Boundary Layer Theory

Why it matters

Boundary layers explain why viscosity affects drag and separation even when most of a flow can be approximated as inviscid. The fluid adjacent to a stationary wall satisfies no slip, while velocity approaches the outer-flow value away from it.

Key ideas

The boundary layer grows downstream as viscous effects spread. It may be laminar, transitional or turbulent; transition depends on disturbances, roughness and pressure gradient, so one critical Reynolds number is not universal. An adverse pressure gradient can lead to separation. Turbulent mixing often delays separation but generally increases skin friction.

Definitions and formulas

For a steady two-dimensional incompressible boundary layer, let U_e be outer velocity and y the wall-normal coordinate.

• Local Reynolds number: Re_x = U_e x/ν, with ν in m²/s.
• Wall shear: τ_w = μ(∂u/∂y)_wall.
• Skin-friction coefficient: C_f,x = τ_w/(ρU_e²/2).
• Displacement thickness: δ* = ∫₀∞(1 − u/U_e)dy.
• Momentum thickness: θ = ∫₀∞(u/U_e)(1 − u/U_e)dy.
• A commonly reported boundary-layer edge is where u reaches about 99% of U_e; this thickness is distinct from δ* and θ.

Worked example: a prescribed velocity profile

For calculation practice, prescribe u/U_e = y/δ for 0 ≤ y ≤ δ and u = U_e above δ. This linear profile is an approximation, not the exact flat-plate solution. Let δ = 0.002 m, U_e = 1 m/s, μ = 0.001 Pa·s and ρ = 1000 kg/m³.

The wall gradient is U_e/δ = 500 s⁻¹, so τ_w = 0.001 × 500 = 0.5 Pa. C_f,x = 0.5/(1000 × 1²/2) = 0.001. Integrating the prescribed profile gives δ* = δ/2 = 1 mm and θ = δ/6 = 0.333 mm.

Answer: 0.5 Pa wall shear, 0.001 local skin-friction coefficient, 1 mm displacement thickness and 0.333 mm momentum thickness for this assumed profile. A local wall-shear value alone does not determine total plate drag; integrate shear over the wetted area.

Common mistakes

• Using the outer velocity as the wall velocity.
• Confusing local and average drag coefficients.
• Applying a zero-pressure-gradient flat-plate correlation to separated flow.
• Assuming no slip means zero velocity gradient or zero wall shear.

Quick check

1. What is velocity at a stationary impermeable wall under no slip? Zero relative to the wall.
2. Does zero wall velocity imply zero shear? No; shear depends on the gradient.
3. What signals incipient separation in the classical steady two-dimensional model? Wall shear approaches zero and near-wall reverse flow may develop downstream.

Reference

NASA: Boundary layer.