Model–Prototype Similarity — Kinematic Similarity Criterion Kinematic similarity between a scale model and its prototype exists when (in addition to geometric similarity) the ratios of velocities and accelerations at corresponding points are equal at corresponding times. Which statement best captures this condition?

Introduction:
Fluid model testing relies on similarity principles to extrapolate results from small-scale models to full-scale prototypes. Kinematic similarity ensures motion patterns—streamlines, velocity directions, and unsteady behavior—are dynamically consistent after applying the scale factors. This question clarifies the precise condition for kinematic similarity.

Given Data / Assumptions:

• Incompressible or weakly compressible flow within the relevant regime.
• Model and prototype are geometrically similar (same shape; scaled size).
• We compare fields at corresponding non-dimensional positions and times.

Concept / Approach:
Kinematic similarity demands that the velocity and acceleration vectors at corresponding points are proportional by constant scale factors. Equivalently, dimensionless velocity and acceleration fields must match, implying equal Reynolds- or Froude-based kinematics for the chosen dominant physics, once geometry is matched. It is weaker than dynamic similarity (which equates force ratios) and stronger than geometric similarity alone.

Step-by-Step Solution:
Require geometric similarity so that corresponding points are well-defined.Impose velocity ratio V_m / V_p = constant at corresponding points (including direction consistency).Impose acceleration ratio a_m / a_p = constant at corresponding points and times.If the governing dimensionless groups controlling kinematics (e.g., Reynolds for viscous dominance, Froude for gravity waves) are matched, these ratio conditions are satisfied.

Verification / Alternative check:
In open-channel flows dominated by gravity, matching Froude number preserves free-surface kinematics; in viscous internal flows, matching Reynolds number preserves the velocity profile shapes.

Why Other Options Are Wrong:

• Equal size and shape: geometric similarity only, insufficient.
• Identical absolute velocities: unnecessary and generally impossible under scaling.
• Equal resultant forces: dynamic similarity, a stronger requirement than kinematic.
• Only identical time scales: ignores geometric and velocity/acceleration requirements.

Common Pitfalls:
Confusing kinematic with dynamic similarity or believing absolute equality (not scaled equality) is required.

Final Answer:
Ratios of velocities and accelerations at corresponding points are equal (with geometric similarity satisfied).