Home >
PYQ Archives >
GATE AE 2019 >
Question 44
Question Statement:
The inviscid, incompressible flow field resulting from a uniform flow past a circular cylinder of radius $R$ is given by potential flow theory. If the pressure coefficient at a point on the surface is $C_p = 0$, the angle $\theta$ (in degrees, with $0 \le \theta \le 90^\circ$) measured from the front stagnation point is _________ degrees.
Official Answer Key:
30.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2019 Q.44 / AE Q.34)**
**1. Intuitive Physical Setup & Given Parameters**:
- Uniform flow $U_\infty$ past a circular cylinder of radius $R$ without circulation (non-lifting cylinder).
- Surface velocity distribution: $V_\theta = -2 U_\infty \sin\theta$.
- Surface pressure coefficient condition: $C_p = 0$.
**2. Step-by-Step Mathematical Derivation**:
1. **Surface Pressure Coefficient Expression**:
From Bernoulli's equation for steady incompressible flow:
C_p = 1 - \left( \frac{V}{U_\infty} \right)^2
On the cylinder surface, $V = |V_\theta| = 2U_\infty \sin\theta$:
C_p(\theta) = 1 - (2\sin\theta)^2 = 1 - 4\sin^2\theta
2. **Setting $C_p = 0$**:
1 - 4\sin^2\theta = 0 \implies 4\sin^2\theta = 1
\sin^2\theta = \frac{1}{4} \implies \sin\theta = \frac{1}{2} \quad (\text{for } 0 \le \theta \le 90^\circ)
3. **Solving for $\theta$**:
\theta = \arcsin(0.5) = 30^\circ
At $\theta = 30^\circ$ (and by symmetry at $150^\circ, 210^\circ, 330^\circ$), the local surface velocity equals freestream velocity ($V = U_\infty$), so static pressure equals freestream static pressure ($C_p = 0$).
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Exact Analytical Angle**: **30.0 degrees** (Acceptable Range: [30, 30]).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **30.0**
- **Cylinder Benchmark Angles**: Stagnation ($C_p = 1$) at $0^\circ, 180^\circ$; Ambient pressure ($C_p = 0$) at $30^\circ, 150^\circ$; Maximum suction ($C_p = -3$) at $90^\circ, 270^\circ$.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.
Launch CBT Mock Test