The fundamental purpose of the Kutta condition in the thin airfoil theory is
________.
Options:
(A) to determine the total strength of the source distribution
(B) to determine the speed of the uniform flow
(C) to incorporate the essential effect of viscosity in the potential flow theory
(D) to incorporate the concept of induced drag in the inviscid theory
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.20)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to identify the fundamental physical purpose of imposing the Kutta condition in thin airfoil theory.
**2. Step-by-Step Mathematical Derivation**:
1. **The Indeterminacy of Pure Potential Flow**:
According to inviscid potential flow theory (Euler equations), any arbitrary value of circulation $\Gamma$ around an airfoil satisfies the governing Laplace equation $\nabla^2 \phi = 0$ and the wall tangency condition $\vec{V} \cdot \vec{n} = 0$.
Without additional physics, the lift force $L' = \rho_\infty V_\infty \Gamma$ is mathematically indeterminate.
2. **Real Fluid Boundary Layer Behavior**:
In any real fluid (even with infinitesimal viscosity $\mu \to 0$), the flow cannot negotiate the infinitely sharp trailing edge with infinite velocity without separating. Viscosity forces the flow to leave smoothly from the upper and lower surfaces at the trailing edge.
3. **Role of the Kutta Condition**:
The Kutta condition dictates that the flow must leave the sharp trailing edge smoothly ($V_{TE}$ finite, zero cross-flow), which uniquely fixes the circulation $\Gamma$ to:
$\Gamma = \pi c V_\infty (\alpha + 2\alpha_{L=0})$
thereby incorporating the indispensable consequence of real-fluid viscosity into an otherwise inviscid mathematical model.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) to determine the total strength of the source distribution**: ❌ **INCORRECT**. Source distributions are determined by camber line slope and thickness distribution.
- **(B) to determine the speed of the uniform flow**: ❌ **INCORRECT**. Free-stream velocity is an independent boundary condition.
- **(C) to incorporate the essential effect of viscosity**: ✅ **CORRECT**. It accounts for boundary layer separation control at the sharp trailing edge, fixing the unique circulation.
- **(D) to incorporate the concept of induced drag**: ❌ **INCORRECT**. Induced drag is a 3D finite wing vortex wake effect; thin airfoil theory is 2D.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: The Kutta condition is the bridge between viscous reality and inviscid potential flow theory. It selects the unique physically realizable circulation from an infinite family of potential flow solutions.
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