AN UPSCALING APPROACH FOR THE BETZ-JOUKOWSKY THEORY

Abstract

The Betz-Joukowsky (BJ) theory is the starting point to study wind turbines because it provides the upper bound of their power at the megascale. However, not all the assumptions that support this theory have been clearly identified in the literature. In this work, an upscaling approach is applied to the mass, linear momentum and mechanical energy equations at the microscopic scale in order to obtain their megascale counterparts. The analysis allows a systematic identification of the assumptions and restrictions involved in the BJ theory, which were found to serve as a guideline for the shape and size of the averaging domain. The derivations show that the actuator disc concept is indispensable in order to recover Froude s theorem. In addition, numerical experiments evidence that the restrictions related to neglecting mass transport and air power at the lateral surfaces of the averaging domain are the hardest ones to satisfy. The results from this work expand \\textcolor{black}{the} current understanding of the BJ theory and open new research lines to improve it by identifying its ancillary assumptions and restrictions.