Turbulence Energy Spectrum Plotter
Turbulence Energy Spectrum Plotter plots the energy spectrum of turbulent flow using Kolmogorov’s -5/3 law with dissipation rate and wave number.
Turbulence Energy Spectrum Overview
The turbulence energy spectrum describes the distribution of kinetic energy across different scales in turbulent flow, governed by Kolmogorov’s -5/3 law in the inertial subrange:
Kolmogorov’s -5/3 Law: \\( E(k) = C \epsilon^{2/3} k^{-5/3} \\)
Where:
- \\(E(k)\\): Energy spectrum (m³/s²)
- \\(k\\): Wave number (1/m)
- \\(\epsilon\\): Energy dissipation rate (m²/s³)
- \\(C\\): Kolmogorov constant (≈ 1.5)
Kolmogorov Length Scale: \\( \eta = \left( \frac{\nu^3}{\epsilon} \right)^{1/4} \\)
The spectrum is valid in the inertial subrange, where \\( k \ll 1/\eta \\).
Example Calculations
Example 1: Water at Low Dissipation
\\(\epsilon = 0.01 \, \text{m}^2/\text{s}^3\\), \\(\nu = 1 \times 10^{-6} \, \text{m}^2/\text{s}\\), \\(k = 100 \, \text{1/m}\\), \\(C = 1.5\\)
Energy Spectrum: \\( E(k) = 1.5 \times 0.01^{2/3} \times 100^{-5/3} \approx 0.000239 \, \text{m}^3/\text{s}^2 \\)
Kolmogorov Length: \\( \eta = \left( \frac{(1 \times 10^{-6})^3}{0.01} \right)^{1/4} \approx 3.162 \times 10^{-5} \, \text{m} \\)
Example 2: Water at High Dissipation
\\(\epsilon = 0.1 \, \text{m}^2/\text{s}^3\\), \\(\nu = 1 \times 10^{-6} \, \text{m}^2/\text{s}\\), \\(k = 100 \, \text{1/m}\\), \\(C = 1.5\\)
Energy Spectrum: \\( E(k) = 1.5 \times 0.1^{2/3} \times 100^{-5/3} \approx 0.000514 \, \text{m}^3/\text{s}^2 \\)
Kolmogorov Length: \\( \eta = \left( \frac{(1 \times 10^{-6})^3}{0.1} \right)^{1/4} \approx 1 \times 10^{-5} \, \text{m} \\)
Example 3: Air Flow
\\(\epsilon = 0.01 \, \text{m}^2/\text{s}^3\\), \\(\nu = 1.5 \times 10^{-5} \, \text{m}^2/\text{s}\\), \\(k = 50 \, \text{1/m}\\), \\(C = 1.5\\)
Energy Spectrum: \\( E(k) = 1.5 \times 0.01^{2/3} \times 50^{-5/3} \approx 0.000492 \, \text{m}^3/\text{s}^2 \\)
Kolmogorov Length: \\( \eta = \left( \frac{(1.5 \times 10^{-5})^3}{0.01} \right)^{1/4} \approx 1.106 \times 10^{-4} \, \text{m} \\)
Example 4: High Viscosity Fluid
\\(\epsilon = 0.05 \, \text{m}^2/\text{s}^3\\), \\(\nu = 0.001 \, \text{m}^2/\text{s}\\), \\(k = 10 \, \text{1/m}\\), \\(C = 1.5\\)
Energy Spectrum: \\( E(k) = 1.5 \times 0.05^{2/3} \times 10^{-5/3} \approx 0.00266 \, \text{m}^3/\text{s}^2 \\)
Kolmogorov Length: \\( \eta = \left( \frac{(0.001)^3}{0.05} \right)^{1/4} \approx 0.002236 \, \text{m} \\)