Operational Mechanisms and Energy Analysis of Variable-Speed Pumping Stations

Abstract

The spatiotemporal uneven distribution of water resources conflicts sharply with human demands, with pumping stations facing efficiency decline due to aging infrastructure and complex hydraulic interactions. This study employs numerical simulation to investigate operational mechanisms in a parallel pump system at the Yanhuanding Yanghuang Cascade Pumping Station. Using ANSYS Fluent 2024 R1 and the SST k-ω turbulence model, we demonstrate that variable-speed control expands the adjustable flow range to 1.17–1.26 m³/s while maintaining system efficiency at 83–84% under head differences of 77.8–79.8 m. Critically, energy losses (δH) at the 90° outlet pipe junction escalate from 3.8% to 18.2% of total energy with increasing flow, while Q-criterion vortex analysis reveals a 63% vortex area reduction at lower speeds. Furthermore, a dual-mode energy dissipation mechanism was identified: at 0.90n₀ speed, turbulent kinetic energy surges by 115% with minimal dissipation change, indicating large-scale vortex dominance, whereas at 0.80n₀, turbulent dissipation rate increases drastically by 39%, signifying a shift to small-scale viscous dissipation. The novelty of this work lies in the first systematic quantification of junction energy losses and the revelation of turbulent energy transformation mechanisms in parallel pump systems. These findings provide a physics-based foundation for optimizing energy efficiency in high-lift cascade pumping stations.

1. Introduction

The spatiotemporal uneven distribution of water resources is a global challenge, leading to a heavy reliance on pumping stations for water transfer. However, pumping systems are among the largest energy consumers in water infrastructure, accounting for a significant portion of operational costs and carbon footprints [1]. To address the disparity between water availability and demand, inter-basin water diversion projects have been widely implemented as a key strategy for reallocating water resources [2]. Within these projects, large-scale centrifugal pumping stations serve a critical function, enabling long-distance water transfer and supporting extensive irrigation and drainage networks [3]. As fundamental elements of such systems, pumping facilities facilitate mechanical lifting across elevation changes, with their operational efficiency directly influencing the overall economic viability of water diversion projects [4].

However, the rated efficiency of most medium- and large-sized centrifugal pumps ranges from 65% to 85% or even lower [5]. Approximately 2% to 3% of global electricity consumption is used for pumping in water supply systems, making the optimization of pumping stations crucial for achieving substantial energy savings [6,7]. The Yanhuanding Yanghuang Cascade Pumping Station in Shaanxi-Gansu-Ningxia is one of China’s pivotal water diversion projects, providing robust support for socioeconomic development and ecological improvement in regions such as Shaanxi, Gansu, and Ningxia [8]. Since its commissioning, the project has cumulatively transferred over 5 billion cubic meters of water, offering essential guarantees for regional poverty alleviation and ecological restoration [9].

In recent years, rising global energy costs and intensifying climate change have prompted nations to focus on optimizing pumping station scheduling to reduce carbon emissions and operational costs [10]. For example, China’s South-to-North Water Diversion Project employs Variable-Frequency Drive (VFD) technology to enhance pumping station efficiency, but challenges such as vibration and cavitation persist due to complex terrain and dynamic demands [2]. Internationally, Brati et al. evaluated different solutions for improving the energy efficiency of pressurized irrigation systems, and compared them with the energy efficiency of conventional irrigation systems; it is verified that the most important factors affecting the efficiency of the system are the type of pump and motor, operating conditions, and the size of special components such as valves, fire hydrants, and pipes [11,12]. However, persistent challenges remain, particularly under ultra-low water level conditions and in achieving high efficiency during multi-pump parallel operation, which demand innovative solutions [3]. Beyond technical hurdles, optimizing pumping station scheduling represents a critical pathway toward achieving carbon neutrality goals, underscoring its broader environmental significance [13]. In response, scholars have explored various optimization strategies, primarily falling into three categories: numerical simulation, intelligent algorithms, and experimental validation. Early research efforts predominantly relied on fixed-speed pumps (FSPs) and genetic algorithms (GAs) for optimizing start-stop strategies; however, these methods often suffered from computational inefficiency and operational response lag [14]. The advent and widespread adoption of variable-frequency technology marked a significant advancement, enabling variable-speed pumps (VSPs) to dynamically adjust rotational speeds to match real-time demand, thereby demonstrating substantial energy-saving potential. For instance, Lamaddalena and Khila [15,16] developed an efficiency-driven control strategy based on network characteristic curves, reporting energy savings of up to 35%. Complementing this, Cimorelli et al. [17] conducted an economic comparison, demonstrating that VSPs offer lower lifecycle costs compared to FSPs. Concurrently, in the realm of intelligent algorithms, techniques such as ant colony optimization (ACO) and particle swarm optimization (PSO) have been applied to multi-objective scheduling problems, seeking to balance the often-competing demands of energy consumption and system reliability [18,19]. Existing methods are prone to local optima in complex pipe networks and lack real-time adaptation to dynamic water levels [20]. Combined rectification schemes (e.g., flow-guiding platforms and vortex elimination plates) can improve flow patterns but rely on high-precision numerical models [21].

