Abstract:
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.