Numerical Simulation of the Flow Field Stabilization of a Pressure-Regulating Device

Abstract

In sprinkler irrigation systems in hilly areas, pressure-regulating devices can improve the uneven pressure distribution on branch pipes and consequently improve the irrigation uniformity. A pressure-regulating device for sprinkler irrigation systems was developed; the variation characteristics of the internal pressure field and velocity field distribution over time, variation law of the elastic diaphragm of the pressure-regulating device with respect to the inlet pressure, and pressure-regulating and flow stabilizing mechanisms were assessed. The results show that the pressure regulation and flow stabilization of the pressure-regulating device were affected by the deformation of the elastic diaphragm. When the elastic diaphragm deforms, the main factors of pressure regulation and flow stabilization in the pressure regulation device gradually shift from the side channel and compensation chamber to the side channel and secondary channel. The channel structure plays a crucial role in the entire fluid system. A comparison and analysis of the changes in cross-sectional pressure and flow velocity of the flow channel under different inlet pressures revealed that increasing the height of the side flow channel baffle and auxiliary flow channel effectively improved the pressure-regulating and flow-stabilizing performance of the pressure-regulating device. The results can be useful to optimize the structure of pressure regulation devices and to solve the problem of excessive pressure deviation in sprinkler irrigation systems in hilly areas.

1. Introduction

Hilly cultivated land in China accounts for more than 50% of the total area and is mainly irrigated by fixed sprinkler irrigation, micro-sprinkler irrigation, and drip irrigation. It is common to use sprinkler irrigation to irrigate the hilly areas [1,2]. The slope in hilly areas may cause significant changes in working pressure within the sprinkler irrigation system that leads to a pressure wave phenomenon in the system. Therefore, it is necessary to improve the water distribution uniformity of sprinkler irrigation systems in hilly areas.

To improve the problem of water distribution uniformity, a pressure-regulating device at the inlet of the sprinkler head was designed. Key structural parameters play an important role in the overall hydraulic performance of the pressure-regulating device. Using hydraulic performance tests, only the hydraulic performance of the entire device can be monitored from a macroscopic viewpoint after the installation of the pressure regulator to obtain the energy dissipation and flow stabilization mechanisms of the pressure regulator under the influence of different structural parameters. However, the influence of structural parameters on the hydraulic performance of pressure-regulating devices from a microscopic viewpoint is also important.

Many studies have used numerical simulations to explore the effects of structural parameters on the hydraulic performance of devices from a microscopic viewpoint [1,2,3,4,5,6]. For example, Wang et al. (2018) determined the structural form and structural parameters of a drip based on simulation results from ANSYS Fluent software which effectively improved the irrigation uniformity [7]. Xu et al. (2020) analyzed the internal compensation principle and hydraulic performance of a pressure compensating irrigator using the ADINA fluid–solid coupling calculation method [8]. Chen et al. (2022) used a mathematical model to predict the flow rate of water irrigators under the influence of different structural parameters, which provided a reference for the rapid design of irrigators with different flow rates [9]. Hou et al. (2022) performed one-way and orthogonal tests for key factors to determine the flow index and reported that sub-channel width, hardness of pressure-compensating elements, and sub-channel depth are the main factors that affect the flow index [10]. Most of these studies focused on pressure regulation in drip irrigation; however, the working pressure in sprinkler irrigation systems is higher than that in drip irrigation systems and very limited research has been conducted on pressure regulation devices for sprinkler irrigation. The pressure-regulating device in sprinkler irrigation systems requires development, and structural parameters must be studied in detail.

Therefore, numerical simulation methods were adopted to design and optimize the structural parameters of sprinkler pressure-regulating devices, internal flow theories under different conditions are obtained, and scientific guidance for the role of pressure-regulating devices in sprinkler systems is provided.

2. Materials and Methods

2.1. Structural Design and Working Principle

A new pressure-regulating device was designed to improve the problem of uneven irrigation uniformity. Figure 1 shows a schematic diagram of the pressure-regulating device structure. The structure included an inlet, an elastic diaphragm, an outlet, a secondary channel, and a side channel. To stabilize the pressure and regulate the flow, when water flows through the pressure-regulating device and impacts the diaphragm, it deforms the diaphragm, leading to changes in the cross-sectional area.

