Effects of Longitudinal and Transverse Travel Direction on the Hydraulic Performance of Sprinkler Machines on Sloping Terrain

Abstract

Background: The influence of varying slope gradients on the hydraulic performance of sprinkler irrigation systems operating on sloping farmland is investigated, with a specific focus on longitudinal and transverse travel directions. Methods: A sprinkler irrigation performance test system for sloping terrain is established, combining theoretical analysis with field experiments. The study systematically examines how different slope conditions affect sprinkler irrigation intensity and uniformity, and identifies the impact of longitudinal and transverse sprinkler machine movement on hydraulic performance under sloping conditions. A three-factor, three-level orthogonal experimental design is adopted, using slope gradient, nozzle pressure, and walking speed as variable factors, with sprinkler irrigation intensity and uniformity as optimization objectives. Results: Results indicate that the influence of slope angle on both sprinkler irrigation intensity and uniformity diminishes along both longitudinal and transverse travel directions. Furthermore, slope gradient exerts the greatest influence on sprinkler irrigation uniformity, followed by walking speed and nozzle pressure. For sprinkler irrigation uniformity, optimal parameters under transverse travel are determined as a 5° slope, 250 kPa pressure, and 30 m/h speed. Under longitudinal travel, optimal sprinkler irrigation uniformity is achieved at a 5° slope, 300 kPa pressure, and 30 m/h speed. Regarding sprinkler irrigation intensity, nozzle pressure and walking speed demonstrate the greatest influence, followed by slope gradient. Optimal intensity for both travel directions is attained at a 5° slope, 300 kPa pressure, and 30 m/h speed. Conclusion: These findings provide theoretical guidance for selecting sprinkler travel patterns and optimizing operational parameters on sloping farmland, offering significant practical implications for enhancing irrigation efficiency and mitigating slope soil erosion.

1. Introduction

Sloping terrain, comprising over one-third of China’s arable land, represents a vital agricultural resource in mountainous and hilly regions. However, steep gradients and severe soil erosion render traditional irrigation methods inefficient, impeding sustainable agricultural development [1]. Sprinkler irrigation has emerged as a critical solution for these areas due to its water-saving potential and adaptability [2]. The undulating topography influences sprinkler system movement, categorized as longitudinal and transverse travel. Interactions between travel direction, slope gradient, and operational parameters may significantly affect sprinkler irrigation intensity and uniformity [3].

Recent research has focused on slope sprinkler irrigation performance. Zhang et al. [4] found that terrain slope significantly altered the distribution and extent of water in their study of the Rainbird LF1200 nozzle(Rainbird Corporation, Glendora, CA, USA). Water distribution becomes increasingly concentrated upslope as slope increases, while throw radius decreases upslope and increases downslope. A derived formula effectively predicts throw radius on slopes. Although slope itself did not significantly affect water application uniformity under the test conditions, pressure and sprinkler spacing had a greater impact on uniformity compared to flat ground. Ge et al. [5] derived radial water distribution curves using interpolation and curve-fitting techniques, enabling the calculation of moving sprinkler water distribution and uniformity coefficients. Issaka et al. [6] investigated the hydraulic performance of impact sprinklers incorporating a fixed water dispersion device under controlled indoor conditions. Key nozzle parameters—including tip geometry, impact angle, diameter, and jet flow immersion depth—significantly governed performance outcomes. A nozzle configuration is recommended for design optimization to enhance water distribution efficiency in low-pressure sprinkler applications. Xu et al. [7] investigate the efficiency of drip irrigation systems. The flow channel structure is optimized through orthogonal experiments, while flow field indicators and sand particle movement characteristics are analyzed using the computational fluid dynamics–discrete element method (CFD-DEM). Through linear regression analysis, the relationship between hydraulic performance and anti-clogging performance is revealed. Tang et al. [8] examine the combined effects of irrigation and slope on key traits of Panax notoginseng. By employing modeling and variable regression analysis, optimal coupling patterns of irrigation and slope for cultivating Panax notoginseng on sloping farmland are comprehensively evaluated. Chauhdary et al. [9] conducted a systematic analysis to examine aspects related to sprinkler irrigation and precision irrigation, identifying key factors influencing the transition of sprinkler irrigation toward precision application methods. Limited research exists on subsurface sprinkler travel modes in sloping terrain or the combined effects of slope, nozzle pressure, and walking speed on irrigation performance. The mechanistic influence of these factors under sloping conditions remains inadequately elucidated, resulting in a lack of theoretical guidance for practical parameter configuration [10].

Existing studies predominantly optimize hydraulic performance for flat-ground systems, with insufficient investigation into the combined effects of travel direction and slope angle on sprinkler systems operating on sloping farmland. Furthermore, parameter adjustments often rely on single-factor analyses, which are insufficient for precision irrigation demands under complex operational scenarios. In contrast to these previous approaches, this study introduces a multi-factorial orthogonal experimental design that simultaneously examines the interactive effects of slope gradient, operating pressure, and travel speed. This methodological framework allows for a systematic analysis of both lateral and longitudinal travel directions, thereby providing a comprehensive optimization strategy for sprinkler irrigation on slopes [11,12].

Terrain undulations, especially slope gradients, are key factors leading to uneven spatial distribution of irrigation water. Multiple irrigation methods are employed on sloping land. Although drip irrigation and micro-irrigation help minimize surface runoff, they are limited by poor adaptability to complex terrain, high system costs, and a tendency toward clogging. In comparison, sprinkler irrigation offers better terrain adaptability and greater cost-effectiveness on slopes. Therefore, a precise analysis of the influence of slope on the hydraulic performance of irrigation systems is essential for providing critical data and theoretical support. The study focuses on sprinkler irrigation systems operating on sloping terrain. Multi-slope experiments under different travel directions are conducted to comparatively analyze the influence of slope on sprinkler irrigation intensity and uniformity under lateral versus longitudinal movement. An orthogonal experimental design quantifies the weighting factors of slope, pressure, and speed on hydraulic performance [13], aiming to reveal multi-parameter synergistic optimization mechanisms. This work provides technical support for the scientific design and efficient operation of slope irrigation systems. Optimizing these systems holds significant application value for irrigation in hilly and mountainous regions and is crucial for enhancing irrigation efficiency and water conservation [14,15,16].

