Numerical simulation of the effects of sandy water on regulator labyrinth channel in micro-sprinkler systems

The effects of sandy water on the W-shaped labyrinth channel of micro-sprinkler irrigation systems with large flowrate were investigated using Computational Fluid Dynamics (CFD). Using ANSYS FLUENT software and different inflow conditions (e.g., pressure, velocity, sediment concentration, and sand particle diameter), internal turbulent multiphase flow and sand deposition were simulated by the Eulerian multiphase flow model. Particle erosion in the labyrinth channel was calculated by the Discrete Phase Model (DPM). The results show that vortex movements and shear actions at the boundary layer cause self-flushing in the channel. The location of sand particle deposits and the turbulent dissipation rate are related to the operating pressure, which is optimal at 300 kPa. The erosion rate of the channel wall is proportional to the inflow sediment concentration but has no obvious relationship with inflow velocity. Based on the movement regulation of sand particles in the labyrinth channel, recommendations on filtration requirements and operating pressure of irrigation systems are proposed.


INTRODUCTION
In a micro-sprinkler irrigation system, laying branch pipes may result in considerable hydraulic loss and pressure differences. This phenomenon causes crops at different locations in the pipeline to receive different amounts of water and nutrients, sprinkler irrigation system are proposed. These results provide additional reference for the stable operation of micro-sprinkler irrigation systems.

Physical model
The modeled geometry for this simulation is based on the channel structure of a pressure-compensated micro-sprinkler irrigator (Shanghai Huawei Co., Ltd, Shanghai, China. Int. Cl. A01G 25/02 (2006.01)) as shown in Figure 1(a). This pressure compensation irrigator consists of a base connected to the capillary, a pressure-compensated flow rate regulator, and a micro-sprinkler. As shown in Figure 1(b), the flow rate regulator contains a pressure compensation chamber, a labyrinth flow channel, and an elastic gasket, which is the key component enabling the pressure compensation function. The rated pressure compensation range of this regulator is 200-400 kPa. Duran et al. (2009) proposed that the occurrence of blockages is independent of whether large-flowrate emitters have a pressure compensation function. Consequently, only the internal flow field of the labyrinth channel (a double-outlet, symmetrical W-shaped arc flow channel structure) was studied. Figure 2 shows a three-dimensional model of the labyrinth channel, constructed by NX.10.0 software, where the diameter of the inlet is 1.9 mm and the length of the channel is 14 mm.

Numerical simulation scheme and boundary conditions
Based on ANSYS Fluent 17.1 software, two simulations of sand movements in the labyrinth channel were carried out. Figure 3 shows a flowchart of the simulation process.
To study the energy dissipation and sand deposition in the channel, transient simulations of water-sand two-phase flow were carried out under different incoming flow conditions. The specific computational setups are listed in Table 1. Solid  particles belong to river sand with a density of 2,500 kg/m 3 . The volume fraction of sand in the water remains far below 5%, indicating that the flow medium is diluted phase flow. Hence, the Eulerian Multiphase Flow model was used to simulate the deposition of sand particles in the channel. Li et al. (2008) found that the flow state inside the channel is turbulent, and suggested that a suitable turbulence model is required to obtain accurate numerical simulation results. Al-Muhammad et al. (2016) compared several turbulence models and found that the results of standard k-ε and Re-Normalization Group (RNG) k-ε models are most consistent with experimental data. Considering the turbulence anisotropy effect and the high strain rate (or the large curvature of the flow surface), the present study used RNG k-ε as viscosity model to improve the accuracy in case of swirling and large curvature. The phase coupled semi-implicit method for pressure linked equations (SIMPLE) scheme was employed for coupling between pressure and velocity. The second-order upwind scheme was  Uncorrected Proof employed for spatial discretization items. The Gidaspow drag model was selected for calculating the drag force between solid phase and liquid phase. Furthermore, to assess the sand erosion in the main flow channel, DPM was used to simulate the movement of sand particles under different inlet velocities. As the labyrinth channel stabilizes the flow rate, different flow rates can be obtained by changing the inlet diameter. Inlet velocities can be calculated according to Equation (1): where V is the inlet velocity (m/s), Q is the rated inlet flow rate of the irrigator (m 3 /s), and r is the radius of the inlet (mm). The sediment concentration of the inlet flow is 10 g/L, i.e., the mass flow rate of sand is 1% of the flow rate Q. The diameters of injected solid particles conform to the Rosin-Rammler distribution, with a range of 0.001-0.1 mm. The relationship between the particle size distribution and the mass fraction satisfies the following functional relationship: where Y d is the mass fraction, d is the particle diameter, d is the mean particle diameter (0.05 mm), and n is the spread parameter (3.5).
Particle erosion rates were monitored at wall boundaries. The erosion rate is defined by Equation (3): where C(d p ) is a function of the particle diameter, set to a constant of 1.8e-9, f (a) represents the function of impact angle, and a is the impact angle of the particles on the wall. b(v) represents the function of relative particle velocity, set to a constant of 2.6 for the simulation, v is the relative velocity of the particle phase to the wall, and A (face) is the area of the cell face at the wall. According to the experimental measurement of Ge et al. (2009), the normal discrete phase reflection coefficient R cn was set to 0.44, and the tangent discrete phase reflection coefficient R ct was set to 0.83.

Computational grid and independent test
The simulation grids were acquired by ANSYS ICEM 17.1 software, which include a simplified structured grid of the main channel for DPM simulation, and an unstructured grid of the whole channel for Eulerian multiphase flow simulation. To ensure calculation accuracy and reduce processor calculation cost, independent tests on the calculation mesh were performed to identify the optimal mesh. In Table 2, five types of meshes with different cell sizes were used to simulate the internal flow of channels under 200 kPa. Outlet velocity was monitored to evaluate the computational accuracy of the optimal mesh. The unstructured Mesh 4 with 3 Â 10 5 grids and the structured Mesh 4 with 2.3 Â 10 5 grids were selected as computational grids for numerical simulations.  Figure 4 shows the general relationship between the flow rate and pressure of the irrigator. At a certain sediment concentration, flow rate values with different sand particle diameters were averaged, as the particle concentration is the main factor causing blockage at a relatively large flow rate of the irrigator (Hadad & Gurka 2013). Under the same pressure level, a greater sand concentration results in a smaller outlet flow of the regulator. The flow rate of the irrigator was tested with sandy water with a sand concentration of 10 g/L. The errors between experimental data and simulation results remain within 10%, indicating calculation accuracy of the numerical simulation. Sigma Plot 14.0 software was used for the linear regression calculation of the operating pressure and the corresponding discharge, which can be expressed by the flow rate-pressure equation, as: where Q represents the flow rate (L h À1 ), and the flow coefficient k is determined by the geometric structure of the emitter flow channel. x represents the flow index, which is a decimal in the range of (0, 1), used to evaluate the hydraulic performance of the emitter (the lower this index, the better the performance of the emitter). Figure 4 shows that an increase in sediment concentration leads to an increase in the discharge coefficient and a decrease in the flow index of the irrigator. A possible reason is that the energy-consuming movements of particles in the channel, such as collision and friction, are stronger at higher concentrations. While such active movements require more energy, they enable more sensitive flow rate adjustments of the irrigator. The flow index of the general labyrinth channel is ∼0.5 (Li & Li 2011) and the flow index of this irrigator is ∼0.35, indicating good energy dissipation performance of the labyrinth channel structure. Figure 5 depicts the relationship between operating pressure and turbulent kinetic energy (TKE) dissipation ratios under different conditions. Energy dissipation in the channel can be reflected by the TKE dissipation ratio, which is defined as the ratio of the difference in TKE between inlet and outlet to the inlet TKE. When the pressure is below 300 kPa, the TKE dissipation ratio remains steady at 0.75 despite pressure changes. Consequently, about 75% of the energy of the flowing medium is converted to internal energy and then dissipated. If the pressure exceeds 300 kPa, the TKE dissipation ratio gradually decreases with increasing pressure. This indicates that the TKE difference between the outlet and the inlet of the flow channel becomes small, and the energy dissipation effect of the flow channel worsens. For a constant pressure condition exceeding 300 kPa, if the concentration of sand particles increases, both the TKE dissipation ratio and the energy dissipation effect decrease. Similarly, the TKE dissipation ratio decreases significantly if the diameter of sand particles increases. Figure 6 shows the vorticity contour in the channel at a pressure of 300 kPa. At the inlet of the flow channel, because of the restricted channel structure, the flow direction of the incoming flow abruptly changes from Z direction to X-Y direction and forms many backflows and vortices. The mixing of flows is an effective energy dissipation behavior at the inlet position. Uncorrected Proof Therefore, the vorticity value in this region is much higher than that of the surrounding area. The existence of vortices as well as the viscous stresses in the boundary layer between fluid and wall cause locally high vorticity values. For example, at curve transition sections, streamlines are smooth and without vortices. However, the vorticity here is highest at around 8-20 (10 3 s À1 ) and water velocity reaches 10-12 m/s, implying that the flow is scouring the wall of the flow channel at high speed. Li et al. (2018) identified flushing of the channel as an effective way to reduce clogging. Therefore, in the flow field of the labyrinth channel, turbulent flow with high vorticity not only causes energy dissipation but also strengthens the selfwashing and reduces sedimentation of sand by flushing the channel wall. Moreover, the vorticity value is proportional to the energy dissipation and inversely proportional to the sediment concentration.