Recent advances in intelligent control include the application of digital twin technology to pumping station optimization. For example, Fan Feng et al. developed a high-precision digital model for a large-scale water diversion project, achieving full automation and significant cost savings [22]. Separately, Hallaji et al. combined BIM with deep learning to create a predictive maintenance framework for sewage pumping stations [23]. While these system-level approaches show promise, their effectiveness depends on an accurate understanding of internal pump hydraulics, particularly energy loss mechanisms and turbulence under variable-speed conditions, which remain poorly characterized. This study addresses this gap through high-fidelity CFD analysis of parallel pump systems, providing the physical insights necessary to develop reliable digital twins for high-lift pumping stations.

However, existing research on pumping stations still exhibits notable limitations. While Lamaddalena and Khila [15] demonstrated the significant energy-saving potential of variable-speed control, their work did not explore the hydraulic loss mechanisms at critical pipeline junctions. Similarly, Cimorelli et al. [17] conducted an economic comparison between variable-speed and fixed-speed pumps but did not investigate the underlying physical mechanisms within the flow field. These knowledge gaps are particularly relevant for high-lift water diversion projects such as Yanhuanding Pumping Station, which has long faced operational challenges including significant flow fluctuations and high energy consumption.

To address these issues, this study employs high-fidelity CFD simulations to investigate the energy loss mechanisms and operational characteristics of a parallel pump system under variable-speed conditions. Specifically, we aim to optimize the matching of flow rates, heads, and rotational speeds at the Yanhuanding Yanghuang Cascade Pumping Station in Shaanxi-Gansu-Ningxia, with the goal of improving operational efficiency and reducing energy consumption. The novelty of this work lies in (a) quantitatively evaluating the energy loss (δH) at the confluence junction and its impact on system efficiency, and (b) revealing vortex dynamics and turbulent energy dissipation patterns under variable-speed operations using Q-criterion and turbulent statistics.

2. Materials and Methods

2.1. Research Object

The Yanhuanding No. 2 Pumping Station, as the core cascade pumping station of the Shaanxi-Gansu-Ningxia Yanhuanding Yanghuang Shared Project, is equipped with 7 units (5 large pumps, 1 small pump, and 1 dedicated irrigation pump). It has a total installed capacity of 19,950 kW, a shaft power of 15,476 kW, and is responsible for high-lift water delivery tasks with a design flow rate of 10.95 m³/s and a total head of 92.5 m (net head of 82.03 m). This study focuses on a system of 6 units (excluding the irrigation unit), composed of 5 large pumps and 1 small pump. The pump groups are divided into three independent units: Pump Group 1 and Pump Group 2 each consist of 1 fixed-frequency large pump and 1 variable-frequency large pump, while Pump Group 3 comprises 1 fixed-frequency large pump and 1 fixed-frequency small pump, corresponding to three water conveyance pipeline systems, as shown in Figure 1. Based on the premise of ignoring the flow effects of inlet and outlet reservoirs, it is assumed that there is no hydraulic interference between pump groups. Therefore, Pump Group 2 is selected as the research object to conduct numerical simulations of variable-frequency speed regulation and flow matching, aiming to optimize computational resource allocation. The study employs self-designed hydraulic models for the large and small pumps (the fixed-frequency and variable-frequency large pumps share the same large pump hydraulic model). The external characteristic curves of the large and small pumps, obtained through CFD simulations, are shown in Figure 2. Although parameter deviations exist compared to the actual engineering models, these align with the core objective of this study: investigating the rotational speed–flow rate matching mechanism under varying water levels.

The overall workflow of the numerical simulation process, from geometric modeling to result analysis, is illustrated in Figure 3.

It is important to note that the findings of this study are based solely on numerical simulations. While the CFD model has been constructed with care and mesh independence was achieved, the lack of experimental or field data for validation remains a limitation. Future work will involve collaborating with the pumping station to obtain operational data for direct comparison and validation of the numerical results, which would further enhance the credibility and practical applicability of the model.

2.2. Grid Generation and Independence Verification

This study employs the Finite Volume Method (FVM), based on conservation laws, to discretize and solve the governing equations [24]. The computational domain consists of a complex flow channel system comprising impellers (2 sets), spiral casings (2 sets), inlet pipelines (2 sets), and outlet pipelines (1 set). Polyhedral meshes are uniformly applied to the fluid domain, with refined mesh division near the impeller and spiral casing regions. To ensure mesh transition, identical mesh scales are adopted at interfaces, and mesh transition zones are set at the outlet of the inlet pipe and the inlet of the outlet pipe to effectively mitigate numerical errors caused by abrupt mesh scale changes [25]. A mesh independence verification was conducted through a four-level mesh scheme (mesh counts: 1.28 million, 1.86 million, 2.54 million, and 3.42 million) by monitoring and comparing key parameters such as flow rate. The results indicate that when the mesh count reaches 2.54 million, all relative errors are below 1.5%, satisfying the engineering accuracy requirements [26].

