Optimal Distribution of the Disk Load: Validity of the Betz–Joukowsky Limit

The popular axial momentum theory relies on the steady, incompressible, axisymmetric, and inviscid flow through a so-called actuator disk. The most important result of this theory is the famous Betz–Joukowsky limit, stating that the maximum power coefficient achievable by an open disk is limited to 16/27. Generally, this value is obtained assuming a priori that the disk is radially uniformly-loaded and the flow is axially one-dimensional. This, however, does not prove that the uniform type is the optimal load, or else that it returns the maximum value of the extracted power. For this reason, this paper preliminary shows that 16/27 is the exact value of the maximum power coefficient of a uniformly loaded disk, even if the flow is not assumed as one-dimensional. Then, it proves, using a calculus of variation approach, that the radially uniform load is optimal. The proof refers to an approximate classical local form of the axial momentum equation. Finally, the paper points out that, since the proof of the Betz–Joukowsky limit relies on this simplifying assumption, the exact evaluation of the optimal radial distribution of the disk load, leading to the maximum value of the power coefficient, is still an open question.