🔄 Eddy Viscosity (Turbulent Viscosity) Calculator
Compare 6 turbulence models side-by-side: k-ε, k-ω, mixing length, Prandtl 1-eq, Baldwin-Lomax, and Smagorinsky LES. Compute νt, μeff, wall damping, and Reynolds stresses. Ref: §7.6, Eq. 7.55.
🔄 Model Comparison Schematic
📝 Configuration
Key Equations:
$\nu_t = C_\mu\dfrac{k^2}{\varepsilon}\quad\text{(k-}\varepsilon\text{)}$
$\nu_t = \dfrac{k}{\omega}\quad\text{(k-}\omega\text{)}$
$\nu_t = \ell^2\left|\dfrac{\partial u}{\partial y}\right|\quad\text{(mixing length)}$
$\nu_t = C_\mu\dfrac{k^2}{\varepsilon}\quad\text{(k-}\varepsilon\text{)}$
$\nu_t = \dfrac{k}{\omega}\quad\text{(k-}\omega\text{)}$
$\nu_t = \ell^2\left|\dfrac{\partial u}{\partial y}\right|\quad\text{(mixing length)}$
📊 Results
Configure inputs and click Compare Models.
ℹ️ About Eddy Viscosity
The Boussinesq hypothesis models turbulent stresses via an eddy (turbulent) viscosity νt. Different models estimate νt differently:
The Boussinesq hypothesis models turbulent stresses via an eddy (turbulent) viscosity νt. Different models estimate νt differently:
- k-ε: two-equation RANS
- k-ω: better near walls
- Mixing length: algebraic, zero-equation
- Smagorinsky: LES subgrid model
📘 Calculation Methodology
Theory
$$\nu_t^{k\text{-}\varepsilon} = C_\mu\frac{k^2}{\varepsilon},\;\;C_\mu=0.09$$
$$\nu_t^{k\text{-}\omega} = \frac{k}{\omega},\;\;\omega=\frac{\varepsilon}{C_\mu k}$$
$$\nu_t^{\ell} = \ell^2\left|\frac{\partial u}{\partial y}\right|$$
$$\nu_t^{1\text{-eq}} = 0.55\sqrt{k}\,\ell$$
$$\nu_t^{\text{Smag}} = (C_s\Delta)^2|S|,\;\;C_s\approx0.1$$
Worked Example
Problem: k=2.5, ε=100, du/dy=500, air
1. k-ε: νt = 0.09×6.25/100 = 5.625×10-3
2. ω = 100/(0.09×2.5) = 444.4
3. k-ω: νt = 2.5/444.4 = 5.625×10-3
4. νt/ν = 5.625e-3/1.46e-5 = 385
5. τturb = 1.225×5.625e-3×500 = 3.45 Pa
1. k-ε: νt = 0.09×6.25/100 = 5.625×10-3
2. ω = 100/(0.09×2.5) = 444.4
3. k-ω: νt = 2.5/444.4 = 5.625×10-3
4. νt/ν = 5.625e-3/1.46e-5 = 385
5. τturb = 1.225×5.625e-3×500 = 3.45 Pa
References
- Launder & Spalding Mathematical Models of Turbulence, 1972.
- Wilcox, D. C. Turbulence Modeling for CFD, 3rd ed.
- Pope, S. B. Turbulent Flows, Cambridge, 2000.
- Gupta, S. C. Applied CFD — §7.6, Eq. 7.55.
- Smagorinsky, J. Mon. Weather Rev., 1963.