Table 1. Grid independence verification summary table.

Grid ID	Number of Grid Cells (N)	Flow Rate (Q [m3/s])	Relative Error (%) (vs. Adjacent Grid)	Recommended?	Remarks
1	1,267,810	4.736	2.65% (vs. Grid 2)	No	Coarse grid, large error
2	1,835,485	4.865	1.79% (vs. Grid 3)	No	Medium grid, moderate error
3	2,541,334	4.955	0.26% (vs. Grid 4)	Yes	Optimal: error <1%, balanced computational cost
4	3,453,146	4.968	-	No	Finest grid, limited gains

summarizes the mesh independence verification. The grid division of Pump Group 2 is shown in Figure 4.

2.3. Governing Equation and Turbulence Model

Numerical simulation is a method where the governing equations of the flow field are discretized to the grid nodes by computational mathematics to obtain the discrete numerical solution [27]. The flow of fluid follows the law of conservation of mass, the law of conservation of momentum, and the law of conservation of energy. The equations are expressed as follows:

The continuity of Equation (1) ensures mass conservation by balancing the inflow and outflow rates within each computational cell:

$${\displaystyle \frac{\partial \overline{u_i}}{\partial x_i}}=0$$

(1)

where $x_i$ is coordinate component, $\overline{u_i}$ is the Reynolds-averaged velocity components along the Cartesian coordinates.

The momentum conservation of Equation (2) governs the fluid motion under the combined effects of pressure gradients, viscous stresses, and turbulence [28]:

$$\rho {\displaystyle \frac{\partial \overline{u_i}}{\partial t}}+\rho \overline{u_j}{\displaystyle \frac{\partial \overline{u_i}}{\partial x_j}}=\rho f_i-{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\mu {\displaystyle \frac{\partial \overline{u_i}}{\partial x_j}}-\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)$$

(2)

Here, ρ is fluid density, t is time coordinate, $\overline{p}$ is time-averaged pressure, μ is kinematic viscosity, $\rho \overline{u_i^{\prime }{u}_j^{\prime }}$ is Reynolds stress for the turbulence flow, and f_i is the body force acting on the unit volume fluid.

The SST k-ω turbulence model (3–4) was selected for its dual capability: it accurately resolves near-wall viscous sublayers through the k-ω formulation, while maintaining robustness in free-shear flows via a blended k-ε approach. This hybrid strategy is critical for capturing energy dissipation mechanisms in high-lift pumping stations, where both wall-bounded turbulence and impeller wake interactions dominate [29]:

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_{j}k\right)-{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_k{\mu }_{\mathrm{t}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]=P_k-{\beta }^{\prime }\rho k\omega$$

(3)

$$\begin{array}{l}{\displaystyle \frac{\partial }{\partial t}}\left(\rho \omega \right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_j\omega \right)=\\ C_{\omega }{P}_k-\beta \rho {\omega }^2+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_{\mathrm{t}}\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+2(1-F_1){\displaystyle \frac{\rho {\sigma }_{\omega 2}}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}\end{array}$$

(4)

where k is turbulence kinetic energy, x_j is spatial coordinate, P_k is production term of k, μ is dynamic viscosity, ω is the specific dissipation rate, μ_t is turbulence kinematic viscosity, C_ω is the coefficient of the production term, F₁ is blending function, σ_ω, β and σ_ω2 are closure parameters.

2.4. Numerical Simulation Method and Definitions

ANSYS Fluent 2024 R1 software is applied to calculate the flow field of Pump Group 2. The medium is water with a density of 998 kg/m³. The turbulence model adopts the SST k-ω model, with Enhanced Wall Treatment for near-wall modeling. ANSYS Fluent 2024 R1 was selected for its robust capabilities in handling complex pipe flow systems and validated turbulence models. Compared to CFX, Fluent offers greater flexibility in customizing boundary conditions for multi-pump parallel systems. The SST k-ω model was adopted due to its superior accuracy in resolving near-wall turbulence and shear stress propagation, critical for capturing energy dissipation in high-lift pumping stations. Based on the operational characteristics of the pumping station, the inlet and outlet boundary conditions of the computational model are set as pressure inlet and pressure outlet, respectively. The outlet pressure is defined as 0 Pa, with specific inlet pressure values configured according to different upstream and downstream head differences. In this study, the head difference is defined as the water level difference between the upstream and downstream reservoirs, calculated by the formula: Head difference = Hupstream − H_downstream. All other walls are defined as no-slip walls. By adjusting the inlet/outlet pressures and the rotational speed of the No. 4 variable-frequency pump, flow field results under different flow conditions are obtained. The two impeller regions are defined as rotating domains with a rotational speed of n = 740 r/min, while the remaining parts are set as stationary domains.

