1 Multi-point Wing Planform Optimization via Control Theory Kasidit Leoviriyakit and Antony Jameson Department of Aeronautics and Astronautics Stanford University, Stanford CA 43 rd Aerospace Science Meeting and Exhibit January 10-13, 2005 Reno Nevada

2 Typical Drag Break Down of an Aircraft ItemCDCD Cumulative C D Wing Pressure120 counts (15 shock, 105 induced) Wing friction45165 Fuselage50215 Tail20235 Nacelles20255 Other15270 ___ Total270 Mach.85 and C L.52 Induced Drag is the largest component

3 Cost Function Simplified Planform Model Wing planform modification can yield larger improvements BUT affects structural weight. Can be thought of as constraints

4 Choice of Weighting Constants Minimizing using Maximizing Range

5 Structural Model for the Wing Assume rigid wing (No dynamic interaction between Aero and Structure) Use fully-stressed wing box to estimate the structural weight Weight is calculated based on material of the skin

6 “Trend” for Planform Modification Increase L/D without any penalty on structural weight by Stretching span to reduce vortex drag Decreasing sweep and thickening wing-section to reduce structural wing weight Modifying the airfoil section to minimize shock Boeing 747 -Planform Optimization Baseline Suggested

7 Redesign of Section and Planform using a Single-point Optimization CLCL C D counts C W counts Boeing (102.4 pressure, 34.6 viscous) 498 (80,480 lbs) Redesigned (78.3 pressure, 38.4 viscous) 464 (75,000 lbs) Baseline Redesign Flight Condition (cruise): Mach.85 CL.45

8 The Need of Multi-Point Design Undesired characteristics Designed Point

9 Cost Function for a Multi-point Design Gradients

10 Multi-point Design Process

11 Review of Single-Point design using an Adjoint method Using 4224 mesh points on the wing as design variables Boeing 747 Plus 6 planform variables -Sweep -Span -Chord at 3span –stations -Thickness ratio Design Variables

12 Optimization and Design using Sensitivities Calculated by the Finite Difference Method f(x)

13 Disadvantage of the Finite Difference Method The need for a number of flow calculations proportional to the number of design variables Using 4224 mesh points on the wing as design variables Boeing flow calculations ~ 30 minutes each (RANS) Too Expensive Plus 6 planform variables

14 Application of Control Theory (Adjoint) Drag Minimization Optimal Control of Flow Equations subject to Shape(wing) Variations GOAL : Drastic Reduction of the Computational Costs (for example C D at fixed C L ) (RANS in our case)

design variables Application of Control Theory One Flow Solution + One Adjoint Solution

16 Sobolev Gradient Continuous descent path Key issue for successful implementation of the Continuous adjoint method.

17 Design using the Navier-Stokes Equations See paper for more detail

18 Test Case Use multi-point design to alleviate the undesired characteristics arising form the single-point design result. Minimizing at multiple flight conditions; I = C D +  C W at fixed C L (C D and C W are normalized by fixed reference area)  is chosen also to maximizing the Breguet range equation Optimization: SYN107 Finite Volume, RANS, SLIP Schemes, Residual Averaging, Local Time Stepping Scheme, Full Multi-grid

19 Single-point Redesign using at Cruise condition

20 Isolated Shock Free Theorem Mach.84 Mach.85 Mach.90 “Shock Free solution is an isolated point, away from the point shocks will develop” Morawetz 1956

21 Design Approach If the shock is not too strong, section modification alone can alleviate the undesired characteristics. But if the shock is too strong, both section and planform will need to be redesigned.

22 3-Point Design for Sections alone (Planform fixed) ConditionMach  /3

23 Successive 2-Point Design for Sections (Planform fixed) ConditionMach  /2 M DD is dramatically improved

24 Lift-to-Drag Ratio of the Final Design

25 C p at Mach 0.78, 0.79, …, 0.92 Shock free solution no longer exists. But overall performance is significantly improved.

26 Conclusion Single-point design can produce a shock free solution, but performance at off-design conditions may be degraded. Multi-point design can improve overall performance, but improvement is not as large as that could be obtained by a single optimization, which usually results in a shock free flow. Shock free solution no longer exists. However, the overall performance, as measured by characteristics such as the drag rise Mach number, is clearly superior.

27 Acknowledgement This work has benefited greatly from the support of Air Force Office of Science Research under grant No. AF F Downloadable Publications