Let us revisit the Equation for the turbulent kinetic energy budget once again. We aim at estimating the dissipation rate. The kinetic energy per unit mass in the large scale turbulence is . The inverse time scale of the transfer of energy is roughly proportional to , where represents the size of the largest eddies or the integral scale. The rate energy supplied to the small eddies, thus, can be estimated as . This argument contributes to describe dissipation rate as

(17.8)

as was also shown before. Now let us consider the production term of the turbulent kinetic energy equation. In order to keep the turbulence going, it demands

(17.9)

It , the conclusion from the above equation is

(17.10)

This is important, as it shows that the correlation between and is strong when and are in the same range of frequencies.

From the definition of turbulent Reynolds number, it becomes possible to obtain the following

	(17.11)
	(17.12)

These equations describe the relationship between the large-scale eddies and the small-scale eddies in terms of the length scale, time scale and the velocity scale.

The various scalings we have developed above could be used to establish the relative importance of the different terms in the turbulent kinetic energy equation. For example

The pressure work term:
by noting that the pressure fluctuations are of the order and local length scale of the flow, which determines the gradients of averaged quantities, should be of the same order as the large eddy size, .

The mean transport of turbulent energy by turbulent motion is .

Transport by viscous stresses

The remaining terms on the right hand side are of the order of . Thus

These estimates show that only the viscous transport of turbulent energy can be neglected at high Reynolds numbers. All other terms have the same order of magnitude and have to be retained in most of flows of practical interest.