Calculation of the induced velocities in lifting line analyses of propellers and turbines

Abstract Lifting line analyses of propellers and horizontal-axis turbines require the axial and circumferential velocities induced by the helicoidal vorticity shed from the blades. These velocities can be found from the analytic solution for a helical vortex of constant radius and pitch due to Kawada and Hardin. This solution, however, involves infinite series of products of Bessel functions and their derivatives, whose evaluation is computationally intensive partly because the number of terms required for a specified accuracy increases without bound as the vortex is approached. We compare three closed-form approximations to the Kawada–Hardin equations. The first, due to Kawada and rediscovered by Lerbs, involves asymptotic expansions for large pitch whereas the second is a more general approximation derived by Wrench and subsequently by Okulov. The last uses additional terms found by Okulov. The three have comparable evaluation times but the third is more accurate. The accuracy of the approximations is assessed for N equispaced and identical helical vortices where N is the number of blades. We provide, for the first time, approximate “remainders” for the Kawada–Hardin equations which allow an assessment of the number of terms in the series required to achieve a specified accuracy. As a test case for assessing the calculations of induced velocity, we consider the design of a hydrokinetic turbine blade to avoid cavitation at two different operating conditions with different vortex pitch. The use of the approximated induced velocities is compared to Prandtl’s well-known tip loss factor. Near the hub and tip the blade shape is altered considerably by the induced velocities and the method used to calculate them.