Energy cascade

Strictly speaking, a cascade requires the energy transfer to be local in scale (only between fluctuations of nearly the same size), evoking a cascading waterfall from pool to pool without long-range transfers across the scale domain.

Somewhere downstream, dissipation by viscosity takes place, for the most part, in eddies at the Kolmogorov microscales: of the order of a millimetre for the present case.

The dynamics at these scales is described by use of self-similarity, or by assumptions – for turbulence closure – on the statistical properties of the flow in the inertial subrange.

A pioneering work was the deduction by Andrey Kolmogorov in the 1940s of the expected wavenumber spectrum in the turbulence inertial subrange.

, may be written down in terms of the fluctuating rates of strain in the turbulent flow and the fluid's kinematic viscosity, v. It has dimensions of energy per unit mass per second.

The energy spectrum of turbulence, E(k), is related to the mean turbulence kinetic energy per unit mass as[2] where ui are the components of the fluctuating velocity, the overbar denotes an ensemble average, summation over i is implied, and k is the wavenumber.

Since diffusion goes as the Laplacian of velocity, the dissipation rate may be written in terms of the energy spectrum as: with ν the kinematic viscosity of the fluid.

In the intermediate range of scales, the so-called inertial subrange, Kolmogorov's hypotheses lead to the following universal form for the energy spectrum: An extensive body of experimental evidence supports this result, over a vast range of conditions.

If δ is the instantaneous displacement of the surface from its average position, the mean squared displacement may be represented with a displacement spectrum G(k) as: A three dimensional form of the pressure spectrum may be combined with the Young–Laplace equation to show that:[8] Experimental observation of this k−19/3 law has been obtained by optical measurements of the surface of turbulent free liquid jets.