Two methods for calculating the velocities induced by a constant diameter far-wake

This paper describes two methods for calculating the velocities induced by the constant diameter helical vortices that occur in the far-wakes of helicopter rotors in hover or vertical flight, propellers, and horizontal-axis wind turbines. The velocities involve infinite integrals which, in general, cannot be integrated analytically. In the first method, each integral is evaluated numerically up to a finite limit and the remaining integral is approximated by an analytical function. The integrand of each function, or "remainder", is found by summation over all of the vortices shed at the same radius. Numerical experiments show the remainders are simplicity and versatility. Furthermore, the remainders can replace numerical integration in some special cases. In the second method, the range of integration is made finite by transforming the integrand into one containing an infinite sum. The sum is approximated using the Euler-Maclaurin formula, and the resulting integral is evaluated numerically. The method is more cumbersome than the first but is robust, very accurate, and should be particularly useful at small vortex pitch.