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

ABSTRACT

The 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.

KEYWORDS:

• Developed turbulence
• law of the wall
• cross-correlation function

Disclosure statement

No potential conflict of interest was reported by the author.