Dynamic similarity for channels dominated by inertia and gravity When inertia and gravity are the primary forces and friction has a minor role, channel design and model–prototype comparison should be based on which dimensionless number?
-
AFroude number
-
BReynolds number
-
CWeber number
-
DMach number
-
EEuler number
Answer
Correct Answer: Froude number
Explanation
Introduction / Context:Choice of the correct similarity parameter is crucial in hydraulic modeling. Free-surface flows such as rivers, spillways, and ship hydrodynamics are governed mainly by inertia and gravity, so the corresponding dimensionless group must capture that balance.
Given Data / Assumptions:
- Free-surface flow conditions; wave and surface effects matter.
- Inertia and gravity dominate; viscous effects are relatively small.
- Geometric similarity between model and prototype is assumed.
Concept / Approach:
The Froude number Fr = V / sqrt(g * L) compares inertial forces to gravitational forces. Dynamic similarity requires Fr_model = Fr_prototype, ensuring homologous behavior of waves, hydraulic jumps, and critical transitions.
Step-by-Step Solution:
Identify dominant forces: inertia and gravity ⇒ use Froude scaling.Set Fr_model = Fr_prototype during design.Use scaled velocity: V_model = V_proto * sqrt(L_model / L_proto).Verification / Alternative check:
Reynolds number governs inertia–viscosity; Weber number inertia–surface tension; Mach number compressibility; Euler number pressure–inertia. None of these match inertia–gravity dominance like Froude.
Why Other Options Are Wrong:
They address different force balances and cannot alone preserve wave-related phenomena in open channels.
Common Pitfalls:
Attempting to match both Reynolds and Froude simultaneously is often impossible; prioritize Froude for free-surface flows.
Final Answer:
Froude number