Sprinkler droplet impact angle affects shear stress distribution on soil surface – a case study of a ball-driven sprinkler

Droplet shear stress is the main cause of soil erosion under sprinkler irrigation, and the effect of droplet impact angle on the shear stress distribution cannot be ignored. In this study, a ball-driven sprinkler was selected to investigate the radial distributions of droplet impact angles under three operating pressures (0.25, 0.30, and 0.35 MPa) and two nozzle diameters (1.9 and 2.2 mm), which are commonly used in agricultural irrigation. The effect of droplet impact angles on the distances from the sprinkler, droplet impact velocities, and shear stresses were analyzed by a 2DVD instrument. Irrespective of the nozzle diameter or operating pressure, the droplet velocities and impact angles near the sprinkler were distributed at 1.0–5.5 m s 1 and 70–90 , respectively, and the droplet shear stress increased with the distance from the sprinkler. Suitable operating pressure and distance from the sprinkler significantly reduced the droplet shear stress. Although the nozzle diameter had a certain effect on the maximum shear stress, the overall effect was insignificant. We developed the models for the radial distribution of droplet shear stresses, which were in good agreement with the measurement. This study proposes a new method for accurately predicating the soil erosion under sprinkler irrigation.


GRAPHICAL ABSTRACT INTRODUCTION
As a high-efficiency water-saving irrigation technique, sprinkler irrigation has been developed, promoted, and applied in China since the 1950s (Yan et al. ). By the end of 2018, China's sprinkler irrigation area has reached 4.4 million hm 2 , accounting for about 20% of the total area under high-efficiency water-saving irrigation. Soil surface sealing or crust formation caused by sprinkler irrigation is common (Chang & Hills a). It not only reduces the water infiltration rate, but also leads to surface runoff. Previous studies have shown that soil particle detachment is the main reason for soil surface sealing (Assouline & Ben-Hur ) and generally related to droplet kinetic energy (Yan et al. ; Caracciolo et al. ). However, some studies (Huang et al. ; Ghadiri & Payne ) showed that from a mechanism perspective, the soil particle detachment from aggregates is caused by external shear stress rather than droplet kinetic energy.
Several studies have been conducted on the stress distribution due to the droplet impact on the soil surface and the soil particle decomposition. Ghadiri & Payne (), based on the water-hammer theory, calculated the compressive stress of falling raindrops and shear stress caused by the flow impact. They found that the lateral shear stress was several times larger than the compressive stress. In another study, Huang et al. () used the Marker and Cell (MAC) numerical technique to simulate the impact of raindrops on a rigid surface. The results showed that the impact pressure distribution was non-uniform. The maximum pressure occurred at the contact circumference, and the rebound velocity on the rigid surface was twice the impact velocity. The studies mentioned above mainly focused on the vertical impact of raindrops, which was quite different from sprinkler irrigation, where droplets hit the soil at oblique angles. Chang & Hills (b) developed a numerical simulation model for studying the pressure and shear stress distribution on a soil surface following sprinkler droplet impact. Simulation analysis indicated that the oblique droplet impact decreased the magnitude of the impact force, but it increased the shear stress, compared with the vertical droplet impact. In another study, Chang & Hills (a) investigated the influence of sprinkler droplet impact angles on soil infiltration through laboratory experiments. They found that the average steady infiltration rates for all soil types increased in the following order of the impact angle: 60 , 45 , and 90 , which implied that the effects of sprinkler droplet impact angle on the shear stress distribution and soil infiltration need to be considered. However, in their research, only three droplet impact angles (90 , 45 , and 60 ) were selected, which was insufficient to reflect the impact of water droplets on the ground at multiple angles. Hence, the effect of droplet impact angle on the shear stress distribution needs to be further investigated.
The sprinkler is an important component of a sprinkler irrigation system, and its performance directly affects the quality of the whole system (Zhu et al. ; Zhang et al. ). With the development of the low-pressure and energy-saving sprinkler irrigation technology, it is critical to develop sprinklers with simple structures at low operating pressure. In recent years, a ball-driven sprinkler that uses a stainless-steel ball to drive the nozzle to rotate and spray has been successfully applied in field crop irrigation (Hui et al. ). However, there are still few reports on the effects of droplet impact angle on the shear stress distribution of a ball-driven sprinkler. In this context, the main objectives of this study were (1) to investigate the radial distributions of droplet impact angles for the ball-driven sprinkler under three operating pressures (0.25, 0.30, and 0.35 MPa) and two nozzle diameters (1.9 and 2.2 mm); (2) to establish the relationships between the droplet impact angle and distance from the sprinkler, droplet velocity, and shear stresses, respectively; and (3) to further develop the models for the radial distribution of droplet shear stresses, validated by experimental data. Figure 1 shows the flowchart of the experiment process.

Experimental setup
The experiment was performed indoors to avoid the impacts of wind. Figure 2 shows the experimental setup.
We used a ball-driven sprinkler with an inlet diameter of 12.7 mm, which is not only simple in its structure and easy to clean, but also has a good application uniformity (Hui et al. ). The components of the sprinkler include a nozzle (diameter of 1.9 or 2.2 mm), a chamber (the space enclosed by the upper and lower parts of the sprinkler

Experimental design
The operating pressure and nozzle diameter of the sprinkler were considered in the radial test of water droplets. Three operating pressures were chosen according to manufacturer's recommendation, namely 0.25, 0.30, and 0.35 MPa. Also, the two nozzle diameters of 1.9 and 2.2 mm that are commonly used in agricultural irrigation were chosen for testing. In total, six trials were performed, each with three replicates.
In each trial, the measuring points were set at 1-m intervals along the radial direction, and the diameters and velocities of the sprinkler droplets falling on each measuring point were monitored by the 2DVD, as shown in Figure 2. The number of the droplets collected at each measuring point was at least 10,000. All the measurements were taken when  the pressure was kept stable. Indoor air and water temperature were approximately 30 and 26 C, respectively, and relative humidity was 60%. The following design standards were adopted in the present experiment: ISO 7749-2 (ISO Standards ) and ISO 15886-3 (ISO Standards ).

Calculation of resultant droplet velocity, impact angle, and shear stress
Since the vertical (V v ) and horizontal (V h ) velocities of a water droplet could be recorded by the 2DVD, the calculation equation for the resultant velocity (V ) of a water droplet was as follows: The impact angle (θ) of a water droplet was obtained by the following equation: where V is the resultant droplet velocity, m·s À1 ; θ is the droplet impact angle, ; V v is the vertical droplet velocity, m·s À1 ; and V h is the horizontal droplet velocity, m·s À1 .
The shear stress (S) of a water droplet in this study was defined as the following formula (Ghadiri & Payne ): where S is the droplet shear stress, N m À2 ; ρ is the droplet density, kg m À3 .

Data analysis
The coefficient of variation (CV) was used to evaluate the dispersion degree of droplet impact angles at different

Radial distribution of droplet impact angles
The droplet impact angle is closely related to the distribution of the droplet shear stress and the soil infiltration (Hattori & Kakuichi ). Figure 5 presents the radial relative fre-    ively. This showed that raising the sprinkler working pressure might contribute to increasing the impact angles of water droplets at a location away from the sprinkler.
This result was not surprising, since higher sprinkler operating pressures generated smaller droplets and faster initial ejecting velocities, which might carry the smaller droplets to a farther location. As shown in Table 1 () also supported this finding.
The effect of nozzle diameter on the droplet impact angle was not significant (P > 0.05, results not presented) at the same operating pressure and measuring point, which was in good agreement with the results reported by Chang & Hills (a). This was partly because larger nozzles generated larger droplet sizes. Thus, the loss ratios of air resistance decreased. Simultaneously, larger nozzles also had less friction losses, which increased the initial ejecting velocities. Hence, the sum of these two effects appeared to be zero, which has also been found by Hills & Gu ().