The rotational speed range of the variable-frequency pump (592–740 r/min) was selected based on the actual operational range of the pumping station and the affinity laws to ensure the pump operates within its safe and efficient envelope. The head differences (77.8 m, 78.8 m, 79.8 m) were chosen according to the typical fluctuation range of the upstream and downstream water levels recorded in the historical operational data of the Yanhuanding Pumping Station.

To facilitate the post-processing and analysis of the numerical results in subsequent sections, the following key parameters are defined and will be utilized throughout this study.

The dimensionless pressure coefficient C_p is adopted to analyze the pressure distribution. It is defined as:

$$C_p={\displaystyle \frac{p-\overline{p}}{\rho g{H}_d}}$$

(5)

where

• p: Local pressure.
• $\overline{p}$: Average pressure at a specific cross-section.
• ρ: Density of the medium.
• g: Gravitational acceleration.
• H_d: Water level difference.

Turbulent kinetic energy (k) is a core indicator for measuring the intensity of turbulent fluctuations, representing the kinetic energy generated by velocity fluctuations (i.e., the difference between instantaneous velocity and mean velocity) per unit mass of fluid. A higher k value indicates more intense turbulent motion in fluid particles, such as in high-turbulence regions around aircraft wings, where k values increase significantly.

$$k={\displaystyle \frac{1}{2}}({u^{\prime }}^2+{v^{\prime }}^2+{w^{\prime }}^2)$$

(6)

where

• k: Turbulent kinetic energy.
• u′: Velocity fluctuation component in the x-direction.
• v′: Velocity fluctuation component in the y-direction.
• w′: Velocity fluctuation component in the z-direction.

The turbulent dissipation rate ε represents the rate at which TKE is converted into thermal energy due to viscous effects per unit mass of fluid, reflecting the dissipation characteristics of turbulent energy. It is defined as:

$$\epsilon =\nu {\left({\displaystyle \frac{\delta {u^{\prime }}_i}{\delta x_j}}\right)}^2$$

(7)

where

• ν: Kinematic viscosity of the fluid.
• ${\displaystyle \frac{\delta {u^{\prime }}_i}{\delta x_j}}$: Fluctuation velocity gradient tensor.

3. Results and Discussion

3.1. External Characteristic Curves

When adjusting the flow rate around 1.05 m³/s, Pump No. 6 can be driven for water extraction. For flow rates below 1.05 m³/s, the variable-frequency Pumps No. 2 and No. 4 are activated. Due to the similarity between Pump Group 1 and Pump Group 2, this study analyzes the rotational speed–flow rate matching under varying water level differences in Pump Group 2, providing foundational data and support for subsequent variable-frequency speed regulation research. Focusing on Pump Group 2, this section investigates the impact of variable-frequency pump speed adjustments on pumping station performance under different water levels. Pump Group 2 consists of two pumps feeding into a single combined pipe via two separate pipelines. The two pumps and pipeline system are treated as an integrated entity, and the characteristic curves of the entire pump-pipe system are plotted. Figure 5 illustrates the flow rate–head curve, flow rate–power curve, and flow rate–efficiency curve of the integrated pump-pipe system under different head differences (between upstream and downstream water levels). And performance characteristics of the fixed-frequency and variable-frequency pump system in Pump Group 2 under different water levels are listed in

Table 2. Performance characteristics of the fixed-frequency and variable-frequency pump system in Pump Group 2 under different water levels.

Head Difference	Q [m3/s]		H [m]		p [kw]		H [%]		n [r/min]
	NO. 3	NO. 4	NO. 3	NO. 4	NO. 3	NO. 4	NO. 3	NO. 4	NO. 3	NO. 4
79.8 m	2.614	1.759	89.18	87.71	2552	1805	89.61	83.82	740	700
78.8 m	2.659	1.714	88.17	86.49	2565	1747	89.64	83.23	740	695
77.8 m	2.703	1.668	87.14	85.26	2577	1688	89.64	82.64	740	690

.

Figure 5 provides critical insights into the system’s performance under variable head differences. As shown in Figure 5a, increasing the head difference from 77.8 m to 79.8 m expands the adjustable flow range of Pump No. 4 from 1.17 m³/s to 1.26 m³/s and shifts the system curves toward lower flows. Figure 5b reveals that although head and input power increase with higher head differences, the high-efficiency zone shifts slightly toward lower flow rates. This provides novel guidance for selecting optimal operating points under varying water levels.

Most significantly, Figure 5c demonstrates that a variable-speed operational range of 700–740 r/min maintains a consistently high system efficiency (83–84%) across all tested head differences, with a broad flow range of 4.37–5.00 m³/s. The stable system efficiency of 83–84% achieved in this study aligns with the findings of Lamaddalena and Khila [15], who reported significant energy savings through variable-speed control in irrigation systems.

