🔬 Kolmogorov Scales Calculator — η, τη, uη
Compute the three Kolmogorov scales (length η, time τη, velocity uη), Taylor microscale λ, integral scale L, scale ratios, Reynolds numbers, and DNS grid estimation. Ref: §7.6.
🔬 Energy Cascade: L → λ → η
📝 Configuration
Key Equations:
$\eta = \left(\dfrac{\nu^3}{\varepsilon}\right)^{1/4}$
$\tau_\eta = \left(\dfrac{\nu}{\varepsilon}\right)^{1/2}$
$u_\eta = (\nu\varepsilon)^{1/4}$
$\eta = \left(\dfrac{\nu^3}{\varepsilon}\right)^{1/4}$
$\tau_\eta = \left(\dfrac{\nu}{\varepsilon}\right)^{1/2}$
$u_\eta = (\nu\varepsilon)^{1/4}$
📊 Results
Configure inputs and click Calculate Scales.
ℹ️ About Kolmogorov Scales
Kolmogorov (1941) showed that the smallest scales of turbulence are uniquely determined by ν and ε. These scales are where viscous dissipation converts turbulent kinetic energy into heat.
Three scales: η (length), τη (time), uη (velocity).
DNS requirement: resolve all scales from L to η, needing ~(L/η)³ grid points.
Kolmogorov (1941) showed that the smallest scales of turbulence are uniquely determined by ν and ε. These scales are where viscous dissipation converts turbulent kinetic energy into heat.
Three scales: η (length), τη (time), uη (velocity).
DNS requirement: resolve all scales from L to η, needing ~(L/η)³ grid points.
📘 Calculation Methodology
Theory
$$\eta=\left(\frac{\nu^3}{\varepsilon}\right)^{1/4},\;\tau_\eta=\sqrt{\frac{\nu}{\varepsilon}},\;u_\eta=(\nu\varepsilon)^{1/4}$$
$$\lambda=\sqrt{\frac{10\nu k}{\varepsilon}},\;L=\frac{k^{3/2}}{\varepsilon}$$
$$\frac{L}{\eta}\sim\text{Re}_L^{3/4},\;N_{\text{DNS}}\sim\left(\frac{L}{\eta}\right)^3\sim\text{Re}_L^{9/4}$$
Worked Example
Problem: ε=100, k=2.5, air (ν=1.46×10⁻⁵)
1. η = (1.46e-5³/100)¼ = 5.5×10⁻⁵ m
2. τη = √(1.46e-5/100) = 3.8×10⁻⁴ s
3. uη = (1.46e-5×100)¼ = 0.035 m/s
4. λ = √(10×1.46e-5×2.5/100) = 6.0×10⁻⁴ m
5. L = 2.5¹·⁵/100 = 3.95×10⁻² m
6. L/η ≈ 718, NDNS ≈ 3.7×10⁸
1. η = (1.46e-5³/100)¼ = 5.5×10⁻⁵ m
2. τη = √(1.46e-5/100) = 3.8×10⁻⁴ s
3. uη = (1.46e-5×100)¼ = 0.035 m/s
4. λ = √(10×1.46e-5×2.5/100) = 6.0×10⁻⁴ m
5. L = 2.5¹·⁵/100 = 3.95×10⁻² m
6. L/η ≈ 718, NDNS ≈ 3.7×10⁸
References
- Kolmogorov, A. N. Dokl. Akad. Nauk SSSR, 30:301, 1941.
- Pope, S. B. Turbulent Flows, Cambridge, 2000.
- Tennekes & Lumley A First Course in Turbulence, MIT Press.
- Gupta, S. C. Applied CFD — §7.6.