Boundary Layer Theory

1. Boundary Layer Concept

It is the thin layer of fluid adjacent to a solid surface where viscous effects are significant.

Outside this layer, the flow can be considered inviscid (ideal).

2. Boundary Layer Thicknesses

• Displacement Thickness (δ^*):

  δ^* = ∫₀^∞ (1 − u/U) dy

  Represents reduction in flow rate due to boundary layer.
• Momentum Thickness (θ):

  θ = ∫₀^∞ (u/U)(1 − u/U) dy

  Accounts for loss of momentum due to boundary layer.
• Energy Thickness (δ_e):

  δ_e = ∫₀^∞ (u/U)(1 − (u/U)²) dy

3. Laminar Boundary Layer

Occurs at low Reynolds numbers (Re < 5 × 10⁵ for flat plates).

δ = 5.0 × (x / √Re_x)

Where:
x = distance from leading edge, Re_x = (ρUx)/μ

4. Turbulent Boundary Layer

Occurs at higher Reynolds numbers (Re > 5 × 10⁵).

δ = 0.37 × x / Re_x^1/5

Flow is chaotic but has higher momentum, which resists separation.

5. Flow Separation

Occurs when the boundary layer detaches from the surface due to adverse pressure gradient.

Separation condition: Wall shear stress → 0 and flow reversal occurs.

Flow separation leads to drag and loss of lift in aerodynamic bodies.

τ_w = μ (∂u/∂y)|_wall → 0