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. 
Kirill A. Kazakov
physics.flu-dyn