The mean velocity profile of near-wall turbulent flow: is there anything in between the logarithmic and power laws?статья
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Аннотация: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 unknownpotential
component with the help of a dispersion relation, a nonlinear integrodifferential
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 themean 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 ∼ y^n, where
y is the distance to the wall, n>0, and the other, U∼ln^p(y/y0), with y0
=const and p1. 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.