A SEMI-IMPLICIT 3-D NUMERICAL MODEL USING SIGAM-COORDINATE FOR NON-HYDROSTATIC PRESSURE FREE-SURFACE FLOWS

A 3-D numerical formulation is proposed on the horizontal Cartesian, vertical sigma-coordinate grid for modeling non-hydrostatic pressure free-surface flows. The pressure decomposition technique and θ semi-implicit method are used, with the solution procedure being split into two steps. First, with the implicit parts of non-hydrostatic pressures excluded, the provisional velocity field and free surface are obtained by solving a 2-D Poisson equation. Second, the theory of the differential operator is employed to derive the 3-D Poisson equation for non-hydrostatic pressures, which is solved to obtain the non-hydrostatic pressures and to update the provisional velocity field. When the non-orthogonal sigma-coordinate transformation is introduced, additional terms come into being, resulting in a 15-diagonal, diagonally dominant but unsymmetric linear system in the 3-D Poisson equation for non-hydrostatic pressures. The Biconjugate Gradient Stabilized (BiCGstab) method is used to solve the resulting 3-D unsymmetric linear system instead of the conjugate gradient method, which can only be used for symmetric, positive-definite linear systems. Three test cases are used for validations. The successful simulations of the small-amplitude wave, a supercritical flow over a ramp and a turbulent flow in the open channel indicate that the new model can simulate well non-hydrostatic flows, supercritical flows and turbulent flows.     

SIGMA-COORDINATE NUMERICAL MODEL FOR SIDE-DISCHARGE INTO NATURAL RIVERS

Due to large topography slopes in natural rivers, pollutant concentration embodies a property of three-dimensional distribution when wastewater is discharged from effluents along the bank. With the sigma coordinate along the vertical dimension fitted to both the moving free surface and the bed topography, a three-dimensional numerical model was developed in the present work to address pollutant transport processes in the above-mentioned cases. To avoid the reduction in accuracy caused by spurious diffusion in the case of steep bottom slopes, a formula for horizontal diffusion in the sigma coordinate system was derived. A case study for the side discharge into a straight open-channel flow shows that numerical results are verified well by experimental data. Furthermore, the present model is also verified by the simulation of discharging wastewater from Fuling Phosphorus Factory effluent into the Three Gorges Reservoir and the agreement between the numerical simulation results and field observation data is satisfactory. The change of the mixing zone scope in the water surface versus the layers along the vertical dimension was also discussed in detail. The study shows that a more realistic calculation for pollutant discharge has been provided by the present model than by the depth-average model which predicts an unrealistically smaller mixing zone.