## 2 Methods: Experiment and Numerical Simulation
### Description of the Research Model  
This section focuses on the analysis of the static properties of the submarine model used in the numerical simulation. The self-built submarine model is shown in Figure 3. The main hull consists of a blunt bow, an axisymmetric parallel middle body, a parabolic stern, while the appendages including stern control surfaces with \"+\" form, the sail and its sailplanes. In order to study qualitatively the effects of propeller thrust and torque on the roll stability, a 7-blade highly skewed propeller is self-designed by referring to relative data of INSEAN E1619 propeller, which is commonly used to the wake behavior analysis of a generic submarine propeller [30]. Notice that the bow, stern, and deck casing section of the model have permeable regions with flooding holes, rather than a conventional envelope body. In other words, the present work takes the influence of flooding regions on the surfacing motion into consideration as truly as possible. Furthermore, since this paper focuses on the response characteristics of emergency rising motion after completing the blowing process, it is necessary to analyze the hydrostatic properties of the model with residual buoyancy. With regard to the numerical simulation of emergency rising, there are three following assumptions to be made:  
* Assuming that the time required for the submarine to blow the ballast tank is very short, the numerical simulation does not model the blowing process of pressure air. At the beginning of the calculation, the model obtains buoyancy force instantaneously, and the variable external force produced by the blowing process does not applied on the submarine. * The model is released with a fixed stermplane angle at the desired speed, it will be not imposed the control force of stermplanes, except the propulsion forces and the rudder control force. * Assuming there is no cavitation effect for the propeller because of its low revolution speed. Based on the above assumptions, the weight of the submarine after blowing off ballast tank is calculated as follows:  
\[m\ =\ m_{\
abla\downarrow}-m_{blowm}-m_{casing} \tag{9}\]  
where, \(m_{\
abla\downarrow}\) is the weight of water that the external hydrodynamic envelope of the submarine displaces, \(m_{blowm}\) is the blown ballast mass and \(m_{casing}\) is the displacement of the permeable region. The position of the center of gravity of the submarine is calculated by:  
\[G\ =\ \frac{m_{\Delta\downarrow}G_{\Delta\downarrow}-m_{blowm}G_{blowm}-m_{casing }G_{casing}}{m} \tag{10}\]  
where, \(G_{\
abla\downarrow}\) is the center of gravity of the submarine envelope displacement, \(G_{blowm}\) is the center of gravity of buoyancy load, \(G_{casing}\) is the center of gravity of the permeable region. The submarine moments of inertia are calculated by:  
\[I\ =\ I_{\
abla\downarrow}-I_{blowm}-I_{casing} \tag{11}\]  
where, \(I_{\
abla\downarrow}\) is the moment of inertia of the submarine envelope displacement, \(I_{blowm}\) is the moment of inertia of the buoyancy load, and \(I_{casing}\) is the moment of inertia of the permeable region. The origin of the body coordinate system \(\mathbf{O_{B}}\) is set to the center of gravity of the submarine. Similar to the adding weight method, the model has residual buoyancy under water by reducing the weight of the model. Then calculate the center of gravity of the model that removes bow, stern, deck permeable regions, and blown ballast load, so does the center of buoyancy. Therefore, the mass centroid of the model in air varies with the change of longitudinal position of the blown ballast. By convention, the center of gravity \((x_{G},y_{G},z_{G})\) of the model with residual buoyancy are defined with respect to (w.r.t.) the midship and model central axis, so is the position of buoyancy and blown ballast load. For instance,  
Figure 3: Diagram of the submarine model with a propeller.