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The mean velocity profile of near-wall turbulent flow

The issue of analytical derivation of the mean velocity profile in a near-wall turbulent flow is revisited in the context of a two-dimensional channel flow. An approach based on the use of dispersion relations for the flow velocity is developed. 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 analyzed 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 near-wall asymptotic of the form U \sim ln^p (y/y0), where y is the distance to the wall, p>0 is arbitrary, and the other, U \sim y^n, with n>0 also arbitrary except in the limit n \to 0 where it turns out to be a universal function of the Reynolds number, n \sim 1/ln(Re). It is proved, furthermore, that given a mean velocity distribution having either asymptotic, one can always construct a cross-correlation function so as to satisfy the obtained equation. These results are discussed in the light of the existing controversy regarding experimental verification of the law of the wall.