In model–prototype testing of hydraulic machines, dynamic similarity is a stronger requirement than geometric similarity. Under which condition do we say that model and prototype have dynamic similarity?

Introduction:
Similarity laws enable prediction of prototype performance from model tests. Three levels exist: geometric (shape), kinematic (motion), and dynamic (forces). Dynamic similarity requires that the ratios of all types of forces match between model and prototype so that the dimensionless parameters governing the flow are equal.

Given Data / Assumptions:

• Hydraulic machine or flow system with inertia, viscous, pressure, gravity, and surface tension forces potentially relevant.
• Objective: accurate force and performance prediction from model to prototype.

Concept / Approach:
Dynamic similarity is achieved when the corresponding dimensionless numbers (e.g., Reynolds, Froude, Weber, Euler numbers) are equal for model and prototype. The options presented do not explicitly state this condition. Identical velocities, equal size/shape, or merely geometric similarity are insufficient; they do not ensure matched force ratios.

Step-by-Step Solution:
1) Identify relevant forces → derive dimensionless groups.2) Impose equality of these groups between model and prototype.3) Recognize that options (a), (b), (c) fail to guarantee force ratio equality.4) Therefore, among given choices, the correct selection is “none of the above”.

Verification / Alternative check:
Example: ship resistance requires Froude similarity; spillway aeration may require Weber too; pumps/turbines often target Reynolds and specific speed similarity.

Why Other Options Are Wrong:
Identical velocities: does not scale forces properly across sizes.

Equal in size and shape: that is identity, not similarity.

Identical shape but different size: only geometric similarity.

Common Pitfalls:
Assuming geometric similarity implies dynamic similarity; ignoring which dimensionless numbers dominate for a given problem.

Final Answer:
none of the above