Turbulent Boundary Layer: Law of the Wall
u⁺=y⁺ (viscous sublayer) · u⁺=(1/κ)ln(y⁺)+B (log layer) · wake region
Law of the Wall: Near a wall, the turbulent boundary layer organizes into distinct regions based on the wall unit y⁺ = y·u_τ/ν, where u_τ = √(τ_w/ρ) is the friction velocity.
Regions: (1) Viscous sublayer y⁺ < 5: u⁺ = y⁺ (laminar, viscous shear dominates). (2) Buffer layer 5 < y⁺ < 30: transition, both viscous and turbulent stresses important. (3) Log-law region y⁺ > 30: u⁺ = (1/κ)ln(y⁺) + B with κ ≈ 0.41 (Kármán) and B ≈ 5.1. (4) Wake region: outer layer, Coles' wake function Π·sin²(πy/2δ).
Universality: The log-law holds for all turbulent wall flows (pipes, channels, flat plates, atmospheric boundary layer) — a remarkable universal structure. Millikan's overlap argument derives it from inner/outer scaling similarity without any turbulence model.
Right panel: Reynolds stress −⟨u'v'⟩⁺ profile and viscous stress ν·du/dy normalized by u_τ².