2.2. Selection of Key Structural Parameters

Key Structural Hydraulic Parameters

Key structural parameters that influence the hydraulic performance of the pressure-regulating device, shown in Figure 2, were selected. According to the working principle of the pressure-regulating device, the following key structural parameters were selected to explore their influence on the hydraulic performance of the pressure-regulating device: the thickness of the elastic diaphragm (T), height of the compensation chamber (H), angle of the channel baffle (β), height of the secondary channel (h), and angle of the secondary channel (α).

2.3. Design and Method of Hydraulic Performance Tests

The sprinkler irrigation test system included a piping system, a centrifugal pump (type: Grundfos CM1-7, Grundfos Pumps (Shanghai) Co., Ltd., Shanghai, China), a pressure gauge (0~600 kPa), an electromagnetic flowmeter (0~20 m³/h), a pressure-regulating device, and an impact sprinkler head (type: PY10, Wangda Co., Ltd., Changzhou, China).

During the hydraulic performance test, the inlet pressure of the pressure-regulating device was adjusted by changing the opening of the inlet valve of the water pipeline, which changed the outlet pressure and outlet flow rate. The installation height of the sprinkler was 1.5 m. The pressure-regulating device inlet pressure range was 0~460 kPa, and 23 groups of data were tested with an interval of 20 kPa between each set of data. Each test was performed to ensure that the flowmeter was stable, the duration was 3 min and repeated 3 times, and the average value was obtained. After the completion of the tests, the outlet pressure and flow rate data were recorded.

2.4. Design and Methos of the Numerical Simulation Experiment

The pressure-regulating device and sprinkler internal flow field were numerically simulated and analyzed using ANSYS Fluent 2020 software. The laws that govern the effects of the changes in structural parameters on the overall internal flow field of pressure-regulating devices were obtained.

2.4.1. 3D Physical Model

Solidworks 3D design software (Solidworks 2023) was used to construct the physical models of the pressure-regulating device and flow field inside the sprinkler, as shown in Figure 3. The physical model of the internal flow field in the pressure-regulating device was divided into two main parts. Figure 4 shows the fluid domain and solid domain of the elastic diaphragm, where the inlet diameter of the sprinkler was 8.3 mm, the diameter of the main nozzle of the sprinkler was 4.2 mm, and the diameter of the sub-nozzle of the sprinkler was 3 mm.

2.4.2. Turbulence Model

The entire internal flow field of the pressure-regulating device was categorized as turbulent. Since the flow inside the pressure regulator is affected by the structural parameters, the RNG k-ε turbulence model was selected as the simulation model [11].

(1)

continuity equation:

$${\displaystyle \frac{\partial u_i}{{\partial x}_i}}=0$$

(1)

(2)

momentum equation:

$${\displaystyle \frac{\partial }{\partial t}}(\rho u_i)+{\displaystyle \frac{\partial }{{\partial x}_j}}(\rho u_i{u}_j)=\rho f_i-{\displaystyle \frac{\partial p}{{\partial x}_i}}+\mu {\displaystyle \frac{{\partial }^2{u}_i}{{\partial x}_i^2}}-{\displaystyle \frac{\partial }{{\partial x}_j}}\rho \overline{(u_i^{\prime }{u}_j^{\prime })}$$

(2)

(3)

turbulent kinetic energy $\mathrm{k}$ equation:

$${\displaystyle \frac{d\left(\rho k\right)}{dt}}+{\displaystyle \frac{d\left(\rho k{u}_i\right)}{dt}}={\displaystyle \frac{\partial }{{\partial x}_j}}({\alpha }_k{\mu }_{eff}{\displaystyle \frac{\partial k}{{\partial x}_j}})+G_k+\rho \epsilon$$

(3)

(4)

turbulent dissipation rate $\mathsf{\epsilon }$ equation:

$${\displaystyle \frac{d\left(\rho \epsilon \right)}{dt}}+{\displaystyle \frac{d\left(\rho \epsilon u_i\right)}{d{x}_i}}={\displaystyle \frac{\partial }{{\partial x}_j}}({\alpha }_k{\mu }_{eff}{\displaystyle \frac{\partial \epsilon }{{\partial x}_j}})+{\displaystyle \frac{\epsilon C_{1\epsilon }^{*}}{k}}{G}_k-C_{2\epsilon }\rho {\displaystyle \frac{{\epsilon }^2}{k}}$$

