## 6 Hydrolplane and Propulsion System Design  
The design of the droplanes is based on the selection of the wing airfoil, which has to be symmetric since the lift is developed in both senses. The main parameters of a typical airfoil are illustrated in Figure 8 where the medium line (2) is the line equidistant, in every point, from the belly and back of the profile and the chord (3) is the line that joints the leading edge (1) and the trailing one (4). In a symmetric section profile, chord and medium lines are coincident. The inclination of the chord with respect to the water surface is the angle of attack (5) [29]. The most commonly adopted airfoils in literature are NACA10 and their derivatives, first developed in the 1940s for airplanes designed both for subsonic and supersonic speed. Among these, the NACA 0015 was selected being commonly used in aeronautic and naval field applications [29; 30; 31; 32; 33]. Figure 8: Wing section nomenclature [29]. 1: Leading edge. 2: Medium line. 3: Chord line. 4: Angle of attack. 5: Trailing edge. Figure 7: Cost analysis conducted on the proposed robot configurations. The meaning of the four digits of NACA 0015 profile is: 0: maximum ordinate of the medium line, with respect to the chord, expressed as a percentage of the chord itself; 0: position, along the chord, of the maximum ordinate of the medium line, expressed in tenths of the chord length; 15: maximum value of thickness distribution, expressed as a percentage of the chord length [29]. The local thickness of the considered airfoil in function of \(c\) (chord length) and \(x\) (distance from the leading edge of a point along the chord expressed as a percentage of the chord length) is expressed by  
\[y_{t}=5 tc\bigg{[}0.2969\sqrt{\frac{x}{c}}\ +\ (-0.1260)\Big{(}\frac{x}{c} \Big{)}\ +\ (-0.3516)\Big{(}\frac{x}{c}\Big{)}^{2}\ +\ 0.2843\Big{(}\frac{x}{c}\Big{)}^{3}\ +\ (-0.1015)\Big{(}\frac{x}{c} \Big{)}^{4}\bigg{]} \tag{1}\]  
and is plotted in Figure 9.  
The NACA0015 reference trend of the lift \(C_{L}\) and drag \(C_{D}\) coefficients as a function of the angle of attack (5 in Figure 8) at varying Reynolds' number \(R_{n}\) were experimentally measured in the past and can be found in the literature [29; 30; 31]. The structure of the proposed robot adopts an aft and a stern hydrofoil with the same geometry as illustrated in Figure 4. Fluid-dynamic analyses are conducted in Solidworks Flow Simulation to evaluate the lift and drag force at various angles of attack and longitudinal velocities in the following cases: (a) computation on the aft and stern hydrofoils when located inside the vehicle, (b) computation on the free wing, and (c) computation on the whole vehicle. The simulations of the whole vehicle are performed exploiting a simplified 3D model (Figure 10) to ensure the convergence of the computations and a restrained simulation time. The hulls and the electric thrusters are considered as a unique body with an equivalent geometry. The battery pack, the control modules and the communication devices are installed inside the two hulls and thus they can be neglected in the fluid dynamic analysis. The effects of the communication cable between the robot and the buoy in terms of increased lift and drag force are not taken into account in this simulation. The mesh adopted for the simulation of the whole vehicle and of the hydrofoil is the same. The volume of the considered fluid is equal to 31 m\({}^{3}\) and is divided into 3,118,905 elements with a logarithmic distribution along the three principal axes. The mesh is thicker on the close surroundings of the robot and along the wake, as illustrated in Figure 11, where a 2D mesh representation along the longitudinal axis is reported for the whole vehicle (a) and the free wing (b). The first part of the simulations is aimed to evaluate and compare the lift (\(C_{L}\)) and drag (\(C_{D}\)) coefficients of the hydrofoils in the free-flow case and when they are located inside the vehicle.