The Analytical Simulation of Youngs's Problem in Comparison to Numerical Solution and Experimental Results of the Trench Flow

In this study the analytical and numerical solutions are used in the steady-state drainage problem in the three rectangular, semicircular, and triangular cross-section trenches. Equations that are derived by conformal mappings and complex potential theory were used for the analytical solution and SEEP/W software was used for the numerical simulation. Furthermore, physical sandbox models are designed for all the three sections in laboratory conditions and the outflows from the drainpipe are measured at 10, 20, 30, 4 and 50 cm depths. According to the three methods, in this study optimum installation depth of drainpipe that has maximum drainage outflow in the both rectangular and semicircular cross-sections is obtained at 40 cm and for the triangular cross-section is 32.5 cm using the analytical solution and 30 cm using the numerical solution and the experimental results. The results are assessed by the differences between the dimensionless discharges obtained from the analytical and numerical solutions and from the experimental conditions using relative root mean square error (RMSE) and coefficient of determination (R2). RMSE in the analytical solution for the triangular section 0.048, the semicircular section 0.061, and the rectangular section 0.071 and the R2 are 0.99, 0.98, and 0.98 and in the numerical solution 0.045, 0.056, and 0.041 for RMSE and 0.87, 0.97, and 0.97 for R2 are obtained, respectively. Comparison of the analytical and numerical solutions results with the experimental results showed that the both methods can simulate seepage flow in the trench with high accuracy. However, the accuracy of numerical solution was more than analytical solution, that it comes from some simplifications and assumptions considered in the analytical solution. In the analytical solution the ponded depth of water is considered small and negligible and the circular perimeter of the drainpipe is taken intoaccount as a line that these are the reasons for the less accurate results of analytical solution. Additionally, the hydraulic of the flow in entrance to the drain pipe is not considered in the analytical solution.