Questions: Dimensional Analysis and Dynamic Similarity

Question 1

A fluid dynamics experiment involves 5 variables (drag force, fluid velocity, object size, fluid density, and fluid viscosity) and 3 fundamental dimensions (mass, length, time). How many independent dimensionless Pi groups does the Buckingham Pi theorem predict?

• A. 5
• B. 3
• C. 2
• D. 8

Answer: (C)

The Buckingham Pi theorem gives n - k = 5 - 3 = 2 dimensionless groups. In this classic problem, the two groups are the drag coefficient C_D = F/(½ρV²L²) and the Reynolds number Re = ρVL/μ. The theorem tells you the count and that the relationship F = f(ρ, V, L, μ) must take the form C_D = f(Re), but experiment or theory is still needed to find the function f.

Question 2

If a scale model matches the Reynolds number of its full-scale prototype, it is expected to be dynamically similar in most relevant respects.

• T. True
• F. False

Answer: False

Dynamic similarity requires ALL relevant Pi groups to match simultaneously. For a ship hull, for instance, both the Reynolds number (viscous effects) and Froude number (gravity/wave effects) matter. Matching Re requires a certain velocity-scale relationship, while matching Fr requires a different one — both conditions cannot be satisfied simultaneously with water as the fluid. In practice, engineers choose which Pi group dominates and accept partial similarity.

Question 3

Dimensional analysis cannot determine the numerical coefficient in a relationship — for example, it can show drag force ∝ ρV²L² but not the exact constant. Why is dimensional analysis still valuable despite this limitation?

Model answer: Dimensional analysis reveals the functional form of relationships and reduces the number of independent variables from n down to n-k dimensionless groups. This dramatically reduces the number of experiments required and ensures results are universal — a curve of C_D vs. Re applies to any sphere in any fluid, not just the specific conditions tested. It also guides scaling: if you change one variable, dimensional analysis tells you how all others must change to preserve similarity.

Without dimensional analysis, testing drag on a sphere in water might yield a table of F vs. V for one sphere size in one fluid — data that cannot be generalized. Expressing the same data as C_D vs. Re collapses all sphere sizes, fluid types, and velocities onto a single curve. The numerical coefficient is determined once by experiment and then applies universally. This is why dimensional analysis is called the 'theory of experiments.'