A quadratic polynomial fitting is applied to the flow rate–head curve of the integrated pump-pipe system. The corresponding quadratic polynomial equations for each head difference are listed in

Table 3. Quadratic polynomial fitting expressions for flow rate–head curves of the pump-pipe system under three water levels.

Head Difference	Quadratic Polynomial
79.8 m	H = 77 + 0.4097 Q2
78.8 m	H = 76 + 0.4097 Q2
77.8 m	H = 75 + 0.4097 Q2

, which describe the relationship between the energy demand (head) and flow rate in the pipeline system. The quadratic coefficients in these equations are identical, while the intercept values correlate with the head difference, calculated as the head difference minus the inherent height of the integrated pump-pipe system. Thus, the flow rate–head fitting curve can be expressed as H = Hp + kQ², where Hp represents the flow-independent head demand (i.e., the pump static head), and k is the resistance coefficient of the pump-pipe system, related to frictional and local resistance losses and proportional to the square of the flow rate. Notably, changes in water level do not alter the value of k.

To further analyze the impact of the flow rate of the No. 4 variable-frequency pump on the No. 3 fixed-frequency pump, a case study under a head difference of 78.8 m is conducted. Figure 6 illustrates the flow rate–head curves of the No. 3 fixed-frequency pump, No. 4 variable-frequency pump, combined No. 3 + 4 pumps, and the pumping station system. Notably, the flow rate in the No. 3 + 4 curve represents the sum of the flow rates from both pumps, while the head is calculated as the weighted average of the two pumps’ heads based on their flow rates. The results show that the flow rate of the No. 4 variable-frequency pump increases with its rotational speed, whereas the flow rate of the No. 3 fixed-frequency pump decreases correspondingly. The head difference between the two pumps and the pumping station system corresponds to the flow loss δH in the inlet and outlet pipes. A quadratic polynomial fitting of δH yields δH = 12.376 − 7.024 Q + 1.162 Q₂, indicating that δH increases with flow rate. When the total system flow rate rises from 2.85 m³/s to 3.46 m³/s, the energy loss proportion attributed to δH increases from 3.8% to 18.2%.

To quantify the flow loss at the outlet pipe junction, this study establishes total pressure monitoring points upstream and downstream of the junction (monitoring positions shown in Figure 7). Figure 8a illustrates the variation in junction loss under dual-pipe confluence conditions as the rotational speed of the No. 4 variable-frequency pump increases. When the speed of Pump No. 4 rises, its flow rate correspondingly increases, intensifying the throttling effect at the junction on the No. 3 fixed-frequency pump. This results in a monotonic upward trend in junction loss for the pipeline associated with Pump No. 3. The flow distribution relationship is modeled via polynomial fitting as Q₃ = 2.92 − 0.1 Q₄ − 0.03 Q₄² Figure 8b, demonstrating that changes in the rotational speed of Pump No. 4 not only directly alter its own flow rate but also significantly affect the flow distribution of Pump No. 3 through flow field interference. Figure 8c displays the efficiency curves of the No. 3 fixed-frequency pump, No. 4 variable-frequency pump, and the pumping station system. As shown, the efficiency of the Pump No. 3 exhibits a nonlinear trend (first increasing then decreasing) within the range of 89.3–89.63%, aligning with its external characteristic curve. In contrast, the efficiency of the Pump No. 4 continuously improves with increasing rotational speed, highlighting the energy-saving advantages of variable-frequency speed regulation. This research reveals the flow-coupling mechanism and energy- efficiency-evolution characteristics between fixed-frequency and variable-frequency pumps in parallel systems, providing a theoretical foundation for optimizing pumping station operations.

3.2. Internal Flow Analysis

According to the analysis of pressure distribution characteristics at the mid-span cross-section of Pump No. 3 and Pump No. 4 shown in Figure 9, under different rotational speed conditions of Pump No. 4, the pressures of both pumps progressively increase along the flow direction. The lowest pressure occurs at the impeller inlet, with a distinct local low-pressure zone near the suction side. The C_p values exhibit a gradient increase from the suction side to the pressure side.

Specific data reveal that the pressure gradient difference ∇C_p (∇C_p = C_{p max} − C_{p min}) at the mid-span cross-section of Pump No. 3 under different speed conditions (1.00n₀ = 740 r/min, 0.94n₀ = 695 r/min, 0.90n₀ = 665 r/min, 0.80n₀ = 592 r/min) is 1.37, 1.31, 1.24, and 1.23, respectively. For Pump No. 4, the corresponding ∇C_p values under the same conditions are 1.38, 1.33, 1.26, and 1.23.

