Computational hydraulic techniques for the Saint Venant Equations in arbitrarily shaped geometry

A numerical model for the one-dimensional simulation of non-stationary free surface and pressurized flows in open and closed channels with arbitrary cross-section will be derived, discussed and applied. This technique is an extension of the numerical model proposed by Casulli and Zanolli [10] for open channel flows that uses a semi-implicit discretization in time and a finite volume scheme for the discretization of the Continuity Equation: these choices make the method computationally simple and conservative of the fluid volume both locally and globally. The present work will firstly deal with the elaboration of a semi-implicit numerical scheme for flows in open channels with arbitrary cross-sections that conserves both the volume and the momentum or the energy head of the fluid, in such a way that its numerical solutions present the same characteristics as the physical solutions of the problem considered. The semi-implicit discretization in time leads to a relatively simple and computationally efficient scheme whose stability can be shown to be independent from the wave celerity √gH. The conservation properties allow dealing properly with problems presenting discontinuities in the solution, resulting for example from sharp bottom gradients and hydraulic jumps. The conservation of mass is particularly important when the channel has a non rectangular cross-section. The numerical method will be therefore extended to the simulation of closed channel flows in case of free-surface, pressurized and transition flows. The accuracy of the proposed method will be controlled by the use of appropriate flux limiting functions in the discretization of the advective terms, especially in the case of large gradients of the physical quantities involved in the problem. In the particular case of closed channel flows, a new flux limiter will be defined in order to better represent the transitions between free-surface and pressurized flows. The numerical solution, at every time step, will be determined by solving a mildly non-linear system of equations that becomes linear in the particular case that the channel has a rectangular cross-section. Careful physical and mathematical considerations about the stability of the method and the solvability of the system with respect to the implemented boundary conditions will be also provided. The study of the existence and uniqueness of the solution requires the solution of a constrained problem, where the constraint expresses that the feasible solutions are physically meaningful and present a non-negative water depth. From this analysis, it will follow an explicit (dependent only on known quantities) and sufficient condition for the time step to ensure the non-negativity of the water volume. This condition is valid in almost all the physical situations without more restrictive assumptions than those necessary for a correct description of the physical problem. Two suitable solution procedures, the Newton Method and the conjugate gradient method, will be introduced, adapted and studied for the mildly non linear system arising in the solution of the numerical model. Several applications will be presented in order to compare the numerical results with those available from the literature or with analytical and experimental solutions. They will illustrate the properties of the present method in terms of stability, accuracy and efficiency