# Numerical Analysis of the Take-Off Performance of a Seaplane in Calm Water
## 3 Numerical Model
### Rigid Body Dynamics  
The forces and moments acting on a seaplane in take-off are shown in Figure 2. \(L_{a}\), \(R_{a}\), \(M_{a}\) are respectively the aerodynamic lift, aerodynamic resistance, and aerodynamic pitching moment acting on CG; \(\triangle\), \(R_{w}\), \(M_{w}\) are the water load, water resistance, and pitching moment acting on CG produced by water, respectively; \(W\) is the weight of the seaplane, \(T\) is the propeller thrust, \(\tau\) is the trim angle, \(d\) is the draft of keel at transom, and \(d_{T}\) is the distance between thrust line and CG.  
A global fixed right-hand Cartesian coordinate system (GS) was used to describe the flow field, defined with the XZ-plane coinciding with the symmetry plane of the seaplane, and the XY-plane paralleled to the calm water surface. The positive x-axis is consistent with the velocity direction and positive z-axis points in the opposite direction of gravity. Besides, a body-fixed right-hand Cartesian coordinate system (BS) locating its origin at the CG was utilized to define the position and attitude of seaplane. The XZ-plane coincides with the symmetry plane of the seaplane. The x-axis is pointing forward in the longitudinal  
Figure 2: The forces and moments acting on a seaplane in take-off.  
axis of seaplane which is parallel to the keel and the positive y-axis points to the port side. The trim angle \(\tau\) is defined as the angle between the x-axis of GS and the x-axis of BS.  
In order to determine the equilibrium trim angle and heave of the seaplane at different speeds, the Dynamic Fluid Body Interaction (DFBI) model was activated to simulate the 2 degree of freedom (2 DOF) motion of the seaplane in response to the forces the fluid exerts and the external forces. Particularly, as we only concerned the forces at final equilibrium state, damping terms were introduced to reduce the convergence time. The governing equations for the 2 DOF motion are written as follows:  
\[m\cdot\frac{\partial w}{\partial t}=F_{fluid,z}+\sum F_{ext,z}-\xi_{w}\cdot w \tag{8}\]  
\[I_{y}\cdot\frac{\partial\omega_{y}}{\partial t}=M_{fluid,y}+\sum M_{ext,y}- \xi_{\omega_{y}}\cdot\omega_{y} \tag{9}\]  
where \(F_{fluid,z}\) and \(M_{fluid,y}\) are the z component of resultant force and the resultant pitching moment acting on the seaplane calculated by the two-phase flow solver. \(F_{ext,z}\) are external forces in z-axis including gravity \(W\) and the z component of propeller thrust \(T_{z}\). \(M_{ext,y}\) is the external pitching moment, that is, the pitching moment caused by propeller thrust \(M_{t}\). \(\xi_{w}\) is the damping factor to the vertical velocity \(w\) and \(\xi_{\omega_{y}}\) is the damping factor to the pitching angular velocity \(\omega_{y}\).  
In summary, a brief workflow of the DFBI module is illustrated in Figure 3. The seaplane was released at initial trim angle and heave, then was free to trim and heave. For each time step, the aerodynamic and hydrodynamic forces acting on the seaplane were calculated by the URANS&VOF solver. The propeller thrust and moment were yielded based on data base and introduced into the DFBI model as external inputs. Then, the forces and moments were brought into Equations (8) and (9) to determine the vertical acceleration and pitching angular acceleration. Hence, the variations in heave and trim angle were yielded and the overset grids were relocated for the next time step. These procedures were repeated until the seaplane reached the equilibrium state.  
Figure 3: Flow chart of the DFBI module.