## 5 Composition Law and Application Enlightenment
### Research Innovation  
This study introduces a data-centric harmonic decomposition framework to analyze hydraulic radial forces on centrifugal pump impellers, advancing beyond traditional holistic CFD models. By integrating high-fidelity CFD datasets with trigonometric fitting, the research reveals that the radial force on a six-blade impeller can be decomposed into periodic components from individual blades, each characterized by identical amplitudes/frequencies but successive 60\({}^{\circ}\) phase shifts corresponding to blade spacing (Figures 16 and 17). Unlike prior works [23; 24; 40] that treat the impeller as an undivided entity, this study demonstrates that centrally symmetric blades (e.g., Blade 1 and Blade 4) cancel out primary harmonic forces (e.g., blade passing frequency components), leaving only constant terms and residual fluctuations. This mechanism highlights the critical role of blade symmetry in suppressing unsteady flow-induced pulsations, a phenomenon not explicitly quantified in conventional CFD-based optimization approaches [25; 26; 27].  
The innovation lies in linking localized blade-wise pressure dynamics to global force characteristics through data-driven decomposition. This approach contrasts with traditional trial-and-error methods (e.g., volume shape modifications [26]) by offering a quantifiable framework to predict force behavior through data analysis. By validating this with CFD and experimental data (Section 2.5), the research establishes a robust methodology for centrifugal pump design, with potential extensions to other turbomachinery requiring vibration control. Recent studies [45; 46] have similarly emphasized the efficacy of CFD-machine learning integration for pump optimization, but this work uniquely focuses on harmonic decomposition to decode force mechanisms. Additionally, literature on blade number effects [47] and fluid-structure interaction contextualizes the symmetry-driven cancellation phenomenon observed here.  
However, this methodology has certain limitations that warrant attention. First, the current framework relies on Reynolds-Averaged Navier-Stokes (RANS) simulations with the _SST_\(k\)-\(\omega\) turbulence model, which may underresolve fine-scale turbulent structures (e.g., small-scale vortices near blade tips). Future work could incorporate large-eddy simulation (LES) to capture unsteady flow features with higher fidelity, particularly in extreme flow conditions like severe cavitation or stall. Second, the study focuses on a six-blade symmetric impeller, and its applicability to asymmetric blade configurations (e.g., odd-numbered blades) remains untested. Extending the harmonic decomposition to such systems would require modifying the phase-shift analysis to account for non-uniform blade spacing. Third, this study represents a theoretical preliminary exploration of force decomposition, and the proposed methodology has not yet been applied to practical impeller optimization design. While the data-driven framework offers a promising basis for reducing radial forces, its effectiveness in real-world applications--such as guiding blade geometry modifications--requires validation through targeted optimization case studies. Future research should apply the harmonic decomposition method to specific impeller designs, quantify the reduction in radial force and vibration levels, and evaluate its engineering utility through prototyping and experimental testing. This step is critical to bridging the gap between theoretical insights and industrial implementation.