A Higher Order Vortex-Lattice Method with a Force-Free Wake

Open Access
Bramesfeld, Goetz
Graduate Program:
Aerospace Engineering
Doctor of Philosophy
Document Type:
Date of Defense:
April 07, 2006
Committee Members:
  • Mark David Maughmer, Committee Chair
  • Barnes Warnock Mccormick Jr., Committee Member
  • Kenneth Steven Brentner, Committee Member
  • Farhan Gandhi, Committee Member
  • Andrew Leonard Belmonte, Committee Member
  • vortex sheet
  • wakes
  • wings
  • potential flow
A higher-order, lifting-surface method is presented that uses elements with distributed vorticity. As a consequence, the newly developed method is highly accurate in force prediction and numerically extremely robust, even when using a relaxed wake model. The accurate prediction is accomplished using lower panel densities than other methods require. The numerical robustness is achieved without compromising the irrotationality assumption, unlike other potential flow methods do that use discrete vortices with solid core models in the wake. The distributed vorticity element of the newly developed method consists of a vortex sheet that holds streamwise vorticity that varies linearly over the element span. Transverse vorticity is concentrated in two vortex filaments that are located along the leading and trailing edge of the element. Their spanwise circulation distributions vary in a parabolic fashion. The circulation of the leading and trailing edge filaments are equal in magnitude, but opposite in orientations. By introducing additional singularities along the edges of the distributed vorticity elements, any extreme velocities associated with the edge singularities of the vortex sheets are removed. The velocity induced by a distributed vorticity element is determined with an analytical expression. One or several spanwise systems of distributed vorticity elements are used to model the lifting surface and the wake that is relaxed using a time-stepping method. There, the shed vorticity forms an essentially continuous vortex sheet. Thus, because of the elimination of point or line singularities, many of the numerical problems are avoided that are encountered with conventional vortex-lattice and panel methods. In addition, the continuous vorticity distribution across the lifting surface yields an accurate load prediction that is relatively insensitive to panel density changes on the lifting surface and in the wake. Consequently, significantly fewer singularity elements are needed to achieve accuracies comparable to other potential flow methods. The subsequent method is a relatively fast tool for determining the location of the free wake and its interaction with complex wing geometries, especially when accurate load predictions are required. The potential of the method is demonstrated with two sample applications. Especially in the case of the formation flight of two aircraft, the relaxed wake model yields performance results that differ to those that are obtained with a fixed wake model.