Isotropic turbulence and the energy cascade

Big eddies hand energy down to little eddies, and viscosity eventually turns it all into heat — Kolmogorov’s $-5/3$ law and the identity of the smallest eddy.

Opening

This is the chapter where the most “turbulence-like” picture in the whole book gets drawn. The five properties listed in Chapter 1 — diffusivity, three-dimensional vorticity, dissipation — finally collapse into a single scenario. After this chapter, the reader should be able to explain why Kolmogorov’s $-5/3$ law has the shape it does from dimensional analysis alone, and to state in one line why turbulence contains a “smallest eddy that cannot be split further.” The outline of why Chapter 10 (DNS) is so expensive also appears here in advance.

Main 1 — Isotropy as an idealization

Isotropy is the property that the statistics of a flow are invariant under rotation. Rotate the coordinate axes by any angle and the mean kinetic energy, the variance, the correlation functions do not change.

Most real-world turbulence is not isotropic. Smoke rising from a chimney stretches upward; near a wall, the direction parallel to the wall stands out. In other words, large scales are almost always anisotropic. But as the eddies become smaller, this directional information tends to be forgotten.

Kolmogorov (A. N. Kolmogorov, 1903–1987) elevated this observation into a hypothesis: at small scales the statistics become locally isotropic (local isotropy). This is not a claim that all turbulence is isotropic from the start; it is a claim that “directional information gets washed out as the cascade descends.” The idealization is what lets us treat the small scales as a single universal picture.

Main 2 — Richardson’s cascade

Lewis Fry Richardson (L. F. Richardson, 1881–1953) left behind a poem in 1922. Paraphrased in plain prose, it says: big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls, and so on down to viscosity in the molecular sense.

That one sentence holds the entire picture of turbulence. Energy enters a flow from the outside always at the large scales — pumps, wings, wind gradients. This energy creates large eddies; large eddies are unstable and break up into smaller eddies; those smaller eddies face the same fate. Energy flows in one direction, from large to small. We call the rate of this flow $\varepsilon$ (epsilon, the energy dissipation rate per unit mass), with units of $\text{m}^2/\text{s}^3$ .

The crucial point is that this cascade is local. An eddy at some intermediate scale $\ell$ receives energy from eddies slightly larger than itself, and hands energy down to eddies slightly smaller. It does not jump scales.

Main 3 — Kolmogorov’s three hypotheses and the $-5/3$ law

In 1941 Kolmogorov organized this picture into three hypotheses:

Energy injection: large eddies inject energy at a rate $\varepsilon$ per unit mass.
Inertial range: in the intermediate range between large and small scales, the statistics depend only on $\varepsilon$ and the wavenumber $k$ (wavenumber, units $1/\text{m}$ , the inverse of an eddy size). Viscosity $\nu$ (nu, the kinematic viscosity, units $\text{m}^2/\text{s}$ ) does not enter directly.
Dissipation range: at sufficiently small scales, viscosity finally consumes the eddies and converts kinetic energy into heat.

Hypothesis 2 is the most powerful. If the inertial-range energy spectrum $E(k)$ — the energy density per unit mass contained near wavenumber $k$ — is a function of only $\varepsilon$ and $k$ , then dimensions alone fix its shape.

Write out the dimensions: $[E] = \text{m}^3/\text{s}^2$ , $[\varepsilon] = \text{m}^2/\text{s}^3$ , $[k] = 1/\text{m}$ . Match the dimensions of $E \sim \varepsilon^a k^b$ on both sides and you get $a = 2/3$ , $b = -5/3$ uniquely. Therefore

E(k) = C \, \varepsilon^{2/3} k^{-5/3}

The constant $C \approx 1.5$ is the Kolmogorov constant. This one line is the most famous result in 20th-century turbulence research, and the $-5/3$ slope in the inertial range has been confirmed in wind tunnels, ocean measurements, and atmospheric measurements alike.

Main 4 — The Kolmogorov length $\eta$

Where does the cascade stop? As eddies become smaller, the local Reynolds number of the eddy, $Re_\ell = u_\ell \ell / \nu$ , decreases. At some threshold size where $Re_\ell \sim 1$ , viscosity overwhelms inertia, and the eddy cannot break up any further — instead it dissipates as heat. We call this size the Kolmogorov length $\eta$ (eta).

$\eta$ is also fixed by dimensional analysis. From $\nu$ and $\varepsilon$ alone there is only one combination with the dimension of length:

\eta = \left( \frac{\nu^3}{\varepsilon} \right)^{1/4}

For room-temperature air ( $\nu \approx 1.5 \times 10^{-5}\,\text{m}^2/\text{s}$ ) with relatively gentle turbulence ( $\varepsilon \approx 10^{-2}\,\text{m}^2/\text{s}^3$ ), $\eta$ is about 0.4 mm. Even the air flow inside a living room carries sub-millimeter smallest eddies. In something like an industrial gas turbine, where $\varepsilon$ is much larger, $\eta$ drops to micrometres.

This number determines the opening sentence of the next chapter: DNS must shrink its grid spacing down to $\eta$ . That is why DNS cost scales roughly as $Re^3$ and earns its reputation as impractical for industrial flows.

In Python

# Build a fake spectrum and measure the slope in the inertial range.
# The answer should land near -5/3 ≈ -1.667.
import numpy as np

# Wavenumbers: 1 to 1000, log-spaced
k = np.logspace(0, 3, 200)

# In the inertial range E ~ k^(-5/3);
# in the viscous range we tack on an exponential roll-off.
eps = 0.01          # energy dissipation rate [m^2/s^3]
C = 1.5             # Kolmogorov constant
E = C * eps**(2/3) * k**(-5/3) * np.exp(-(k / 600)**2)

# Fit a straight line in log-log over the inertial range only
mask = (k >= 5) & (k <= 100)
slope, intercept = np.polyfit(np.log10(k[mask]), np.log10(E[mask]), 1)
print(f"inertial-range slope = {slope:.3f} (theory -1.667)")

# Check the Kolmogorov length too.
nu = 1.5e-5         # kinematic viscosity of air [m^2/s]
eta = (nu**3 / eps)**(1/4)
print(f"Kolmogorov length eta = {eta*1e3:.3f} mm")

If the printed slope lands near $-1.667$ and $\eta$ comes out under 1 mm, you have touched both of the cascade’s central equations with your own hands.

To the next chapter

Chapter 10: Direct Numerical Simulation (DNS) quantifies why the $\eta$ we met here becomes the direct cause of DNS’s computational blow-up, walking through the grid-point scaling explicitly. With the cascade picture in mind, the trade-offs between DNS, LES, and RANS will appear much more sharply.