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 ________.
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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.
Step 1: List all variables and their dimensions.
$V \to LT^{-1}$, $\rho \to ML^{-3}$, $\mu \to ML^{-1}T^{-1}$, $c \to L$, and $\alpha$ is already dimensionless (a ratio/angle).
Step 2: Form the dimensionless group from the dimensional variables using the exponent method.
Look for a combination $\rho^a V^b c^d \mu^{-1}$ that is dimensionless. Matching mass: $a = 1$. Matching time: from $\mu^{-1}$ giving $+1$ power of $T$, and $V^b$ giving $-b$, balance gives $b=1$. Matching length: $-3a + b + d + 1 = 0 \Rightarrow -3+1+d+1=0 \Rightarrow d=1$. So the group is
\[ \Pi_1 = \frac{\rho V c}{\mu} = Re \]
This confirms, from first principles rather than by counting, that the 4 dimensional quantities $V, \rho, \mu, c$ collapse into exactly one dimensionless number.
Step 3: Bring back the parameter that was already dimensionless.
$\alpha$ never needed reduction, so it stands on its own as a second independent parameter, $\Pi_2 = \alpha$. The full aerodynamic response (lift, drag, pressure distribution) at low Reynolds number depends only on the pair $(\Pi_1, \Pi_2) = (Re, \alpha)$.
Step 4: Size a full factorial test grid over the two parameters.
A full factorial design that sweeps 10 equally spaced levels of $Re$ against 10 equally spaced levels of $\alpha$ needs one run for every combination:
\[ N_{min} = 10 \times 10 = 100 \]
Step 5: Sanity check against the wrong options.
If someone forgot to combine $V, \rho, \mu, c$ into $Re$ first, they would think there are 5 raw parameters and plan $10^5 = 1,00,000$ runs (option D), or 4 raw parameters giving $10^4=10,000$ runs (option C); both waste effort because many of those raw combinations repeat the same $Re$. Treating everything as one parameter (option A, 10 runs) under-samples, since $\alpha$ genuinely changes the flow independently of $Re$ and must be swept separately.
Only the reduced pair $(Re, \alpha)$ gives the true minimum.
\[ \boxed{N_{min} = 100} \]
