Transonic Airfoil Design

1. Transonic Airfoil Design Preliminary Presentation AE 4903, Fall 2002 Sriram Rallabhandi Georgia Institute of Technology

2. Presentation Outline • Need and motivation for transonic airfoil design • Important aspects of Supercritical airfoils • Analysis method • Full potential equation • Euler/Navier Stokes formulation • Validation of analysis method • Design Methodology • Some results • Conclusions and Future work

3. Need and Motivation • Off-the-shelf airfoils cannot always be used • Limitations of Existing airfoils • Catalogued airfoils may not be suited for revolutionary designs • May not match the requirements of intended application • Geometry very important for Aerodynamics • A theoretical design procedure allows: • Greater geometric flexibility • An automated computation of airfoil shape • Economic exploration of many airfoil concepts in the initial design stages • A repeatable design method rather than a cut and trail approach

4. Aspects of Supercritical airfoils • Low shock drag losses at Transonic speed • Shock stands at a significantly aft location • Upper surface has a pressure plateau • Extremely steep pressure recovery towards the aft of upper surface • Lower surface has roughly constant negative pressure coefficient

5. Analysis Method 1:Full potential Equation Brief overview: • Construct a smooth grid with adequate resolution near the airfoil surface • Initialize φ to 0.0 and compute metrics at all reqd. points • Discretize the FPE equation into a sparse matrix form using finite difference formulation • Solve for φ at the next time step using any of the efficient solution techniques • Compute Γ and φ at boundaries and far-field • Compute loads and Cp using φ

6. FPE (Contd.) • O-grid used because it accurately follows rounded bodies without much skewness • Conformal mapping methods are used • O-grid is better suited for subsonic flows: Outer boundary tends to match the region of influence of solid body

7. Euler/N-S formulation • System of Equations • Roe’s flux difference splitting used for Inviscid fluxes • Central differencing used for Viscous fluxes • No-slip boundary condition at wall imposed • Free-stream boundary conditions at outer boundaries • Riemann invariants proved to be causing a limit cycle behavior and so were not used for viscous cases • 2-step Explicit time integration scheme used

8. Euler/N-S (Contd.) • C-grid provides good clustering control for viscous problems • Elliptic PDEs used to generate a grid over the airfoil

9. Validation of Analysis method (FPE) NACA 0012, M=0.63, Alpha = 2.0 NACA0012, M=0.72, Alpha = 0.0

10. Validation (Euler/N-S) NACA 0012, M=0.63, Alpha = 2.0 Log Residual plot

11. Validation (Euler/N-S) NACA 0012, M=0.72, Alpha = 0.0 Log Residual plot

12. Design Methodology • Navier-Stokes solver: slow and time consuming • Sometimes limit cycle behavior observed and flux limiter had to used for convergence • Numerical optimization method used with FPE analysis method • Airfoil surface approximated as a polynomial • Squared difference of Cp calculated and Target Cp is used as objective function for minimization • Care taken to close the airfoil trailing edge and leading edge • Some intermediate airfoil shapes and Cp distributions were also observed

13. Results Cp distribution and Airfoil shape after a few iterations

14. Results

15. Results In this figure, the Angle of attack was fixed at 0.0

16. Conclusions and Future Work • Analysis tool and its capabilities have been explored • The design method has been tested for new airfoil generation • Have to generate a target Cp distribution for a supercritical airfoil to design • Involves understanding the performance requirements of the airfoil • Extend to 3-D wings if time and computational effort permits

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