A planar vortex sheet has a local circulation strength distribution $\gamma(x) = 2 U_\infty \left(1 - \frac{x}{c}\right)$ along a chord from $x = 0$ to $x = c$. The total circulation $\Gamma$ around the entire sheet is:

(A) $\Gamma = U_\infty c$

(B) $\Gamma = 2 U_\infty c$

(C) $\Gamma = \frac{1}{2} U_\infty c$

(D) $\Gamma = 0$

Answer Key:

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### 1. Strategy & Theoretical Principles By definition, the total circulation $\Gamma$ of a vortex sheet is the line integral of its vortex strength density $\gamma(x)$ along the chord: $$\Gamma = \int_0^c \gamma(x) \, dx$$ ### 2. Analytical Formulation & Calculation Given $\gamma(x) = 2 U_\infty \left(1 - \frac{x}{c}\right)$: $$\Gamma = 2 U_\infty \int_0^c \left(1 - \frac{x}{c}\right) \, dx$$ Evaluating the integral: $$\int_0^c \left(1 - \frac{x}{c}\right) \, dx = \left[ x - \frac{x^2}{2c} \right]_0^c = c - \frac{c}{2} = \frac{c}{2}$$ Multiply by $2 U_\infty$: $$\Gamma = 2 U_\infty \left(\frac{c}{2}\right) = U_\infty c$$ ### 3. Distractor Forensics - **Option A ($U_\infty c$):** Correct total circulation. - **Option B ($2 U_\infty c$):** Forgot the triangular integration factor $1/2$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Total circulation is the area under the $\gamma(x)$ distribution curve. - **Standard Reference:** Anderson, J.D., *Fundamentals of Aerodynamics*, Chapter 4.

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