Kinetic energy of the mean flow

Generally, at high Reynolds numbers, viscous dissipation is small when compared to the energy delivered by the mean flow to turbulence. This estimate is presented below; it can be conducted by comparing the last two terms on the right side of Equation (14.7). a quantity, such as may be expressed in terms of base flow variables or the turbulent scales. It follows that which has a dimension of the reciprocal of time, may be expressed in terms of convection variables or diffusion variables.

Where and L represent characteristic velocity and length scales representing streamwise convection. The term is a measure for the turbulent fluctuations and is the thickness of turbulent boundary-layer. This is also called as integral scale which is much larger than the Kolmogorov length scale . Now (conducting a purely order-of-magnitude estimate, without regard for the sign),

Subsequently the term in Equation (14.7) becomes

or

The ratio of the rate of energy delivered to turbulence and viscous dissipation can be determined as

The quantity is a turbulence Reynolds number, and since it is related to the scales of the base flow, it is large compared to unity. Thus, at high Reynolds numbers, the kinetic energy associated with turbulent flow is much more than that of viscous dissipation. Hence, it can be said that at high Reynolds number, Turbulence extracts much more energy from the mean flow than viscous dissipation is able to do.