2. Materials and Methods

2.1. Experimental Setup and Slope Design

A schematic diagram of a sprinkler machine for hilly-mountainous terrain (Figure 1), illustrating its specific structure and components. The sprinkler head employed in this experiment was a DWPT-type flat spray nozzle with an equivalent outlet diameter of 4.88 mm, which demonstrates a stable correlation between pressure and flow rate. The irrigation system was supplied by a crawler-type sprinkler irrigation machine, providing reliable climbing capability and stable irrigation performance especially under sloped terrain conditions. The experimental site was selected on representative sloping farmland with varying gradients, prioritizing the investigation of slope effects on sprinkler hydraulic performance [17,18,19]. To comprehensively study system performance under different slope conditions, five slope gradients (5°, 10°, 15°, 20°, and 25°) were established for both transverse (parallel to contours) and longitudinal (normal to contours) travel directions. These gradients represent common slope ranges in agricultural practice, enhancing the applicability of the findings. The physical configuration of the hilly terrain sprinkler system is shown in Figure 2, demonstrating its capability for slope experiments and excellent gradeability. Travel direction selection considered topographic characteristics and irrigation requirements. Transverse travel (parallel to contours) and longitudinal travel (normal to contours) simulate operational methods encountered under actual field conditions. This approach enables comprehensive evaluation of sprinkler performance across diverse operating scenarios, informing optimal design and operational strategies for slope irrigation systems [20,21,22].

The hydraulic performance of a lateral-move sprinkler irrigation system operating on hilly terrain was investigated. The system was equipped with six nozzles. A grid array of rain gauges (inner diameter: 20 cm, height: 15 cm) was positioned along the travel direction. The field layout for the slope sprinkler irrigation experiment is presented in Figure 3, showing the in situ testing scenario and the arrangement of rain gauges on the sloping field. Under static test conditions, the first row of gauges was maintained at a distance of 2 m from the sprinkler pipeline. The rain gauge matrix featured six rows spaced longitudinally at 1 m intervals and ten columns spaced transversely at 0.5 m intervals (five columns positioned laterally on each side of the travel path). Following the completion of each irrigation event on the slope, hydraulic performance parameters were determined by measuring the collected water volume within each gauge. All tests were conducted under highly stable meteorological conditions. Wind speed was monitored in real time using anemometers to ensure windless conditions were selected, thereby minimizing the interference of wind with water droplet distribution trajectories. All experiments were carried out at the same site to maintain consistent temperature and humidity ranges, effectively isolating the influence of extraneous environmental variables. Systematic errors arising from environmental conditions were mitigated, allowing strict control over factors that could affect water distribution patterns beyond the primary variables under investigation. Prior to each experiment, the sprinkler system traversed a designated non-test zone to stabilize nozzle pressure, irrigation flow rate, and travel speed. This procedure prevented inaccuracies caused by transient hydraulic fluctuations and ensured that actual operating parameters aligned closely with the predefined target values. Prior to the experiment, all rain gauges underwent rigorous static calibration. The measurement accuracy procedure was as follows: Rainwater collected in the rain gauge was poured into a graduated cylinder with a measurement accuracy of 2 mL. The observer maintained eye level with the water surface in the graduated cylinder and recorded the measured value. To ensure data reliability, three independent replications were performed for each test condition, with random errors mitigated using the arithmetic mean method. Schematic diagrams detailing the experimental configurations for transverse travel (parallel to contour lines) and longitudinal travel (normal to contour lines) are provided in Figure 4 and Figure 5. Specifically, transverse travel refers to the movement of the sprinkler system parallel to the contour lines of the slope, i.e., horizontal displacement. Longitudinal travel indicates that the machine moves along the direction of the steepest slope, corresponding to ascending or descending the incline directly.

2.2. Parameter Selection Principles

Based on preliminary experiments and relevant data, key control parameters and experimental ranges influencing the hydraulic performance of translating sprinkler irrigation systems are established [23,24,25,26]. Hydraulic performance encompasses both sprinkler irrigation intensity and uniformity. The application rate governs soil infiltration and runoff potential by evaluating the amount of water applied per unit time, while sprinkler irrigation uniformity reflects the spatio-temporal consistency of water distribution to crops, assessing its evenness across the field [27,28]. Slope gradient critically influences hydrodynamic behavior. The primary distinction between sloping and flat terrain resides in gradient, which substantially modifies the gravitational component of water flow direction, thereby affecting irrigation water distribution. Gradient further governs runoff volume, soil erosion risk, and sprinkler irrigation uniformity, making it a primary design consideration for sloping-land sprinkler systems. When gradients exceed 15°, accelerated surface flow occurs, and applied water becomes runoff-prone. Crops on steeper slopes may receive insufficient moisture, whereas lower-lying areas experience localized waterlogging, resulting in uneven water distribution and reduced irrigation efficiency. Gradients below 5° inadequately demonstrate slope irrigation impacts; consequently, three test gradients (5, 10, and 15°) are selected [29,30,31,32].

