Problem Statement
A flat plate of 2m width and 4m length is kept parallel to air flowing at 5m/s. Determine the length of plate over which the boundary layer is laminar and shear stress at the location where boundary layer ceases to be laminar. Take ρ of air = 1.208 kg/m³ and υ of air = 1.47×10⁻⁵ m²/s.
Given Data
Solution Approach
To solve this problem, we’ll first calculate the Reynolds number at the end of the plate to determine if the flow transitions from laminar to turbulent. Then, we’ll find the critical distance where transition occurs, and calculate the boundary layer thickness and shear stress at that location.
Calculations
Identifying Flow Regime
Step 1: Calculate the Reynolds number at the end of the plate (x = L).
Since Re > 5×10⁵ (the critical Reynolds number for a flat plate), the flow transitions from laminar to turbulent somewhere along the plate.
Step 2: Calculate the distance from the leading edge where transition occurs.
Therefore, the boundary layer is laminar up to 1.47 m from the leading edge.
Step 3: Calculate the boundary layer thickness at the transition point.
The boundary layer thickness at the transition point is 10.39 mm.
Step 4: Calculate the skin friction coefficient at the transition point.
Step 5: Calculate the shear stress at the transition point.
Laminar boundary layer length: 1.47 m
Shear stress at transition point: 0.01418 N/m²
Detailed Explanation
Boundary Layer Transition Mechanism
When fluid flows over a flat plate, a boundary layer develops where the velocity gradually increases from zero at the wall (due to the no-slip condition) to the free stream velocity. The boundary layer starts as laminar near the leading edge and transitions to turbulent as the Reynolds number increases along the plate.
Significance of Critical Reynolds Number
The critical Reynolds number Rex = 5×10⁵ marks the typical transition point from laminar to turbulent flow for a flat plate with a smooth leading edge in a low turbulence free stream. This transition significantly affects:
- Skin friction (increases in turbulent region)
- Heat transfer rates (higher in turbulent region)
- Boundary layer thickness (grows more rapidly in turbulent region)
Physical Interpretation of Results
In this problem, we found that:
- The first 1.47 m of the 4 m plate experiences laminar flow
- The remaining 2.53 m experiences turbulent flow
- At the transition point, the boundary layer thickness is 10.39 mm
- The shear stress at transition is 0.01418 N/m²
Laminar Boundary Layer Equations
For a laminar boundary layer on a flat plate, we used the Blasius solution approximations:
- Boundary layer thickness: δ = 5x/√(Rex)
- Local skin friction coefficient: Cf = 0.664/√(Rex)
- Wall shear stress: τ = (1/2)CfρU²
Engineering Applications
Understanding boundary layer behavior and transition is crucial for:
- Aerodynamic design of aircraft wings and fuselages
- Design of ship hulls to minimize drag
- Cooling of electronic components
- Drag reduction strategies in various fluid flow applications
- Heat exchanger design and optimization
Practical Considerations
In real-world applications, several factors can affect boundary layer transition:
- Surface roughness (can trigger earlier transition)
- Pressure gradient (adverse gradients promote earlier transition)
- Free stream turbulence intensity
- Surface temperature (in compressible flows)
- Surface curvature
This analysis provides valuable insights into the fundamental behavior of boundary layers on flat plates and serves as a foundation for more complex flow scenarios encountered in engineering practice.