(4)

(5)

turbulent viscosity equation:

$${\mu }_t=\rho C_u{\displaystyle \frac{k^2}{\epsilon }}$$

(5)

where u is the velocity component of the fluid; x is a coordinate vector; $\rho$ is the density of the fluid, m³/kg; fi is the mass force along direction I, N/kg; P is the pressure of the fluid, kPa; $U_{eff}=\mu +{\mu }_t$, ${\mu }_t$ is the turbulent viscosity coefficient of the fluid; $\mu$ is the coefficient of dynamic viscosity of the fluid; $C_{\mu }$ = 0.0845; ${\alpha }_k={\alpha }_s=1.39$; $G_k$ is the turbulent kinetic energy production term; $\rho \overline{(u_i^{\prime }{u}_j^{\prime })}$ is the average Reynolds stress; k is the turbulent energy; $\epsilon$ is the turbulent dissipation rate. $C_{1\epsilon }^{*}=C_{1\epsilon }-{\displaystyle \frac{\eta (1-\frac{\eta }{{\eta }_o})}{1+\beta {\eta }^3}}, \eta =\sqrt{2S_{ij}.S_{ij}}{\displaystyle \frac{k}{\epsilon }}$, $S_{ij}={\displaystyle \frac{1}{2}}({\displaystyle \frac{{\partial u}_j}{{\partial x}_i}}+{\displaystyle \frac{{\partial u}_i}{{\partial x}_j}})$, ${\eta }_o$ = 4.377. $\beta$ = 0.012, $C_{1\epsilon }$ = 1.42, $C_{2\epsilon }$ = 1.68.

2.4.3. Structure Setup

The deformations of the internal elastic diaphragm of the designed pressure-regulating device in this work are small- and medium-sized deformations, so the Mooney–Rivlin model was selected [12]. The formula to obtain the modulus of elasticity of the elastic diaphragm is:

$$E=6(C_1+C_2)$$

(6)

The relationship between shore hardness HA and the modulus of elasticity E is:

$$E=\left\{{\displaystyle \frac{15.75+2.15H_A}{100-H_A}}\right\}$$

(7)

The relationship between material constants ${ C}_1$ and ${ C}_2$ is:

$$C_1=0.25C_2$$

(8)

2.4.4. Meshing

The quality of the mesh should be fully considered in the numerical simulation process, so the mesh should be divided using the correct division method. Based on the set nozzle structural parameters, ANSYS ICEM 2020 software was used to mesh the flow field inside the pressure-regulating device. Due to the complexity of the flow structure in the pressure regulator, meshing was performed in an unstructured tetrahedral manner. According to the principle of operation of the pressure-regulating device, a fluid–solid coupling contact surface was provided between the upper and lower surfaces of the elastic diaphragm and the pressure-regulating device. A contact surface was provided between the outlet of the pressure-regulating device and the inlet of the nozzle. The grid setup part and the entire pressure-regulating device are shown in Figure 5.

2.4.5. Boundary Condition Setting

The pressure-regulating device inlet boundary condition was set by the pressure inlet, and the outlet boundary condition was set by the pressure outlet. Since the outlet of the device was interconnected with the atmosphere, the outlet pressure was simulated as atmospheric static pressure (1 atm). The contact surface between elastic diaphragm and fluid domain of the device was set as a fluid–solid coupling surface, and all wall surfaces were set as no-slip wall surfaces. The water flow in the pressure-regulating device was a viscous incompressible fluid.

2.4.6. Simulation Reliability Verification

To verify the reliability of the numerical simulation of the pressure-regulating device, the simulation results were compared and analyzed with the experimental test results. Numerical simulation was used for calculation and analysis. The graphs of the relationship between outlet flow and working pressure of the pressure-regulating device are shown in Figure 6.

Figure 6 shows that under different inlet pressure conditions, the errors between simulation results and test results were less than 10%, and the simulated values were within the permissible range of error compared with the test results. The results show that the adopted numerical simulation model is highly accurate and can be used to investigate the mechanism of regulating and stabilizing the flow in pressure-regulating devices in depth.

