# Cavitator Design for Straight-Running Supercavitating Torpedoes
## 2 Cavitator Design Procedure
In the case of a short-range homing torpedo with a sonar system, the cone-type cavitator may be a good choice because it is much easier to accommodate the sonar system. After the cavitator type is determined, the optimal cavitator size that can ensure the minimum overall drag and basic motion stability of the torpedo should be determined. An algorithm was developed to determine the conditions of straight level flight with variation in the cavitator size and to calculate the overall drag of the torpedo in straight level flight. In this study, the motion scheme of stationary planing along the lower internal cavity surface was considered for straight level flight, as depicted in Figure 2, because the lowest hydrodynamic drag occurs with this scheme owing to the smallest wetted part [7]. During this motion, it is assumed that there are no roll and yaw motions owing to the vertical fins and an automatic feedback control system, and the horizontal fins are supposed to be retracted into the vehicle to minimize the hydrodynamic drag and interference with the supercavity. Therefore, only the longitudinal vertical plane motion of the vehicle was considered on the basis of these assumptions. A free-body diagram of a supercavitating vehicle in a longitudinal vertical plane is shown in Figure 3. The equations of force and moment equilibrium for straight level flight can be expressed as  
\[F_{zc}+F_{zp}+F_{zb}=G-T\sin\alpha_{B} \tag{3}\] \[F_{xc}+F_{xp}+F_{xf\_u}+F_{xf\_l}=T\cos\alpha_{B}\] \[M_{c}+M_{p}+M_{f}+M_{b}=0\]  
where \(F_{xc}\), \(F_{zc}\), and \(M_{c}\) represent the hydrodynamic drag, lift, and the sum of their moments acting on the cavitator, respectively. \(F_{xp}\), \(F_{zp}\), and \(M_{p}\) represent the hydrodynamic drag, lift, and the sum of their moments on the wetted part of the vehicle by planing, respectively. \(F_{zb}\) and \(M_{b}\) represent the hydrostatic buoyancy force and its moment on the wetted part, respectively. \(F_{xf\_u}\), \(F_{xf\_l}\), and \(M_{f}\) represent the hydrodynamic drag force and the sum of their moments acting on the upper and lower vertical fins, respectively. \(G\) represents the vehicle weight, \(T\) is the thrust (which equals the overall drag of the vehicle), \(\alpha_{B}\) is the trim angle of the vehicle, and \(\delta_{c}\) is the inclination angle of the cavitator. Figure 2: Motion scheme for straight level flight. To determine the conditions of straight level flight, the equations of force and moment equilibrium were solved iteratively until the following practical convergence criteria were satisfied by varying the diameter of the cavitator (\(d_{c}\)), its angle of inclination (\(\delta_{c}\)), and the trim angle of the vehicle (\(\alpha_{B}\)):  
\[ \begin{split}\left|d_{\text{lift}}\right|\leq 1\%\ \&\ \left|d_{\text{mom}}\right|\leq 1\%\\ d_{\text{lift}}\equiv\frac{\left(F_{x_{c}}+F_{z_{\text{ip}}}+F_{z _{\text{iq}}}-Tsin_{B}\right)-G}{G}\times 100\\ d_{\text{mom}}\equiv\frac{\left(M_{p}+M_{f}+M_{b}\right)-M_{c}}{M_{ c}}\times 100\end{split}  \tag{4}\]  
where \(d_{\text{lift}}\) and \(d_{\text{mom}}\) represent the difference between the total lift force of the vehicle and its weight, and that between the total moment on the cavitator and the total moment on the wetted part of the vehicle, respectively. A flowchart of the iteration process is shown in Figure 4. To solve the equations of force and moment equilibrium, the cavity shapes and hydrodynamic forces were predicted using existing mathematical models, which are described in the following section. After the conditions of straight level flight were determined, the optimal cavitator size that minimizes the overall drag of the vehicle was finally determined by comparing the overall drag calculated under each straight level flight condition.