1. Unsteady Viscous Lift Frequency Response Using The Triple Deck Theory
Haithem Taha and Amir Rezaei
Mechanical and Aerospace Engineering
University of California, Irvine
AIAA Science & Technology Forum & Exposition, 8-12 Jan 2018, Kissimmee, Florida
Haithem Taha

2. Classical Theory of Unsteady Aerodynamics
Is there a fundamental issue/flaw with it? What is the reason behind the meager state of flutter predictability?
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?
Haithem Taha

3. Flutter = Unsteady Aerodynamics + Structural Dynamics
The Unsteady Kutta Condition
Reviews/Seminal efforts:
Sears, AIAA (1976)
Crighton, Ann. Rev. of Fluid Mech. (1985)
Research Flurry in the 1970s and 1980s:
Orszag and Crow (1970), Basu and Hancock (1978), Daniels (1978), Satyanarayana and Davis (1978), Bass et al. (1982), …
Due to the failure of predicting flutter speed in the 1950s and 1960s:
Woolston (1951), Rott and George (1955), Abramson and Chu (1958, 1959, 1967), Henry (1961), Chu (1962), Shen and Crimi (1965), …
Flutter = Unsteady Aerodynamics + Structural Dynamics
It’s a fundamental flaw and not a higher-order effect!
Recent Dissatisfaction with the Kutta condition in the low Re, high frequency bio-inspired flight: Ansari et al. (2006), Pitt Ford and Babinsky (2013), Hemati et al. (2014), ….
Haithem Taha

4. 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 (1970s):
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
Goldstein (1930)
Prandtl (1904): Boundary-Layer Equations
Blasius (1908): Similarity Solution

5. Development of a Viscous Unsteady Theory
Unsteady Extension of the Triple Deck = Viscous Extension of the Classical Unsteady Theory
Triple Deck Boundary Layer Theory (1970s):
Messiter, SIAM J. Appl. Math. (1970)
Stewartson, Proceedings of the Royal Society of London (1970)
Non zero alpha: Brown and Stewartson, JFM (1970)
Solution: Chow and Melnik, Conference on Numerical Methods in Fluid Dynamics (1976)
Haithem Taha

6. Development of a Viscous Unsteady Theory
Unsteady Extension of the Triple Deck = Viscous Extension of the Classical Unsteady Theory
Unsteady Triple Deck Theory
Brown and Daniels, JFM (1975): High frequency (k>>1) linearized solution.
Brown and Cheng, JFM (1981): Relatively low frequency k=O(1).
𝑑 𝑑𝑡 term  0 up to 1st order
Potential Flow Solution
Satisfying the Kutta Condition
Viscous Contribution
Trailing Edge Singularity
Steady problem by Brown & Stewartson (1970)
𝐵 𝑣 = 𝐵 𝑣 𝑈 2
Haithem Taha

7. Development of a Viscous Unsteady Theory
Unsteady Extension of the Triple Deck = Viscous Extension of the Classical Unsteady Theory
ℎ 𝑡 , 𝛼(𝑡)
Linear Dynamics G(k)
𝛼 𝑠 (𝑡)
𝐵 𝑠 (𝑡)
𝜋 𝐵 v (𝑡) -
𝐶 𝐿 𝑐 (𝑡) X +
Airfoil Motion +
Triple Deck Viscous Nonlinearity
Theodorsen’s Linear Dynamics 2𝜋𝐶(𝑘)
𝛼 3/4 (𝑡)
Linear Differential Operator
Potential
flow
Describing Function
Haithem Taha

8. URANS Computational Results for a Pitching NACA 0012
Bass et al. water tunnel experiment (1982): “a 30 deg phase lag in C(k) provides improved agreement between theory and experiment.”
Haithem Taha

9. Physical Illustrations
Viscosity Induced Lag
- Stokes second problem
𝑢 𝑦,𝑡 =𝑈 𝑒 −𝑦/𝛿 cos 𝜔𝑡− 𝑦 𝛿
𝛿= 2𝜈/𝜔
𝜁 𝑦,𝑡 =− 𝜕𝑢 𝜕𝑦 = 2 𝑈 𝛿 𝑒 −𝑦/𝛿 cos 𝜔𝑡− 𝑦 𝛿 − 𝜋 4
Lag in Circulation Development
Haithem Taha

10. Physical Illustrations
Lag in Circulation Development and the Kutta Condition
The Kutta Condition ∆ 𝑃 𝑇𝐸 =0
Potential Flow (Outside BL): 𝑃 𝐴 = 𝑃 𝐵
∆ 𝑃 𝑇𝐸 =0→ 𝑃 𝐴 −∆ 𝑃 𝐴 = 𝑃 𝐵 −∆ 𝑃 𝐵
Outside of the BL:
𝑃 𝐴 𝜌 𝑈 𝐴 2 + 𝜕 𝜙 𝐴 𝜕𝑡 = 𝑃 𝐵 𝜌 𝑈 𝐵 2 + 𝜕 𝜙 𝐵 𝜕𝑡
𝑈 𝐴 2 −𝑈 𝐵 2 + ∆ 𝑃 𝐴 −∆ 𝑃 𝐵 𝜌 =− 𝜕 𝜙 𝐴 −𝜙 𝐵 𝜕𝑡 𝛤
Vorticity flux out of the BL =− 𝛤
0 𝛿 𝐴 𝜁𝑢𝑑𝑦 + − 𝛿 𝐵 0 𝜁𝑢𝑑𝑦 =− 𝛤
BL Theory (curved surface):
0 𝛿 𝐴 𝜁𝑢𝑑𝑦 = 0 𝛿 𝐴 𝜕𝑢 𝜕𝑦 +𝜅𝑢 𝑢𝑑𝑦 = 𝑈 𝐴 ∆ 𝑃 𝐴 𝜌
𝑈 𝐴 2 −𝑈 𝐵 2 + ∆ 𝑃 𝐴 −∆ 𝑃 𝐵 𝜌 =− 𝛤
If ∆ 𝑃 𝐴 =∆ 𝑃 𝐵 =0, 𝑈 𝐴/𝐵 = 𝑈 ∞ ± 1 2 𝛾 𝑇𝐸 → 𝑈 ∞ 𝛾 𝑇𝐸 =− 𝛤
Haithem Taha

11. Physical Illustrations
Lag in Circulation Development and the Kutta Condition
𝑈 𝐴 2 −𝑈 𝐵 2 + ∆ 𝑃 𝐴 −∆ 𝑃 𝐵 𝜌 =− 𝛤
Haithem Taha

12. Conclusions
Need for relaxing the Kutta condition and developing a novel framework for unsteady aerodynamics.
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.
Lag in Circulation Development and the Kutta Condition.
Haithem Taha

13. Acknowledgment UCI Virginia Tech: Amir Rezaei Prof. Muhammad Hajj
Prof. Craig Woolsey
Prof. Saad Ragab
wake

14. Unsteady Viscous Lift Frequency Response Using The Triple Deck Theory
Thank You!
Haithem Taha and Amir Rezaei
Mechanical and Aerospace Engineering
University of California, Irvine
AIAA Science & Technology Forum & Exposition, 8-12 Jan 2018, Kissimmee, Florida
Haithem Taha