3. Results

3.1. Characterization of the Overall Flow Field of the Pressure-Regulating Device

3.1.1. Pressure Distribution

The pressure distributions inside the pressure-regulating device at different moments are shown in Figure 7. The red circled showed the compensation chamber and the diaphragm change part. The pressure distribution at the inlet of the regulating device was relatively uniform. The elastic diaphragm deformed gradually by increasing the time steps until it reached in full contact with the cam when the deformation of the elastic diaphragm reached its peak value. When the elastic diaphragm was in full contact with the tab, the area of the pressure compensation chamber was almost zero, which led to a dramatic pressure increase in the secondary channel area. This change indicates that the differential pressure regulation within the device gradually shifted from the compensation chamber to the side channel. This change can help maintain the stability of the outlet pressure.

3.1.2. Flow Rate Distribution

The distribution of the flow rate inside the pressure-regulating device at different moments is shown in Figure 8. The flow rate distribution at the inlet of the pressure-regulating device was relatively uniform. The elastic diaphragm was deformed by gradually increasing the time steps until it was in contact with the tab. The deformation of the elastic diaphragm reached peak value, and the maximum flow rate in the device was 16.4 m/s. When the elastic diaphragm was in full contact with the tabs, water flowed through the side channels directly into the secondary channel and exacerbated the degree of turbulence in the secondary channel.

3.2. Characterization of Elastic Diaphragm Deformation

To investigate the deformation characteristics of the solid domain when the pressure-regulating device works, an elastic diaphragm with a hardness of 65 HA was selected as an important benchmark for the parameter of this solid simulation. Equations (6)–(8) were used to calculate the material constants C1 and C2, and the modulus of elasticity, which were 0.1481, 0.5923, and 4.4429 MPa, respectively. The deformation states of the elastic diaphragm at different inlet pressures are shown in Figure 9.

The deformation of the elastic diaphragm in the Z direction under different inlet pressures is shown in Figure 10. When the inlet pressure variation interval was 0~116.5 kPa, the maximum deformation of the elastic diaphragm increased with increasing inlet pressure. When the inlet pressure was 116.5 kPa, the elastic diaphragm was in a state of maximum deformation, and the deformation at the center of the elastic diaphragm was 6.05 mm. When the inlet pressure change interval was 116.5~400 kPa, the elastic diaphragm deformation experienced small fluctuations, relatively close to 6 mm. When the diaphragm was gradually moved towards the tab, the area of the pressure compensation chamber gradually reduced to 0. The pressure regulation function was transferred from the pressure compensation chamber portion to the secondary channel.

3.3. Side Channel Flow Characteristics

3.3.1. Parameters of the Numerical Simulation Model

The side channel and secondary channel are the important parts of the pressure regulating device; thus, the numerical simulation analysis of these parts can provide a reference for designing the pressure-regulating device. In order to investigate the influence of internal structural parameters on the internal flow field of the pressure-regulating device, the height and angle of the secondary channel and the channel baffle were selected as the simulation factors and the single-factor control variable method was adopted. The simulation scheme of the structural parameters for six groups of a prototype of the pressure-regulating device is shown in

Table 1. Simulation schemes.

Model	Elastic Diaphragm Thickness/mm	Compensation Chamber Height/mm	Channel Baffle Angle/°	Secondary Channel Height/mm	Secondary Channel Angle/°
1	1	6	0	3	0
2	1	6	0	4	45
3	1	6	40	3	90
4	1	6	40	4	45
5	1	6	80	5	45
6	1	6	80	5	90

.

3.3.2. Selection of Monitoring Points for the Side Channel Simulation

To more accurately obtain the effects of structural parameters on the internal flow channel of the pressure-regulating device, 25 monitoring surfaces were divided between the inlet and outlet of the lateral flow channel of the pressure-regulating device. The effects of different structural parameters on the internal flow rate and pressure drop variation at different inlet pressures were obtained. The monitoring surfaces were numbered and located as shown in Figure 11.