Nozzle pressure governs droplet atomization quality. Investigating pressure effects on atomization across varying slopes is essential for optimizing sloping-land sprinkler system performance. Enhanced atomization promotes finer, more uniform aerial water dispersion, increasing droplet-air interfacial contact and reducing droplet descent velocity, thus improving sprinkler irrigation uniformity and infiltration. On slopes, effective atomization adapts to complex topography, ensuring complete water coverage. Poor atomization yields large droplets that cause immediate runoff or uneven slope distribution, compromising irrigation quality. Pressures exceeding 300 kPa elevate per-unit-area energy consumption, and droplet kinetic energy may exceed crop tolerance thresholds. Conversely, pressures below 200 kPa reduce flow coefficients and atomization angles, failing to meet crop water demands. Thus, three pressure levels (200, 250, and 300 kPa) are tested.

Walking speed governs both irrigation coverage area and water distribution per unit time while also influencing operational efficiency and energy use. Higher speeds improve equipment efficiency by covering larger areas within equivalent durations. Analysis of speed-dependent hydraulic performance enables identification of optimal velocities that enhance system performance and economic returns while maintaining irrigation quality. However, speeds surpassing 60 m/h impose excessive mechanical loads on system components, accelerating wear and energy consumption. Excessively slow speeds (<30 m/h) reduce operational efficiency and increase per-unit-area irrigation costs. Accordingly, three speed gradients (30, 45, and 60 m/h) are selected.

This study therefore optimizes the effects of slope (i), nozzle pressure (p), and walking speed (v) on sprinkler irrigation uniformity and application rate for mobile sprinkler systems in hilly terrain. These three parameters serve as experimental factors, with slopes tested at 5°, 10°, and 15°; pressures at 200, 250, and 300 kPa; and speeds at 30, 45, and 60 m/h. The hydraulic performance factor levels are presented in

Table 1. Factor level table.

Level	Factor
	A	B	C
	Slope/(°)	Nozzle Pressure/(kPa)	Walking Speed/(m/h)
1	5	200	30
2	10	250	45
3	15	300	60

.

2.3. Orthogonal Test Program Design

Compared with full-factorial experimentation, orthogonal design reduces the required number of trials exponentially while preserving experimental completeness, significantly enhancing efficiency. This study employed an L₉(3³) orthogonal array—a three-factor, three-level design requiring only nine experimental runs to comprehensively evaluate all factor-level interactions. The experimental matrix is presented in

Table 2. Orthogonal Test Protocol Table.

Test Number	A	B	C
1	1	1	1
2	1	2	2
3	1	3	3
4	2	1	2
5	2	2	3
6	2	3	1
7	3	1	3
8	3	2	1
9	3	3	2

.

Analysis combined range analysis and analysis of variance (ANOVA). Range analysis quantifies factor influence magnitude on response variables, where the range value R directly indicates effect strength. Factors demonstrating larger R values receive priority consideration in optimization. ANOVA was performed using statistical software to conduct multifactorial significance testing. At α = 0.05, p > 0.05 indicates nonsignificant factor effects, p ≤ 0.05 denotes statistically significant effects, and p ≤ 0.01 identifies highly significant effects.

2.4. Hydraulic Performance Calculation Formulas

(1) Calculation formula of sprinkler irrigation intensity

Sprinkler irrigation intensity (I, mm/h) represents the depth of water applied per unit time. Following irrigation trials, water depth is determined gravimetrically using cylindrical collectors (inner diameter: 20 cm). The calculation is:

$$\begin{array}{c}I={\displaystyle \frac{\Delta M}{A\times t}}\times 10\times 3600\end{array}$$

(1)

where I is the sprinkler irrigation intensity, mm/h; $\Delta M$ indicates the volume change of water collected by the rain barrel before and after the test, cm³; A is the cross-sectional area of the rain barrel receiving port, cm²; t is the sprinkler irrigation time, s; Note: The unit of I is mm/h. The measured value is multiplied by 10 to convert cm to mm and by 3600 to convert s to h, resulting in the appropriate dimensional transformation.

(2) Calculation formula of sprinkler irrigation uniformity

Sprinkler irrigation uniformity is quantified using CU:

$$\begin{array}{c}Cu=\left(1-{\displaystyle \frac{{\sum }_{i=1}^n\left|x_i-\overline{x}\right|}{n\overline{x}}}\right)\times 100\%\end{array}$$

(2)

where $Cu$ is the sprinkler irrigation uniformity, %; $x_i$ represents the measured water depth of the i-th measurement point, mm; $\overline{x}$ is the average water depth of all measurement points, mm; n is the total number of rain barrels deployed.

3. Results and Analysis

3.1. Effect of Slope Factor on Hydraulic Performance

3.1.1. Influence of Different Slopes of Transverse Travel on Sprinkler Irrigation Intensity

Figure 6 displays the average sprinkler irrigation intensity measured by rain gauges in each column under varying slopes. At a 5° slope, the average sprinkler irrigation intensity decreases laterally from left (lower elevation) to right (higher elevation), declining from 17.1 mm/h in Column 1 to 15.9 mm/h in Column 10, resulting in an expanded difference of 1.2 mm/h. As the slope increases to 15°, this gradient intensifies, with Column 1 recording 15.4 mm/h and Column 10 registering 13.1 mm/h, resulting in an expanded difference of 2.3 mm/h. In summary, lateral position exhibits a negative correlation with sprinkler irrigation intensity under identical slope conditions, where lower elevations consistently demonstrate higher intensity than higher elevations. Furthermore, increased slope significantly amplifies spatial variability in sprinkler irrigation intensity, and the overall average intensity decreases from 16.6 mm/h at 5° to 14.3 mm/h at 15°.

Experiments investigating sprinkler irrigation under varying slope conditions—using a sprinkler machine traversing slopes—demonstrated significant spatial variability in sprinkler irrigation intensity. During sprinkler machines transverse travel direction, slope gradients influence water trajectories from lateral sprinklers, generating distinct component forces. This induces elevation-dependent variations in operating pressure at sprinkler heads and produces asymmetric wetting patterns. Specifically, left-side (downslope) nozzles operate closer to ground level, where gravitational and slope effects increase static pressure and discharge rates. Conversely, right-side (upslope) nozzles at higher elevations experience pressure reduction and diminished discharge. Consequently, sprinkler irrigation intensity distribution becomes spatially heterogeneous across the lateral plane.

