A numerical proof of the Betz–Joukowsky limit

According to the Betz–Joukowsky limit, the maximum power coefficient of a wind turbine is 16/27. This limit is obtained assuming a priori that the disk is radially uniformly-loaded. In other words, the proof of the limit does not evaluate the optimum type of the load distribution, but it directly speculates that radially uniform load is the optimal one. Therefore, it should be questioned if a radially varying load with a power coefficient higher than 16/27 exists, which overrules the Betz–Joukowsky limit. For this reason, the paper presents, for the first time, an original numerical optimisation strategy aimed at proving, with a prescribed accuracy level, the validity of the Betz–Joukowsky limit, i.e., at verifying whether or not the optimal load distribution is uniform. The proposed strategy combines a Computational-Fluid-Dynamic-Actuator-Disk model with a classical gradient-based optimisation algorithm. The former aims at computing the objective function (i.e., the power coefficient), whereas the latter is devoted to the detection of the optimum. The procedure assumes that the load distribution is a real analytic function, while the optimisation variables are the coefficients of its Taylor polynomial. The proposed methodology is thoroughly verified, and its overall accuracy level is quantified. Up to an 8th-degree Taylor polynomial, the outcome of the numerical optimisation procedure differs from the Betz–Joukowsky solution by a quantity equal to the uncertainty of the adopted computational strategy. In other words, within the given accuracy range, the analysis confirms the validity of the Betz–Joukowsky limit based on the radially uniform load. Finally, it should be stressed that if the actual optimum differed from the Betz–Joukowsky limit by an amount less than the accuracy of the adopted numerical method, then that optimum could not be detected by the proposed procedure.