Kolmogorov Length Scale Calculator

What is the Kolmogorov Length Scale?

The Kolmogorov length scale (η) is the smallest length scale in turbulent flow. At scales smaller than η, viscous dissipation dominates and turbulent eddies cannot exist. It represents the "bottom" of the energy cascade—the point where kinetic energy is converted to internal energy.

**Physical Interpretation:** Imagine stirring a cup of coffee. Large swirls break down into smaller swirls, which break into even smaller ones. Eventually, you reach a size so small that viscosity smooths everything out—that's the Kolmogorov scale. For typical flows, η is on the order of micrometers to millimeters.

\eta = ((\nu^3)/(\epsilon))^1/4

The Energy Cascade and Dissipation

Turbulent flows exhibit a cascade of energy from large scales to small scales:

**Large Scales (Integral Scale):** - Energy is injected (e.g., by mean flow gradients) - Characteristic length: L (integral length scale) - Characteristic velocity: u' (turbulent velocity scale)

**Inertial Subrange:** - Energy transfers from large to small scales - No production or dissipation (just transfer) - Follows Kolmogorov's -5/3 power law: E(k) \propto k^-5/3

**Small Scales (Dissipation Range):** - Viscous dissipation converts kinetic energy to heat - Characteristic length: \eta (Kolmogorov scale) - Characteristic velocity: u_\eta = (\nu\epsilon)^1/4

**The Dissipation Rate:** The dissipation rate (ε) represents how fast kinetic energy is converted to heat: \epsilon = \nu \overline((\partial u_i')/(\partial x_j))^2

For isotropic turbulence: \epsilon ≈ (u'^3)/(L)

\epsilon ≈ (u'^3)/(L)

Derivation of the Kolmogorov Scale

Kolmogorov's insight was dimensional analysis. At the smallest scales:

**Relevant parameters:** - Kinematic viscosity: \nu (m²/s) - Dissipation rate: \epsilon (m²/s³)

**Dimensional analysis:** We need a length scale from \nu and \epsilon: [\eta] = [\nu]^a [\epsilon]^b

Solving: \eta \propto \nu^3/4 \epsilon^-1/4

**The Reynolds Number at Kolmogorov Scale:** Re_\eta = (u_\eta \eta)/(\nu) = \frac(\nu\epsilon)^1/4 · (\nu^3/\epsilon)^1/4\nu = 1

Remarkably, the Reynolds number based on Kolmogorov scales is always unity! This confirms that at this scale, inertial and viscous forces are balanced.

Re_\eta = 1

Relationship to Other Turbulent Scales

The Kolmogorov scale relates to other important turbulent scales:

**Taylor Microscale (λ):** \lambda = √(\frac10\nu k)\epsilon

Relationship: \lambda ≈ √(15) · \eta ≈ 3.87\eta

The Taylor microscale is larger than the Kolmogorov scale and represents the scale where velocity gradients are significant.

**Integral Length Scale (L):** L = \frack^3/2\epsilon

Relationship: (L)/(\eta) = Re_L^3/4 where Re_L = (u'L)/(\nu)

For high Reynolds numbers, the separation between integral and Kolmogorov scales is enormous. At Re_L = 10^6, L/\eta ≈ 10^4!

(L)/(\eta) = Re_L^3/4

Practical Applications

**CFD Mesh Requirements:** - Mesh must resolve down to Kolmogorov scale for DNS (Direct Numerical Simulation) - For Re = 10^6, need ~10^12 grid points—computationally prohibitive! - This is why RANS and LES models exist—they don't resolve all scales

**Turbulence Modeling:** - k-ε model: Uses \eta implicitly through dissipation rate - LES: Resolves scales larger than filter width (typically >> η) - DNS: Must resolve all scales including η

**Mixing and Chemical Reactions:** - Small-scale mixing occurs at Kolmogorov scale - Fast chemical reactions require resolution of η - Scalar mixing time: \tau_\eta = (\nu/\epsilon)^1/2

**Experimental Measurements:** - Hot-wire anemometry can measure down to ~10η - PIV (Particle Image Velocimetry) typically resolves ~100η - Laser Doppler Velocimetry can reach ~η

N_DNS \propto Re_L^9/4

Typical Values

**Atmospheric Boundary Layer:** - \epsilon ≈ 10^-3 m²/s³ - \nu ≈ 1.5 × 10^-5 m²/s - \eta ≈ 1 mm

**Pipe Flow (Re = 10⁵):** - \epsilon ≈ 0.1 m²/s³ - \eta ≈ 0.1 mm

**Laboratory Grid Turbulence:** - \epsilon ≈ 10^-4 m²/s³ - \eta ≈ 0.5 mm

**High-Speed Jets:** - \epsilon ≈ 10^4 m²/s³ - \eta ≈ 10 μm

\eta \sim 0.1 \text to 1 \text mm (typical flows)

Worked Example

Inputs

• kinematicViscosity: 1.5e-5
• dissipationRate: 0.1

Result: η = 0.000116 m (116 μm)

Explanation

**Problem Setup:** Consider turbulent air flow with kinematic viscosity ν = 1.5×10⁻⁵ m²/s and dissipation rate ε = 0.1 m²/s³.

**Step 1: Calculate Kolmogorov Length Scale** η = (ν³/ε)^(1/4) = ((1.5×10⁻⁵)³ / 0.1)^(1/4) η = (3.375×10⁻¹⁵ / 0.1)^(1/4) = (3.375×10⁻¹⁴)^(1/4) η = 1.16×10⁻⁴ m = **116 μm**

**Physical Interpretation:** The smallest turbulent eddies in this flow are about 116 micrometers in size. Below this scale, viscous forces dominate and convert kinetic energy directly into heat. This is typical for moderate Reynolds number pipe or channel flows.

**Comparison:** For comparison, a human hair is about 50-100 μm thick, so these smallest eddies are roughly the size of a hair diameter!

Second Scenario

Inputs

• kinematicViscosity: 1.5e-5
• dissipationRate: 0.075

Result: η = 0.000116 m (116 μm)

Explanation

This scenario uses different inputs (kinematicViscosity = 1.5e-5, dissipationRate = 0.075) to show how changing one variable affects the kolmogorov length scale result. Run the calculator above with these values to get the exact updated output with step-by-step work.