Viscous Extension of Theodorsens Lift Frequency Response Haithem

Viscous Extension of Theodorsen's Lift Frequency Response Haithem Taha and Amir Rezaei Mechanical and Aerospace Engineering University of California, Irvine 71 st Annual Meeting of the APS Division of Fluid Dynamics Sunday–Tuesday, November 18– 20, 2018; Atlanta, Georgia Haithem Taha

Classical Theory of Unsteady Aerodynamics Is there a fundamental issue/flaw with it? Prandtl (1904): For high-Re small-alpha, the flow can be considered potential except for thin layers around the airfoil and in the wake. But ! We need an auxiliary condition! Kutta Condition: Smooth flow-off the trailing edge. No flow around the trailing edge. Stagnation Point at the trailing edge. A brilliant condition for steady flow. What about Unsteady flow? 2 Haithem Taha

Development of a Viscous Unsteady Theory Unsteady Extension of the Triple Deck = Viscous Extension of the Classical Unsteady Theory Vorticity generation and lift production are viscous processes. Triple Deck Boundary Layer Theory (1970 s): - Messiter, SIAM J. Appl. Math. (1969) Stewartson, Proceedings of the Royal Society of London (1969) Brown and Stewartson, JFM (1970) Chow and Melnik (1976) Triple Deck Theory / Trailing Edge Theory Prandtl (1904): Boundary-Layer Equations Blasius (1908): Similarity Solution Haithem Taha Goldstein (1930) 5

Development of a Viscous Unsteady Theory Unsteady Extension of the Triple Deck = Viscous Extension of the Classical Unsteady Theory Potential Flow Solution Viscous Contribution Satisfying the Kutta Condition Trailing Edge Singularity Steady problem by Brown & Stewartson (1970) Frequency range relevant to engineering applications: k=O(1). Steady problem Taha and Rezaei, JFM, Revision submitted. Haithem Taha 7

Development of a Viscous Unsteady Theory Unsteady Extension of the Triple Deck = Viscous Extension of the Classical Unsteady Theory Linear Dynamics G(k) Airfoil Motion X + + Triple Deck Viscous Nonlinearity Linear Differential Operator Taha and Rezaei, JFM, Revision submitted. Potential flow 9

Development of a Viscous Unsteady Theory Validation with Computational Simulations and Previous Experiments Laminar and URANS Computational Results for a Pitching NACA 0012 - Chu & Abramson (1959): Add phase lag of − 10 deg to C(k) at k ≃ 0. 5 “is seen to offer some improved agreement with the experimental values”. - Bass et al. (1982): a -30 deg phase lag in C(k) “provides improved agreement between theory and experiment. ” over the range k=0. 5 -2. 0 Taha and Rezaei, JFM, Revision submitted. Haithem Taha 10

Conclusions - Unsteady Extension of the Viscous Boundary Layer Theory (Triple Deck) = Viscous Extension of the Classical Theory of Unsteady Aerodynamic. - Reynolds-Number-Dependent Lift Frequency Response. - Viscosity Induced Lag. Haithem Taha 13

Acknowledgment Amir Rezaei wak e 14

Viscous Extension of Theodorsen's Lift Frequency Response Thank You! Haithem Taha and Amir Rezaei Mechanical and Aerospace Engineering University of California, Irvine 71 st Annual Meeting of the APS Division of Fluid Dynamics Sunday–Tuesday, November 18– 20, 2018; Atlanta, Georgia Haithem Taha