🌀 Prandtl Mixing Length Calculator — Turbulent Boundary Layer
Compute the Prandtl mixing length ℓ with Van Driest wall damping, law-of-the-wall velocity profile (u⁺ vs y⁺), eddy viscosity νt, and Reynolds shear stress τturb. Reference: §7.5, §7.6.
🌀 Turbulent Boundary Layer Schematic
📝 Configuration
Key Equations:
Prandtl mixing length:
$\ell = \kappa\,y$
Van Driest damping:
$\ell = \kappa\,y\left[1 - \exp\!\left(-\dfrac{y^+}{A^+}\right)\right]$
Reynolds stress:
$\tau_{\text{turb}} = \rho\,\ell^2\left(\dfrac{\partial u}{\partial y}\right)^2$
Prandtl mixing length:
$\ell = \kappa\,y$
Van Driest damping:
$\ell = \kappa\,y\left[1 - \exp\!\left(-\dfrac{y^+}{A^+}\right)\right]$
Reynolds stress:
$\tau_{\text{turb}} = \rho\,\ell^2\left(\dfrac{\partial u}{\partial y}\right)^2$
📊 Results & Visualization
Configure inputs and click Calculate.
ℹ️ About Prandtl Mixing Length
The Prandtl mixing length hypothesis (1925) models turbulent momentum transport by analogy with molecular transport. The mixing length ℓ represents the average distance a turbulent eddy travels before mixing with adjacent fluid.
Near the wall, ℓ ∝ y (linear growth), but the Van Driest damping function reduces ℓ in the viscous sublayer. In the outer region, ℓ is capped at ~0.09δ (Clauser's model).
The law of the wall (u⁺ vs y⁺) provides the universal velocity profile used with this model.
The Prandtl mixing length hypothesis (1925) models turbulent momentum transport by analogy with molecular transport. The mixing length ℓ represents the average distance a turbulent eddy travels before mixing with adjacent fluid.
Near the wall, ℓ ∝ y (linear growth), but the Van Driest damping function reduces ℓ in the viscous sublayer. In the outer region, ℓ is capped at ~0.09δ (Clauser's model).
The law of the wall (u⁺ vs y⁺) provides the universal velocity profile used with this model.
📘 Calculation Methodology
Theory
$$\ell_{\text{Prandtl}} = \kappa\,y \quad(\kappa \approx 0.41)$$
$$\ell_{\text{VD}} = \kappa\,y\left[1-\exp\!\left(-\frac{y^+}{A^+}\right)\right] \quad(A^+\approx 26)$$
$$\ell_{\text{outer}} = 0.09\,\delta \quad\text{(Clauser)}$$
$$\tau_{\text{turb}} = \rho\,\ell^2\left(\frac{\partial u}{\partial y}\right)^2,\quad \nu_t = \ell^2\left|\frac{\partial u}{\partial y}\right|$$
$$\text{Law of wall: } u^+ = \frac{1}{\kappa}\ln y^+ + B \quad(B\approx5)$$
Worked Example
Problem: Air at U∞=30 m/s, δ=50 mm, y=5 mm
ρ=1.225, μ=1.789×10⁻⁵
1. Re_δ = 1.225×30×0.05/1.789e-5 = 102,850
2. Cf = 0.027×Re-1/7 ≈ 0.00455
3. τ_w = ½×Cf×ρ×U² ≈ 2.51 Pa
4. u_τ = √(τ_w/ρ) ≈ 1.43 m/s
5. y⁺ = 0.005×1.43/1.46e-5 ≈ 490 → log-law region
6. ℓ_VD ≈ κy(1-e⁻¹⁹) ≈ κy ≈ 2.05 mm
7. ℓ_outer = 0.09×50 = 4.5 mm → ℓ = min = 2.05 mm
ρ=1.225, μ=1.789×10⁻⁵
1. Re_δ = 1.225×30×0.05/1.789e-5 = 102,850
2. Cf = 0.027×Re-1/7 ≈ 0.00455
3. τ_w = ½×Cf×ρ×U² ≈ 2.51 Pa
4. u_τ = √(τ_w/ρ) ≈ 1.43 m/s
5. y⁺ = 0.005×1.43/1.46e-5 ≈ 490 → log-law region
6. ℓ_VD ≈ κy(1-e⁻¹⁹) ≈ κy ≈ 2.05 mm
7. ℓ_outer = 0.09×50 = 4.5 mm → ℓ = min = 2.05 mm
Assumptions & References
Assumptions: Incompressible, steady, fully developed turbulent BL. Zero or mild pressure gradient. Boussinesq eddy-viscosity hypothesis. Equilibrium turbulence (production ≈ dissipation).
- Schlichting, H. Boundary Layer Theory, Springer.
- Pope, S. B. Turbulent Flows, Cambridge, 2000.
- Gupta, S. C. Applied CFD — §7.5, §7.6.
- Van Driest, E. R. NACA TN 3145, 1956.