A thin symmetric airfoil is equipped with a plain trailing-edge flap of chord fraction $c_f / c = 0.20$. In thin airfoil theory, the flap hinge angle $\theta_f$ is defined by $\cos \theta_f = 1 - 2(c_f/c) = 1 - 2(0.20) = 0.60$ (so $\theta_f = \arccos(0.60) \approx 0.9273\text{ rad}$). The theoretical flap lift effectiveness parameter $\frac{\partial C_l}{\partial \delta}$ (in $\text{rad}^{-1}$) is given by $2(\pi - \theta_f + \sin \theta_f)$. The value of $\frac{\partial C_l}{\partial \delta}$ is closest to:

(A) 6.03 rad^-1

(B) 4.28 rad^-1

(C) 2.51 rad^-1

(D) 1.85 rad^-1

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles In classical thin airfoil theory, deflecting a plain trailing-edge flap by an angle $\delta$ modifies the camber line slope over the aft portion $\theta_f \le \theta \le \pi$ where $\cos \theta_f = 1 - 2(c_f/c)$. The lift coefficient increment produced by flap deflection $\delta$ is: $$\Delta C_l = \left[ 2(\pi - \theta_f) + 2 \sin \theta_f \right] \delta = 2(\pi - \theta_f + \sin \theta_f) \delta$$ Hence, the flap effectiveness derivative is: $$\frac{\partial C_l}{\partial \delta} = 2(\pi - \theta_f + \sin \theta_f)$$ ### 2. Analytical Formulation & Calculation Given: - $\cos \theta_f = 0.60$ - $\theta_f = \arccos(0.60) \approx 0.927295\text{ rad}$ - $\sin \theta_f = \sqrt{1 - (0.60)^2} = \sqrt{0.64} = 0.80$ 1. Compute bracketed expression: $$\pi - \theta_f + \sin \theta_f = 3.141593 - 0.927295 + 0.80 = 2.214298 + 0.80 = 3.014298$$ 2. Multiply by 2: $$\frac{\partial C_l}{\partial \delta} = 2 \times 3.014298 \approx 6.0286\text{ rad}^{-1} \approx 6.03\text{ rad}^{-1}$$ (Notice this approaches the theoretical thin airfoil lift-curve slope $2\pi \approx 6.28\text{ rad}^{-1}$ as flap chord approaches the full chord). ### 3. Distractor Forensics - **Option A (6.03 rad^-1):** Correct thin airfoil flap effectiveness. - **Option B (4.28 rad^-1):** Forgot factor of 2 or used $c_f/c = 0.10$. - **Option C (2.51 rad^-1):** Evaluated without $\pi$ term. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Ensure $\theta_f$ is substituted in radians (not degrees) when adding to $\pi$ and $\sin \theta_f$. - **Standard Reference:** Anderson, J.D., *Fundamentals of Aerodynamics*, Chapter 4 (Flapped Airfoils).

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