Regarding hydraulic networks, head loss varies with slope inclination, predominantly affecting truss-supported branch pipes aligned with the slope gradient. This orientation creates pressure differentials between the lowest (first) and highest (last) nozzles. At 15° slopes, pressure at elevated nozzles decreases substantially relative to lower nozzles, whereas contour-parallel main pipes exhibit minimal pressure fluctuation. This occurs because water flowing upward along branch pipes must overcome both the gravitational component parallel to the slope and localized terrain resistance, progressively increasing head loss. Downslope nozzles conversely benefit from gravitational potential energy, resulting in minor pressure variations. Furthermore, steeper slopes accentuate the conversion of gravitational potential to kinetic energy, intensifying pressure instability across pipelines and nozzles and thereby exacerbating cumulative head loss.

3.1.2. Influence of Different Slopes of Transverse Travel on Sprinkler Irrigation Uniformity

Sprinkler irrigation uniformity under varying slope gradients was determined during lateral movement tests of the irrigation machine across slopes at nozzle pressures of 200 kPa and 250 kPa. This analysis elucidates the variation in machine uniformity with slope gradient and compares performance between pressure regimes. Sprinkler irrigation uniformity under transverse travel at varying pressures and slope gradients is presented in Figure 7.

During lateral movement at 250 kPa nozzle pressure, sprinkler irrigation uniformity decreases progressively with increasing slope. Maximum uniformity (86.2%) occurs at 5°, declining to a minimum (62.3%) at 25°. Similarly, at 200 kPa, uniformity decreases from 81.2% (5° slope) to 57.3% (25° slope). The sprinkler irrigation uniformity varies significantly with slope angle during lateral system movement. Increased slope consistently results in decreased Cu values. At 5° slopes, Cu values remain relatively high, indicating favorable uniformity. Conversely, at 25° slopes, Cu values exhibit substantial reduction—declining by 24.0 percentage points at 250 kPa and 23.9 points at 200 kPa compared to 5° conditions. Lower operating pressure (200 kPa) consistently yields poorer uniformity across all slopes relative to 250 kPa operation.

This phenomenon occurs because spray trajectory is governed by initial velocity, ejection angle, and gravitational acceleration. Uphill nozzles must counteract the downslope gravitational component, reducing effective spray range. Conversely, downhill nozzles experience gravitational acceleration, extending their range. As slope increases, the velocity component parallel to the slope intensifies, exacerbating water distribution asymmetry. Consequently, the elliptical wetting pattern undergoes deformation: the uphill sector contracts due to gravitational resistance, while the downhill sector expands through gravitational assistance. This imbalance in truss-end wetting pattern overlap directly reduces the sprinkler irrigation uniformity.

3.1.3. Influence of Different Slopes of Longitudinal Travel on Sprinkler Irrigation Intensity

Sprinkler irrigation intensity variation was investigated during longitudinal machine movement (parallel to slope contours) across gradients at nozzle pressures of 200 kPa and 250 kPa. Figure 8 presents the corresponding intensity values under these pressure-slope combinations. Under both pressures, sprinkler irrigation intensity decreases with increasing slope angle. At 250 kPa, intensity declines from 17.5 mm/h (5° slope) to 13.1 mm/h (25° slope). Similarly, at 200 kPa, intensity decreases from 15.2 mm/h to 11.1 mm/h over the same slope range. These results demonstrate a significant negative correlation between sprinkler irrigation intensity and slope angle during longitudinal operation.

From an irrigation mechanics perspective, longitudinal uphill movement engages a dual-action mechanism. First, the downslope gravitational component opposes upward spray trajectories, inducing flow obstruction and velocity reduction. Given the direct relationship between sprinkler irrigation intensity and flow velocity, this velocity decline proportionally reduces sprinkler irrigation intensity. Second, on the downslope spray side, gravitational acceleration extends throw distance but causes water overshoot beyond target areas, resulting in effective application volume loss. The synergistic action of these factors systematically reduces net sprinkler irrigation intensity.

Terrain-mediated mechanisms further explain this behavior: On gentle slopes (e.g., 5°), minimal vertical gravitational components permit stable flow regimes that promote uniform water distribution and complete infiltration, thereby attenuating intensity reduction. At steeper slopes (>15°), amplified slope-parallel gravitational components accelerate surface runoff and prolong droplet retention on inclined surfaces. Under extreme slopes (25°), gravitational forces progressively dominate over initial spray kinetic energy, transitioning flow dynamics toward terrain-potential-energy control. This shift precipitates marked declines in sprinkler irrigation intensity.

3.1.4. Influence of Different Slopes of Longitudinal Travel on Sprinkler Irrigation Uniformity

Sprinkler irrigation uniformity during longitudinal machine movement (parallel to slope contours) across varying gradients was determined at nozzle pressures of 200 kPa and 250 kPa. This analysis quantifies slope-induced uniformity changes, with corresponding diagrams presented in Figure 9. At 250 kPa nozzle pressure, the sprinkler irrigation uniformity decreases from 85.8% (5° slope) to 63.4% (25° slope). Similarly, at 200 kPa, Cu declines from 84.8% to 58.3% over identical slope conditions. These trends demonstrate that sprinkler irrigation uniformity decreases progressively with increasing slope during longitudinal operation.

The experimental results establish terrain slope as the primary factor influencing Cu values under longitudinal movement. Increasing slope angles consistently reduce Cu, with favorable uniformity (Cu > 84%) observed at 5° slopes. Conversely, 25° slopes induce substantial uniformity degradation, exhibiting 22.4 and 26.5 percentage-point reductions at 250 kPa and 200 kPa, respectively. Lower operating pressure (200 kPa) consistently yields poorer uniformity across all slopes compared to 250 kPa operation.

