Question

For a two-dimensional incompressible flow over a flat plate, the laminar boundary layer thickness at a distance \( x \) from the leading edge is \( \delta \). If \( Re_x \) is the Reynolds number defined based on length scale \( x \), \[ \frac{\delta}{x} \propto \_\_\_\_\_\_. \]

• A. \( Re_x^{-1/2} \)
• B. \( Re_x^{-1} \)
• C. \( Re_x^{-3/2} \)
• D. \( Re_x^{-2} \)

Hint:

For laminar boundary layers, the boundary layer thickness is proportional to \( x \cdot Re_x^{-1/2} \), where \( Re_x \) is the Reynolds number based on the distance from the leading edge.

Answer 1

The Correct Option is A

For a laminar boundary layer in a two-dimensional incompressible flow, the boundary layer thickness \( \delta \) is related to the Reynolds number \( Re_x \) by the following empirical relationship: \[ \delta \propto x \cdot Re_x^{-1/2} \] Thus, the ratio \( \frac{\delta}{x} \) is proportional to \( Re_x^{-1/2} \): \[ \frac{\delta}{x} \propto Re_x^{-1/2} \] Step 1: Reynolds number definition
The Reynolds number \( Re_x \) at a distance \( x \) from the leading edge is defined as: \[ Re_x = \frac{\rho u x}{\mu} \] where \( \rho \) is the fluid density, \( u \) is the flow velocity, and \( \mu \) is the dynamic viscosity. Step 2: Relation between boundary layer thickness and Reynolds number
From the theory of boundary layers in laminar flow, we know that the thickness of the boundary layer is inversely proportional to the square root of the Reynolds number, as shown above. Conclusion:
Therefore, the ratio \( \frac{\delta}{x} \) is proportional to \( Re_x^{-1/2} \).