1. 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

2. 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

3. 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

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

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

6. 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

7. 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

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

9. 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

10. Approach
Circulatory Contribution
Versus
Haithem Taha

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

12. Frequency Response
Haithem Taha

13. 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

14. Frequency Response
Change with the mean angle of attack
Haithem Taha

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