7.4 Kolmogorov's theory

Kolmogorov's theory of turbulence

Kolmogorov's theory provides a statistical framework for describing how energy moves through turbulent flows, from the largest eddies down to the smallest ones where viscosity converts kinetic energy into heat. Developed by Andrey Kolmogorov in 1941, it introduced the concept of the energy cascade along with similarity hypotheses that predict universal behavior at small scales. The theory's most famous result is the five-thirds law for the energy spectrum in the inertial subrange.

This section covers the energy cascade, the Kolmogorov microscales, the two similarity hypotheses, the five-thirds law, and the theory's limitations.

Energy cascade in turbulent flows

Turbulent flows contain a hierarchy of eddies spanning a wide range of sizes. Energy enters the flow at the largest scales, set by the geometry and boundary conditions (think of the diameter of a pipe or the wingspan of an aircraft). These large eddies are unstable and break apart into smaller eddies, which themselves break into still smaller ones.

This successive transfer of energy from large scales to small scales is the energy cascade. The key point: during this transfer through intermediate scales, viscosity plays almost no role. Energy simply passes through. Only at the very smallest scales do viscous forces become significant, and there the kinetic energy is finally dissipated as heat.

Richardson's famous rhyme captures it well: "Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity."

Kolmogorov length scale

The Kolmogorov length scale $\eta$ is the size of the smallest eddies in a turbulent flow, the scale at which viscous dissipation dominates. It's defined as:

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

where $\nu$ is the kinematic viscosity and $\epsilon$ is the mean energy dissipation rate per unit mass.

At scales near and below $\eta$, the flow is smooth and the eddies are isotropic (statistically the same in all directions). For a rough sense of magnitude: in atmospheric turbulence, $\eta$ is on the order of 1 mm, while in a laboratory water channel it might be around 0.1 mm.

Kolmogorov time scale

The Kolmogorov time scale $\tau_\eta$ represents the characteristic turnover time of the smallest eddies:

$\tau_\eta = \left(\frac{\nu}{\epsilon}\right)^{1/2}$

You can think of this as how long it takes the smallest eddy to "rotate" once before viscosity dissipates it. At this scale, the local Reynolds number is of order unity, meaning inertial and viscous forces are in balance.

Kolmogorov velocity scale

The Kolmogorov velocity scale $v_\eta$ is the characteristic velocity of the smallest eddies:

$v_\eta = (\nu \epsilon)^{1/4}$

Notice that all three Kolmogorov microscales depend only on $\nu$ and $\epsilon$. This is not a coincidence; it follows directly from the first similarity hypothesis.

Kolmogorov's first similarity hypothesis

Also called the local isotropy hypothesis, this states:

The large-scale flow might be highly anisotropic (e.g., a jet, a boundary layer, a wake), but by the time energy cascades down to the small scales, all "memory" of the large-scale geometry is lost. The small-scale statistics become universal, the same regardless of how the turbulence was generated.

This is why the Kolmogorov microscales are built from just two parameters. The large-scale details don't matter at these scales.

Kolmogorov's second similarity hypothesis

This hypothesis focuses on the inertial subrange, the intermediate range of scales that are much smaller than the energy-containing eddies but much larger than the dissipative eddies:

Viscosity is irrelevant here because these scales are still too large for viscous effects to matter. The only parameter controlling the dynamics is the rate at which energy passes through: $\epsilon$. This hypothesis leads directly to the five-thirds law.

Kolmogorov's five-thirds law

The most celebrated result of the theory. In the inertial subrange, the energy spectrum follows:

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

where:

• $E(k)$ is the energy spectral density at wavenumber $k$
• $\epsilon$ is the mean energy dissipation rate
• $C \approx 1.5$ is the Kolmogorov constant (determined experimentally)
• $k$ is the wavenumber (inversely proportional to eddy size)

The $-5/3$ exponent can be derived from dimensional analysis. Since $E(k)$ in the inertial subrange can depend only on $\epsilon$ and $k$ (by the second similarity hypothesis), the only dimensionally consistent combination gives the $-5/3$ power law.

This scaling has been confirmed across an enormous range of turbulent flows, from wind tunnels to the atmosphere to the ocean.

Energy spectrum in the inertial subrange

The inertial subrange sits between the energy-containing range (large scales) and the dissipation range (small scales). Its defining features:

• Energy is transferred through these scales at a roughly constant rate $\epsilon$, with negligible dissipation
• The energy spectrum follows the $k^{-5/3}$ power law
• Neither the large-scale geometry nor the viscosity significantly affects the dynamics here

For the inertial subrange to exist, the Reynolds number must be high enough that there is a wide separation between the largest and smallest scales. At low Reynolds numbers, the energy-containing and dissipation ranges overlap, and no clear inertial subrange appears.

