⚡ Turbulent Kinetic Energy (k) Calculator
Compute TKE k = ½(u'²+v'²+w'²), turbulence intensity, dissipation rate ε, Kolmogorov/Taylor/integral scales, eddy viscosity νt, turbulent Reynolds numbers, and anisotropy tensor. Reference: §7.6.
⚡ Energy Cascade Schematic
📝 Configuration
Key Equations:
$k = \frac{1}{2}\left(\overline{u'^2}+\overline{v'^2}+\overline{w'^2}\right)$
$\nu_t = C_\mu\dfrac{k^2}{\varepsilon}\quad(C_\mu=0.09)$
$\eta = \left(\dfrac{\nu^3}{\varepsilon}\right)^{1/4}$
$k = \frac{1}{2}\left(\overline{u'^2}+\overline{v'^2}+\overline{w'^2}\right)$
$\nu_t = C_\mu\dfrac{k^2}{\varepsilon}\quad(C_\mu=0.09)$
$\eta = \left(\dfrac{\nu^3}{\varepsilon}\right)^{1/4}$
📊 Results & Visualization
Configure inputs and click Calculate.
ℹ️ About TKE
Turbulent kinetic energy k measures the energy in velocity fluctuations. It is the cornerstone of RANS turbulence models (k-ε, k-ω).
The energy cascade: large eddies (scale L) → inertial subrange (E∝κ⁻⁵/³) → dissipation at Kolmogorov scale η.
Turbulence intensity TI = urms/U is a key input for CFD simulations.
Turbulent kinetic energy k measures the energy in velocity fluctuations. It is the cornerstone of RANS turbulence models (k-ε, k-ω).
The energy cascade: large eddies (scale L) → inertial subrange (E∝κ⁻⁵/³) → dissipation at Kolmogorov scale η.
Turbulence intensity TI = urms/U is a key input for CFD simulations.
📘 Calculation Methodology
Theory
$$k = \tfrac{1}{2}\left(\overline{u'^2}+\overline{v'^2}+\overline{w'^2}\right)$$
$$\varepsilon = C_\mu^{3/4}\frac{k^{3/2}}{L},\quad \nu_t = C_\mu\frac{k^2}{\varepsilon}$$
$$\eta=\left(\frac{\nu^3}{\varepsilon}\right)^{1/4},\;\lambda=\sqrt{\frac{10\nu k}{\varepsilon}},\;L=\frac{k^{3/2}}{\varepsilon}$$
$$\text{Re}_t=\frac{k^2}{\nu\varepsilon},\;\text{Re}_\lambda=\frac{u_{rms}\lambda}{\nu}$$
Worked Example
Problem: u'=1.5, v'=1.2, w'=1.0 m/s, U=30 m/s
Air: ρ=1.225, μ=1.789e-5, δ=50mm
1. k = 0.5(2.25+1.44+1.00) = 2.345 m²/s²
2. urms = √(2×2.345/3) = 1.250 m/s
3. TI = 1.250/30 = 4.17%
4. L = 0.09×0.05 = 4.5 mm
5. ε = 0.09⁰·⁷⁵ × 2.345¹·⁵ / 0.0045 = ~130 m²/s³
6. η = (ν³/ε)⁰·²⁵ ≈ ~50 μm
Air: ρ=1.225, μ=1.789e-5, δ=50mm
1. k = 0.5(2.25+1.44+1.00) = 2.345 m²/s²
2. urms = √(2×2.345/3) = 1.250 m/s
3. TI = 1.250/30 = 4.17%
4. L = 0.09×0.05 = 4.5 mm
5. ε = 0.09⁰·⁷⁵ × 2.345¹·⁵ / 0.0045 = ~130 m²/s³
6. η = (ν³/ε)⁰·²⁵ ≈ ~50 μm
Assumptions & References
Assumptions: Incompressible, high-Re turbulence. Isotropic dissipation. Equilibrium cascade. Boussinesq eddy-viscosity. Standard k-ε constants (Cμ=0.09).
- Pope, S. B. Turbulent Flows, Cambridge, 2000.
- Tennekes & Lumley A First Course in Turbulence, MIT Press.
- Gupta, S. C. Applied CFD — §7.6.
- Kolmogorov, A. N. Dokl. Akad. Nauk SSSR, 1941.