Question

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 \(N_{min}\) while taking 10 equally-spaced test values of each independent parameter of the problem in a suitable range, then \(N_{min}\) is ________.

• A. 10
• B. 100
• C. 10,000
• D. 1,00,000

Hint:

Use the Buckingham Pi theorem to reduce the 4 dimensional variables (speed, density, viscosity, chord) to the Reynolds number; the angle of attack is already dimensionless. Two independent parameters mean \(10^2\) runs.

Answer 1

The Correct Option is B

Step 1: Identify the physical variables involved.
The experiment studies the low speed aerodynamic behaviour of an airfoil section. The variables that control the result are the freestream speed \(V\), the freestream density \(\rho\), the freestream viscosity \(\mu\), the airfoil chord \(c\), and the angle of attack \(\alpha\). That gives 5 operational parameters in total.

Step 2: Separate dimensional and non-dimensional variables.
Four of these variables, \(V\), \(\rho\), \(\mu\), and \(c\), carry physical dimensions built from mass, length and time (M, L, T). The angle of attack \(\alpha\) is already a pure number (it is a ratio, measured in radians or degrees), so it does not need any further reduction.

Step 3: Apply the Buckingham Pi theorem to the dimensional variables.
The Buckingham Pi theorem states that the number of independent dimensionless groups equals the number of dimensional variables minus the number of fundamental dimensions they use. Here there are 4 dimensional variables (\(V\), \(\rho\), \(\mu\), \(c\)) and 3 fundamental dimensions (M, L, T), so the count of independent dimensionless groups is
\[ 4 - 3 = 1 \]
This single group is the Reynolds number, \(Re = \dfrac{\rho V c}{\mu}\). So instead of separately varying \(V\), \(\rho\), \(\mu\) and \(c\), the entire aerodynamic behaviour at a given angle of attack depends only on \(Re\).

Step 4: Count the true number of independent governing parameters.
Adding back the already-dimensionless angle of attack \(\alpha\), the flow field is fully controlled by just 2 independent non-dimensional parameters: \(Re\) and \(\alpha\).

Step 5: Build the minimum test matrix.
The plan is to take 10 equally spaced test values of each independent parameter. Since there are only 2 truly independent parameters (\(Re\) and \(\alpha\)), and every combination of the two must be tested to map out the characteristics, the minimum number of runs is
\[ N_{min} = 10 \times 10 = 10^2 = 100 \]

Step 6: Why the other options are wrong.
Option (A), 10, would be correct only if there were a single independent parameter, but the angle of attack cannot be folded into the Reynolds number, so one test sweep is not enough.
Options (C) 10,000 (\(=10^4\)) and (D) 1,00,000 (\(=10^5\)) come from testing the 4 or 5 raw dimensional variables independently without first combining \(V\), \(\rho\), \(\mu\) and \(c\) into the Reynolds number. That approach wastes runs, because many different combinations of \(V\), \(\rho\), \(\mu\), \(c\) give the same \(Re\) and therefore the same flow behaviour. Dimensional analysis is exactly what removes this redundancy.

Final Answer:
The minimum number of test runs is 100. \[ \boxed{N_{min}=100} \]