In 1941 Andrei N. Kolmogorov published a paper in which he derived a formula for the energy spectrum of turbulence. This spectrum gave the distribution of energy among turbulence vortices as function of vortex size. In this work Kolmogorov founded the field of mathematical analysis of turbulence. There were other attempts at such analysis before but never such a striking result.

It is now recognized that Kolmogorov's result is not empirically correct but nevertheless it is a magnificent piece of applied mathematical analysis. This material presents an abreviated form of Kolmogorov's work.

The Navier-Stokes equations which describe the dynamics of the flow of viscous fluids can be into a spectral form in which the variables are the wave numbers for vortices of various sizes. The wave number k of a vortex of spatial dimension L is given by

k = 2π/L

The spectral form of the Navier-Stokes equations indicate that energy can be transferred from two wave numbers, k₁ and k₂, to a wave number k₃ only if

k₃ = k₁ + k₂

This is called the Selection Rule.

There are two extremes of the operation of the Selection Rule. The wave numbers of interacting vortices can be approximately equal, k₁ ≅ k₂, in which case

k₃ ≅ 2k₁

This is the type of interaction Kolmogorov assumed.

The other extreme is where one wave number is very small compared to the other, say k=δ≅0, in which case energy is transferred to k₃=k₁+δ, a nearby wave number.

Kolmogorov envisioned a process in which mixing occurs over a range of wave numbers, say from k_min to k_max. The turbulent mixing transfers energy to the higher wave numbers. Over some range, say k_max to k_ν the viscous dissipation of energy is not important. Beyond k_ν the spectrum is affected by the viscosity of the fluid. The range over which viscous effects are not important can be called the inertial range. Kolmogorov derived an formula for the energy spectrum over the inertial range.

Kolmogorov's Energy Spectrum

Kolmogorov's analysis deduced that E, the energy density per unit wave number should depend only upon the wave number k and ψ, the rate of energy dissipation per unit volume. The upper limit of the inertial range k_ν should depend only upon the molecular viscosity ν and ψ.

The dimensions of these variables are:

[k] = 1/L
[ψ] = L²/T³
[ν] = L²/T
[E] = L³/T²

Thus if E(k,ψ) = Ck^αψ^β for some constant C dimensional compatibility requires

L³ = L^-αL^2β
so -α+2β = 3
T^-2 = T^-3β
so
-3β = -2
and hence
β = 2/3
and thus
α = 4/3 - 4 = -5/3

Thus Kolmogorov's energy spectrum is

E(k,ψ) = Ck^-5/3ψ^2/3

The wave number, k_ν, at which viscosity makes energy dissipation effects significant is a function only of molecular viscosity ν and the rate of energy dissipation per unit volume, ψ. Thus if k_ν=Dν^γψ^δ for some constant D, dimensional compatibility requires

L^-1 L^2γL^2δ
or 2γ + 2δ = -1
and
T⁰ = T^-γT^-3δ
-γ - 3δ = 0 → γ=-3δ
and hence
-6δ + 2δ = -4δ = -1
so
δ=1/4
and
γ=-3/4

Thus

k_ν = Dν^-3/4ψ^1/4

The energy spectrum is the relation between E and k with ψ held fixed. Below is shown an illustration of this relation.

This relationship brings to mind L.F. Richardson's little poem

The energy spectrum in terms of the scale of the vortices is found by noting that if F(L) is the energy spectrum in terms of the scale L of a vortex then

∫E(k)dk = ∫E(k)(dk/dL)dL = ∫F(L)dL
so
F(L) = E(k)(dk/dL).

This means that

F(L) = Ck^-5/3(-2π/L²) = C'L^5/3L^-2
= C'L^-1/3

Thus the alternate version of Kolmogorov's spectrum is

F(L) = C'L^-1/3

This is shown in the diagram below.

Thus the energy density is lower for the larger vortices and energy is more concentrated in the smaller scale vortices.