3.3.3. Side Channel Pressure Field and Its Distribution Characteristics

As an important part of the device to realize the energy dissipation and flow stabilization, the performance of the side channel can be analyzed via numerical simulation. This can provide a reference for pressure-regulating device design. The models were numerically simulated using the Fluent software, and the pressure distributions in the side channel under different inlet pressures were obtained. The results after post-processing are shown in Figure 12. The energy consumption in the device increased by increasing inlet pressure under different inlet pressure conditions and this is the mechanism of the pressure regulator that achieves steady flow. As shown in Figure 12, under different inlet pressure conditions, the local distribution of high pressure was basically identical. The high-pressure region was mainly distributed at the tooth tips and corners, as shown in the red circle in Figure 12. The low-pressure region was mainly distributed at the top of the tooth. This distribution occurred because the flow in the side channel was affected by the localized head loss due to the sudden change in cross-sectional area of the overflow. A significantly larger pressure gradient was also produced. The sudden change in the cross-sectional area of the overflow mainly occurred in the channel baffle, because the channel baffle made the channel cross-section of the flow suddenly decrease. The local head loss of water flow at the channel baffle increased. These results imply that properly adjusting the angle of the channel baffle can improve the pressure-regulating device’s ability to dissipate energy and stabilize the flow.

To reveal the characteristics of the pressure regulation inside the flow channel of the pressure-regulating device, the average pressure of each cross-section inside the flow channel was monitored, and monitoring results are shown in Figure 13. After the water flowed into the side channel, the average pressure tended to increase and subsequently decrease. Monitoring surfaces 1–3 were mainly located at the inlets of the side channel, and there was a rapid upward trend. Starting from monitoring surface 4, the pressure at the monitoring surface for each model gradually decreased. A comparative analysis of the average pressure distribution curves of models 5 and 6 reveals that, for monitoring surfaces 9–16, the energy dissipation was greater, and the average pressure value was lower. Pressure in the other models did not significantly change at these locations. As shown in Figure 13, the average pressure drop of the monitoring surface was mainly distributed in the channel baffle and channel diversion areas, i.e., monitoring surfaces 4–25, which accounted for approximately 98% of the entire channel. When the inlet pressure reached 400 kPa, the pressure difference in the secondary channel accounted for more than 30% of the total regulated pressure. Among them, the pressure drop was greater at monitoring surfaces 20–25, and the energy dissipation was faster. This can be attributed to symmetrical water flow in the side channel at the confluence, which allows the two water streams to converge and mix, causing the gradual decline in the pressure inside the regulating device along the direction of flow. The change trend in the average pressure at the monitoring surfaces for models 1, 3, and 5 showed that the outlet pressure of these models approached zero, indicating that the excess pressure of the water flow has been eliminated at the outlet of the regulator and the water flow has become stable. While the other models had some pressure at the outlet, which indicated that the hydraulic performance of the other models was somewhat inferior in the comparison. The results revealed that the pressure drop throughout the flow channel occurred more significantly in model 5 and the change in the pressure was relatively stable under different inlet conditions. This indicated that model 5 was the most effective in terms of regulating the pressure within the device as compared to the other models.

3.3.4. Side Channel Velocity Field and Its Distribution Characteristics

The velocity distributions and local velocity vector distributions in the side channel under different inlet pressure conditions are shown in Figure 14 and Figure 15. As shown in Figure 14, the flow velocity distribution inside the device exhibited a high degree of consistency under different inlet pressure conditions. According to different flow velocity magnitudes, the flow field inside the flow channel can be divided into three main regions: the main flow region, the low-velocity region, and the vortex region. The main flow region was the area with highest speeds, as shown in zone C in Figure 15. The low-velocity region was dominated by low and negative velocities, as shown in zones A and B in Figure 15. There was a clear boundary between these two zones. The flow rate in zone C was significantly higher than that in zones A and B. The maximum flow velocity was mainly concentrated in the middle region between zones A and B, where the side channels bent to form an eccentric, circular region. When the water flowed through the flow channel of the device, the sudden change in the cross-section of the inner flow channel caused continuous water flow convergence and mixing that led to energy exchange and dissipation, as shown in the red circles in Figure 15 The main flow zone had a high flow velocity, and the area of the main flow zone in the side channels gradually increased with increasing inlet pressure. The lower flow velocities in zones A and B were affected by the higher flow velocities in the main flow zone, and a larger swirling backflow formed near the bends in the flow channel. Among them, a small part of the water flow impacted by the main flow region continuously returned to the main flow region. The other was stranded in the low-velocity region and mixed with the new inflow of water to form a vortex, maintaining rotational kinetic energy. Energy was constantly drawn and converted into thermal energy dissipation. The outside of the vortex rotated in the same direction as the water flow, accelerating and promoting water flow, whereas the inside of the vortex rotated in the opposite direction, and drag forces slowed the water flow. These vortices caused turbulence in the water flow and energy dissipation, which reduced the energy loss in the flow path. The maximum kinetic energies of turbulence at inlet pressures of 200 kPa, 300 kPa, and 400 kPa were 1.72, 2.34, and 3.59 times greater, respectively, than that of the turbulence at an inlet pressure of 100 kPa. The inlet pressure of the pressure-regulating device increased, which can effectively enhance the turbulent energy inside the side flow channel and the energy transfer capacity of the flow channel.

