Turbulent Boundary Layer — Log-Law & Overlap Region
Von Kármán constant, viscous sublayer, buffer layer, and outer flow
5000Re_τ
0.41κ
30Log-law y⁺ start
—u⁺(y⁺=100)
—C_f × 10³
Law of the wall: u⁺ = y⁺ (viscous sublayer, y⁺ < 5); u⁺ = (1/κ)ln(y⁺) + B (log-law, 30 < y⁺ < 0.3Re_τ); κ ≈ 0.41, B ≈ 5.1. Buffer layer (5 < y⁺ < 30): transition, fit by van Driest formula or Spalding's law. Outer layer: velocity defect (U_e − u)/u_τ = -1/κ · ln(y/δ) + const (wake function). Skin friction: C_f = 2(u_τ/U_e)² ≈ 2[κ/ln(Re_τ)]². The overlap region (matching inner+outer) is the origin of the log law.