Relationship between droplet impact angle and distance from the sprinkler
To further determine the relationship between droplet impact angle and distance from the sprinkler, the radial dis- According to regression analysis results, the average droplet impact angle and the distance from the sprinkler showed a good linear relationship (the mean of R 2 under all working conditions was 0.850, Figure 6) for three operating pressures and two nozzle diameters, which could be expressed by Equation (4): where θ a is the average droplet impact angle, ; l is the distance from the sprinkler, m; a and b are the coefficients in the equation.

Relationship between droplet impact angle and droplet velocity
Droplet impact velocity is a key factor affecting the droplet shear stress, and a high droplet velocity has a higher potential to detach soil particles. Figure  With the gradual increase of the distance, the distribution range of droplet velocities expanded continually, but the impact angles of some water droplets began to  Corrected Proof their linear equations were expressed as follows: where V a is the average droplet velocity, m s À1 ; c and d are the coefficients in the equation.
After removal of the coefficients c and d, using a stepwise linear regression procedure, the relationships of θ a with p and V a for the two nozzles were determined using the following equations: For the nozzle diameter of 1.9 mm, θ a ¼ 24:30pV a À 11:44V a À 73:60p þ 113:90 (12) For the nozzle diameter of 2.2 mm, θ a ¼ 10:90pV a À 7:00V a À 29:70p þ 99:08 (13)

Relationship between droplet impact angle and droplet shear stress
Droplet shear stress is a major contributor to soil surface crusting (Ghadiri & Payne ). Figure 9 presents the relationships between droplet impact angles and shear stresses for different sprinkler working conditions. Obviously, a greater distance from the sprinkler resulted in an increased droplet shear stress, and the peak shear stresses of all irrigation treatments appeared at the end of the spray jet (10 m).  10,798.4, 9,682.9, and 7,246.1 N m 2 , respectively. These data revealed that the maximum shear stresses at 2.2 mm were above those at 1.9 mm, and the maximum values gradually decreased as operating pressures increased. This finding fully explained that increasing the sprinkler pressure or decreasing the nozzle size could effectively lessen the upper limit of droplet shear stresses, although the effect of nozzle size on the impact angle distribution was not significant. The main reason for this phenomenon was that the decrease of the nozzle size reduced droplet impact velocities, and therefore the shear stresses correspondingly tapered off, although the droplet impact angle was only slighlty changed (Ghadiri & Payne ).
Based on Figure 9, with a larger droplet impact angle, the shear stress decreased, irrespective of the nozzle diameter and the operating pressure. This implied that the droplet impact angle was closely related to the shear stress, which was in line with the results reported by Chang & Hills (a, b). Through regression analysis, we found a strong quadratic relationship between the droplet shear stress and impact angle, with an average R 2 value of 0.916 ( Figure 9). The quadratic equation could be expressed as follows: where f, g, and h are the coefficients in the equation.
Due to the good linear relationships between p and coefficients f, g, and h, the equations of S with p and θ for the two nozzles could be calculated by Equations (15) and (16), respectively.
For the 1.9 mm diameter, the equation was as follows: S ¼ À8p Á θ 2 þ 10:61θ 2 þ 1207p Á θ À 1740:70θ For the 2.2 mm diameter, the equation was as follows: Model for the radial distribution of droplet shear stresses Scientific prediction of the radial distribution of droplet shear stresses is of great significance for optimizing sprinkler irrigation systems and reducing soil erosion risk. Figure 10 presents the distribution models of aver-    (17) and (18).
Our findings provide a new method for accurately predicating the soil erosion under sprinkler irrigation, and a basis for optimizing the working parameters of the sprinkler irrigation system. However, due to the unique structure of the ball-driven sprinkler, it may not be suitable for other typical sprinkler types, such as the impact sprinkler, fixed spray plate sprinkler and rotating spray plate sprinkler, etc., so further research is deserved. Besides, this study does not involve the relationship between the droplet shear stress and soil infiltration rate, which still needs to be determined.