# Numerical Study on Aerodynamic Characteristics of High-Speed Planing Trimaran
## 2 Materials and Methods
### CFD Condition Setting and Accuracy Verification  
#### 2.2.1 Governing Equations and Turbulence Models  
The maximum speed in the study is 25 m/s. Although the speed is very high for the planing craft, it is still far lower than the acoustic velocity. It belongs to low speed flow, and the change of air density can be ignored. Therefore, the incompressible air flow is selected in the study.  
The simulation of the flow field is essentially to solve the N-S equation. The equation is as following:  
\[\frac{\partial V}{\partial t}+(V\cdot\
abla)V=f-\frac{1}{\rho}\
abla p+\frac{ \mu}{\rho}\
abla^{2}V \tag{2}\]  
During simulation, the governing equation of incompressible viscous flow is described by the Reynold-averaged Navier-Stokes equations (RANSE), which are the most widely used method in engineering. The equation is described as follows:  
\[\frac{\partial\overline{u_{i}}}{\partial t}+\overline{u_{j}}\frac{\partial \overline{u_{i}}}{\partial x_{j}}+\frac{\partial}{\partial x_{j}}\left( \overline{\overline{u_{i}^{\prime}}u_{j}^{\prime}}\right)=-\frac{1}{\rho} \frac{\partial\overline{p}}{\partial x_{i}}+v\frac{\partial\overline{2^{-}u_{ i}}}{\partial x_{i}\partial x_{j}}, \tag{3}\]  
where \(\overline{u_{i}}\),\(\overline{u_{j}}\) denote the time-averaged velocity components;\(u_{i}^{\prime}\),\(u_{j}^{\prime}\) are the fluctuations of the velocity components; \(\overline{p}\) is the time-averaged pressure; \(\rho\) is dynamic viscosity coefficient; t is time; and \(x_{i}\), \(x_{j}\) are unit vectors in directions of \(i\) and \(j\). The Reynolds number in the study is very high, reaching the order of \(10^{6}\), far exceeding the minimum standard of turbulence, so the turbulence model is selected in the study. The standard k-\(\epsilon\) turbulence model is selected in the study, which has a wide range of application and reasonable calculation accuracy in the flow field calculation, and has been widely used in the calculation of engineering flow field. The equation is described as follows:  
The transported variable of turbulent kinetic energy, \(k\), is defined as follows:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{j}}\Bigg{[}\left(\mu_{1}+\frac{\mu_{t}}{ \sigma_{k}}\right)\frac{\partial k}{\partial x_{j}}\Bigg{]}+\mu_{t}\frac{ \partial u_{j}}{\partial x_{i}}\Bigg{(}\frac{\partial u_{j}}{\partial x_{i}} +\frac{\partial u_{i}}{\partial x_{j}}\Bigg{)}-\rho\epsilon, \tag{4}\]  
Figure 4: Longitudinal position.  
The second transported variable \(\varepsilon\) is defined as follows:  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}}( \rho\varepsilon u_{i})=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu_{1}+ \frac{\mu_{t}}{\sigma_{\varepsilon}}\bigg{)}\frac{\partial\varepsilon}{ \partial x_{j}}\Bigg{]}+C_{1}\mu_{t}\frac{\varepsilon}{k}\frac{\partial u_{j} }{\partial x_{i}}\Bigg{(}\frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u _{i}}{\partial x_{j}}\Bigg{)}-C_{2}\rho\frac{\varepsilon^{2}}{k}, \tag{5}\]  
where \(\mu_{1}\) is the laminar eddy viscosity coefficient, and \(C_{1}\), \(C_{2}\), \(\sigma_{k}\), \(\sigma_{\varepsilon}\) are empirical constants.