This indicates that as the rotational speed decreases (to 0.80n₀ condition), the internal pressure variations in both pumps become more uniform, whereas the pressure gradient reaches its maximum at the rated speed (1.00n₀). Notably, although the rotational speed of Pump No. 3 remains unchanged, its pressure distribution is significantly influenced by the speed variations in Pump No. 4. As the rotational speed of Pump No. 4 decreases, the pressure gradient (∇C_p) across both impellers shows a significant reduction (e.g., from 1.37 to 1.23 for Pump No. 3). This homogenization of pressure can be attributed to the reduced dynamic head and lower flow velocities resulting from speed regulation. According to the Bernoulli principle, lower velocities lead to higher static pressures, thereby mitigating the extreme pressure difference between the suction and pressure sides of the blades. Notably, even the fixed-speed Pump No. 3 exhibits a more uniform pressure distribution, indicating that its operating point is shifted by the hydraulic interference from the variable-speed pump, moving it away from the high-flow, high-pressure-gradient condition. In the tongue region (Zone A), the high-pressure zone of Pump No. 3 contracts as the speed of Pump No. 4 decreases, while the high-pressure zone of Pump No. 4 in Zone A expands significantly with its own speed increase. This phenomenon can be explained by dynamic flow changes: when the speed of Pump No. 4 decreases, the increased flow velocity in Pump No. 3 leads to a pressure drop (Bernoulli effect), while the reduced flow rate of Pump No. 4 causes a pressure rise in its Zone A.

Figure 10 shows the turbulent kinetic energy (TKE) distribution of Pump No. 3 and Pump No. 4 under different rotational speeds, revealing the TKE distribution patterns at the mid-span cross-sections of the two pumps under varying operational conditions. Within Pump No. 3, the high-TKE core regions are primarily concentrated in the laminar-to-trailing-edge flow development zones on the suction side of the blades. As the rotational speed of Pump No. 4 decreases, the diffusion range of these regions exhibits a significant expansion trend. Under the 1.00n₀ condition, the TKE distribution inside the impeller of Pump No. 4 displays a quasi-symmetric topological structure, showing spatial characteristics markedly similar to those of Pump No. 3. In contrast, the 0.94n₀ condition induces localized high-TKE concentration zones near the leading edge and tongue region of Pump No. 4’s blades, while extensive low-TKE bands emerge within three adjacent impeller flow passages. Notably, as the rotational speed of Pump No. 4 decreases to 0.90n₀, the turbulent fluctuation intensity in the impeller outlet region significantly intensifies; when the speed decreases to 0.80n₀, the high-TKE region at the outlet of Pump No. 4’s runner further develops, significantly impacting the systematic hydraulic efficiency decline. The rotational speed adjustment of Pump No. 4 exerts a pronounced influence on the turbulent fluctuation intensity distribution in its Zone B, specifically manifesting as an increase in TKE within this region as the speed decreases.

3.3. Flow Field Analysis at the Outlet Pipe Junction

The outlet pipeline features a pipe junction as a distinctive characteristic of the parallel pump system, which is analyzed here.

Figure 11 presents the pressure, turbulent kinetic energy (TKE), velocity contours, and velocity vector diagrams of the pump group’s outlet pipeline under different rotational speeds, revealing the evolution of flow characteristics at the mid-span cross-section of the outlet pipe under varying operational conditions. At rotational speeds of 1.00n₀, 0.94n₀, 0.90n₀, and 0.80n₀, the pressure gradient difference ∇C_p progressively decreases to 0.36, 0.29, 0.21, and 0.16, respectively. This indicates that as the rotational speed reduces from 1.00n₀ to 0.80n₀, the pressure distribution within the outlet pipe becomes more uniform. Flow analysis demonstrates that after intense vertical collisions at the pipeline junction, the fluid forms velocity vectors directed toward the upper left of the diagram. Simultaneously, constrained by the pipe walls, upward high-speed flow regions develop. This flow separation phenomenon causes Zone C to exhibit typical low-velocity, low-pressure, and high-TKE characteristics. Notably, as the rotational speed of Pump No. 4 decreases, the high-TKE clusters in Zone C show a significant contraction trend. This arises because the reduced speed of Pump No. 4 decreases its pipeline flow rate, while the flow rate in Pump No. 3’s pipeline relatively increases, effectively weakening the collision intensity of fluids at the junction.

Vortex structures at the junction were analyzed using the Q-criterion. Figure 12 illustrates the vortex structure diagrams and velocity streamlines at the junction under different rotational speeds, where Q = 1000 s⁻² was adopted to identify vortex structures, revealing the flow field variations under different speed conditions. Notably, under the 1.00n₀ condition, the confluence of inflows from Pump No. 3 and Pump No. 4 generates large vortices in the outlet pipe, with the vortex distribution area occupying nearly one-third of the pipe cross-section. This induces significant local backflow, substantially impacting the flow field. The Q-criterion analysis reveals a dramatic 63% contraction in the vortex core area at the pipeline junction when the speed reduces to 0.80n₀. This suppression of large-scale vortices is a direct consequence of the reduced momentum of the incoming flows from both pipelines. With lower speeds, the violent collision and shear layer at the 90° junction are alleviated, preventing the formation and sustenance of energetic vortices. The mitigation of these vortices is critically important as they are a major source of hydraulic loss (δH), which was quantified to consume up to 18.2% of the total energy in Section 3.1. This finding physically explains the energy loss mechanism and underscores the detrimental effect of the junction design.

