JOURNAL OF AIRCRAFT Vol. 39, No. 3, May– June 2002 Hybrid Inverse Airfoil Design Method for Complex Three-Dimensional Lifting Surfaces Ashok Gopalarathnam ¤ North Carolina State University, Raleigh, North Carolina 27695 and Michael S. Selig † University of Illinois at Urbana—Champaign, Urbana, Illinois 61801 A method is presented for inverse design of airfoils for complex three-dimensional wings in incompressible ow. The method allows for prescription of inviscid velocity distributions over different cross sections of the wing in a multipoint fashion. A hybrid approach is used to determine the shapes of the wing cross sections that satisfy the design specications. The airfoils forming the cross sections of the wing are generated using an inverse code for isolated airfoil design. A three-dimensional panel method is then used to obtain the velocity distributions over the resulting wing. The isolated airfoil velocity distributions are then used as design variables in a multidimensional Newton iteration method to achieve the design specications on the wing. The method is particularly useful for complex geometries such as junctures, where three-dimensional and interference effects have to be accounted for in the design process. A key feature of the design method is a scheme to avoid using the panel method for sensitivity computations for the Newton iterations. This scheme not only results in signicant reductions in computation time but also enables the integration of any readily available three-dimensional analysis code in executable form. Examples are shown to demonstrate the usefulness of the method. Nomenclature C l = airfoil lift coefcient, chord D 1 c = airfoil/section chord F = vector containing the residuals J = Jacobian matrix n = number of design variables V = airfoil/section velocity nondimensionalized by the freestream velocity v = desired value for the velocity difference over a segment ® = angle of attack, deg 1V = velocity difference over a segment normalized by the freestream velocity ± x = vector containing the corrections to the design variables Subscripts i = index of design variable for the Newton iteration r = property associated with the wing root t = property associated with the wing tip w = property associated with the wing 2 D = property associated with an airfoil in isolation 3 D = property associated with a cross section of a three-dimensional lifting surface Introduction A HYBRID method has been developed for wing design. The method uses an approach similar to the one developed by the Presented as Paper 99-0401 at the AIAA 37th Aerospace Sciences Meet- ing and Exhibit, Reno, NV, 11– 14 January 1999; received 29 March 1999; revision received 29 January 2002; accepted for publication 1 February 2002. Copyright c ° 2002 by Ashok Gopalarathnam and Michael S. Selig. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0021-8669/02 $10.00 in correspondenc e with the CCC. ¤ Assistant Professor, Department of Mechanical and Aerospace Engineer- ing, Box 7910; ashok g@ncsu.edu. Member AIAA. † Associate Professor, Department of Aeronautical and Astronautical En- gineering, 306 Talbot Laboratory, 104 S. Wright Street; m-selig@uiuc.edu. Senior Member AIAA. authors for inverse design of multielement airfoils 1 in potential ow, which was subsequently extended to handle viscous inverse design. 2 This multielement airfoil design method uses a novel hybrid ap- proach by coupling an isolated airfoil design code 3;4 (PROFOIL) with a two-dimensional panel method. The inverse airfoil design code is used to design the elements of the multielement airfoil in isolation, and the panel method is then used to analyze the result- ing multielement airfoil. An integral boundary-layer method is then used to obtain the boundary-layer developments on the individual elements. By using a multidimensional Newton iteration procedure, the design variables associated with the isolated airfoil design are adjusted to achieve desired velocity and boundary-layer prescrip- tions on the elements of the multi-element airfoil. A key feature of the method is a scheme for rapid computation of the Newton itera- tion sensitivities without using the panel method. This scheme was developed based on the observation of similarities 1 in the changes in velocity distributions for an airfoil in isolation and as a part of the multielement airfoil. The hybrid approach as well as the sensitiv- ity computation scheme enable rapid, interactive design even with several multipoint viscous prescriptions. 2 Several experiences with successful applications of the isolated airfoil design method 5;6 and the more recent hybrid approach for rapid, interactive multielement airfoil design 7 have naturally prompted the question of whether such an approach can be used for designing complex three-dimensional wings. This paper presents the extension of the hybrid approach to three-dimensional aero- dynamic systems. More specically, the current work describes a potential-ow method for designing the shapes of wing cross sec- tions to achieve multipoint prescriptions in velocity distributions. For this purpose the isolated airfoil design code 3;4 (PROFOIL) is used to generate the airfoils that form the cross sections of the three- dimensional aerodynamic system, and a three-dimensional analysis method (the panel method PMARC/CMARC 8;9 inthis paper)isused to analyze the resulting wing system. The variables associated with the design of the airfoils in isolation are then adjusted to achieve the prescriptions on the wing cross sections. The paper shows that, as with the multielement airfoil design problem, there are similar- ities in the velocity-distribution changes for an airfoil in isolation and as part of a complex three-dimensional system. This similarity is used to an advantage to avoid repeated panel method analyses for the sensitivity computation via nite differencing. Instead, the 409