General Points - Transport and Coherent Structures in Wall Turbulence - page 7

stress (which, more specifically, are ), introduce six

unknowns, for which, a priori , we have no additional
equations. The onl y possibility, then, is to link them to the
shearing terms

− ρ uu

i j

by way of considerations that are
usually phenomenological. The issue of turbulent flows in
general, and wall turbulence in particular, lies precisely in
the modeling of these terms, which enables us to close the
system of equations.

∂∂ Ux

i j

1.5. Exact relations in a fully developed turbulent
channel flow

We will now lay out a few exact solutions in the case of a
fully developed 2D turbulent channel flow. These solutions
will enable us to link the wall shear stress to the distribution
of the Reynolds stresses and clearly establish the reason why
turbulence increases transfers at the wall. The flow is
homogeneous along the streamwise x and spanwise z
directions, which gives us . The channel is

considered t o be infinite. Consequently, the s panwise
ve lo ci ty is

∂∂ x = ∂∂ z = 0

. For reasons of continuity, and

W = 0 V = 0

()

. The Reynolds equations along the streamwise and
wall-normal directions then assume the exact forms:

UUy =

1 ∂ P Uuv

x

∂

2

∂

0 =− + ν −

2

ρ ∂ ∂ y ∂ y

[1.13]

∂∂
ρ∂

1 P vv

y

0 =− −

∂ y

We can see, from this last equation, that the pressure is
not solely a function of x , but rather t here are variations
along y , induced by the gradient of vv . This is the first
difference from a laminar Poiseuille flow. Integration of