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 FA = πCAAoU 2 . (109)− 2 The drag coefficient CA has both pressure and skin friction components, and hence area Ao 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 FA 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)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 Axx along the vehicle (slender) x-axis (forward), and Azz 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 Mm = (Azz − Axx)U 2 sin 2∂ (110)−2 � −(Azz − Axx)U 2∂. Azz > Axx for a slender body, and the negative sign indicates a negative pitch with respect to the vehicle’s pitch axis. The added mass terms Azz and Axx can be estimated from analytical expressions (available only for regular shapes such as ellipsoids), from numerical calculation, or from slender body approximation (to follow). 8.3 Separation Moment In a viscous fluid, flow over a streamlined body is similar to that of potential flow, with the exceptions of the boundary layer, and a small region near the trailing end. In this latter area, a helical vortex may form and convect downstream. Since vortices correlate with low pressure, the effect of such a vortex is stabilizing, but it also induces drag. The formation of the vortex depends on the angle of attack, and it may cover a larger area (increasing the stabilizing moment and drag) for a larger angle of attack. For a small angle of attack, the transverse force Fn can be written in the same form as for control surfaces: 1 Fn = πCnAoU 2 (111)2 1 �Cnπ ∂AoU 2 .� 2 �∂ With a positive angle of attack, this force is in the positive z-direction. The zero-∂ drag force FA is modified by the vortex shedding: 1 Fa = πCaAoU 2 , where (112) C−2 a = CA cos 2 δ. The last relation is based on writing CA(U cos δ)2 as (CA cos2 δ)U 2, i.e., a decomposition using apparent velocity.8.4 Net Effects: Aerodynamic Center 37 8.4 Net Effects: Aerodynamic Center The Munk moment and the moment induced by separation are competing, and their mag-nitudes determine the stability of a hullform. First we simplify: Fa = ρa−Fn = ρn∂ Mm = ρm∂. −Each constant ρ is taken as positive, and the signs reflect orientation in the vehicle reference frame, with a nose-up angle of attack. The Munk moment is a pure couple which does not depend on a reference point. We pick a temporary origin O for Fn however, and write the total pitch moment about O as: M = Mm + Fnln (113) = (−ρm + ρnln)∂. where ln denotes the (positive) distance between O and the application point of Fn. The net moment about O is zero if we select ρm ρln = , (114) n and the location of O is then called the aerodynamic center or AC. The point AC has an intuitive explanation: it is the location on the hull where Fn would act to create the total moment. Hence, if the vehicle’s origin lies in front of AC, the net moment is stabilizing. If the origin lies behind AC, the moment is destabilizing. For self-propelled vehicles, the mass center must be forward of AC for stability. Similarly, for towed vehicles, the towpoint must be located forward of AC. In many cases with very streamlined bodies, the aerodynamic center is significantly ahead of the nose, and in this case, a rigid sting would have to extend at least to AC in order for stable towing. As a final note, since the Munk moment persists even in inviscid flow, AC moves infinitely far forward as viscosity effects diminish. 8.5 Role of Fins in Moving the Aerodynamic Center Control surfaces or fixed fins are often attached to the stern of a slender vehicle to enhance directional stability. Fixed surfaces induce lift and drag on the body: 1 L = πAf U 2Cl(∂) � ρL∂ (115)2 1 D = πAf U 2Cd ρD (constant) −2 � −38 8 STREAMLINED BODIES These forces act somewhere on the fin, and are signed again to match the vehicle frame, with ρ > 0 and ∂ > 0. The summed forces on the body are thus: X = Fa − D cos ∂ + L sin ∂ (116)| | | |� −ρa − ρD + ρL∂2 Z = Fn + L cos ∂ + D sin ∂| | | |� ρn∂ + ρL∂ + ρD∂. All of the forces are pushing the vehicle up. If we say that the fins act a distance lf behind the temporary origin O, and that the moment carried by the fins themselves is very small (compared to the moment induced by Llf) the total moment is as follows: M = (−ρm + ρnln)∂ + (ρL + ρD)lf∂. (117) The moment about O vanishes if ρm = ρnln + lf(ρL + ρD). (118) The Munk moment ρm opposes the aggregate effects of vorticity lift ρn and the fins’ lift and drag ρL + ρD. With very large fins, this latter term is large, so that lf might be very small; this is the case of AC moving aft toward the fins. A vehicle with excessively large fins will be difficult to turn and maneuver. Equation 118 contains two length measurements, referenced to an arbitrary body point O. To solve it explicitly, let lfn denote the (positive) distance that the fins are located behind Fn; this is likely a small number, since both effects usually act near the stern. We solve for lf: ρm + ρnlfn ρlf = . (119) n + ρL + ρD This is the distance that AC is located forward of the fins, and thus AC can be referenced to any other fixed point easily. Without fins, it can be recalled that ρm ρln = . n Hence, the fins