Computation of Rotorcraft Wake Geometry using NURBS

Abstract

This thesis contains the results of research in the area of rotorcraft aerodynamics with a focus on method development related to the vortical wake generated by rotor blades. It is applied to a vortex tube representation of the wake (using a single NURBS surface) and a simplified filament wake model where only the tip vortices are taken into account. A complete wake model would include all vorticity (both shed and trailed) released by the blades along their entire span. Here, the wake model is reduced to the most important part: the vorticity released at the tips of the blades. In the first part of the thesis (Chapters 2–4), a vortex filament method is developed that uses Non-Uniform Rational B-Spline (NURBS) curves as the basis for the geometric description of vortex filaments. NURBS have the advantage that they are very suitable to describe curved geometry. As a consequence, fewer elements are required to get an accurate description of a curved vortex filament when compared to a discretization that uses straight line segments for which an analytical solution of the Biot-Savart law is available that gives the velocity induced by one such segment. In this thesis, an alternative method is used. The Biot-Savart law is regularized using velocity smoothing functions and the induced velocity is computed by automatic integration, using adaptive quadrature. This has the advantage that the contribution due to the local curvature at an evaluation point located on a vortex filament is taken into account automatically, which is in contrast with methods using straight line segments that require an additional correction. While validating the NURBS-based vortex methods, it was observed that the regularization of the analytical solution of the Biot-Savart law for a straight line segment that is used in the literature contains an error. A convergence analysis of the regularization shows that for evaluation points located near the centre line of a line segment beyond its endpoints, the induced velocity value does not converge towards the analytical reference, even though no asymptotes are encountered for the induced velocity as the centre line is approached. An improved correction is proposed that rectifies this error. The consequence of using the original correction in a vortex filament method are investigated using a single viscous vortex ring. The second part (Chapters 6–8) concerns the development of an explicit, multi-step wake relaxation algorithm for use in the rotorcraft trim task. The relaxation algorithm follows from a description of the trim problem, which states that vorticity released from the rotor at the same point (radius and azimuth angle) belongs to the same streamline. It is assumed that a streamline is a smooth curve that can be modelled using a cubic NURBS curve. By reversing the equation to compute the derivative along a NURBS curve at a fixed parametric point, an interpolation algorithm follows that can reconstruct a streamline given its starting point and derivative data along the curve. In Chapter 6, the method is applied to a vortex tube approximation of the rotor wake that is modelled using a single NURBS surface. The long computing times and the inability to find a trimmed solution in forward flight are the main reasons why the trim method has been adapted for a vortex filament wake model in Chapter 7. The discrete nature of a filament wake model when compared to the continuous nature of a vortex tube model, means that a small modification is necessary to make the trim method compatible with a filament wake. It is related to the position along the streamline curves where velocity is computed and where it is required to update the streamline. Harmonic interpolation is introduced as a means to improve the efficiency of the relaxation method. Viscous diffusion and the effect of stretching on the core sizes are taken into account as well. The trim method is put through its paces using an isolated rotor in hover, first out of ground effect, then in ground effect and finally, in forward flight. In Chapter 8, the filament wake trim methodology is coupled to a structural and aerodynamic model of helicopter rotor and a boundary element method to account for the influence of non-planar ground effect. The rotor is trimmed in hover above a finite ground plane that serves as a simple representation of the flight deck of a frigate.