Sand deposition characteristics in the labyrinth channel
Pressure is one of the important factors affecting the distribution of sand particles in the labyrinth channel. The contours of the sand phase volume fraction under different pressure conditions are compared, to assess the relationship between pressure and sand deposition. It is found that 300 kPa is the critical pressure at which the location of sand deposition changes. Figure 7(a) shows the sand distribution characteristics in the channel for pressures below 300 kPa. Sand particles are mainly settled along the outer side of the bend and the straight channel, and sand deposition is proportional to sediment concentration. At the same concentration, the sediment distribution in the channel is similar, but smaller particle sizes are more likely to cause clogging. Since these small sand particles also have lower kinetic energy, once they enter the low-velocity zone, they cannot escape, resulting in clogging. Figure 7(b) shows the sand distribution in the channel for pressures above 300 kPa, where sand particles are mainly deposited in the downstream face of the bent channel, which is completely different. At high  Uncorrected Proof pressure levels, the kinetic energy of the flow is large. Thus, there is an obvious speed difference in the inner bend of the channel. The velocity of the upstream face exceeds that of the downstream face. Consequently, a low-velocity zone forms at the downstream face, where sand particles are deposited. Figure 8 shows the single emitter clogging rate with different sandy water conditions under different pressures. According to the international standard, the single emitter clogging rate I can be calculated by Equation (5): As I ! 0.25, blockage occurs inside the emitter flow channel. Figure 8 shows that the clogging rate I is related to the operating pressure, and 300 kPa is the critical pressure threshold, determining whether blockage occurs. If the operating pressure remains below 300 kPa, the occurrence of clogging inside the channel depends on the conditions of the incoming flow (i.e., both sediment concentration and sand diameter). If the flowing medium is sandy water with high sediment concentration, flow with smaller sand diameters achieves a higher blockage rate, and blockage is more likely to occur in the channel. The possible reason for this phenomenon is that the kinetic energy of small sand particles is weak. After entering the lowvelocity region, sand particles do not have enough kinetic energy to escape. Thus, their deposition in the velocity dead zone leads to a risk of blockage. If the flowing medium is sandy water with low sediment concentration, flow with a larger sand diameter results in a greater blockage rate. The possible reason is that at low sediment concentration, small sand particles are directly washed out of the channel because of their better flow behavior, whereas large sand particles are more likely to deposit inside the channel because of gravity and collision. However, when the operating pressure exceeds 300 kPa, the clogging rates at any operating conditions remain below 0.25, i.e., clogging is less likely to occur in the channel.
In summary, the location of sand accumulation in the labyrinth channel of the emitter can be obtained through numerical simulation. The probability of the blockage of a single emitter under different sandy water conditions can be calculated. The clogging rate can be used to evaluate the optimal operating pressure and the critical filtering standard of the irrigation system.

