GATE AE 2019 • Q44 NAT • 2 Marks

Aerodynamics & Fluid Mechanics

Potential Flow Theory

Subtopic: Flow Past a Circular Cylinder & Surface Pressure Coefficient

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