## 3 Setting and Verification of Numerical Simulation Strategy
### Kappel Propeller Modeling  
Parametric modeling is the process of controlling the shape of the model using various parameters, and the difference between the Kappel tip rake propeller and conventional propeller is mainly that the rake distribution changes, and it is expected that the energy loss caused by the spire tip flow and tail vortex shrinkage can be effectively reduced by optimizing the blade tip geometry. In this paper, the NACA0066 (a = 0.8, mod) is selected as the basis for the design and modeling of the Kappel tip rake propeller, and the geometric 3D coordinate conversion of the Kappel propeller is changed on the basis of the traditional propeller 3D coordinate conversion [18] as follows:  
\[\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]=\left[\begin{array}{c}X_{S}+C(S-0.5)\sin\varphi\cdot\cos \phi-f\cos\varphi\cos\phi\mp 0.5t\cos\varphi\cos\phi\\ r\cos\theta+C(S-0.5)\sin\varphi\cdot\sin\phi-f\cos\varphi\sin\phi\pm 0.5t\cos \varphi\sin\phi\\ r\sin\theta\pm 0.5t\sin\varphi\end{array}\right] \tag{8}\]  
where \(X\), \(Y\), and \(Z\) are the three-dimensional coordinate values of the points on the blade surface, \(r\) is the radius of the propeller at the corresponding section, \(X\)s is the rake, \(C\) is the blade chord length, \(S\) is the dimensionless length distribution of two-dimensional airfoil in the chord direction, \(\Phi\) is the pitch, \(\theta\) is the skew,\(f\) and \(t\), respectively, represent the camber and the thickness, and \(\varphi\) is the angle between the tangent of the distribution curve of rake and the axis of abscissa.  
In this paper, the radial distribution of rake is adjusted by changing the tip rake \(X_{S}\)/D, taking \(X_{S}\)/D = 0.000, 0.025, 0.050, 0.075, 0.100, and 0.125, and using the B-spline curve to design the distribution of rake along the propeller radius r/R, as shown in Figure 1, corresponding to the Kappel tip rake propellers Kap00, Kap01, Kap02, Kap03, Kap04, and Kap05, respectively. It should be noted here that the radial distribution of rake is consistent with that of the conventional end plate propeller before 0.9R through the fourth-order cubic B-spline control points, and begins to change to varying degrees after 0.9R. Numbers 1-5 in Figure 1 represent the 5 control points used for curve fitting.  
The geometry of the Kappel tip rake propeller is shown in Figure 2. It can be seen from the figure that the degree of blade tip bending increases with the increase in rake, and the blade tip bending of Kap05 propeller is the most obvious. Other major geometrical parameters of the Kappel tip rake propeller are shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Diameter (mm)**} & **Pitch Ratio at** & \multirow{2}{*}{**Skew (\({}^{\circ}\))**} & **Projected Area** & **Hub Diameter** \\  & **0.7R** & & **Ratio** & **Ratio** \\ \hline
250 & 1.0338 & 16 & 0.4788 & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Other major geometrical parameters of the Kappel tip rake propeller.  
Figure 1: The radial distributions of rake \(X_{S}\)/D belonging to the six types of propellers [19].