Compared to lateral movement, longitudinal operation exhibits distinct droplet trajectory dynamics. For uphill nozzles, the initial droplet velocity vector opposes the slope gradient, extending flight duration while reducing horizontal throw distance. Conversely, downhill nozzles discharge droplets with velocity vectors aligned to the slope, increasing throw distance and inducing forward displacement of overlapping spray patterns. Concurrently, gravitational acceleration promotes downslope water migration along the terrain, creating moisture distribution asymmetry. This manifests as lower-elevation over-saturation and upper-elevation under-irrigation. Such distributional non-uniformity intensifies progressively with slope inclination, resulting in significant degradation of the sprinkler irrigation uniformity.

3.2. Optimization of Hydraulic Performance by Different Factors Under Orthogonal Test

3.2.1. Orthogonal Test Analysis of Hydraulic Performance Under Transverse Travel

Through orthogonal design experiments conducted using a sprinkler irrigation system, the hydraulic performance of slope farmland under lateral movement was assessed. Range analysis was employed to investigate the influence of slope gradient, sprinkler pressure, and walking speed on slope sprinkler irrigation performance. Analysis of the orthogonal test results indicates that the range value R (i.e., the maximum difference between the average indicator values across levels of a given factor) directly reflects the magnitude of factor influence. A larger R value signifies a greater impact on the indicator; consequently, such factors are prioritized in experimental design.

The range analysis results for hydraulic performance are presented in

Table 3. Analysis table of orthogonal test results of transverse travel.

Test No.	A	B	C	Sprinkler Irrigation Uniformity (%)	Sprinkler Irrigation Intensity (mm/h)
1	1	1	1	89.8	15.7
2	1	2	2	84.5	12.5
3	1	3	3	74.1	10.7
4	2	1	2	76.5	10.5
5	2	2	3	73.8	8.8
6	2	3	1	79.1	19.3
7	3	1	3	64.2	8
8	3	2	1	76.4	14.6
9	3	3	2	68.6	11.9
Optimal Uniformity	1	2	1	92.2	13.5
Optimal Intensity	1	3	1	86.6	22.6
$\overline{K_1}$	82.80	76.83	81.77
$\overline{K_2}$	76.47	78.23	76.53
$\overline{K_3}$	69.73	73.93	70.70
R (sprinkler irrigation uniformity)	13.1	4.3	11.1
$\overline{K_1}$	13.0	11.4	16.5
$\overline{K_2}$	12.9	12.0	11.6
$\overline{K_3}$	11.5	14.0	9.2
R (sprinkler irrigation intensity)	1.5	2.6	7.3

Note: where $\overline{K_i}$ (factor level indicator, i = 1, 2, 3) is the average value of the test results when the influencing factors in column j (j = A, B, C) are taken at level i in Table 3, R is the extreme difference value of column j, and R = max ($\overline{K_1}\overline{,K_2},\overline{K_3}$) − min ($\overline{K_1,}\overline{K_2}\overline{{,K}_3}$).

. For sprinkler irrigation uniformity, slope is identified as the most influential factor, followed by walking speed, with nozzle pressure having the least impact. The order of influence is A > C > B. Under the condition that the sprinkler irrigation uniformity standard is 85%. The optimal combination for achieving higher sprinkler irrigation uniformity is A₁B₂C₁ (achieved optimal sprinkler irrigation uniformity of 92.2%), which corresponds to a slope of 5°, a nozzle pressure of 250 kPa, and a walking speed of 30 m/h. In contrast, walking speed exerts the strongest effect on sprinkler irrigation intensity, followed by nozzle pressure, while slope shows minimal influence. The significance ranking is C > B > A. Horizontal optimization suggests that configuration A₁B₃C₁ provides the best performance for sprinkler irrigation intensity under the test conditions. Horizontal optimization yields the optimal parameter combination A₁B₃C₁ (achieve the optimal sprinkler irrigation intensity of 22.6 mm/h), corresponding to a slope of 5°, nozzle pressure of 300 kPa, and walking speed of 30 m/h. This combination yielded optimal water application performance in slope farmland field trials, providing a scientific basis for sprinkler irrigation machine parameterization.

By plotting evaluation indicator-factor change curves, this study investigates the patterns of influence of nozzle pressure and walking speed on hydraulic performance under slope conditions

Regarding sprinkler irrigation uniformity (Figure 10a), a decrease is observed as the slope increases when the sprinkler moves horizontally across the slope. Due to gravity, water experiences additional forces on the sloping terrain, inducing preferential downhill flow of droplets. This leads to uneven water distribution and reduced overall sprinkler irrigation uniformity. Under sloped conditions, sprinkler irrigation uniformity first increases and then decreases with increasing nozzle pressure. Higher pressure improves water atomization and promotes a more uniform spray fan, which enhances water distribution on the slope. However, beyond a certain pressure, the water stream becomes overly concentrated, forming larger droplets or jets that reduce aerial dispersion and worsen uniformity. Furthermore, sprinkler irrigation uniformity decreases as traveling speed increases on slopes. This decline is due to variations in discharge and spray range among individual sprinklers, leading to diminished overall sprinkler irrigation uniformity across the irrigated area.

For sprinkler irrigation intensity (Figure 10b), a decrease is seen with increasing slope. Water emitted from nozzles on both sides is affected by the slope, introducing directional components that cause asymmetry in nozzle pressure and spray coverage. The sprinkler irrigation intensity increases with higher nozzle pressure, as elevated pressure raises flow velocity and discharge rate, resulting in more water applied per unit time per unit area. In contrast, the application rate decreases with increasing traveling speed. A higher speed expands the coverage area per unit time, while the sprinkler discharge remains constant, reducing the amount of water delivered per unit area.

