# A CFD Study of the Hydrodynamic Characteristics of an Autonomous Underwater Helicopter
## 2 Methodology and Numerical Approach  
### Methodology  
The AUH operates almost at the seabed, completing tasks and missions that require motion stability, communication, charging, etc. In deep-sea exploration, many factors influence the disk-shaped hull, so we will attempt to achieve high maneuverability. Based on previous studies, the high maneuverability of the AUH is usually obtained by optimizing the geometry and thruster system. The AUH, equipped with two thrusters, experiences challenges in maintaining a steady orientation in the water. Conversely, employing four thrusters enables the AUH to modify and regulate its posture with greater efficacy, hence assuring steady motion. The hovering-type AUV features a symmetrical design and eight thrusters that provide enhanced stability; however, it occupies more space and results in increased energy losses. To increase the installation area for sensors and minimize energy consumption, the AUH is designed with four propellers, comprising two vertical and two horizontal propellers. The disk-shaped hull is defined as a body of revolution, with height and diameter being two critical dimensions for the profile of the AUH's hull. The Non-Uniform Rational B-Splines (NURBSs) include a curved B-spline and Bezier. It is convenient for making the design of curvature. As shown in Figure 1 and detailed in Table 1, the parameters used to model the AUH in ANSYS Fluent are defined. The AUH's parametric profile is sketched with three control points on a quarter of its outline. \begin{table}
\begin{tabular}{c c} \hline
**Parameter.** & **Value** \\ \hline h1 & 100 mm \\ \hline h2 & 160 mm \\ \hline h3 & 300 mm \\ \hline h4 & 400 mm \\ \hline L1 & 125 mm \\ \hline L2 & 250 mm \\ \hline L3 & 375 mm \\ \hline L4 & 1000 mm \\ \hline \end{tabular}
\end{table} 
Table 1: Geometric characteristics of AUH utilized for computation. Figure 1: The schematic illustration is employed for calculations. Reynolds-Averaged Navier-Stokes (RANS) equations offer the most efficient approach for analyzing viscous flow. The \(k\)-\(\varepsilon\) or \(k\)-\(\omega\) two-equation turbulence model is a turbulence model utilized in industrial applications. They are to perform a numerical analysis of hydrodynamic coefficients for AUV [31; 32]. The governing equations encompass the conservation of mass, momentum, and Reynolds-Averaged Navier-Stokes (RANS) equations, which are employed to analyze the single-phase fluid flow surrounding the autonomous underwater helicopter and are expressed as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho\cdot u_{ i})=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{j}}\big{(} \rho u_{i}u_{j}\big{)}=\frac{\partial p}{\partial x_{i}}+\frac{\partial}{ \partial x_{i}}\Bigg{[}\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{i}}{ \partial x_{i}}\right)\Bigg{]}+\frac{\partial}{\partial x_{j}}\Big{(}-\rho \overline{u_{i}^{\prime}u_{j}^{\prime}}\Big{)} \tag{2}\]  
where \(u_{i}\) and \(u_{j}\) are the mean average flow velocities, \(u_{i}^{\prime}\) and \(u_{j}^{\prime}\) are fluctuation velocities, \(\rho\) is the density of water, \(p\) is the pressure, \(\mu\) is the fluid dynamic viscosity, \(\delta_{ij}\) is the Kronecker delta function, and \(-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\) is the Reynolds stress tensor, which shows the turbulence effect. The Reynolds stress for incompressible flow is expressed as follows:  
\[-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}=\mu_{t}\left(\frac{\partial u_{i }}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)-\frac{2}{3} \left(\rho k+\mu_{t}\frac{\partial u_{i}}{\partial x_{i}}\right)\delta_{ij} \tag{3}\]  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\varepsilon} \tag{4}\]  
where \(C_{\mu}\) is a dimensionless constant.