Turbulent Boundary Layer — Log-Law of the Wall

Prandtl's mixing-length theory predicts that near a wall, turbulent mean velocity follows a universal logarithmic profile: U⁺ = (1/κ)ln(y⁺) + B. The viscous sublayer (U⁺ = y⁺) transitions to the log-law at y⁺ ≈ 30. This "law of the wall" holds across Reynolds numbers from 10³ to 10⁸.

Viscous sublayer: U⁺ = y⁺ (y⁺ < 5)  |  Buffer: 5 < y⁺ < 30  |  Log-law: U⁺ = (1/κ)ln y⁺ + B (y⁺ > 30)
0.41
5.0
1000
0
u_τ/U_∞:  |  δ⁺ = Re_τ:  |  Cf: