Kinetic energy of fluctuations

In a steady, homogeneous, pure shear flow, all averaged quantities except are independent of position. In such a situation, Equation (15.1) results in

(15.3)

Referring to the scale relations used in the earlier sub-section, we can write

For a large value or large Reynolds numbers, we obtain

Which states that the fluctuating strain rate is much larger than the mean strain rate. Comment: Note that at the scale of the fluctuations, the production mechanism of turbulence has a sign opposite to that of viscous dissipation, and they are of comparable magnitude. Thus, an equilibrium is possible between these mechanisms, particularly when convection and diffusion of the velocity fluctuations on the scale of the mean flow are negligible. An alternative picture of equilibrium can be imagined in terms of a power spectrum of the velocity fluctuations. At equilibrium the power spectrum is broad-band with a certain universal shape, indicating steady flow of energy from the large to the small scales. An energy cascade is then said to be realized in the turbulence field.