Question

It is desired to estimate the aerodynamic drag $D$ on a car traveling at a speed of 30 m/s. A one-third scale model of the car is tested in a wind-tunnel following the principles of dynamic similarity. The drag on the scaled model is measured to be $D_m$. The ratio $D/D_m$ is .......... (rounded off to 1 decimal place).

Hint:

For dynamic similarity, drag scales with frontal area. If length ratio is $n$, drag ratio is $n^2$.

Answer 1

Step 1: Drag force formula.
For dynamically similar flows (same $Re$, $M$): \[ D \propto \tfrac{1}{2} \rho V^2 A \] where $A =$ reference area (frontal area). Step 2: Scaling law.
For geometrically similar bodies: \[ A \propto L^2 \] Here, full-scale car : model scale = $1 : 1/3$ → $L/L_m = 3$. \[ \frac{A}{A_m} = \left( \frac{L}{L_m} \right)^2 = 3^2 = 9 \] Step 3: Velocity scaling.
Given dynamic similarity, velocity ratio = 1 (since both full-scale and model at $V=30$ m/s and fluid is same). Step 4: Drag ratio.
\[ \frac{D}{D_m} = \frac{A}{A_m} \left(\frac{V}{V_m}\right)^2 = 9 \times 1^2 = 9 \] \[ \boxed{9.0} \]