blade element momentum equations, robust solution methodology, guaranteed convergence, BEM


The blade element momentum equations, though conceptually simple, can be challenging to solve reliably and efficiently with high precision. These requirements are particularly important for efficient rotor blade optimization that utilizes gradient-based algorithms. Many solution approaches exist for numerically converging the axial and tangential induction factors. These methods all generally suffer from a lack of robustness in some regions of the rotor blade design space, or require significantly increased complexity to promote convergence. The approach described here allows for the blade element momentum equations to be parameterized by one variable: the local inflow angle. This reduction is mathematically equivalent, but greatly simplifies the solution approach. Namely, it allows for the use of one-dimensional root-finding algorithms for which very robust and efficient algorithms exist. This paper also discusses an appropriate arrangement of the equation and corresponding bounds for the one-dimensional search—intervals that bracket the solution and over which the function is well-behaved. The result is a methodology for solving the blade element momentum equations with guaranteed convergence and at a superlinear rate.

Original Publication Citation

Ning, A., “A Simple Solution Method for the Blade Element Momentum Equations with Guaranteed Convergence,” Wind Energy, Vol. 17, No. 9, Sep. 2014, pp. 1327–1345. doi:10.1002/we.1636

Document Type

Peer-Reviewed Article

Publication Date


Permanent URL






Ira A. Fulton College of Engineering and Technology


Mechanical Engineering

University Standing at Time of Publication

Assistant Professor