Shahriar Khosravi and David W. Zingg
Institute for Aerospace Studies, University of Toronto, 4925 Dufferin Street, Toronto, Ontario, Canada M3H 5T6
Email: shahriar.khosravi@mail.utoronto.ca
Phone: (416) 667-7887
Induced drag is typically 40% of the total drag of a commercial aircraft in cruise [1]. Thus, it is important to explore concepts with the potential to reduce induced drag. This has motivated many researchers in the past to study winglets. However, conclusions made vary depending on the design problem considered and the level of physical detail that the models used have been able to capture. This means that more work needs to be done in order to understand the fundamental tradeoffs in the design of wings with winglets. This study attempts to contribute to this objective through the application of high-fidelity aerostructural optimization.
The aerostructural optimization framework used in this work consists of six main components: 1) a multiblock flow solver for the Euler equations [2], 2) a finite-element structural solver for the analysis of the structure [3], 3) a mesh movement technique based on the linear elasticity equations for moving the aerodynamic grid during aerostructural analysis and optimization [4], 4) a surface-based free-form deformation technique for moving the structures mesh during optimization [5], 5) a B-spline parameterization method for geometry control which is coupled with the linear elasticity mesh movement technique [4], and 6) the gradient-based optimizer SNOPT [6, 7] with gradients calculated using the coupled discrete-adjoint method [5].
The baseline wing is based on the Boeing 737-900 with the RAE 2822 airfoil. The geometric parameterization and design variables are shown in Figure 1. The structural layout used is shown in Figure 2. The structural design variables are the thickness values of the structural components. All optimization cases are initiated with an initially planar wing, but the optimizer has the freedom to create a nonplanar feature at the wingtip by varying the geometric design variable that controls the cant angle. Two configurations are considered: winglet-up and winglet-down. The initially planar wing has a cant angle of zero. For a winglet-up configuration, the optimizer is allowed to vary the cant angle between 0 and +90°. Similarly, the optimizer can create a winglet-down feature by varying the cant angle between 0 and -90°.
There are two load conditions: cruise and 2.5g. The cruise Mach number is equal to 0.785 at an altitude of 35,000 feet. The Mach number at the 2.5g load condition is equal to 0.798 at an altitude of 12,000 feet. The 2.5g load condition sizes the structure. The objective function is of the form
J = (beta*(D/D_0))+((1-beta)*(W/W_0))
where D is the drag in cruise and W is the weight of the wing. D_0 and W_0 are the respective initial values. Three values for beta have been chosen: 0.1, 0.5, and 1.0.
Figure 3 shows the Pareto fronts of optimal designs for the winglet-up and winglet-down configurations. The optimizer did not create a winglet-up feature for any of the objective functions even though it did have the freedom to do so. Furthermore, a winglet-down configuration emerged only when β is greater than or equal to 0.5. Figure 3 indicates that the winglet-down configuration provides a drag reduction of approximately 4% in comparison to the other configurations.
References
[1] Kroo, I., “Drag Due to Lift: Concepts for Prediction and Reduction,” Annual Review of Fluid Mechanics, Vol 33, No. 1, 2001, pp. 587-617.
[2] Hicken, J. E. and Zingg, D. W., “Parallel Newton-Krylov Solver for the Euler Equations Discretized Using Simultaneous-Approximation Terms,” AIAA Journal, Vol. 46, No. 11, 2008, pp. 2773-2786.
[3] Kennedy, G. J. and Martins, J. R. R. A., “A parallel finite-element framework for large-scale gradient-based design optimization of high-performance structures,” Finite Elements in Analysis and Design, Vol. 87, September 2014, pp. 56-73.
[4] Hicken, J. E., and Zingg, D. W., “Aerodynamic Optimization Algorithm with Integrated Geometry Parameterization and Mesh Movement,” AIAA Journal, Vol. 48, No. 2, 2010, pp. 400-413.
[5] Zhang, Z. J., Khosravi, S., and Zingg, D. W., “High-Fidelity Aerostructural Optimization with Integrated Geometry Parameterization and Mesh Movement,” 56th AIAA /ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, No. 2022522, Kissimmee, Florida, January 2015.
[6] Gill, P. E., Murray, W., and Saunders, M. A., “SNOPT: An SQP Algorithm for Large-Scaled Constrained Optimization,” SIAM Journal on Optimization, Vol. 12, 1997, pp. 979-1006.
[7] Perez, R. E., Jansen, P. W., and Martins, J. R. R. A., “pyOpt: A Python-Based Object Oriented Framework for Nonlinear Constrained Optimization,” Structures and Multidisciplinary Optimization, Vol. 45, No. 1, 2012, pp. 101-118.
Topics: Aerodynamic optimization and uncertainty analysis methods; Multidisciplinary Analy
