Turbulent Pipe Flow — Reynolds Stress Profile

Von Kármán's log-law of the wall: U⁺ = (1/κ)ln(y⁺) + B. Explore viscous sublayer, buffer layer, and log-law region. Reynolds stresses from the k-ε mixing length model.

Mean Velocity Profile

U⁺ (log-law)
Viscous sublayer
DNS data approx

Reynolds Stress Budget

-⟨u'v'⟩/u_τ²
⟨u'²⟩/u_τ²
⟨v'²⟩/u_τ²

Parameters

U⁺ = y⁺ (y⁺ < 5)
U⁺ = 5ln(y⁺) - 3.05 (buffer)
U⁺ = (1/κ)ln(y⁺) + B (log law)
κ ≈ 0.41, B ≈ 5.2
Re_τ = 1000
δ_ν = R/Re_τ
U_CL⁺ ≈ 0.00

Turbulence Intensity

TKE production and dissipation profiles. Peak near y⁺≈12.