Synthesizing the above analysis of the outlet pipeline, it is evident that the right-angle junction causes severe fluid collisions, leading to pronounced flow separation. The resulting vortex zones, characterized by low velocity, low pressure, and high TKE, compress the flow passage, adversely affecting the flow field. Moreover, this phenomenon intensifies with increasing flow rate. While variable-frequency pump speed regulation can partially mitigate it, a fundamental solution remains unattained. Further optimization requires redesigning the outlet pipeline layout, including adjustments to pipe diameter and junction angle.

3.4. Statistical Analysis of Turbulent Kinetic Energy and Turbulent Dissipation Rate

This section conducts quantitative statistics on the distribution characteristics of turbulent kinetic energy (TKE) and turbulent dissipation rate (TDR) across various components of the pumping station based on numerical simulation results, aiming to reveal localized features of turbulent energy transfer and dissipation. The physical meaning and mathematical expression of TKE and TDR have been introduced in Section 2.4.

High-dissipation-rate regions typically correspond to rapid energy dissipation by small-scale vortices.

For the wall surfaces of pumping station components, TKE and TDR are statistically analyzed via area-based summation:

$$k_{surf}={\sum }_{i=1}^f{k}_i A_i , {\epsilon }_{surf}={\sum }_{i=1}^f{\epsilon }_i A_i$$

(8)

For the fluid domains of components, TKE and TDR are statistically analyzed via volume-based summation:

$$k_{volu}={\sum }_{i=1}^c{k}_i V_i , {\epsilon }_{volu}={\sum }_{i=1}^c{\epsilon }_i V_i$$

(9)

where

• k_surf: Sum of TKE on component wall surfaces.
• ε_surf: Sum of TDR on component wall surfaces.
• k_volu: Sum of TKE within component fluid domains.
• ε_volu: Sum of TDR within component fluid domains.

Surface statistics reveal the impact of flow separation, reattachment, and wall friction on turbulence. For instance, high k_surf regions may correspond to flow separation points, while high ε_surf regions reflect small-scale dissipation near the viscous sublayer. Volume statistics identify dominant energy transfer mechanisms. A high k_volu/ε_volu ratio indicates that large-scale vortices dominate the energy cascade, whereas a low ratio suggests that small-scale dissipation prevails. Combined surface-volume analysis comprehensively evaluates the spatial distribution of energy transfer mechanisms.

Figure 13 illustrates the TKE and TDR distribution characteristics in the impellers and spiral casings of Pump No. 3 and Pump No. 4. The results show that during variable-speed operation of Pump No. 4 (740→592 rpm), the wall TDR (ε_surf) of the impellers exhibits significantly opposite evolution trends: when the rotational speed of Pump No. 4 decreases from 1.00n₀ to 0.80n₀, ε_surf of Pump No. 3 increases by 16.4% (reaching 1.28 × 10⁶ m⁴/s³), while ε_surf of Pump No. 4 itself decreases by 37% (to 0.686 × 10⁶ m⁴/s³). This phenomenon reveals the speed-coupling effect in parallel systems: the speed reduction in Pump No. 4 decreases the total system flow, forcing the fixed-speed Pump No. 3 to operate at a lower flow point, intensifying flow separation and energy losses in its impeller; meanwhile, the turbulence intensity of Pump No. 4 weakens due to reduced rotational speed, optimizing its internal flow pattern.

During variable-speed operation, the mainstream regions of the impellers exhibit a distinct dual-mode turbulent energy transfer characteristic:

When the speed of Pump No. 4 decreases to 0.90n₀, the impeller TKE (k_volu) surges by 115% (0.089 → 0.192 m⁵/s²), while TDR (ε_volu) increases marginally by 8% (6895 → 7444 m⁵/s³). The energy retention time (k/ε) extends from 12.9 s to 25.7 s, revealing an efficient energy cascade dominated by large-scale vortices. When the speed further decreases to 0.80n₀, TDR increases drastically by 39% (6895 → 9599 m⁵/s³), surpassing the growth rate of TKE, indicating a shift to small-scale viscous dissipation dominance.