While preliminary range analysis identifies key factors influencing the hydraulic performance of the sprinkler irrigation system, it does not precisely determine the significance of each factor’s impact on the performance indicators. Therefore, variance analysis using the SPSS27 (IBM SPSS Statistics 27) data processing system is employed to calculate the significance coefficients of each parameter.

According to the variance analysis for lateral sprinkler irrigation uniformity (

Table 4. ANOVA for transverse travel effects on sprinkler irrigation uniformity in sloping terrain.

Source of Variance	Sum of Squared Deviations	Degree of Freedom	Mean Square	F-Value	p
Slope	256.187	2	128.093	49.015	0.020
Nozzle Pressure	28.860	2	14.430	5.522	0.153
Walking Speed	183.887	2	91.943	35.182	0.028

), the p-values for slope and walking speed are 0.020 and 0.028, respectively, both below the 0.05 significance threshold. Statistical significance is typically defined as p ≤ 0.05, with p < 0.01 considered highly significant. This indicates that both slope and walking speed exhibit a significant influence (0.01 < p < 0.05) on sprinkler irrigation uniformity. Conversely, the p-value for nozzle pressure (0.153) exceeds the significance threshold. The variance analysis for lateral sprinkler irrigation intensity (

Table 5. ANOVA for transverse travel effects on sprinkler irrigation intensity in sloping terrain.

Source of Variance	Sum of Squared Deviations	Degree of Freedom	Mean Square	F-Value	p
Slope	4.029	2	2.014	1.299	0.435
Nozzle Pressure	10.909	2	5.454	3.516	0.221
Walking Speed	84.362	2	42.181	27.194	0.035

) reveals a p-value of 0.035 for walking speed, indicating a statistically significant influence (0.01 < p < 0.05) on sprinkler irrigation intensity. In contrast, the p-values for slope (0.435) and nozzle pressure (0.221) exceed the significance threshold. In summary, slope primarily affects uniformity; its impact on intensity is comparatively minor.

3.2.2. Orthogonal Test Analysis of Hydraulic Performance Under Longitudinal Traveling

Orthogonal design experiments employing a sprinkler irrigation system were conducted to evaluate the hydraulic performance of slope farmland under longitudinal movement. The influence of slope gradient (A), nozzle pressure (B), and walking speed (C) on sprinkler performance was investigated using extreme value analysis (range analysis). For sprinkler irrigation uniformity (

Table 6. Analysis table of orthogonal experiment results for longitudinal travel.

Test No.	A	B	C	Sprinkler Irrigation Uniformity (%)	Sprinkler Irrigation Intensity (mm/h)
1	1	1	1	83.3	11
2	1	2	2	81.5	13.4
3	1	3	3	79.2	17.9
4	2	1	2	70.4	9.5
5	2	2	3	64.9	11.3
6	2	3	1	79.1	20.8
7	3	1	3	58.2	7.5
8	3	2	1	72.2	13.1
9	3	3	2	70.7	17.9
Optimal Performance	1	3	1	90.3	21.7
$\overline{K_1}$	81.3	70.6	78.2
$\overline{K_2}$	71.5	72.9	74.2
$\overline{K_3}$	67.0	76.3	67.4
R (sprinkler irrigation uniformity)	14.3	5.7	10.8
$\overline{K_1}$	14.1	9.3	15.0
$\overline{K_2}$	13.9	12.6	13.6
$\overline{K_3}$	12.8	18.9	12.2
R (sprinkler irrigation intensity)	1.3	9.6	2.8

): The range values (R) directly reflect the extent of each factor’s influence.

Slope gradient is the most influential factor affecting sprinkler irrigation uniformity, followed by walking speed, while nozzle pressure has the least impact. The order of factor importance is A > C > B. In contrast, nozzle pressure plays the dominant role in influencing sprinkler irrigation intensity, with walking speed being the second most significant factor. Slope gradient has only a minor effect. The importance ranking for intensity is B > C > A. Optimal Parameter Combination: Given the positive correlation between the average indicator values and performance, horizontal optimization identified the optimal parameter combination A1B3C1 for both sprinkler irrigation uniformity and intensity (the optimal sprinkler irrigation uniformity and sprinkler irrigation intensity reached 90.3% and 21.7 mm/h). This corresponds to specific operating conditions: a slope gradient of 5°, nozzle pressure of 300 kPa, and walking speed of 30 m/h. This combination demonstrated superior water application performance in slope irrigation trials, providing a scientific basis for sprinkler irrigation machine parameterization.

Evaluation index-factor change curves are plotted to investigate how variations in nozzle pressure, walking speed, and longitudinal slope conditions affect the hydraulic performance of sprinkler irrigation machines. The curves for sprinkler irrigation uniformity and intensity are presented in Figure 11.

Regarding sprinkler irrigation uniformity (Figure 11a), uniformity decreases as slope increases. Gravity causes water flow to spray preferentially downslope, resulting in uneven surface water distribution. This unevenness leads to overwetting in lower areas and insufficient moisture in upper areas, reducing overall uniformity. Under slope conditions, sprinkler irrigation uniformity increases with rising nozzle pressure. Increased pressure mitigates static head loss-induced pressure reduction under longitudinal slopes. Conversely, increasing walking speed decreases sprinkler irrigation uniformity on slopes. Higher walking speed accelerate drift loss, causing sprayed water to move outside the intended irrigation area.

