Question

For dynamic similarity between model and prototype involving viscous and gravitational forces, which dimensionless numbers must be simultaneously matched?

• A. Reynolds and Mach numbers
• B. Reynolds and Froude numbers
• C. Euler and Mach numbers
• D. Weber and Strouhal numbers

Hint:

Whenever gravitational forces are critical in fluid dynamics (e.g., wave action or free surfaces), always pair the Froude number with the Reynolds number.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the specific pair of dimensionless numbers that must be matched to ensure dynamic similarity between a model and a prototype when fluid viscosity and gravity are the dominant forces.

Step 2: Key Formula or Approach:
To achieve complete dynamic similarity, the ratios of dominant forces must be identical between the model and the prototype.
- Viscous forces are characterized by the Reynolds number ($Re$):
\[ Re = \frac{\rho v L}{\mu} \]
- Gravitational forces are characterized by the Froude number ($Fr$):
\[ Fr = \frac{v}{\sqrt{g L}} \]

Step 3: Detailed Explanation:

• In physical systems where gravity and viscosity are both critical (such as liquid agitation in unbaffled vessels with free vortex formation, or open channel flow), both forces affect fluid movement simultaneously.

• To scale-up or model these systems accurately, both the Reynolds number (inertial-to-viscous force ratio) and the Froude number (inertial-to-gravitational force ratio) must be matched between the model and the prototype.

• Mach number: Important in gas dynamics involving compressibility effects.

• Weber number: Important in surface tension-dominated flows.

• Euler number: Compares pressure force to inertial force.

Step 4: Final Answer:
Therefore, the Reynolds and Froude numbers must be simultaneously matched.