# Performance Evaluation of a Novel Propulsion System for the Spherical Underwater Robot (SURIII)
## 3 Hydrodynamic Analysis
### Related Parameters  
\(M_{RB}\) is received by weighing the robot in air. The SURIII is 6.9 kg, so the 3D model is:  
\[M_{RB}=\left[\begin{array}{ccc}m&0&0\\ 0&m&0\\ 0&0&I_{zz}\end{array}\right]=\ \left[\begin{array}{ccc}6.9&0&0\\ 0&6.9&0\\ 0&0&0.1615\end{array}\right] \tag{8}\]\(M_{A}\) is the mass of a fluid sphere, so its 3D model is:  
\[M_{A}=\left[\begin{array}{ccc}16.75&0&0\\ 0&16.75&0\\ 0&0&0.3864\end{array}\right] \tag{9}\]  
The coefficients of linear damping force can be expressed as:  
\[D_{1}v=C_{1}\times diag\{u,v,w,r\} \tag{10}\]  
The initial conditions stipulate that the temperature of water is 20 \({}^{\circ}\)C. At this temperature, the linear viscous coefficient is \(C_{1}\) = 1 \(\times\) 10\({}^{-4}\). The _diag_ {_u,v,w,r_} is the velocity matrix of the SURIII, where \(u\) is the velocity of the surge motion, \(v\) is the velocity of the sway motion, \(w\) is the velocity of heave motion, \(r\) is the rotation velocity of the yaw motion. These two parameters are both very small, so \(D_{1}v\) can be ignored. The Equation (7) may be expressed as:  
\[(M_{RB}+M_{A})\dot{v}+D_{q}(v)v=\tau \tag{11}\]  
where \(D_{q}(v)v\) equals to the resistance of the robot underwater,  
\[D_{q}(v)v=F_{d}=\frac{1}{2}C_{d}\rho V^{2}A \tag{12}\]  
where \(F_{d}\) is the drag of water, \(C_{d}\) is the drag coefficient, \(\rho\) is the density of the fluid, \(V\) is the relative speed between SURIII and fluid in different motions, \(A\) is the effective cross-sectional area of SURIII.  
For surge motion, the SURIII could be approximately as a sphere. In this case, \(C_{d}\) only associated with the Reynolds number [25] and its formula is:  
\[R_{e}=\frac{uD}{v} \tag{13}\]  
where \(R_{e}\) is the Reynolds number, \(u\) is the velocity of the surge motion, \(D\) is the diameter of the robot, \(v\) is the kinematic viscosity of the water and \(\
u=1\times 10^{-6}\) Pa\(\cdot\)s at the temperature of 20 \({}^{\circ}\)C. The experimental maximum velocity of surge motion is 0.15 m/s [11], the diameter of the robot is 0.4 m, so the \(R_{e}\) equals to 6 \(\times\) 10\({}^{4}\) and this value is more than 10\({}^{4}\) and less than 3 \(\times\) 10\({}^{5}\). The drag coefficient in surge motion can be selected from Table 2[26; 27]. If \(R_{e}\) equals to 6 \(\times\) 10\({}^{4}\), \(C_{d}\) is 0.4.  
In heave motion, the robot cannot be regarded as a sphere. On the one hand, for the case shown in Figure 11a, there are four gaps for the water-jet thrusters and the fluid will flow across these gaps, so that the robot cannot be classified as a closed sphere in this motion. On the other hand, for the case shown in Figure 9, the edge of hull cannot be ignored for heave motion, so that the two hemispheres  
\begin{table}
\begin{tabular}{l c c c} \hline \hline \(R_{e}\) & \(R_{e}<10^{4}\) & \(10^{4}<R_{e}<3\times 10^{5}\) & \(3\times 10^{5}<R_{e}<10^{6}\) \\ \hline \(C_{d}\) & \(24/R_{e}+6.48\times R_{e}{}^{-0.573}+0.36\) & 0.4 & 0.4 \\ \(C_{d}\) & \(30/R_{e}+0.46\) & 0.46 & 0.46 \\ \(C_{d}\) & \(24/R_{e}+(1+0.0654\,R_{e}{}^{2/3})^{2/3}\) & 0.4 & 0.40 \\ \(C_{d}\) & \((0.325+(0.124+24/R_{e}{}^{1/2}))\) & - & - \\ \(C_{d}\) & \((0.63+4.8\times R_{e}{}^{-0.5})^{2}\) & 0.4 & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Relationship between Reynolds number and drag coefficient for the sphere.  
cannot form a smooth sphere and the diameter of the hull is transformed to 46 cm. According to the Equation (12), the \(C_{d}\) can be expressed as:  
\[C_{d}=\frac{2F_{d}}{\rho V^{2}A} \tag{14}\]  
In Equation (14), the maximum experimental velocity of heave motion is 0.083 m/s [11]. The maximum cross-sectional area can be obtained by the projection of 3D model in the vertical direction, so \(A\) is 0.1661 m\({}^{2}\). The drag of water cannot be calculated in this case, thus the CFD simulation is conducted to get the estimation of it.