Dissipation range of energy spectrum

Below the Kolmogorov length scale (high wavenumbers beyond the inertial subrange), you enter the dissipation range. Here:

• Viscous forces dominate and convert kinetic energy into heat
• The energy spectrum drops off much faster than $k^{-5/3}$, typically showing an exponential decay
• The exact shape of the spectrum in this range is not universal and depends on the specific flow

The transition from the inertial subrange to the dissipation range occurs around $k \sim 1/\eta$.

Universal equilibrium range

The universal equilibrium range encompasses both the inertial subrange and the dissipation range. Across this entire range, the turbulence statistics are determined by just $\epsilon$ and $\nu$.

• In the inertial subrange (the upper portion), only $\epsilon$ matters (viscosity is negligible)
• In the dissipation range (the lower portion), both $\epsilon$ and $\nu$ matter

The term "equilibrium" refers to the balance between the energy received from larger scales and the energy either passed to smaller scales or dissipated. The term "universal" reflects the prediction that these statistics are the same for all turbulent flows at sufficiently high Reynolds numbers.

Limitations of Kolmogorov's theory

Kolmogorov's 1941 theory (often called K41) rests on assumptions that don't always hold:

• Homogeneity and isotropy: Real turbulent flows are often inhomogeneous (e.g., near walls) and anisotropic (e.g., in shear flows). The theory applies strictly only to the small scales, and even there, local isotropy can fail at moderate Reynolds numbers.
• Intermittency: K41 assumes the energy dissipation rate $\epsilon$ is uniform in space and constant in time. In reality, dissipation is highly intermittent, concentrated in thin sheets and filaments. This causes measurable deviations from the predicted scaling exponents, especially for higher-order statistics.
• High Reynolds number requirement: A clear inertial subrange only develops when the Reynolds number is high enough to separate the large and small scales. Many practical flows don't meet this condition.

Intermittency in turbulent flows

Intermittency refers to the fact that intense turbulent activity is not spread evenly through the flow. Instead, the energy dissipation rate fluctuates strongly in both space and time, with rare but extreme bursts of activity.

This has concrete consequences:

• The probability distributions of velocity increments develop heavy tails (they're not Gaussian), especially at small scales
• The scaling exponents for higher-order structure functions deviate from the K41 predictions. For example, K41 predicts the $p$-th order structure function scales as $r^{p/3}$, but experiments show systematic departures for $p > 3$
• These deviations grow more pronounced at smaller scales and for higher-order statistics

Refined similarity hypotheses

Kolmogorov himself, along with Oboukhov, proposed corrections in 1962 (often called K62 or the refined similarity hypotheses) to address intermittency:

1. Replace the global mean dissipation rate $\epsilon$ with a locally averaged dissipation rate $\epsilon_r$, averaged over a region of size $r$
2. Assume that the statistics of velocity increments at scale $r$, conditioned on $\epsilon_r$, still follow the original K41 scaling
3. Model the fluctuations of $\epsilon_r$ as log-normal (Kolmogorov and Oboukhov's specific proposal) or using other intermittency models (e.g., the multifractal model)

These refinements modify the scaling exponents and provide better agreement with experimental data, particularly for higher-order statistics. However, the exact form of the intermittency corrections remains an active area of research.

Experimental validation of Kolmogorov's theory

The five-thirds law is one of the most thoroughly tested predictions in fluid dynamics:

• Grid turbulence in wind tunnels produces nearly isotropic turbulence and shows clear $k^{-5/3}$ scaling in the inertial subrange
• Atmospheric turbulence measurements (e.g., by Grant, Stewart, and Moilliet in tidal channels, 1962) provided early and compelling confirmation over several decades of wavenumber
• Pipe and channel flows show the $-5/3$ spectrum at high Reynolds numbers, though the inertial subrange is narrower

Where the theory falls short:

• Higher-order statistics (skewness, flatness of velocity derivatives) show clear departures from K41 predictions, consistent with intermittency
• The dissipation range spectrum varies between flows, confirming it is not universal
• At moderate Reynolds numbers, the inertial subrange may be too narrow to observe clean $-5/3$ scaling

Applications of Kolmogorov's theory

• Turbulence modeling: The energy cascade concept underpins subgrid-scale models in Large Eddy Simulation (LES), where only the large eddies are resolved and the effect of smaller eddies is modeled. It also informs closure models in Reynolds-Averaged Navier-Stokes (RANS) simulations, such as the $k$-$\epsilon$ model.
• Turbulent mixing: Predicting how scalars (temperature, chemical species, pollutants) mix in turbulent flows requires understanding the cascade and the scales at which molecular diffusion acts. This matters for combustion, atmospheric dispersion, and ocean mixing.
• Aerodynamics and hydrodynamics: Estimating turbulent energy dissipation helps predict drag on aircraft, wind turbines, and ship hulls. The Kolmogorov scales also guide mesh resolution requirements in computational fluid dynamics.
• Atmospheric and oceanic science: The energy cascade framework extends to geophysical turbulence, though with important modifications (e.g., the inverse energy cascade in 2D turbulence). These ideas are central to weather prediction and climate modeling.