A 2D, Roe-Upwind finite volume model is built based on unstructured triangular meshes, and is applied to numerical simulation of discontinuous flows and tidal bore on irregular bottom topography successfully.The model is based on conservation forms of the shallow water equations. The computational domains are discretised into a set of unstructured triangular meshes, which can fit complicated geometries such as bank and islands more precisely. Define each triangular cell as a control volume, integrate the equations in the volume, and store the average of the conserved variables at the center of the volume, then a local Riemann problem is formed at the interfaces of the volume. So the model has the characteristics of discontinuous solutions, the discreteness and numerical calculation of the equations can be solved with the method of discontinuous solutions.In the space discreteness of the model, the numerical flux and the reconstructions of the conserved variables at the interfaces of control volumes are the form of Roe's approximate Riemann solver and the 2-order upwind approach, which called the Roe-Upwind finite volume model. Balancing flux gradients and source terms in equations makes the model can solve the shallow water flows and discontinuous flows with variable depth. So the harmonious Roe-Upwind finite volume model is constructed finally.The 2D, Roe-Upwind finite volume model is applied to numerical simulation of some classic discontinuous flow examples to prove the validity and applicability. The examples include Stoker problem, 2D dam-break problem, oblique hydraulic jump problem, landslide bore problem and mixed flow problem. All the computed results are good agreement with the analytic solutions, and present steep jump without non-physical oscillations near the shock.The model is applied to numerical simulation of tidal bore successfully. Firstly, construct an ideal estuary model with the shape of funnel, analyze the changes of the tidal bore with the factors like the tide intensity, river depth, relaxation rate of estuary and river topography through numerical calculation, and conclude laws from many