Wake Expansion and the Finite Blade Functions for Horizontal-Axis Wind Turbines

This paper considers the effect of wake expansion on the finite blade functions in blade element/momentum theory for horizontal-axis wind turbines. For any velocity component, the function is the ratio of the streamtube average to that at the blade elements. In most cases, the functions are set by the trailing vorticity only and Prandtl’s tip loss factor can be a reasonable approximation to the axial and circumferential functions at sufficiently high tip speed ratio. Nevertheless, important cases like coned or swept rotors or shrouded turbines involve more complex blade functions than provided by the tip loss factor or its recent modifications. Even in the presence of significant wake expansion, the functions derived from the exact solution for the flow due to constant pitch and radius helical vortices provide accurate estimates for the axial and circumferential blade functions. Modifying the vortex pitch in response to the expansion improves the accuracy of the latter. The modified functions are more accurate than the tip loss factor for the test cases at high tip speed ratio that are studied here. The radial velocity is important for expanding flow as it has the magnitude of the induced axial velocity near the edge of the rotor. It is shown that the resulting angle of the flow to the axial direction is small even with significant expansion, as long is the tip speed ratio is high. This means that blade element theory does not have account for the effective blade sweep due to the radial velocity. Further, the circumferential variation of the radial velocity is lower than of the other components.