Concurrently, the spiral casing shows continuous energy accumulation characteristics, with TKE peaking at 0.228 m⁵/s² (+48%) under the 592 rpm condition, becoming the primary source of system pressure pulsations. The statistical analysis of TKE and TDR uncovers a dual-mode energy dissipation mechanism within the impellers. At 0.90n₀, the surge in TKE with only a marginal increase in TDR suggests that the energy is primarily stored in large-scale, coherent turbulent structures that dissipate slowly (longer k/ε time). This is typical of flow fields dominated by large vortices containing significant kinetic energy. Conversely, at 0.80n₀, the drastic jump in TDR indicates a regime shift where energy is rapidly dissipated into heat by small-scale, viscous eddies. This likely occurs because the further reduced Reynolds number promotes viscous effects, and the large-scale structures break down more efficiently into dissipative scales. This trade-off reveals a fundamental challenge in variable-speed operation: while lower speeds can improve global flow patterns (e.g., pressure homogenization, vortex suppression), they may aggravate near-wall viscous losses, which limits the overall efficiency gains. This dual characteristic aligns closely with the turbulence structure decomposition observed by Chen et al. using TR-PIV technology, where large-scale structures under high-velocity gradients continuously fragment into dissipative-scale structures via shear effects. [30].

3.5. Comparison with Empirical Head Loss Calculation

In this study, Computational Fluid Dynamics (CFD) simulations were conducted to analyze the flow characteristics at a T-junction with a 90° angle. The CFD results indicated that the pressure losses in the run and branch were 4.7556 m and 5.193 m, respectively. Corresponding loss coefficients (K-values) were calculated as 0.61 for the run and 1.09 for the branch. These values fall within the typical ranges provided in engineering handbooks, specifically Crane Technical Paper No. 410 and Idelchik’s Handbook of Hydraulic Resistance, which report K-values of 0.3–0.4 for the run and 1.1–1.7 for the branch [31,32]. This alignment suggests that the CFD simulations accurately represent the flow behavior at the T-junction, see

Table 4. CFD vs. engineering handbook K-value comparison.

Flow Path	CFD Loss hmh_m (m)	CFD Loss Coefficient K	Engineering Handbook Typical K Range	Remarks
Run	4.7556	0.61	0.3–0.4	Matches typical value
Branch	5.193	1.09	1.1–1.7	Matches typical value

.

4. Conclusions

Based on high-fidelity numerical simulations, this study systematically investigates the operational mechanisms and energy characteristics of a parallel pump system under variable-speed conditions at the Yanhuanding Yanghuang Cascade Pumping Station. The primary goal was to optimize the matching of flow rates, heads, and rotational speeds to enhance operational efficiency and reduce energy consumption. The main findings are summarized as follows:

Adjusting the rotational speed of variable-frequency pumps significantly expands the adjustable flow range of the pump group (1.17–1.26 m³/s) while maintaining high operational efficiency. Under varying head differences (77.8–79.8 m), the pump units consistently operated within a high-efficiency range of 82.64% to 89.64%. Notably, the variable-speed operation of Pump No. 4 enabled individual pumps to sustain efficiencies above 82% across the entire adjusted flow range. However, in parallel systems, increasing the rotational speed of the variable-frequency pump reduces the flow rate of the fixed-frequency pump due to hydraulic interaction. Furthermore, energy losses (δH) at the outlet pipe junction increase substantially with higher flow rates, with the proportion of energy loss rising from 3.8% to 18.2%, highlighting the significance of junction design in system efficiency.

The comprehensive analysis of turbulent kinetic energy (TKE) and turbulent dissipation rate (TDR) reveals a dual-mode energy dissipation mechanism within the impellers. When the speed of Pump No. 4 is reduced to 0.90n₀, a surge in TKE (115% increase) with only a marginal rise in TDR (8%) indicates an energy cascade dominated by large-scale, coherent vortices, characterized by an extended energy retention time. In contrast, a further reduction to 0.80n₀ triggers a regime shift, where a drastic 39% increase in TDR signifies the dominance of small-scale viscous dissipation, rapidly converting turbulent energy into heat. This fundamental trade-off—between beneficial large-scale flow improvements and aggravated near-wall viscous losses—imposes a practical lower limit for speed regulation.

The study demonstrates that variable-speed regulation serves as an effective tool for optimizing parallel pump systems. By strategically reducing pump speed, flow momentum at the junction is lowered, leading to suppressed vortex formation and more uniform pressure distribution. These improvements collectively contribute to an expanded operational flow range. The observed opposite evolution of wall TDR in the two pumps (Pump No. 3: +16.4%; Pump No. 4: −37%) further underscores the strong hydraulic coupling in parallel systems. Thus, an optimal operational speed exists that balances the reduction in large-scale hydraulic losses (δH) against the intensification of small-scale dissipation.

Looking forward, future studies should focus on integrating adaptive control algorithms, such as fuzzy PID or reinforcement learning, to enable dynamic pump speed adjustments in response to real-time hydraulic data. This approach could further enhance energy efficiency and operational flexibility, providing a robust framework for the intelligent management of cascade pumping stations.