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# lec8 - 8 8.1 STREAMLINED BODIES Nominal Drag Force A...

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8 STREAMLINED BODIES 8.1 Nominal Drag Force A symmetric streamlined body at zero angle of attack experiences only a drag force, which has the form 1 F A = πC A A o U 2 . (109) 2 The drag coeﬃcient C A has both pressure and skin friction components, and hence area A o is usually that of the wetted surface. Note that the A -subscript will be used to denote zero angle of attack conditions; also, the sign of F A is negative, because it opposes the vehicle’s x -axis. 8.2 Munk Moment Any shape other than a sphere generates a moment when inclined in an inviscid ﬂow. d’Alembert’s paradox predicts zero net force , but not necessarily a zero moment. This Munk moment arises for a simple reason, the asymmetric location of the stagnation points, (Continued on next page)

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36 8 STREAMLINED BODIES where pressure is highest on the front of the body (decelerating ﬂow) and lowest on the back (accelerating ﬂow). The Munk moment is always destabilizing, in the sense that it acts to turn the vehicle perpendicular to the ﬂow. Consider a symmetric body with added mass components A xx along the vehicle (slender) x - axis (forward), and A zz along the vehicle’s z -axis z (up). We will limit the present discussion to the vertical plane, but similar arguments can be used to describe the horizontal plane. Let represent the angle of attack, taken to be positive with the nose up this equates to a negative pitch angle δ in vehicle coordinates, if it is moving horizontally. The Munk moment is: 1 M m = ( A zz A xx ) U 2 sin 2 (110) 2 ( A zz A xx ) U 2 ∂. A zz > A xx for a slender body, and the negative sign indicates a negative pitch with respect to the vehicle’s pitch axis. The added mass terms A zz and A xx can be estimated from analytical expressions (available only for regular shapes such as ellipsoids), from numerical calculation, or from slender body approximation (to follow).
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lec8 - 8 8.1 STREAMLINED BODIES Nominal Drag Force A...

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