Question Statement:
An experimental study is planned to map out the low-Reynolds number
incompressible steady two-dimensional aerodynamic characteristics of a
promising novel airfoil. The operational parameters of the problem are the speed,
density and viscosity of the freestream, the chord of the airfoil and its angle of
attack. If the objective is to achieve this with the minimum number of test runs
Nmin while taking 10 equally-spaced test values of each independent parameter of
the problem in a suitable range, then Nmin is ________.
Options:
-
(A) 10
-
(B) 100
-
(C) 10,000
-
(D) 1,00,000
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.43)**
**1. Intuitive Physical Setup & Given Parameters**:
We want to map the 2D aerodynamic characteristics of an airfoil in low-Reynolds number incompressible steady flow:
- Independent operational parameters listed: speed ($V$), density ($\rho$), viscosity ($\mu$), chord ($c$), angle of attack ($\alpha$).
- Test protocol: taking 10 equally spaced test values of each independent dimensionless parameter over a suitable range.
- We need to determine the **minimum number of test runs** $N_{\text{min}}$ by applying dimensional analysis.
**2. Step-by-Step Dimensional Analysis Derivation**:
1. **Buckingham $\pi$-Theorem Application**:
- The aerodynamic coefficients of interest are dimensionless lift, drag, and moment:
$C_L, C_D, C_m = f(V, \rho, \mu, c, \alpha)$
- For steady 2D incompressible flow, dimensional analysis reduces the 5 parameters ($V, \rho, \mu, c, \alpha$) having 3 fundamental dimensions (Mass, Length, Time) to:
$\text{Number of independent non-dimensional parameters} = 5 - 3 = 2$
- These two independent non-dimensional parameters are:
1. **Reynolds number**: $Re = \frac{\rho V c}{\mu}$
2. **Angle of attack**: $\alpha$ (which is already non-dimensional)
2. **Minimum Number of Test Runs ($N_{\text{min}}$)**:
- To fully map the operational domain over 10 values of each independent governing non-dimensional group:
- 10 distinct values of Reynolds number $Re$
- 10 distinct values of angle of attack $\alpha$
- The total number of required grid points in parameter space is:
$N_{\text{min}} = 10 \times 10 = 100\text{ test runs}$
- Without dimensional analysis, varying all 5 individual dimensional parameters would have required $10^5 = 100,000$ runs, illustrating the immense power of aerodynamic scaling.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 10**: ❌ **INCORRECT**. Only covers 1 parameter.
- **(B) 100**: ✅ **CORRECT**. Exactly $10^2 = 100$ runs across the two independent dimensionless parameters ($Re$ and $\alpha$).
- **(C) 10,000**: ❌ **INCORRECT**. Corresponds to 4 parameters ($10^4$).
- **(D) 1,00,000**: ❌ **INCORRECT**. Brute-force dimensional testing of all 5 parameters ($10^5$), completely ignoring dynamic similarity and Buckingham $\pi$-theorem.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: In 2D low-speed aerodynamics, all dimensional parameters collapse into exactly two dimensionless groups: $Re$ and $\alpha$. Hence, 10 samples each require $10 \times 10 = 100$ tests.