Numeričko modeliranje tokova koje karakteriše nagla lokalna promena dubine i protoka

Abstract

Discontinuous flows are the flows in which large gradients arise in both the flow depth and the discharge due to specific external and/or internal boundary conditions. With reference to the discontinuity and through the discontinuity itself, the basic hypotheses, upon which the Saint-Venant shallow water equations are based, are violated. The problems in mathematical and numerical modelling of discontinuous flows arising from this violation, and the specific structure of the appropriate numerical schemes used in solving such problems are discussed in this Thesis. Three approaches to mathematical modelling of discontinuous flows: shock-fitting, through method and pseudoviscosity method, are presented first. They are followed by a review of numerical methods used in numerical modelling of free-surface flows with discontinuities: the Method of characteristics, the Godunov method, the Finite Difference Methods based on the flux-splitting technique, and the Finite Element Method. Close attention is paid to the approach and the scheme that allows for weak solutions, i.e. to the through method and the Finite Difference Method based on the explicit MacCormack scheme. The proposed scheme is verified using: 1° the analytical solutions, 2° the author's measurement performed on a laboratory rig and 3° the available measurements of the other authors. Since the quality of the computational results highly depends on: 1° the form under which the governing equations are written (conservation or non- conservation form), 2° the conservative property of the numerical scheme used, and 3° the diffusive/dispersive character of the scheme, great attention is paid to the analysis of the conservation property of the MacCormack scheme and the problems regarding the quasi-physical effects of the scheme, such as spurious oscillations in the vicinity of the discontinuity. Finally, a case study is included to indicate the performance of the proposed model in the engineering practice.