For sprinkler irrigation intensity (Figure 11b), during longitudinal movement, the gravitational component acting downslope opposes the upward spray direction. This opposition impedes water flow movement, resulting in relatively reduced sprinkler irrigation intensity. However, under slope conditions, sprinkler irrigation intensity increases with higher nozzle pressure. Elevated pressure increases the initial velocity of ejected water, extending spray distance and broadening coverage. Additionally, greater nozzle pressure increases the water volume discharged per unit time, leading to higher water application per unit area and consequently increased sprinkler irrigation intensity. For instance, water ejected under high pressure possesses greater kinetic energy, enabling projection over longer distances, expanding the irrigated area, and enhancing intensity. Sprinkler irrigation intensity decreases as walking speed increases. Higher walking speed enlarges the area covered per unit time. With constant water discharge, the water volume per unit area decreases, reducing sprinkler irrigation intensity. For example, rapid sprinkler movement delivers less water per unit area, whereas slower movement increases unit-area water volume and corresponding intensity.

Analysis of variance (ANOVA) in SPSS determines factor significance for sprinkler irrigation performance. Significance coefficients quantify each parameter’s influence. Sprinkler irrigation uniformity (

Table 7. ANOVA for longitudinal travel effects on sprinkler irrigation uniformity in sloping terrain.

Source of Variance	Sum of Squared Deviations	Degree of Freedom	Mean Square	F-Value	p
Slope	321.496	2	160.748	166.100	0.006
Nozzle Pressure	49.496	2	24.748	25.572	0.038
Walking Speed	177.709	2	88.854	91.813	0.011

): Slope (p = 0.006), nozzle pressure (p = 0.038), and walking speed (p = 0.011) all exhibit a statistically significant influence (p < 0.05) on sprinkler irrigation uniformity during longitudinal movement. Sprinkler irrigation intensity (

Table 8. ANOVA for longitudinal travel effects on sprinkler irrigation intensity in sloping terrain.

Source of Variance	Sum of Squared Deviations	Degree of Freedom	Mean Square	F-Value	p
Slope	2.727	2	1.363	6.492	0.133
Nozzle Pressure	140.827	2	70.413	335.302	0.003
Walking Speed	11.207	2	5.603	26.683	0.036

): Nozzle pressure (p = 0.003) and walking speed (p = 0.036) exhibit a significant influence (p < 0.05) on intensity, while slope shows no statistical significance (p = 0.133). Critically, slope primarily affects sprinkler irrigation uniformity; its impact on intensity is comparatively minor.

4. Discussion

This study establishes a slope irrigation experimental platform to systematically reveal the influence of slope on hydraulic performance (sprinkler irrigation uniformity and intensity) during both lateral and longitudinal sprinkler travel, along with the underlying optimization mechanisms.

Study the differences in hydraulic performance responses to slope gradients under transverse and longitudinal walking modes. Transverse travel (parallel to contour lines): Significant differences in hydraulic performance response to slope gradients are observed between transverse and longitudinal travel modes. During transverse travel, sprinkler irrigation intensity increases in low-lying areas, with the disparity between high- and low-elevation areas becoming more pronounced as the slope gradient increases. This phenomenon primarily results from gravitational forces causing sprayed water to accumulate downslope. Lateral pipes arranged along the slope induce pressure differences between the first and last sprinklers. Furthermore, upon leaving the sprinkler, water droplet trajectories are deflected downslope by gravity. Concurrently, lower wind resistance downslope (or potential influence from downslope airflow) contributes to increased water accumulation in lower-elevation areas. Steeper slopes intensify this water flow convergence, exacerbating distribution non-uniformity characterized by water deficit uphill and surplus downhill. Longitudinal travel (perpendicular to contour lines): In contrast, during longitudinal movement, hydraulic performance exhibits an overall decreasing trend with increasing slope. This occurs because sprinkler rows arranged along the slope cause the sprinkler irrigation strip (or branch pipe) to lie at an angle on the slope surface. The sprayed water undergoes distortion along the slope, deforming the elliptical wetted area pattern. The wetted area contracts uphill (due to gravitational pullback) and extends downhill (due to gravity-assisted diffusion). Consequently, the water reception time per unit area decreases. Additionally, a portion of the water flow is directed more rapidly towards the slope bottom, reducing residence time and water accumulation directly beneath the sprinkler head, thereby decreasing overall hydraulic performance. This mechanism contrasts sharply with the localized hydraulic performance increases observed during transverse travel.

Orthogonal experiments and analysis of variance further quantify the relative influence of key factors. Slope is confirmed as the primary significant factor affecting sprinkler irrigation uniformity, with its influence exceeding that of nozzle pressure and walking speed. This result underscores the critical importance of controlling slope (or selecting flatter terrain) to ensure irrigation uniformity in slope operations. For sprinkler irrigation intensity, nozzle pressure and walking speed exert more significant influence, while slope impact is relatively minor. Notably, optimal parameter combinations are identified for different optimization objectives (sprinkler irrigation uniformity or intensity) and travel directions. A key commonality is observed: at a 5° slope and 30 m/h walking speed, combined with appropriate pressure (250 kPa or 300 kPa), optimal sprinkler irrigation uniformity or intensity is achieved in both lateral and longitudinal travel modes. The selected equipment and parameters represent widely adopted standard configurations in this field, based on preliminary experimental results. This setup constitutes a typical research scenario, enabling the investigation of the relationship between key performance indicators and operating parameters under controlled and standardized slope irrigation conditions. It provides an initial, reproducible benchmark for system optimization, offering direct reference value for subsequent studies under similar conditions. This study utilized an orthogonal experimental design to balance experimental efficiency and economic feasibility. Although repeated trials were conducted to enhance reliability, the limited number of experiments may have reduced statistical power. Therefore, the effects of factors with p-values approaching the significance threshold of 0.05 should be interpreted with caution. Future studies could further verify these results by increasing the sample size or employing more predictive modeling approaches, such as response surface methodology.

