The Behavior of Fixed Point Iteration and Newton-Raphson Methods in Solving the Blade Element Momentum Equations

There is a substantial body of ongoing research improving the Blade Element Momentum (BEM) theory and applying it to the optimization of wind turbine rotors. Both of these developments challenge the suitability of fixed point iteration schemes being applied to advanced BEM models. This article explores the mathematical behavior of the BEM equations, with special attention to the application of numerical methods. Under special conditions, multiple solutions will exist when the airfoil is stalled. This situation gives increased uncertainty, where uncertainty in airfoil behavior is already high. This also demonstrates that there could be circumstances where the wake state has weak dependence on blade state. Fixed point iteration and Newton-Raphson numerical methods are investigated in this paper. Both methods will become unstable under certain conditions. The investigation shows that the Newton-Raphson method has well defined conditions for instability in terms of design variables and airfoil properties. By comparison, the fixed point function used here exhibits instability over a larger range.