A multiple surrogate-based optimization strategy in conjunction with an evolutionary algorithm has been employed to optimize the shape of a simplified hydraulic turbine diffuser utilizing three-dimensional Reynolds-averaged Navier–Stokes computational fluid dynamics solutions. Specifically, the diffuser performance is optimized by changing five geometric design variables to maximize the average pressure recovery factor for two inlet boundary conditions with different swirl, corresponding to different operating modes of the hydraulic turbine. Polynomial response surfaces and radial basis neural networks are used as surrogates, while a hybrid formulation of the NSGA-IIa evolutionary algorithm and a ϵ-constraint strategy is applied to construct the Pareto front from the two surrogates. The proposed optimization framework drastically reduces the computational load of the problem, compared to solely utilizing an evolutionary algorithm. For the present problem, the radial basis neural networks are more accurate near the Pareto front while the response surface performs better in regions away from it. By using a local resampling updating scheme the fidelity of both surrogates is improved, especially near the Pareto front. The optimal design yields larger wall angles, nonaxisymmetrical shapes, and delay in wall separation, resulting in 14.4% and 8.9% improvement, respectively, for the two inlet boundary conditions.

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SUMMARYNumerical optimization techniques in ow design are often used to ÿnd optimal shape solutions, regarding, for instance, performance, ow behaviour, construction considerations and economical aspects. The present paper investigates the possibilities of using these techniques in the design process of a hydropower plant. This is realized by optimizing the shape of an existing sharp heel draft tube and validating the result with previously performed experiments. The actual shape optimization is carried out with the response surface methodology, by maximizing the average pressure recovery factor and minimizing the energy loss factor. The result from the optimization shows that it is possible to ÿnd an optimal solution on rather coarse grids. The location of the optimum is similar to the experiments, but the improvements are unexpectedly small. This surprising result indicates that the simulated ow ÿeld does not completely act as the real ow, which may be a result of the applied inlet boundary conditions, insu cient turbulence models and=or the steady ow assumption.

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