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Hyperbolic systems with geometrical source terms arise in various applications. One such application is fluid flow with cross section area variations, which arise naturally in the study of fluid flow phenomena in ducts, pipes, shock tubes, and nozzles. In such applications the source term may have low regularity, which causes numerical difficulty particularly when steady state solutions are computed. The fix for this numerical difficulty is to approximate the source term using decomposition and upwinding, thus creating balance between numerical flux gradient and numerical source term. In this paper we introduce extension of the Q-schemes proposed in [1] to nozzle flow equations, i.e., Euler equations with geometrical source term.

Problem of computing streamlines of water flow inside the Francis turbine is solved using two-dimensional potential flow model and finite element method. The entire computational algorithm is implemented through developed numerical objects. The resulting software needs simple geometrical input, creates appropriate finite element mesh and computes streamlines efficiently and accurately, so it presents a good tool for turbine flow analysis.

The shape optimization method in Computational Fluid Dynamics applications using Genetic Algorithm and commercial turbulent flow modeling software FLUENT is presented. Two test cases are solved: the optimum nozzle design and the optimum Francis turbine spiral casing design. The shape of the nozzle and turbine spiral casing are parameterized by cubic Bezier splines, the coordinates of control points being optimizing parameters. In the nozzle case, the energy losses, defined as the difference of total mechanical energy flux through inflow and outflow cross-sections, are minimized. The objective functional for spiral casing shape optimization is defined as the surface averaged value of the magnitude of the difference between calculated and desired velocities at the outlet, so as to give as homogenized spiral casing outflow as possible.

In the paper a model of liquid-gas mixture flow in pipelines is presented. The result enables simulation of transient and cavitation phenomena so it provides a tool for prediction and prevention of pollution due to damages on pipelines as well as an environmentally improved pipeline design. Particularly an application to a pipeline Plase-Opatija in the coastal zone is presented in the paper. The applied methods are mathematical and numerical. The mathematical model consists of an initial-boundary value problem for a hyperbolic conservation law system, which is derived from the Kraneneburg's model for gas-liquid mixture flow in pipes. These hyperbolic conservation laws together with boundary conditions are then numerically solved using the essentially non-oscillatory schemes developed by Shu, Osher et al. The proposed cavitation model is tested on the Kranenburg's test example. The test shows excellent characteristics of the model, in fact it is superior to Kranenburg's numerical results. The model is applied to a real pipeline in order to predict cavitation phenomenon and improve pipeline change design.