Turbulent boundary layer structure: three regions — viscous sublayer (y⁺ < 5), buffer layer (5–30), log layer (y⁺ > 30).
u⁺ = y⁺ (viscous) | u⁺ = (1/κ) ln y⁺ + B (log layer)
where y⁺ = yu*/ν and u⁺ = u/u*. Von Kármán constant κ ≈ 0.41 is
universal. Roughness shifts the log law downward by ΔB = (1/κ) ln(1 + k⁺·0.3). The upper panel shows the physical boundary layer profile; right panel shows the classic semi-log plot.