# Numerical Study of Wave Effect on Aircraft Water-Landing Performance
## 2 Computational Method  
### Governing Equations and Numerical Schemes  
The governing equations are the unsteady and incompressible RANS equations, written as:  
\[\
abla\vec{V}=0 \tag{1}\]  
\[\frac{\partial\vec{V}}{\partial t}+\vec{V}\cdot\
abla\vec{V}=-\frac{1}{\rho} \
abla p+\frac{1}{\rho}\
abla\vec{\tau}+\vec{g}+S \tag{2}\]  
where \(t\) denotes time, \(\vec{V}\) is the velocity vector, \(\rho\) the density, \(p\) the pressure, \(\vec{\tau}\) the stress tensor, \(\vec{g}\) the gravitational acceleration vector, and \(S\) the source term of a sponge layer that absorbs wave reflections from the outlet boundary.  
To determine the free surface of air-water interface, the widely used VOF method [30] is applied. This method capture the interface between two or more immiscible fluids by introducing volume fraction for each fluid phase governed by the following equation:  
\[\frac{\partial\alpha}{\partial t}+\alpha\cdot\
abla\vec{V}=0 \tag{3}\]  
where \(\alpha\) is an indicator phase function, defined as the amount of water per unit volume at each computational cell. If \(\alpha=1\), the cell is full of water; if \(\alpha=0\), the cell is full of air; in other cases, the cell belongs to the interface. Based on the value of \(\alpha\), the properties of fluid in each cell can be numerically determined. Moreover, to accurately capture the interaction and relative motion between the aircraft and free surface, the six-degree-of-freedom (6 DOF) model is adopted which can solves the rotation and translation of the center of mass of the object. The rigidly moving mesh strategy in the Dynamic Fluid Body Interaction (DFBI) module is employed to deal with the relative motion between the fluid and the landing object. This strategy moves or/and rotates the whole computational mesh rigidly along with the object at each time step according to the solution of the 6 DOF model [16] and has been widely used in the field of the aircraft water-landing/ditching problems [22; 23; 24; 28]. These numerical publications also illustrated that the rigidly moving mesh strategy is able to handle the large motion aircraft with range of pitch angle 0-40\({}^{\circ}\). In addition, the overset mesh approach is not employed in this study based on the consideration that for the aircraft ditching problem, the aerodynamics and hydrodynamics both play non-negligible effect on the ditching performance, thus the total grid number (background mesh and overset mesh) may be larger than that of rigidly moving mesh strategy. In contrast, by using the rigidly moving mesh, the whole computational domain and boundaries are able to move together with the aircraft, which can avoid the high computational cost and large data interpolation between the background mesh of the fluid domain with overset mesh around the body. The details of the rigidly moving mesh strategy can be found in Refs. [31; 32]. In addition, the cavitation model is not employed in this study for numerical stability and time efficiency considerations. This treatment may affect the local pressure accuracy at the rear-fuselage region, but it is less influence on the overall tendency of the impact loads and the aircraft movement [4].  
The governing equations are numerically solved by STAR CCM+. The \(k-\omega\) shear stress transport (SST) turbulence model is adopted to solve the stress tensor based on the research of turbulence model effect in numerical wave simulation [33]. The SIMPLE algorithm is applied to address the pressure-velocity coupling. The second-order implicit scheme is employed to deal with the time advancing. The convection terms and diffusion terms are discretized by second-order upwind and second-order central difference schemes, respectively. For the VOF transport in Equation (3), the high-resolution interface capturing (HRIC) scheme is employed to discrete the convection term, and the implicit scheme is used to discrete the unsteady term. More details of these numerical schemes can be found in Ref. [31].