GATE AE 2020 • Q21 NAT • 1 Mark Aerodynamics & Fluid Mechanics

Aerodynamics & Fluid Mechanics

Basic Fluid Mechanics

Subtopic: Viscous Flow & Boundary Layer Equations

Question Statement:

Velocity of an airplane in the body-fixed axes is given as $[u, v, w] = [100, -10, 20]$ m/s. The sideslip angle $\beta$ (in degrees) is _________ (round off to two decimal places).

Options:

  • (A) The governing parameter increases monotonically with Atmospheric Flight Mechanics
  • (B) The governing parameter remains invariant across the boundary
  • (C) The parameter satisfies the conservation equation for Atmospheric Flight Mechanics
  • (D) The parameter depends exclusively on viscous dissipation

Official Answer Key:

-5.60

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2020 Q.21)** **1. Intuitive Physical Setup & Given Parameters**: - Body-fixed velocity components: - Axial component: $u = 100\text{ m/s}$ - Lateral component: $v = -10\text{ m/s}$ - Normal component: $w = 20\text{ m/s}$ **2. Step-by-Step Flight Mechanics Derivation**: 1. **Total Airspeed ($V$)**: V = \sqrt{u^2 + v^2 + w^2} = \sqrt{100^2 + (-10)^2 + 20^2} = \sqrt{10000 + 100 + 400} = \sqrt{10500} \approx 102.4695\text{ m/s} 2. **Definition of Sideslip Angle ($\beta$)**: \sin\beta = \frac{v}{V} = \frac{-10}{102.4695} \approx -0.09759 3. **Evaluating $\beta$ in Degrees**: \beta = \arcsin(-0.09759) \approx -5.599^\circ \approx -5.60^\circ (Or with small-angle approximation $\beta \approx \frac{v}{u} = \frac{-10}{100} = -0.1\text{ rad} \approx -5.73^\circ$). Exact definition $\arcsin(v/V)$ evaluates precisely to **-5.60 degrees**, falling squarely in the official acceptable interval **[-5.62, -5.57]**. **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Exact Evaluation**: **-5.60 degrees** (Official Range: [-5.62, -5.57]). **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **-5.60** (Acceptable Range: **-5.62 to -5.57**) - **Key Formula**: $\beta = \arcsin\left(\frac{v}{\sqrt{u^2+v^2+w^2}}\right)$.
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