Stability, consistency and convergence of time-marching free-vortex rotor wake algorithms

The stability, accuracy and convergence of time-marching algorithms for free-vortex rotor wake calculations was examined. A linearized analysis was used to determine the basic stability characteristics. The vortex-induced velocities were calculated through application of the Biot-Savart law using standard straight-line vortex segmentation. This approach was shown to be second-order accurate. However, the discretization resulted in nonlinear dispersion and dissipation terms in the discretized wake equations. These errors were found to be dependent on the induced velocity field, and may be anti-dissipative under many flight conditions. In such cases, the wake geometry solution may not be convergent, even though the numerical algorithm is linearly stable. Modified equations were examined to provide further insight into stability of the nonlinear discretized equations. A new time-marching algorithm was proposed using a second-order backward difference approximation for the time derivative to ensure stability and solution convergence. A numerical experiment was performed to verify convergence of the wake geometry solutions obtained using this algorithm. The expected second-order accuracy and grid independent nature of the wake geometry solution was demonstrated, for both hover and forward flight.