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This paper suggests a method of simultaneous multi-objective shape optimization of hydraulic turbine runner and draft tube (DT) with the objective to increase turbine efficiency in wide range of operating points (OPs). Runner and DT are the main sources of energy losses in hydraulic turbines. Coupling runner and DT in computational fluid dynamics (CFD) analysis enables correct statement of boundary conditions for efficiency evaluation, while simultaneous variation of these components allows more flexible adjustment of flow passage geometry. Detailed runner parameterization with 28 free geometrical parameters and DT parameterization with nine free parameters are given. Optimization problem is solved using multi-objective genetic algorithm (MOGA). For each variation of runner and DT shapes, flow field in wicket gate (WG), runner, and DT is simulated using steady-state Reynolds-Averaged Navier–Stokes (RANS) equations with k-e turbulence model. Energy-based boundary conditions are used for the calculations, allowing determination of efficiency of the whole turbine in correspondence with International Electrotechnical Commission (IEC) standard. Formulations of multiple OP efficiency objective functions and constraints are discussed in detail. To demonstrate the advantages of simultaneous runner and DT variation, two optimization problems are solved for a medium specific speed Francis turbine. Namely, single runner and coupled “runner–DT” optimizations are carried out. It is shown that optimized runner–DT geometry outperforms the result of single runner optimization by about 0.3% in terms of average efficiency, showing the potential of the developed approach to improve multiregime turbine characteristics in practical design optimization problems.

Three-dimensional model of fracture propagation is proposed. The model simultaneously accounts rock deformation in the vicinity of a fracture and a cavity, fluid flow inside the fracture and its propagation in the direction that is selected by a growth criterion. The results of the sensitivity analysis of model solution to the variation of model parameters are presented.

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