Stokes number of particles in a labyrinth channel
The track of particles in turbulent flow is related to the inertia and expansion capabilities of particles, which can be reflected by the Stokes number (Crowe et al. 1988). The Stokes number is proportional to the ratio of the particle aerodynamic response time (t A ) and the fluid time (t F ), which is expressed as: where r p is the particle density, d p is the particle diameter, m l is the liquid dynamic viscosity, V is the flow velocity, and D is the hydraulic diameter. If S t , 1, particles will follow the flow very well; if S t 1, particles will deviate from the following trajectory; if S t . 1, particles can penetrate turbulent flow structures, and the inertial force plays a dominant role in the motion of particles (Liu et al. 2019). Uncorrected Proof Figure 9 shows the Stokes number of the particle swarm at the inlet of the channel according to DPM simulation. The result shows that the standard deviation of the Stokes number of sand between different inlet velocity conditions is not significant, i.e., remains within 0.13. If the diameter of sand particles exceeds 0.04 mm (which accounts for ∼70% of the particle swarm), the Stokes number of sand exceeds 1, implying that such sand particles do not move well with the flow in the flow channel. The followability of sand particles in the fluid affects the trajectory of particles. Therefore, it can be assumed that during the flow process, the probability for collision between sand particles and channel increases with a large Stokes number. This also implies that the erosive effect of sand particles on the wall of the channel is mainly caused by sand particles with a diameter exceeding 0.04 mm.

Erosion on the wall of the labyrinth channel
In the labyrinth channel, the bend section has a direct collision with the flow, and the erosion will be more severe here than at the straight section. Therefore, DPM simulation focused on monitoring the erosion rate on the wall of the arc channel. The results are presented in Tables 3 and 4. Table 3 shows the relationship between flow velocity and erosion rate on the channel wall. These results reflect that changing the inlet velocity has less effect on the sand erosion rate. If the sand concentration is 10 g/L, the average erosion rate remains around 3.66 Â 10 À8 kg/m 2 -s, independent of the inlet velocity. Table 4 shows an obvious proportional relationship between sediment concentration and erosion rate. With increase in sand concentration, the erosion on the channel wall increases because more solid particles collide with the channel surface. However, according to the research of Zheng et al. (2008), with increase in sediment concentration, the interaction between solid particles will consume considerable kinetic energy. Therefore, the erosion rate will not change significantly. To reduce erosion of the labyrinth channel surface and prolong the service life of the emitter, the concentration of impurities in the pipeline needs to be controlled, without having to adjust the operating flow rate of the micro-sprinkler irrigation system. Furthermore, the materials of the emitter should be adjusted by the corresponding water source quality to strengthen the adaptability of the emitter.
Figure 9 | Stokes number of particle swarms at the inlet of the channel, where the diameters of particles follow an R-R distribution, and the sand concentration is 10 g/L.

CONCLUSIONS
This study evaluated the flow characteristics of sandy water in a W-shaped labyrinth channel through CFD numerical simulation. The distribution of accumulated sand and erosion problems in the channel were investigated under different flow conditions. The simulation data show that because of the effects of the labyrinth flow channel structure on sandy water, the reflow mixing behavior and shear action inside the channel increase energy dissipation and reduce the risk of sand accumulation. The optimal operating pressure of the micro-sprinkler irrigation system is estimated from the deposition distribution and the erosion rate of sand by numerical simulation. Applying this optimal operating pressure reduces the sand deposition and erosion in the labyrinth channel while ensuring the basic hydraulic performance of the emitter. In this specific case study, the optimal operating pressure for the micro-sprinkler emitter is 300 kPa. The erosion of sand particles on the channel surface is proportional to the sediment concentration and independent of the flow velocity. Micro-sprinkler irrigation systems should formulate a relevant filtration standard of sand concentration rather than restrict the diameter of sand particles, as this reduces the configuration requirements of the filtration device and the system construction cost. The data and conclusions of this research are all drawn from numerical simulation, and corresponding experimental research may be implemented in the future to assess the actual flow conditions inside the labyrinth channel.