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Unformatted text preview: 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 ζm a ( x ), where the nominal added mass value for a cylinder is β ζm a = π d 2 ζx. (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 z b ( x,...
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.154 taught by Professor Michaeltriantafyllou during the Fall '04 term at MIT.
- Fall '04