Question Statement:
Consider steady, incompressible, inviscid flow past two airfoils shown in the figure.
The coefficient of pressure at the trailing edge of the airfoil with finite angle, shown
in figure (I), is CPI while that at the trailing edge of the airfoil with cusp, shown in
figure (II), is CPII. Which one of the following options is TRUE?
Scientific schematic diagram reproduced from official IIT paper
Options:
-
(A) CPI < 1, CPII < 1
-
(B) CPI = 1, CPII = 1
-
(C) CPI = 1, CPII < 1
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(D) CPI < 1, CPII = 1
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2024 Q.24)**
**1. Kutta Condition at Trailing Edges**:
For steady, incompressible, inviscid potential flow over an airfoil:
- **Figure (I) - Finite Trailing Edge Angle ($\theta > 0$)**:
The flow cannot turn through a sharp corner with finite angle; hence the trailing edge must be a stagnation point ($V = 0$).
By Bernoulli's equation, at a stagnation point where $V = 0$:
$$C_{P, I} = 1 - \left(\frac{V}{U_\infty}\right)^2 = 1 - 0 = 1$$
- **Figure (II) - Cusp Trailing Edge ($\theta = 0$)**:
For a smoothly merging cusp, upper and lower surface tangents are parallel at the trailing edge. The flow leaves smoothly at a finite, non-zero exit speed ($V > 0$).
Consequently:
$$C_{P, II} = 1 - \left(\frac{V}{U_\infty}\right)^2 < 1$$
**2. Conclusion**:
$$C_{P, I} = 1, \quad C_{P, II} < 1$$
**3. Official Master Key**:
- **Official Answer Key**: **C** ($C_{P, I} = 1, C_{P, II} < 1$)
- **Exam Takeaway**: Kutta condition imposes zero velocity (stagnation point, $C_p = 1$) at a finite-angle trailing edge, but non-zero velocity ($C_p < 1$) at a cusped trailing edge.