Kolmogorov Theory of Turbulence and Beyond

Turbulence is a phenomenon in fluid dynamic systems characterized by the chaotic changes in velocity field. Several examples are there to describe the phenomenon of turbulence including smoke rising from cigarettes, terrestrial atmospheric circulation, jet exhaust from a nozzle and flow over a golf ball. The breakdown of the orderly flow of liquid makes it a difficult subject of study. The study involves people from many disciplines such as applied mathematics, physics, chemistry and engineering. The curiosity emerged as to how the energy is dissipated in a turbulent flow.

Flow visualization of a turbulent jet, made by laser-induced fluorescence. The jet exhibits a wide range of length scales, a prerequisite for the appearance of an energy cascade in the turbulence modeling. (Credit: Wikipedia)

In 1941, the Russian scientist Andrei N Kolmogorov derived a formula for the energy spectrum of turbulence which gives the distribution of energy among turbulence vortices as a function of vortex size. The dynamics of the flow is described by the Navier-Stokes equation and once written in the spectral form, the variables are wave numbers for vortices of various sizes. The wavenumber for a vortex of spatial dimension L is written as

k = 2π/L

The energy can be transferred from two wave numbers k1 and k2 to a wave number k3 if

k3 = k1 + k2

Kolmogorov deduced that the energy density per unit wave number should depend only upon the wave number (k) and the rate of energy dissipation per unit volume (ψ) which gives a relation as,

E = C k-5/3 ψ2/3

Here, C is a constant. The problem is pretty obvious for highly viscous flows, however, for flows with vanishingly small viscosity, the energy dissipation is utterly obscure.

Turbulence in the tip vortex from an airplane wing. (Credit: Wikipedia)

The Recent publication from J. I. Cardesa, A V Martin, and J Jiménez reconfirmed the Kolmogorov claim. They verified the concept of energy cascade and found in their simulation result that the bigger vortices break down to smaller ones and the process leads to the dissipation of energy.

Energy-eddies at four different scales (Δ) for the same instant in a numerical simulation of turbulence in a periodic cube. (Credit: J. I. Cardesa et al. Science)

Even the low viscous fluids, like gases, quickly convert the kinetic energy into heat and slow down when turbulence occurs. Turbulence spreads the energy into smaller eddies which increase the local viscosity. The process is much like the friction in the solid world. The increased local viscosity enhances the resistance between layers of fluid and thus dissipate the energy as heat.


  1. Cardesa, J. I., Vela-Martín, A. & Jiménez, J., Science Vol. 357, Issue 6353, pp. 782-784.
  2. David Castelvecchi, Nature, 548, 383 (2017).
  3. https://en.wikipedia.org/wiki/Energy_cascade.

Rakesh Moulick, Ph.D., is presently working as an Assistant Professor at Lovely Professional University, Punjab, India. Dr. Moulick is a theoretical plasma physicist. His research interest lies in the study of Plasma sheath, Particle in cell techniques, Fluid dynamics, molecular dynamics etc. Apart from having a dignified research honor, he takes a keen interest in popularizing science and lures as a good science communicator.