Question

Consider model testing where \( \lambda \) is the prototype-to-model length scale ratio. Let \( v_p \) and \( v_m \) denote the corresponding fluid kinematic viscosities. If Froude and Reynolds similarities are maintained between the prototype and model, then which one of the following is correct?

• A. \( v_m = \lambda^{-3/2} v_p \)
• B. \( v_m = \lambda^{3/2} v_p \)
• C. \( v_m = \lambda^{2/3} v_p \)
• D. \( v_m = \lambda^{-2/3} v_p \)

Hint:

1. For model testing, ensure both Froude and Reynolds similarities are satisfied for accurate scaling.
2. Use \( \lambda \) to scale physical properties, where \( \lambda \) is the length scale ratio.
3. Apply the scaling relations systematically: \( v_m = v_p \lambda^{-1/2} \) and \( \nu_m = \nu_p \lambda^{-3/2} \).

Answer 1

The Correct Option is A

Step 1: Understand Froude and Reynolds similarities. - Froude similarity ensures the ratio of inertial to gravitational forces is maintained, given by \( \text{Fr} = \frac{v}{\sqrt{gL}} \), where \( v \) is velocity, \( g \) is gravitational acceleration, and \( L \) is characteristic length. - Reynolds similarity ensures the ratio of inertial to viscous forces is maintained, given by \( \text{Re} = \frac{vL}{\nu} \), where \( \nu \) is the kinematic viscosity. Step 2: Derive velocity scale using Froude similarity. From Froude similarity: \[ \frac{v_p}{\sqrt{gL_p}} = \frac{v_m}{\sqrt{gL_m}} \] Simplify to find the velocity ratio: \[ \frac{v_p}{v_m} = \sqrt{\frac{L_p}{L_m}} = \sqrt{\lambda} \] Thus, \[ v_m = \frac{v_p}{\sqrt{\lambda}} = v_p \lambda^{-1/2}. \] Step 3: Relate kinematic viscosities using Reynolds similarity. From Reynolds similarity: \[ \frac{v_p L_p}{\nu_p} = \frac{v_m L_m}{\nu_m} \] Substitute \( L_p = \lambda L_m \), \( v_m = v_p \lambda^{-1/2} \): \[ \frac{v_p (\lambda L_m)}{\nu_p} = \frac{(v_p \lambda^{-1/2})L_m}{\nu_m}. \] Simplify to find: \[ \nu_m = \nu_p \lambda^{-3/2}. \] Conclusion: The kinematic viscosity ratio between the model and prototype is given by \( \nu_m = \lambda^{-3/2} \nu_p \), which matches option \( \mathbf{(A)} \).