# Design and Analysis of the High-Speed Underwater Glider with a Bladder-Type Buoyancy Engine
## 2 Analysis of UG
### Maximum Speed Analysis  
The ratio of lift and drag generated by the angle of attack (AoA) of the UG during gliding affects the stability of the pitching posture of the UG. As shown in Figure 5, when the angle of attack changes, the resultant force of lift and drag changes accordingly, and the posture of the UG needs to be corrected to achieve equilibrium. In Figure 5, the UG's bow direction is to the right.  
As indicated by the CFD analysis results, Figure 6 summarizes the lift and drag coefficients according to the angle of attack. It was introduced in the performance optimization considering the part where the ratio of lift and drag is optimal during the glide of the UG. As shown in Figure 6, when the angle of attack of 0-10\({}^{\circ}\) is considered, the increase in lift is more pronounced than the increase in drag.  
The path angle of the UG can be expressed as shown in Equation (3) based on the ratio of lift and drag:  
\[\zeta=-\tan^{-1}\biggl{(}\frac{D}{L}\biggr{)} \tag{3}\]  
Here, \(D\) is drag force, \(L\) is lift force, and \(\zeta\) is glide path angle. Figure 7 shows the relationship between the angle of attack and the glide angle based on the calculated coefficients.  
Figure 5: Action of lift and drag force.  
Figure 6: Simulated lift and drag coefficients.  
The speed of the UG according to the maximum ballast control amount and the glide angle can be expressed by Equations (4)-(6) [11]:  
\[V=\left(\frac{m_{0}g\sin\zeta}{\frac{1}{2}\rho C_{D}(\zeta(\alpha))}\right)^{1/2} \tag{4}\]  
\[V_{z}=\left(\frac{m_{0}g\sin\zeta}{\frac{1}{2}\rho C_{D}(\zeta(\alpha))}\right)^ {1/2}\sin\zeta \tag{5}\]  
\[V_{x}=\left(\frac{m_{0}g\sin\zeta}{\frac{1}{2}\rho C_{D}(\zeta(\alpha))}\right) ^{1/2}\cos\zeta \tag{6}\]  
where \(V\) is glide speed, \(V_{Z}\) is depth rate, \(V_{X}\) is horizontal speed, \(m_{0}\) is excess mass, \(g\) is gravity, \(C_{D}\) is drag coefficient, \(\rho\) is density, \(\alpha\) is angle of attack, and the calculation results are illustrated in Figure 8.  
In Equation (4), the angle at which the horizontal speed (\(V_{X}\)) becomes the maximum is the point at which \(\cos\ \zeta\ \sin^{1/2}\zeta\) is the maximum. As shown in Figure 9, the glide angle reaches its maximum at \(\cos\ \zeta\ \sin^{1/2}\zeta=0.657\  
Figure 8: Relationship between glider velocity and path angle.  
Figure 7: Relationship between the angle of attack and path angle.  
Equation (7) expresses the maximum speed of the UG as a function of the ballast control amount. The velocity relational expression based on the displacement and the maximum ballast control ratio (\(n_{b}=m_{0}/\rho V\)) was derived, and the relation is illustrated in Figure 10.  
\[V_{x}=\left(\frac{m_{0}g\sin\zeta}{\frac{1}{2}\rho C_{D}(\zeta(\alpha))}\right)^ {1/2}\times 0.657 \tag{7}\]  
The design displacement in this study is 108 kg, the target speed is 2 knots (1.028 m/s), the ballast control ratio is 0.012, and the control amount is 0.6 kg. The maximum speed for the maximum ballast control amount is depicted in Figure 11.  
Figure 10: Relationship between maximum velocity and glider volume.  
Figure 9: Relationship between \(\cos\ \zeta\ \sin^{1/2}\zeta\) and path angle.