🌊 Boundary Layer Analysis Calculator
Laminar and turbulent boundary layer properties over a flat plate with transition detection and profile visualization.
📝 Configuration
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:
$C_D = 0.074/Re_L^{0.2} - A/Re_L$
$\delta = 5.0 x / \sqrt{Re_x}$
$\delta^* = 1.7208 x / \sqrt{Re_x}$
$C_f = 0.664 / \sqrt{Re_x}$
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 the inputs and click Calculate to see results.
ℹ️ About Boundary Layers
A boundary layer forms when viscous fluid flows over a surface. The velocity transitions from zero at the wall (no-slip) to the free-stream velocity U∞.
Key concepts:
• Laminar BL: Smooth, ordered layers; lower skin friction; Blasius solution
• Turbulent BL: Chaotic mixing; higher skin friction; fuller velocity profile
• Transition: Occurs at Re_x ≈ 5×10⁵ for smooth plates
• δ (thickness): Distance where u = 0.99 U∞
• δ* (displacement): Mass flow deficit due to BL
• θ (momentum): Momentum deficit; key for drag calculation
A boundary layer forms when viscous fluid flows over a surface. The velocity transitions from zero at the wall (no-slip) to the free-stream velocity U∞.
Key concepts:
• Laminar BL: Smooth, ordered layers; lower skin friction; Blasius solution
• Turbulent BL: Chaotic mixing; higher skin friction; fuller velocity profile
• Transition: Occurs at Re_x ≈ 5×10⁵ for smooth plates
• δ (thickness): Distance where u = 0.99 U∞
• δ* (displacement): Mass flow deficit due to BL
• θ (momentum): Momentum deficit; key for drag calculation
📘 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:
$$\text{Laminar (Re}_x < 5 \times 10^5\text{): } \delta = \frac{5 x}{\sqrt{Re_x}}, \quad C_f = \frac{0.664}{\sqrt{Re_x}}$$
$$\text{Turbulent (Re}_x \ge 5 \times 10^5\text{): } \delta = \frac{0.38 x}{Re_x^{1/5}}, \quad C_f = \frac{0.059}{Re_x^{1/5}}$$
Worked Engineering Example
Problem Statement:
Air ($ u = 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 $Re_x$:
$$Re_x = \frac{U x}{ u} = \frac{10 \times 0.2}{1.5 \times 10^{-5}} = 1.333 \times 10^5$$ 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.
Air ($ u = 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 $Re_x$:
$$Re_x = \frac{U x}{ u} = \frac{10 \times 0.2}{1.5 \times 10^{-5}} = 1.333 \times 10^5$$ 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.