To reveal the details in the modeled flow paths, this work analyzed the flow path cross-sectional velocities for each pressure regulator model under different inlet pressure conditions, and the results are shown in Figure 16. Figure 13 shows that the velocity trend is consistent with the pressure trend. The overall monitoring surface flow velocity map in Figure 16 shows that two peaks and one trough occurred in all mean flow velocity profiles, and the higher peaks mainly occurred at monitoring surfaces 9–11 and 16–17. The peak-and-valley values occur because of the different inlet pressures, and the peak-and-valley values are proportional to the inlet pressure; a greater inlet pressure corresponds to more pronounced peak-and-valley values of the mean flow rate. When the model structure changed, the difference between the mean flow velocities at the monitoring surfaces for each inlet pressure condition also changed. The difference in the mean flow rate in model 5 under different inlet pressure conditions was smaller than that in the other five groups of models. The preset pressure of the pressure-regulating device of model 5 parameters was 200–225 kPa; at this time, the flow index was 0.3152 [13]. These findings indicate that improving the side channel baffles and secondary channel heights can effectively improve the performance of the pressure-regulating device. Thus, the problem of excessive pressure deviation at the inlet of the sprinkler in systems in hilly areas is improved.

4. Discussion

Uniform irrigation spraying has a significant effect on crop growth in hilly areas. The problem of excessive pressure deviation at the inlet of the sprinkler system in hilly areas can be solved by a pressure-regulating device. The channel structure of the pressure-regulating device plays an important role in performance. In this study, a pressure-regulating device for a sprinkler irrigation system was designed, and the principle of the pressure regulation function was revealed through numerical simulation.

According to the analysis of the flow field, model 5 had the best performance, which is consistent with the conclusions of Zhao (2023) [13]. The preset pressure of the pressure-regulating device under the parameters of model 5 was 200–225 kPa, and the flow index was 0.3152 [13]. Thus, the experimental results of model 5 demonstrated the best performance.

The methods in this study are different from the research methods of drip irrigation emitters [14,15]. The main difference between the pressure-regulating device of the sprinkler irrigation system and the drip irrigation system was the principle of pressure regulation. In the pressure regulation of this study, due to the large pressure range of the sprinkler system, different structures were required to achieve the regulation process [16,17]. The numerical simulation method was used to verify the effect of pressure regulation in this study, which was similar to research methods in other fields [18]. The difference was that the changes in the pressure field and velocity field, as well as the switching of flow channels, need to be investigated during the pressure regulation process. The verification of the application effect of pressure-regulating devices in sprinkler irrigation systems still needs to be continuously improved in future research.

5. Conclusions

(1)

Numerical simulations of the pressure-regulating device reveal that the device has essentially identical pressure and flow rate distributions at different moments. The deformation of the elastic membrane significantly impacts the implementation of pressure regulation and flow stabilization mechanisms. When the degree of deformation of the elastic membrane increased, the main pressure regulation and flow stabilization mechanism of the pressure-regulating device gradually shifted from the side channel and compensation chamber to the side channel and auxiliary channel.

(2)

The optimized model 5 had better pressure distribution and flow rate distribution than the other models under different pressure conditions, which indicates that improving the side channel baffles, secondary channel heights, and secondary channel angle can effectively improve the performance of the pressure-regulating device in regulating and stabilizing the flow.