Dimensional analysis and similarity

Boundary layers separate when near-wall momentum runs out

Viscosity matters in a thin region next to a wall even when the outer flow is fast. An adverse pressure gradient asks near-wall fluid to move toward higher pressure; the slow layer can reverse and detach.

Balance terms in the boundary layer

Along a flat wall, streamwise inertia scales as U²/L and wall-normal viscous diffusion as νU/δ². Balancing gives δ/L ~ Re^{-1/2}. Separation occurs when wall shear τ_w = μ(∂u/∂y)_wall falls to zero and then changes sign; the no-slip condition remains at the wall while the near-wall flow reverses.

At U = 5 m/s, L = 0.2 m in water, Re = 10⁶ and δ estimate is 0.2 mm. A pressure rise downstream can thicken this layer and cause separation even at the same Re.

A thin layer can control a large force

Drag, mixing, and pressure recovery depend on boundary-layer state. Surface roughness may trigger transition; the simple δ scaling does not predict turbulent thickness or exact separation location.

Work it: Sketch velocity profiles before and after separation. For U = 2 m/s, L = 0.1 m, ν = 10⁻⁶ m²/s, estimate δ and identify one measurement that would detect separation.