The mean velocity profile of near-wall turbulent flow: is there anything in between the logarithmic and power laws?

Abstract

ABSTRACTThe question of possible analytical forms for the mean velocity profile in a near-wall turbulent flow is addressed. An approach based on the use of dispersion relations for the flow velocity is developed in the context of a two-dimensional channel flow. It is shown that for an incompressible flow conserving vorticity, there exists a decomposition of the velocity field into rotational and potential components, such that the restriction of the former to an arbitrary cross section of the channel is a functional of the vorticity and velocity distributions over that cross section, while the latter is divergence-free and bounded downstream thereof. By eliminating the unknown potential component with the help of a dispersion relation, a nonlinear integro-differential equation for the flow velocity is obtained. It is then analysed within an asymptotic expansion in the small ratio v*/U of the friction velocity to the mean flow velocity. Upon statistical averaging in the lowest nontrivial order, this equation relates the mean velocity to the cross-correlation function of the velocity fluctuations. Analysis of the equation reveals existence of two continuous families of solutions, one having the power-law near-wall asymptotic U ∼ yn, where y is the distance to the wall, n > 0, and the other, U ∼ ln p(y/y0), with y0 = const and p ≥ 1. In the limit of infinite channel height, the exponent n turns out to be asymptotically a universal function of the Reynolds number, n ∼ 1/ln Re, whereas p → 1. Thus, the logarithmic profile (p = 1) is found to be a member of the power-log family whose members with p > 1 are intermediate between the power- and logarithmic-law profiles with respect to their slopes at large y. These results are discussed in the light of the existing controversy regarding experimental verification of the law of the wall.