Optimal design of savonius wind turbines using ensemble of surrogates and CFD analysis

Abstract

Current study presents fluid flow analysis using CFD and a surrogate based framework for design optimization of Savonius wind turbines. The CFD model used for the study is validated with results from a physical model in water tunnel experiment. Four variables that best define blade geometry are considered and a feasible design space consisting of different combinations of these variables that provide positive overlap ratio is identified. The feasible space is then sampled with Latin hyper cube design of experiment. Numerical simulations utilizing K-epsilon turbulence model are performed at each point in the Design of Experiments to obtain coefficient of performance and weighted average surrogate (WAS) is fitted to them. Novelty of the current work is the use of WAS for design of savonius turbine. The WAS is an ensemble of surrogates that consists of polynomial response surface, kriging and radial basis functions. Error metrics reveal that WAS performs better compared to any surrogate individually thus avoiding misleading optima and eliminates surrogate dependent optima. WAS is used to explore the design space and perform optimization with limited number of CFD analyses. It is observed that at the optimal profile, there is more power on the rotors and primary recirculation in the immediate downstream of rotor is high, enforcing maximum momentum on turbine.

Appendices

Appendix 1

The mesh convergence information for the computer model used in the validation study is provided below in Table 7. Two meshes were generated for the same profile with ±10% difference in number of quad elements and the Cp values were compared. It is observed that there is ~ 4% change in Cp value when the mesh size is decreased and ~1% increase when the mesh cells are increased by 10%. Hence further study is performed by increasing the number of mesh cells by 75% and still the observed Cp value increases by ~1%. Therefore, the mesh size corresponding to iteration number 3 is used in the work.

Table 7 Mesh convergence information

Appendix 2

The weighted average surrogate (WAS) is formulated as a weighted sum of the three individual approximation methods. The weights are calculated in such a way that they (a) reflect the confidence in each individual surrogate and (b) filter out adverse effects associated with individual surrogates which represent the sample data well, but predict poorly at designs not included in the sample data. WAS can be expressed as follows:

$$ {\widehat{y}}_{WAS}(x)=\sum \limits_{i=1}^N{q}_i(x){\widehat{y}}_i(x)={Q}^T(x)\widehat{y}(x) $$

(12)

$$ \sum \limits_{i=1}^N{q}_i(x)=1 $$

(13)

where ${\widehat{y}}_{WAS}(x)$ is the predicted response by the WAS model, q_i(x) is the weight associated with the ith surrogate at x and and ${\widehat{y}}_i(x)$ is the predicted response by the ith surrogate. The various functions that could be used for assigning weights for WAS are shown in Table 8. We use the Best PRESS in this work.

Table 8 Different forms of weights for WAS