## 3 Validation of the Proposed Numerical Methods
### Experimental Results of the Flow around a Vertical Plane
#### 3.2.2 Comparison of the Time-History Forces and Moments  
The focus of this study is primarily on the temporal variations of the lateral force (\(F_{y}\)), rolling moment (\(M_{x}\)), and yawing moment (\(M_{z}\)) experienced by the submarine hull during vertical plane flow. There are varying levels of noise interference due to factors such as trailer body vibration and force-induced fluctuations in the submarine support components. To reduce this, the data were processed using the Savitzky-Golay filter [30]. The Savitzky-Golay filter essentially employs a moving window and implements the least square method in the time domain to achieve optimal fitting. This filtering technique effectively eliminates noise while preserving the shape and width of the signal. If the window size is set as \(2n+1\), each measurement point can be represented by \(x=\{-n,-n+1,\ldots,0,\ldots,n-1,n\}\). When fitting the data within the window using an \(m-1\) degree polynomial is of interest, the polynomial can be expressed as follows:  
\[y=a_{0}+a_{1}x+\cdots+a_{m-1}x^{m-1}, \tag{14}\]  
where \(a_{i},j\in[0,\ m-1]\) represents the coefficients of the polynomial. With the aforementioned settings, a system of \(m\) linear equations consisting of \(2n+1\) equations can be obtained, which is represented by the following:  
\[Y_{(2n+1)\times 1}=X_{(2n+1)\times m}\cdot A_{m-1}+E_{(2n+1)\times 1} \tag{15}\]  
A solution to the aforementioned equations requires \(2n+1>m\). The coefficient vector can be obtained using the least square method:  
\[A=\left(X^{T}\cdot X\right)^{-1}\cdot X^{T}\cdot Y \tag{16}\]  
After obtaining the coefficient vector (\(A\)), the smoothed value is appropriately evaluated using this vector \(\hat{Y}=X\cdot A\). In this data processing, a larger window size can smooth longer signal cycles but may result in a slower filter response. A higher polynomial order can better fit the curve of the signal, but it may also introduce excessive noise or excessively smooth the signal. Therefore, to balance the signal smoothness and preservation of signal features, the window size was set equal to 21, and the polynomial order was considered to be 3. As an example, the experimental data for the lateral force (\(F_{y}\)) at a velocity of \(w=1\) m/s has been presented in Figure 13.  
In comparing the time-domain data of \(F_{y}\), \(M_{x}\), and \(M_{z}\) of the numerical calculations and those of the experimental results, the vertical flow results were selected for three speeds in the experimental speed range (0.5, 1, and 1.6 m/s). Such comparison studies in the time-domain are provided in Figures 14-16. Due to the randomness of vortices, obtaining consistent time-domain data between experimental results and CFD calculations is somewhat difficult. Therefore, a comparison study was performed from the point of view of the force curve and amplitude. By looking  
Figure 13: Comparison between the raw and filtered data of the lateral force \(F_{y}\) at \(w=1.0\) m/s. at Figures 14-16, regarding comparing the results at \(w=0.5\) m/s, the changing trends of the experimental results of \(F_{y}\) and \(M_{z}\) basically coincide with the simulation time-domain curves, but the amplitude errors are relatively large due to the fluctuation variations. Since \(M_{x}\) has a relatively small value, the experimental data fluctuate quite a lot. With increasing speed, the variable trends of \(F_{y}\), \(M_{x}\), and \(M_{z}\) experimental results are substantially consistent with the simulation time-domain curves, but there are still certain amplitude deviations. Table 5 provides the amplitudes of different directions of the considered major forces at different speeds. From the results of Table 5, it can be seen that, with increasing speed, the error lessens to some extent. The reasons for these errors can be analyzed from two aspects. On the one hand, when the speed is low, the force characteristics are not obvious, and the sensor is hardly capable of accurately measuring.