欧拉法融合拉格朗日法高效模拟灌溉二维地表水运动规律

大规模现代化农业灌溉管理下，为实现快速高效地获知灌溉水运动及其分布的目的，该文基于二维浅水方程组的欧拉-拉格朗日混合型表达形式，提出了一种高效简洁的欧拉-拉格朗日混合解法。该解法的基本物理变量被严格地定义在欧拉型非结构化三角形有限体积单元格上，且变量在单元格之间呈现出阶梯分布状态，以精准地捕捉各类地表浅水波动并有效地保持质量守恒性；由于控制方程中不存在水运动的对流梯度项（或位置加速度项），仅通过拉格朗日迹线追踪的形式获得未知与已知时间步之间的变量关系，故与广泛应用的欧拉解法相比，离散格式表达式极为简洁易用；在地表水运动的干湿边界处，地表水位梯度项被做了修正，以严格地保证各物理量之间的数值平衡，进而能高精度的模拟整个畦田内的地表水流推进/消退全过程。为验证模型的模拟性能，选取一种高效的欧拉解法（非迭代型全隐式标量耗散有限体积法）求解二维浅水方程组做为对比模型，基于3个典型畦灌试验的实测数据，从模拟精度、质量守恒性和计算效率3个方面，对比分析了2种数值解法的性能。结果表明，2种解法在模拟精度方面相差无几，且欧拉-拉格朗日混合解法比欧拉解法具有更好的质量守恒性；在计算效率方面，欧拉-拉格朗日混合解法比欧拉解法的效率提高了约5.3倍。故该文提出的二维浅水方程组的欧拉-拉格朗日混合解法，更适用于二维灌溉地表水运动的模拟分析。

Euler-Lagrange hybrid numerical simulation for two-dimensional surface water flow in irrigation

Abstract: A simple and efficient Euler-Lagrange hybrid numerical solution for 2-dimensional shallow water equations was proposed in this study. To validate the simulation performance of the proposed numerical model, an efficient Euler solution (scalar-dissipation finite-volume method with non-iterative and fully implicit time scheme) was selected as comparative solution. At the same time, 3 typical basin irrigation experiments were carried out. The validation procedure concerned 3 aspects: simulation accuracy, mass conservation and computational efficiency. The corresponding index were average relative error and Nash-Sutcliffe efficiency coefficient between the observed and simulate data, water quantity balance error and run time in personal computer. The validated results showed that, the average relative errors and Nash-Sutcliffe efficiency coefficient between the observed and simulated data for the proposed Euler-Lagrange hybrid and selected Euler solutions were very similar. Thus, the accuracy of proposed Lagrange solution was very high. In terms of the mass conservation, the proposed Euler-Lagrange hybrid solution was slightly higher than the selected Euler solution, which means a very good performance in mass conservation of the proposed Euler-Lagrange hybrid solution. In computational efficiency, the proposed Euler-Lagrange hybrid solution was 5.3 times higher than the selected Euler solution due to the former had no advection gradient term (or displacement acceleration). Consequently, the proposed Euler-Lagrange hybrid solution for 2-dimensional shallow water equations in basin irrigation was an efficient and simple numerical tool for analysis on irrigation water flow, especially in the condition of large-scale intensive agricultural cultivation. The proposed Euler-Lagrange hybrid numerical solution for 2-dimensional shallow water equations exhibited obvious physical meanings: 1) The basic state variables such as water depth and discharge were strictly defined on the triangle spatial cell and the variable values were ladder distribution, which could accurately capture every shallow water waves and was the basic requirement of modern numerical analysis. 2) The advection gradient (or displacement acceleration) was not included in the governing equations, which resulted in very simple spatial scheme and could be easily applied by user. 3) The water level gradient term was corrected at the dry-wet boundary by means of judging the relationship between the water level/surface relative elevation in the current and its adjacent spatial cells. After this correction, the surface water advance and recession processes could be accurately simulated in the whole domain. Compared with the classical Euler numerical solution, the proposed Euler-Lagrange hybrid numerical solution avoided advection gradient terms, which was the main problem in computational fluid dynamics due to its extremely strong nonlinearity. This characteristic largely declined the numerical solution difficulty and thus the resulting spatial-temporal algebraic system presented concise formulation, which considerably simplified the calculation and greatly reduced the application difficulty. Compared with the common Lagrange numerical solution, the proposed hybrid solution could preserve the physical conservation due to its strict state variable definition of Finite-Volume method. Meanwhile, the proposed hybrid numerical solution did not capture the movement trajectory of water flow particles, and thus could maintain high computational efficiency and could easily set the initial and boundary conditions. Consequently, the proposed Euler-Lagrange hybrid numerical solution for 2-dimensional shallow water equations in this study exhibited the advantage of both Euler and Lagrange numerical solutions. Additionally, the commonly applied Lagrange numerical solution, such as smoothed particle hydrodynamics, required to define massive of spatial particles to represent water flows. Therefore, the Lagrange numerical solution could easily simulate the large deformation and strong nonlinear processes, such as wave breaking and splashing. By contrast, the Euler state variable definition method in this study actually lost this advantage. However, the surface irrigation process could not commonly include these phenomenon. Thus, the method of coupling Euler and Lagrange numerical solutions in this study exhibited well numerical performance.