Turbulent Boundary Layer & Log-Law of the Wall

Universal velocity profile structure in wall-bounded turbulent flow

Law of the Wall

The time-averaged velocity profile near a wall collapses to a universal form in wall units:

u⁺ = y⁺ (viscous sublayer, y⁺ < 5)
u⁺ = (1/κ)ln(y⁺) + B (log layer, y⁺ > 30)

Where u⁺ = u/u*, y⁺ = y·u*/ν, κ ≈ 0.41 (von Kármán), and B ≈ 5.0 (smooth wall).

u* = √(τ_w/ρ)

The log layer spans the "overlap region" between viscous and outer scales — it is one of the most robust results in turbulence. Velocity fluctuations in this region scale as u'rms ∝ u*.

Viscous sublayer (u⁺=y⁺)
Log-law fit
Actual profile (DNS fit)
Turbulent eddies