🌊 Boundary Layer Analysis Calculator

Laminar and turbulent boundary layer properties over a flat plate with transition detection and profile visualization.

Flat Plate (y = 0) U_∞ (Flow) δ(x) Growth Laminar Transition Re_cr Turbulent Leading Edge (x=0)

Flat Plate Boundary layer Growth

Viscous friction near the plate slows down the incoming stream, establishing a boundary layer that expands downstream. The boundary transitions from laminar Blasius profiles to chaotic, highly diffusive turbulent profiles at the transition point.

• Laminar Growth: Characterized by smooth fluid layers, expanding proportionally to $\sqrt{x}$.
• Transition Zone: Triggers when the Reynolds number $Re_x$ exceeds the critical threshold ($Re_{cr}$).
• Turbulent Layer: Rapid mixing increases thickness ($\propto x^{4/5}$) and wall shear stress.

📝 Configuration

Flow Conditions

Fluid Properties

Plate Geometry

Transition

Laminar (Blasius):
$\delta = 5.0 x / \sqrt{Re_x}$
$\delta^* = 1.7208 x / \sqrt{Re_x}$
$C_f = 0.664 / \sqrt{Re_x}$

Turbulent (1/7 power law):
$\delta = 0.37 x / Re_x^{0.2}$
$\delta^* = \delta / 8$
$C_f = 0.0592 / Re_x^{0.2}$

Mixed drag:
$C_D = 0.074/Re_L^{0.2} - A/Re_L$

📊 Results & Visualization

Configure inputs and click Analyze Boundary Layer to view results.

📘 Calculation Methodology

Mathematical Model & Theory

Fluid flow over a flat plate creates a boundary layer due to shear stresses. The boundary layer thickness ($\delta$) and skin friction coefficient ($C_f$) are modeled by Blasius (laminar) and empirical turbulent equations:

$$\text{Laminar (Re}_x < Re_{cr}\text{): } \delta = \frac{5 x}{\sqrt{Re_x}}, \quad C_f = \frac{0.664}{\sqrt{Re_x}}$$ $$\text{Turbulent (Re}_x \ge Re_{cr}\text{): } \delta = \frac{0.37 x}{Re_x^{1/5}}, \quad C_f = \frac{0.0592}{\Re_x^{1/5}}$$

Assumptions

  • Steady, uniform, and incompressible free-stream flow conditions.
  • Two-dimensional flow over a perfectly flat, zero-thickness plate with no pressure gradients ($dp/dx = 0$).
  • Rigid body with smooth surface finish (no surface roughness corrections).
  • Transition is model-driven directly at $Re_{cr}$.

Academic References

  1. White, F. M.: Fluid Mechanics, McGraw-Hill, 8th Edition.
  2. Schlichting, H.: Boundary-Layer Theory, McGraw-Hill.
  3. Blasius, H.: Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. Phys.

Worked Engineering Example

Problem Statement:
Air ($\nu = 1.5 \times 10^{-5}$ m²/s) flows over a flat plate at 10 m/s. Find the boundary layer thickness at a distance of 0.2 m from the leading edge.

Step-by-step Solution:
1. Calculate local Reynolds number $Re_x$:
$$Re_x = \frac{U_\infty x}{\nu} = \frac{10 \times 0.2}{1.5 \times 10^{-5}} = 1.333 \times 10^5$$ 2. Since $Re_x < 5 \times 10^5$, flow is laminar:
$$\delta = \frac{5 x}{\sqrt{Re_x}} = \frac{5 \times 0.2}{\sqrt{1.333 \times 10^5}} = \frac{1.0}{365.15} = 0.00274 \text{ m} = 2.74 \text{ mm}$$
Final Result:
The boundary layer thickness is 2.74 mm.