ζx, t + ζt): µ(x + ζx, t + ζt) = µ(x, t) + ζt + ζx + h.o.t.,�t �x and noting that ζx = −Uζt. The fluid is convected downstream with respect to the body. 9.4 Differential Force on the Body If the local transverse velocity is wn(x, t), then the differential inertial force on the body here is the derivative (following the fluid) of the momentum: d ζF = [ma(x, t)wn(x, t)] ζx. (127)− dt Note that we could here let the added mass vary with time also – this is the case of a changing cross-section! The lateral velocity of the point zb(x) in the body-reference frame is �zb = w − xq, (128)�t wsuch that n = w − xq − U∂. (129) Taking the derivative, we have ζF � � = [ma(x, t)(w(t) − xq(t) − U∂(x, t))] . ζx − �t − U �x We now restrict ourselves to a rigid body, so that neither ma nor ∂ may change with time. ζF � = ma(x)(−w˙ + xq˙) + U [ma(x)(w − xq − U∂)] . (130)ζx �x� 42 9 SLENDER-BODY THEORY 9.5 Total Force on a Vessel The net lift force on the body, computed with strip theory is � xN Z = ζF dx (131) xT where xT represents the coordinate of the tail, and xN is the coordinate of the nose. Ex-panding, we have � xN � xN Z = ma(x) [−w˙ + xq˙] dx + U [ma(x)(w − xq − U∂)] dx xT xT �x | x=xN= −m33 w˙ − m35q˙ + Uma (x)(w − xq − U∂)x=xT . We made use here of the added mass definitions � xN m33 = ma(x)dx xT � xN m35 = xma (x)dx.− xT Additionally, for vessels with pointed noses and flat tails, the added mass ma at the nose is zero, so that a simpler form occurs: Z = −m33 w˙ − m35q˙ − Uma(xT )(w − xT q − U∂(xT )). (132) In terms of the linear hydrodynamic derivatives, the strip theory thus provides Zw˙= −m33 Zq˙= −m35 Zw = −Uma (xT ) Zq = UxT ma(xT ) Z∂(xT ) = U2 ma(xT ). It is interesting to note that both Zw and Z∂(xT ) depend on a nonzero base area. In general, however, potential flow estimates do not create lift (or drag) forces for a smooth body, so this should come as no surprise. The two terms are clearly related, since their difference depends only on how the body coordinate system is oriented to the flow. Another noteworthy fact is that the lift force depends only on ∂ at the tail; ∂ could take any value(s) along the body, with no effect on Z.� � � � � 9.6 Total Moment on a Vessel 43 9.6 Total Moment on a Vessel A similar procedure can be applied to the moment predictions from slender body theory (again for small ∂): � xN M = − xT xζF dx � xN = x [ma(x)(w − xq − U∂)] dx xT �t − U �x � xN � xN = xma(x)( ˙w − xq˙)dx − U x [ma(x)(w − xq + U∂)] dx. xT xT �x Then we make the further definition � xN 2 m55 = x ma(x)dx, xT (note that m35 = m53) and use integration by parts to obtain M = −m35 w˙ − m55q˙ − Uxma (x)(w − xq − U∂)x=xN +x=xT|� xN U ma(x)(w − xq − U∂)dx. xT The integral above contains the product ma(x)∂(x), which must be calculated if ∂ changes along the length. For simplicity, we now assume that ∂ is in fact constant on the length, leading to UmM = −m35 w˙ − m55q˙ + UxT ma(xT )(w − xT q − U∂) + 33 w + Um35q − U2 m33∂.