A theory for fully developed turbulent pipe and channel flows is proposed which
extends the classical analysis to include the effects of finite Reynolds number. The
proper scaling for these flows at finite Reynolds number is developed from dimensional
and physical considerations using the Reynolds-averaged Navier–Stokes equations.
In the limit of infinite Reynolds number, these reduce to the familiar law of the wall
and velocity deficit law respectively.
The fact that both scaled profiles describe the entire flow for finite values of
Reynolds number but reduce to inner and outer profiles is used to determine their
functional forms in the ‘overlap’ region which both retain in the limit. This overlap
region corresponds to the constant, Reynolds shear stress region
(30 < y+ < 0.1R+
approximately, where R+ = u*R/v).
The profiles in this overlap region are logarithmic,
but in the variable y + a where a is an offset. Unlike the classical theory, the additive
parameters, Bi, Bo, and log coefficient,
1/κ, depend on R+. They are asymptotically
constant, however, and are linked by a constraint equation. The corresponding friction
law is also logarithmic and entirely determined by the velocity profile parameters, or
vice versa.
It is also argued that there exists a mesolayer near the bottom of the overlap
region approximately bounded by 30 < y+ < 300 where there is not the necessary
scale separation between the energy and dissipation ranges for inertially dominated
turbulence. As a consequence, the Reynolds stress and mean flow retain a Reynolds
number dependence, even though the terms explicitly containing the viscosity are
negligible in the single-point Reynolds-averaged equations. A simple turbulence model
shows that the offset parameter a accounts for the mesolayer, and because of it a
logarithmic behaviour in y applies only beyond y+ > 300, well outside where it has
commonly been sought.
The experimental data from the superpipe experiment and DNS of channel flow
are carefully examined and shown to be in excellent agreement with the new theory
over the entire range 1.8 × 102 < R+ < 5.3 × 105.
The Reynolds number dependence of all the parameters and the friction law can be determined from the single empirical
function, H = A/(ln R+)α for α > 0,
just as for boundary layers. The Reynolds number
dependence of the parameters diminishes very slowly with increasing Reynolds
number, and the asymptotic behaviour is reached only when R+ [Gt ] 105.