# Effect of Waves on the Behavior of Emergent Buoyantly Rising Submarines Using CFD
## 2 Theory
### Solid Body EOM  
The solid-body EOM [9; 10] are based on [PERSON]'s second law: \(Force=Mass\times Acceleration\). The six equations (three for translation and three for rotation) are as follows: Axial translation (along the x-axis)  
\[m\left[\dot{u}-v\tau+wp-x_{G}\left(q^{2}+r^{2}\right)+y_{G}\left( pq-\dot{r}\right)+z_{G}\left(pr+\dot{q}\right)\right]\] \[=X_{H}\left(t,\mathbf{y}\right)-\left(W-B\right)\sin\theta\] (2a) Lateral translation (along the y-axis) \[m\left[\dot{v}-wp+ur-y_{G}\left(r^{2}+p^{2}\right)+z_{G}\left( qr-\dot{p}\right)+x_{G}\left(qp+\dot{r}\right)\right]\] \[=Y_{H}\left(t,\mathbf{y}\right)-\left(W-B\right)\cos\theta\sin\varphi\] (2b) Normal translation (along the z-axis) \[m\left[\dot{w}-uq+vp-z_{G}\left(p^{2}+q^{2}\right)+x_{G}\left( rp-\dot{q}\right)+y_{G}\left(rq+\dot{p}\right)\right]\] \[=Z_{H}\left(t,\mathbf{y}\right)-\left(W-B\right)\cos\theta\cos\varphi\] (2c) Rolling rotation (around the x-axis) \[I_{x}\dot{p}+\left(I_{z}-I_{y}\right)qr-\left(\dot{r}+pq\right)I _{xz}+\left(r^{2}-q^{2}\right)I_{yz}\] \[+\left(pr-\dot{q}\right)I_{xy}+m\left[y_{G}\left(\dot{w}-uq+vp \right)-z_{G}\left(\dot{v}-wp+ur\right)\right]\] \[=K_{H}\left(t,\mathbf{y}\right)+\left(y_{G}W-y_{B}B\right)\cos \theta\cos\varphi\] \[-\left(z_{G}W-z_{B}B\right)\cos\theta\sin\varphi \tag{2d}\]  
Figure 1: Coordinate systems used in the simulation.  
Pitching rotation (around the y-axis)  
\[I_{y}\dot{q}+\left(I_{x}-I_{z}\right)rp-\left(\dot{p}+qr\right)I_{ xy}+\left(p^{2}-r^{2}\right)I_{zx}\] \[+\left(qp-\dot{r}\right)I_{yz}+m\left[z_{G}\left(\dot{u}-vr+wq \right)-x_{G}\left(\dot{w}-uq+vp\right)\right]\] \[=M_{H}\left(t,\mathbf{y}\right)-\left(x_{G}W-x_{B}B\right)\cos \theta\cos\varphi\] \[-\left(z_{G}W-z_{B}B\right)\sin\theta \tag{2e}\]  
Yawing rotation (around the z-axis)  
\[I_{z}\dot{r}+\left(I_{y}-I_{z}\right)rp-\left(\dot{q}+rp\right)I _{yz}+\left(q^{2}-p^{2}\right)I_{xy}\] \[+\left(rq-\dot{p}\right)I_{zx}+m\left[x_{G}\left(\dot{v}-wp+ur \right)-y_{G}\left(\dot{u}-vr+wq\right)\right]\] \[=N_{H}\left(t,\mathbf{y}\right)+\left(x_{G}W-x_{B}B\right)\cos \theta\sin\varphi\] \[+\left(y_{G}W-y_{B}B\right)\sin\theta \tag{2f}\]  
where \(\left(x_{G},y_{G},z_{G}\right)\) are the coordinates of the center of gravity (\(CG\)) in the mid-ship axes; \(\left(x_{B},y_{B},z_{B}\right)\) are the coordinates of the center of buoyancy (\(CB\)) in the mid-ship axes; \(W\) is the weight of the submarine; \(B\) is the buoyancy of the submarine; \(I=\left(I_{x},I_{y},I_{z},I_{xy},I_{yz},I_{zx}\right)\) denotes the moment of inertia; \(F_{H}\) denotes the hydrodynamic forces and moments acting on the submarine. Note that \(W\), \(CG\), and \(I\) are constants throughout the simulation; \(B\) and \(CB\) depend on the volume of the submarine that emerges from the water; and \(F_{H}\) was evaluated through CFD simulations.  
The velocity in Equation (2) is defined by body-fixed coordinates, and the position and posture of the submarine are defined by inertial (Earth-fixed) coordinates. The following equations express the relationship between the body-fixed and inertial variables [9; 10]:  
\[\dot{\xi}= u\cos\theta\cos\psi+v\left(\sin\varphi\sin\theta\sin\psi-\cos \varphi\sin\psi\right)\] \[+w\left(\sin\varphi\sin\psi+\cos\varphi\sin\theta\cos\psi\right) \tag{3a}\] \[\dot{\eta}= u\cos\theta\sin\psi+v\left(\cos\varphi\cos\psi+\sin\varphi\sin \theta\sin\psi\right)\] \[+w\left(\cos\varphi\sin\theta\sin\psi-\sin\varphi\cos\psi\right)\] (3b) \[\dot{\zeta}=-u\sin\theta+v\cos\theta\sin\varphi+w\cos\theta\cos\varphi\] (3c) \[\dot{\varphi}=p+\left(r\cos\varphi+q\sin\varphi\right)\tan\theta\] (3d) \[\dot{\theta}=q\cos\varphi-r\sin\varphi\] (3e) \[\dot{\psi}=\frac{r\cos\varphi+q\sin\varphi}{\cos\theta} \tag{3f}\]  
The submarine has 12 state variables (as shown in Equation (1)) and 12 differential equations that are defined in Equations (2) and (3); thus, the set of equations is complete.