The sprinkler nozzles employed in this experiment were not equipped with pressure regulators, thereby focusing specifically on examining the fundamental effects of slope gradient itself. Pressure regulators, when installed upstream of sprinklers, function to compensate for pressure fluctuations induced by variations in terrain and elevation. This study, conducted under uncompensated conditions, systematically reveals—for the first time—the fundamental interactions between slope gradient, travel patterns, and operational parameters. Understanding these elementary influences provides a theoretical basis for optimizing compensated system designs. Furthermore, given that a significant proportion of sprinkler equipment currently in widespread use across many regions still operates without pressure regulators, the findings of this study offer direct guidance for the operation and management of such systems on sloped terrain. Based on these results, the findings will be compared and discussed in relation to systems equipped with pressure regulators in order to evaluate and optimize pressure compensation performance under complex slope conditions.

The scientific value of establishing “near-windless” conditions lies in our successful creation of a controlled experimental environment, which allowed us to systematically reveal—for the first time—the isolated effects of slope itself, as well as its interactions with pressure and travel speed, on the hydraulic performance of sprinkler irrigation systems on slopes. The findings obtained under these conditions can serve as benchmark data for the hydraulic performance of sprinklers in sloped environments. These findings provide a direct theoretical basis for the scientific operation of slope irrigation systems. In practice, depending on soil-water conservation needs (e.g., minimizing runoff in low-lying areas) or uniformity requirements, either longitudinal or lateral travel modes can be prioritized, and the optimized operating parameters identified here (particularly slope and speed control) should be implemented. Beyond its advantages in hydraulic performance, the lateral travel mode (along contour lines) is also more energy-efficient than the longitudinal travel mode. This is because it eliminates the need for repeated ascent and descent maneuvers against gravitational resistance, thereby leading to lower overall operating costs. This approach significantly enhances water use efficiency in slope irrigation and effectively mitigates the risk of soil erosion caused by uneven water distribution or excessive intensity, offering substantial practical value for sustainable agricultural irrigation on sloped farmland. The optimized parameter combination established in this study offers a technical pathway for achieving water and energy conservation in slope irrigation. Its direct economic significance lies in the potential to enhance crop yield and quality through improved irrigation uniformity—without increasing, and possibly even reducing, water and energy consumption—via optimized resource allocation. Sprinkler irrigation uniformity is a critical factor directly affecting crop yield stability and irrigation and fertilizer efficiency. Excessive application rates can exceed the soil infiltration capacity, leading to significant surface runoff and soil erosion. Therefore, this research is oriented toward providing a soil and water conservation solution for sloping farmland irrigation. The aim is to achieve efficient water use and uniform crop growth without inducing soil erosion, thereby supporting the overarching goals of ensuring food security and promoting sustainable agricultural development.

This study is conducted on experimental slopes, primarily considering three factors: slope, pressure, and speed. Actual field conditions present greater complexity, with factors such as wind direction, wind speed, soil infiltration characteristics, and crop canopy also significantly influencing sprinkler water distribution and infiltration. The optimal parameter combinations derived have not yet been validated under complex field conditions, such as those involving wind effects, crop interference, and spatial variability in soil properties. Consequently, their effectiveness in practical agricultural applications may require context-specific adjustments. Future research will incorporate temporal variability and long-term environmental effects by implementing continuous field-based monitoring across various seasons and meteorological conditions. A comprehensive evaluation of the optimized parameters under real agricultural conditions should be conducted, and a calibration relationship between experimental data and field trial results must be established. This approach aims to validate the generalizability of our laboratory results and develop more robust predictive models. Incorporates these variables to conduct experiments under more realistic conditions or employs numerical simulations. This research aims to incorporate the developed model with crop-specific water requirement models to investigate stage-specific irrigation strategies for sloped terrains. The ultimate objective is the development of an intelligent decision support system capable of integration with modern farm management platforms. Such a system would enable variable-rate irrigation based on real-time topographic, soil, and crop data. This approach is expected to comprehensively improve water use efficiency and enhance environmental sustainability in agricultural systems on slopes.

5. Conclusions

This study systematically investigates slope angle effects on sprinkler irrigation hydraulic performance during lateral and longitudinal machine movements through theoretical analysis and experimental validation. Orthogonal experiments reveal synergistic optimization mechanisms among slope angle, nozzle pressure, and walking speed. Key findings are:

(1) During lateral movement on slopes, average sprinkler irrigation intensity increases with decreasing elevation, with the differential widening at steeper slopes. Longitudinal movement shows sprinkler irrigation intensity decreasing with increasing slope. Both movement modes exhibit reduced sprinkler irrigation uniformity with increasing slope angles.

(2) Orthogonal testing identifies slope angle as the primary factor affecting sprinkler irrigation uniformity, followed by walking speed and nozzle pressure. Optimal lateral movement parameters (5° slope, 250 kPa pressure, 30 m/h speed) achieve peak sprinkler irrigation uniformity. For longitudinal movement, maximum sprinkler irrigation uniformity occurs at 5° slope, 300 kPa pressure, and 30 m/h speed.

(3) Slope demonstrates less pronounced influence on sprinkler irrigation intensity than nozzle pressure and walking speed. Analysis of variance confirms walking speed and nozzle pressure significantly affect sprinkler irrigation intensity, with slope secondary. Identical optimal parameters (5° slope, 300 kPa pressure, 30 m/h speed) maximize intensity for both movement modes.

(4) The optimal parameter combinations derived from orthogonal experiments inform the configuration of operational parameters for sprinkler irrigation systems on sloping farmland, offering a valuable reference for researchers working under comparable conditions. These optimized parameters aid in regulating irrigation application rates on slopes, thereby mitigating critical challenges such as surface runoff and soil erosion. The implications extend beyond water conservation and yield improvement to encompass ecological and environmental protection. The patterns uncovered in this study establish a theoretical foundation for the future development of variable-rate sprinkler irrigation systems tailored to sloping terrains. Such systems will facilitate truly precision irrigation on slopes, ultimately enhancing the efficiency of water resource utilization.