Works by Barenblatt and others have shown that besides the logarithmic law of the wall — the limit for infinite Reynolds numbers — there exist power-law solutions, which "are dependent" on the Reynolds number. In 1996, Cipra submitted experimental evidence in support of these power-law descriptions. This evidence itself has not been fully accepted by other experts. In 2001, Oberlack claimed to have derived both the logarithmic law of the wall, as well as power laws, directly from the Reynolds-averaged Navier–Stokes equations, exploiting the symmetries in a Lie group approach. However, in 2014, Frewer et al. refuted these results. Below the region where the law of the wall is applicable, there are other estimations for friction velocity. In the region known as the viscous sublayer, below 5 wall units, the variation of formula_17 to formula_18 is approximately 1:1, such that:  where,  This approximation can be used farther than 5 wall units, but by formula_21 the error is more than 25%. In the buffer layer, between 5 wall units and 30 wall units, neither law holds, such that: with the largest variation from either law occurring approximately where the two equations intercept, at formula_25. That is, before 11 wall units the linear approximation is more accurate and after 11 wall units the logarithmic approximation should be used, though neither are relatively accurate at 11 wall units. The mean streamwise velocity profile formula_17 is improved for formula_27 with an eddy viscosity formulation based on a near-wall turbulent kinetic energy formula_28 function and the van Driest mixing length equation. Comparisons with DNS data of fully developed turbulent channel flows for formula_29 showed good agreement.