Expanding Wake Induction Effects on Thrust Distribution on a Rotor Disk

Wake expansion is necessary to satisfy conservation of mass when induction in flow through a wind turbine rotor disk is substantial. An iterative model is developed to calculate this expansion in an axisymmetric wake generated by an idealized unyawed disk operating at high tip speed ratios. The model satisfies all conservation constraints and boundary conditions, and predicts induction in the wake flow normal to the rotor disk plane due to the sheet of vorticity convected downwind along the surface of the wake. Expansion effects on induction in flow through the rotor disk and in the wake ale noted. The effect of skewing the axis of the wake out of alignment with the axis of a yawed rotor disk is approximated with a simple cross flow model which yields an expanding skewed wake. The sheet of vorticity convected downwind along the surface of this wake is found to significantly alter the induction in flow through the rotor disk from that predicted when wake expansion is neglected with the cylindrical wake approximation. This alteration is in the direction of levelizing the induction and resulting thrust distributions on the yawed disk. This in turn reduces the forward shift of the center of thrust from the axis of a yawed rotor, and this reduction may be very substantial at higher levels of induction and hence wake expansion. The effects of wake expansion on the level of thrust appear to be of second order compared to the effects on center of thrust. These findings are likely to be of particular importance to small wind turbines which are designed to yaw out of the wind as required at higher wind speeds to prevent undesirable overshoots in power and loads.