In external flow over a smooth flat plate, beyond what minimum local Reynolds number (Re_x) does the boundary layer typically transition to turbulent, under standard conditions?

Introduction:
Reynolds number is a dimensionless ratio that compares inertial to viscous forces in a fluid flow. For external flow over a flat plate, it is common to characterize transition from laminar to turbulent boundary layers using a critical local Reynolds number, Re_x, based on distance from the leading edge. Knowing the approximate threshold for turbulence helps in estimating skin friction, heat transfer, and mass transfer rates in engineering design.

Given Data / Assumptions:

• Smooth flat plate with uniform approach velocity.
• Local Reynolds number definition: Re_x = (rho * U * x) / mu.
• Standard, low free-stream turbulence, no strong pressure gradient.

Concept / Approach:
Experiments show that on a smooth, nominally two-dimensional flat plate with low free-stream turbulence, transition commonly begins when Re_x is of the order of 3 × 10^5 to 10^6. A widely used design value is about 5 × 10^5. Above this threshold, small disturbances amplify, causing the laminar boundary layer to transition to turbulent, which increases skin friction and heat transfer coefficients but also enhances mixing near the wall.

Step-by-Step Solution:

Verification / Alternative check:
Flat-plate correlations for friction factor and Nusselt number often piecewise combine laminar and turbulent formulas with transition near Re_x ≈ 5 × 10^5. Wind tunnel and water channel measurements support this conventional benchmark under low-disturbance conditions.

Why Other Options Are Wrong:

• 5 × 10^-5: Far too small; transition requires very large Re_x, not a tiny number.
• 10^6: Possible in some conditions, but 5 × 10^5 is the standard minimal value typically cited.
• 500: Characteristic of creeping or laminar regimes, not transition for external plates.

Common Pitfalls:
Assuming a single universal critical value regardless of surface finish, pressure gradient, or free-stream turbulence. Transition is sensitive; use 5 × 10^5 as a design reference but validate for specific environments.

Final Answer:
5 × 10^5