## 3 Results and Discussion  
### Effects of the Rotational Speed  
The decay law of swirl intensity of spiral flow induced by the hole cleaning device (\(\alpha=0^{\circ}\)) is shown in Figure 4 under four different speeds, and the decay equation was obtained by fitting and regression displayed in Table 3. Swirl intensity was used to charac \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mesh Numbers** & **Pressure Drop (Pa/m)** & **Error (\%)** \\ \hline
522,100 & 1284.3 & 6.21\% \\
784,950 & 1363.2 & 0.45\% \\
1,036,950 & 1367.0 & 0.17\% \\
1,396,200 & 1369.3 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Sensitivity of grid size to simulation results. Figure 3: Comparison between simulated and experimented cuttings bed height. terize the swirl intensity in the spiral flow field, also known as swirl number \(S\), which is defined as follows [12; 25]:  
\[S=\frac{\int_{0}^{R}{u}{w}{\rho}_{1}{r}^{2}dr}{R\int_{0}^{R}{u}^{2}{\rho}_{1}rdr} \tag{21}\]  
where \(u\) is the axial velocity and \(w\) is the circumferential velocity. In the process of blade rotation, the spiral flow decays following the exponential law [26; 27; 28], and its decay equation is expressed as follows:  
\[S=S_{0}e^{-\beta L/D_{h}} \tag{22}\]  
where \(S_{0}\) is the swirl intensity at the end section of the blade, that is, the initial swirl intensity; \(\beta\) is the decay coefficient; \(L\) is the distance; and \(D_{h}\) is the hydraulic diameter. The initial swirl intensity \(S_{0}\) increases as the rotational speed increases. When the rotational speed is 200 rpm, the initial swirl intensity \(S_{0}=0.361\), which is 3.72 times that of 80 rpm, but the increasing rate gradually slows down at a high rotational speed. For example, at 80, 120, and 160 rpm, the increasing values of the initial swirl intensity \(S_{0}\) are 0.075, 0.082, and 0.107, respectively, for each 40 rpm increase in rotational speed. The decay coefficient of spiral flow is more significant when the rotational speed is more considerable. The research results show that the decay coefficient at 200 rpm is about 41.6% higher than that of a rotational speed at 80 rpm, as shown in Table 3. The reason for this is that the energy loss caused by wall friction and fluid viscosity is also significant at a high rotational speed. Figure 5 shows the time-averaged fluid's circumferential velocity distribution under different rotational speeds. The circumferential velocity of the fluid behind the suction  
Figure 4: Decay law of swirl intensity of spiral flow induced by the hole cleaning device (\(\alpha=0^{\circ}\)) along the flow direction under different rotational speeds (numerical results). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rotational** & **Decay Equation** & **Initial Swirl** & **Decay** & **\(\mathbf{R^{2}}\)** \\
**Speed rpm** & & **Intensity \(S_{0}\)** & **Coefficient \(\beta\)** & **\(\mathbf{R^{2}}\)** \\ \hline
80 & \(S_{n}=0.097e^{-0.012L/D_{h}}\) & 0.097 & 0.012 & 0.993 \\
120 & \(S_{n}=0.172e^{-0.013L/D_{h}}\) & 0.172 & 0.013 & 0.982 \\
160 & \(S_{n}=0.254e^{-0.015L/D_{h}}\) & 0.254 & 0.015 & 0.990 \\
200 & \(S_{n}=0.361e^{-0.017L/D_{h}}\) & 0.361 & 0.017 & 0.975 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Decay equation of spiral flow induced by the hole cleaning device (\(\alpha=0^{\circ}\)) under different rotational speeds (numerical results). surface of the blade is large under the action of a rotation. The circumferential velocity intensity and distribution range of fluid increase with the rotational speed increase. The maximum tangential velocity exceeds 1.35 m/s as the rotational speed is 200 rpm. The main principle of the hole cleaning device is to reduce the thickness of the cuttings bed and to transfer cuttings to the high side of the wellbore by stirring and shearing the cuttings bed at the low side of the wellbore.