The classical lifting line optimization of a horizontal-axis turbine is described in this paper. The theory is based on the classical helical vortex model of constant pitch to determine the optimum blade circulation distribution for maximum power extraction from a fluid stream in steady flow. A vortex lattice method in combination with the induction factor method for computing the induced velocities is used to solve the optimization equations. Optimum power curves, including the effect of viscous drag are presented. The application of the theory to the determination of optimum chord and pitch distributions is illustrated in a simple design example of a marine current turbine.
The problem of finding the conditions under which a horizontal-axis turbine extracts the maximum power from a fluid stream is of great importance both in the wind and marine current fields. For rotors with blades of high aspect ratio, the lifting line theory offers a suitable model for the lifting action of the blades which inherently takes into account the finite number of blades. For propellers, Betz (1919) derived the conditions to be satisfied by the potential flow problem for an optimum circulation distribution, which leads to minimum kinetic energy losses in the propeller slipstream. Goldstein (1929) solved the potential flow problem to obtain the optimum circulation distributions. For marine propellers the lifting line model evolved to a well-established tool for the hydrodynamic design of these systems. Among others, this was due to the work of Lerbs (1952), who extended the design method to moderately loaded non-optimum wake-adapted propellers by applying the induction factor method of calculating induced velocities due to helical vortices. There is an extensive body of literature on the lifting line model for marine propellers and the interested reader may be referred to the book of Breslin and Andersen (1994).