## 4 Results and Discussion
### CFD Results  
To further elucidate the relationship between the hydraulic radial force and the circumferential pressure asymmetry on the impeller, the mid-plane pressure distributions under five flow conditions (0.8\(Q_{d}\) to 1.2\(Q_{d}\)) are analyzed, as shown in Figure 24. These  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Direction** & \begin{tabular}{c} **Flow** \\ **Condition** \\ \end{tabular}  & \begin{tabular}{c} **\(a\)** \\ **[N]** \\ \end{tabular}  & \begin{tabular}{c} **\(\omega\)** \\ **[rad/s]** \\ \end{tabular}  & \ \begin{tabular}{c} **\(\Phi\)** \\ **[rad]** \\ \end{tabular}  & \begin{tabular}{c} **\(b\)** \\ **[N]** \\ \end{tabular}  & \(R^{2}\) \\ \hline \multirow{4}{*}{X} & 0.8\(Q_{d}\) & 28.722 & 303.687 & 1.172 & \(-\)0.185 & 0.986 \\  & 0.9\(Q_{d}\) & 30.719 & 303.687 & 1.172 & 0.698 & 0.989 \\  & 1.0\(Q_{d}\) & 32.852 & 303.687 & 1.161 & 1.651 & 0.986 \\  & 1.1\(Q_{d}\) &pressure contours provide direct insights into the spatial non-uniformity of the flow field, which is the primary driver of hydraulic radial force generation. The following observations are drawn from Figure 24:  
1. Pressure Trend with Flow Rate: As the flow rate increases from \(0.8Q_{d}\) to \(1.2Q_{d}\), the overall static pressure magnitude within the impeller and solute exhibits a gradual decrease. This aligns with the pump's characteristic performance curve (see Figure 6), where the head declines with increasing flow rate due to the combined effects of hydraulic losses (e.g., friction and turbulence) and reduced energy conversion efficiency in the impeller-volute interaction. 2. Pressure Uniformity at Rated Flow: Around the rated flow condition (i.e., \(1.0\,Q_{d}\)), the pressure distribution demonstrates optimal circumferential symmetry. The interaction  
Figure 24: Mid-plane static pressure distributions under different flow conditions: (**a**) \(0.8Q_{d}\); (**b**) \(0.9Q_{d}\); (**c**) \(1.0Q_{d}\); (**d**) \(1.1Q_{d}\); and (**e**) \(1.2Q_{d}\). between the impeller blades and the volute tongue is minimized, leading to balanced pressure gradients around the impeller periphery. This aligns with the earlier findings in Sections 4.1-4.4, where the hydraulic radial force around \(1.0Q_{d}\) exhibited minimal fluctuations and a near-constant angle. 3. Increased Non-Uniformity at Off-Design Conditions: When the flow deviates from the rated point, the pressure asymmetry intensifies. However, distinct pressure distribution characteristics are observed between high-flow (e.g., \(1.2Q_{d}\)) and low-flow (e.g., \(0.8Q_{d}\)) conditions. Specifically, under the \(1.2Q_{d}\) condition, the pressure non-uniformity becomes significantly more pronounced compared to the \(0.8Q_{d}\) case. This discrepancy can be partially attributed to the amplified hydraulic radial forces on the impeller at higher flow rates, as illustrated in Figure 8a, where the intensified circumferential pressure gradients directly correlate with the elevated force magnitudes. While the CFD-derived pressure contours qualitatively illustrate the flow non-uniformity, they lack the temporal resolution to quantitatively evaluate the dynamic evolution of the hydraulic radial force. This limitation underscores the significance of the data-centric approach adopted in this study. By leveraging high-fidelity CFD datasets, the time-resolved hydraulic radial forces are systematically decomposed, revealing periodic patterns and harmonic cancellations between symmetric blades (Section 4.3). Trigonometric fitting and FFT analysis further quantified the amplitude-phase relationships, enabling a deeper understanding of the force composition. Such insights are unattainable through conventional flow field visualization alone. To advance the suppression of hydraulic radial forces, future work should integrate advanced data mining techniques (e.g., proper orthogonal decomposition or machine learning) with mechanistic interpretations of CFD-predicted flow structures.