Efficient Free Wake Calculations Using Analytical/Numerical Matching and Far-Field Linearization

A hybrid analytical and numerical method called Analytical/Numerical Matching (ANM) is used to calculate the velocity field of rotor wake vortex filaments. A simple discretization of the Biot-Savart law is used to obtain an efficient but low resolution numerical calculation of the velocity field. A fictitious large vortex core is introduced and the discretized velocity field is calculated accurately for a smooth large core filament. At points close to the filament, a local analytical correction based on the Biot-Savart law for a curved vortex filament is employed to remove the nearfield error and add the correct nearfield velocity contribution associated with the actual core size. Because this correction is computationally inexpensive, the overall approach is both very efficient and very accurate. The accuracy of the ANM method has been studied in detail for vortex rings. Free wake solutions are obtained much faster using the ANM method in comparison to other methods with comparable accuracy. Because ANM is functionally simple, it is possible to obtain derivatives of the velocity expressions in closed form using a symbolic manipulation program. By computing derivatives of the velocity field the free wake calculation has been linearized. The linearized form is computationally efficient since it avoids large numbers of repetitive Biot-Savart law calculations. Linearized free wake problems are solved by a time marching method and compared with full nonlinear results. A special mathematical approach to obtain solutions for linearized periodic wake motion by a method that does not involve time marching is also discussed. This approach finds periodic solutions directly by inversion of a large matrix.