Geometrically-Exact Extension of Theodorsen’s Frequency Response Model Haithem E Taha University of California, Irvine Zhimiao Yan and Muhammad R Hajj Virginia Tech AIAA Science & Technology Forum & Exposition, 5-9 Jan 2015, Kissimmee, Florida Haithem Taha

Quasi-Steady Aerodynamics Gaps in the Literature of Unsteady Aerodynamics 𝑘 < 0.1 𝑘 > 0.1 𝛼<15−20° 𝛼>25° Quasi-Steady Aerodynamics 2D: * Theodorsen * Wagner * Schwarz and Sohngen * Peters 3D: * RT Jones * Reissner * ULLT * UVLM Nonlinearity of the flow Prominent Unsteadiness Efficiency and Computational Burden There is a need to develop an aerodynamic model that captures the nonlinearity and unsteadiness with a feasible computational burden. Haithem Taha

Objective Develop an aerodynamic model that captures the flow nonlinearity in an unsteady fashion with a feasible computational burden in a compact form. Colleagues Sharing a similar Objective: Drs. Granlund and Ol at AFRL Drs. Ramesh, Gopalarathnam and Edwards at NCSU Dr. Eldredge at UCLA Dr. Rowley at Princeton

Previous Efforts Extension of Duhamel Principle to Arbitrary Non-conventional Lift Curves wake Taha et al., Aerosp. Sci Technol., 2014. Haithem Taha

Previous Efforts Issues Taha et al., Aerosp. Sci Technol., 2014. wake Haithem Taha

Previous Efforts 𝑥 =𝑓(𝑥,𝑢) Issues 𝑥 =𝑓(𝑥,𝑢) Nonlinearity of the Input-Output Map ( 𝐶 𝐿 −𝛼): √ Nonlinearity of the Flow Dynamics : x Taha et al., Aerosp. Sci Technol., 2014. Haithem Taha

Approach Extend Theodorsen’s model relaxing Geometrically-Exact Extension Theodorsen 1938 Extend Theodorsen’s model relaxing Flat wake Small angle of attack Small disturbances to the mean flow components Time-invariant free-stream. A semi-analytical, geometrically-exact, unsteady potential flow model is developed for airfoils undergoing large amplitude maneuvers. Haithem Taha

Approach Non-circulatory Contributions: Jukowsky Transformation, Non-circulatory Contribution Non-circulatory Contributions: - Non-circulatory Contributions - Circulatory Contributions Versus Haithem Taha

Approach 𝑞 𝜃 NC 𝜃=0 + 𝑞 𝜃 C 𝜃=0 =0 Kutta Condition and the Circulatory Contribution Cannot satisfy the Kutta Condition (Finite Velocity at the trailing edge). So, invoke an additional component: Circulatory Contribution Kutta Condition: 𝑞 𝜃 𝜃=0 =0 𝑞 𝜃 NC 𝜃=0 + 𝑞 𝜃 C 𝜃=0 =0 Each time step, we shed a vortex. Use Kutta condition to determine the strength of the newly shed vortex: no need to solve a linear system like the UVLM. Calculate loads using the unsteady Bernoulli’s equation. Add suction if needed. Update the locations of the discrete vortices. Haithem Taha

Approach Circulatory Contribution Versus Haithem Taha

Validation Eldredge et al., AIAA, 2009. Ramesh et al., Theor. Comput. Fluid Dyn., 2013. Haithem Taha

Frequency Response Haithem Taha

Frequency Response Change with the mean angle of attack At each 𝛼 0 , we weeps the k-range. At each k, 𝐻 is adjusted such that k 𝐻 = 5 ° → Linearized Dynamics about 𝛼 0 - Calculate the circulatory loads and determine the phase and relative magnitude with the quasi-steady loads at the same 𝛼 0 . Haithem Taha

Frequency Response Change with the mean angle of attack Haithem Taha

Geometrically-Exact Extension of Theodorsen’s Frequency Response Model Thank You! Haithem E Taha Mechanical and Aerospace Engineering University of California, Irvine