Effect of tangential derivative in the boundary layer on time averaged energy dissipation rate

We show that for shear driven flows the energy dissipation rate per unit volume is dominated by a function of the energy dissipation rate in the boundary layer due to the tangential derivatives of the tangential velocities or the tangential derivatives of the normal velocity. Hence if the energy dissipation rate in a (thick enough) boundary layer due to the tangential derivatives of the tangential velocities or the tangential derivatives of the normal velocity is small, the energy dissipation rate per unit volume has to be small as well. This leads to a possible explanation of the sub-Kolmogorov bound to the energy dissipation rate per unit volume for shear driven flows observed in laboratory data. However, the smallness of these tangential derivatives in the boundary layer in the averaged sense still needs to be proved rigorously from the Navier–Stokes equations. We hope that our result will stimulate discussion on modeling, numerical simulation and laboratory experiment on these tangential derivatives in the boundary layer.