Coupled simulation and validation with fully implicit time scheme for free-surface-pressurized water flow in pipe/channel

Abstract: In the irrigation water distribution system including pipe and cannel as well as control valve/gate, water flow presents both free surface and pressurized flows. Saint-Venant equations are often applied to discribe the free-surface-pressurized water flow in pipe/channel by means of Preissmann slot approach and then four-point implicit finite difference and vector-dissipation finite-volume approaches with explicit-time scheme are applied to simulation of unsteady flow in pipe/channel. However, it is very different for gravity diffusion wave in free-surface water flow and pipe elastic wave in pressure water flow, which induces different constraint on time step size, low computational efficiency and large water balance error in the modes based on these 2 numerical solutions. To solve these problems, Saint-Venant equations was applied to describe the free surface and pressure water flows in irrigation water distribution system, conjunctive with the Preissmann slot approach. Then a scalar-dissipation finite-volume scheme was developed to spatially discretize all terms of the governing equations. This scheme exhibited more simple expression and was more suitable to written computational code than the four-point implicit finite difference approach and vector-dissipation finite-volume approaches. On the basis of the spatial scheme, a fully implicit time scheme was implemented to temporally discretize all terms of the governing equations to result in a nonlinear algebraic equation system. To efficiently solve this nonlinear algebraic equation system, a dual time approach was introduced, which included real- and pseudo-time steps, to make a linearization. The advantage of the dual time approach was the existence of a ratio between real- and pseudo-time steps. The value of the ratio could be automatically adjusted according to the known pipe water flow conditions and then the coefficient matrix of the algebraic equation system could maintain diagonally dominant all the time. In such case, the absolute convergence could be achieved whether free surface or pressurized flow was in pipe according to numerical analysis theory. As a result, a fully coupled model of free-surface-pressure flow for irrigation water distribution system was proposed. A standard physical test, which strictly controlled the initial and boundary conditions under the indoor condition, was firstly applied to validate the performance of the proposed model. The validated results showed that the proposed model could well simulate the free surface and pressurized water flow processes, which was similar to vector-dissipation finite-volume approach and better than four-point implicit finite difference approach. Meanwhile, the water balance error of the proposed model was only 0.16%. By contrast, the error values of the models based on four-point implicit finite difference and vector-dissipation finite-volume approaches were 2.1% and 1.2%, respectively. The computational efficiency of the proposed model was 1.3 and 5.2 times higher than the existing 2 models. Furthermore, a field experiment was performed in Hebei Yehe irrigation area, April 5, 2013. On the basis of the field observed data, the proposed model still exhibited better performance than the 2 existing models. In details, the water balance error of the proposed model was only 0.016%, by contrast, 2.68% and 1.35% for the 2 existing models. The efficiency of the proposed model was still 1.3 and 5.2 times higher than the existing 2 models. Consequently, the proposed model overcomes the disadvantages of the existing models and is s