The UMIST-N Near-Wall Treatment Applied to Periodic Channel Flow

CHAPTER 3. CHANNEL FLOW 35
varies only in y. The shear stress may be expressed as
so Equation 3.5 can be rewritten as
d 〈U〉
τ(y) = ρν − ρ 〈uv〉 (3.17)
dy
dτ
dy
It follows from Equation 3.18 that dτ
= d 〈P 〉
dx
(3.18)
dy is constant. Let τw be the shear stress
at the wall, such that τ| y=0 = τw. Also, τ| y=δ = 0 since d〈U〉
dy
y=δ
= 0 and
〈uv〉| y=δ = 0. With these boundary conditions on τ, Equation 3.18 may be
integrated to yield
d 〈P 〉
dx
= −τw
δ
(3.19)
Reτ may be defined from τw, the geometry of the channel, and the properties
of the fluid according to
Reτ =
δ τw
ν ρ
It is of interest, based on Equations 3.19 and 3.20, that only one of
(3.20)
d〈P 〉
, τw, dx
or Reτ must be specified in order to completely define the flow. Usually, Reτ
is the quoted parameter.
Another Reynolds number is the bulk Reynolds number, based on average
velocity
Re♭ =
where U indicates a spatial average.
U (2δ)
ν
A further Reynolds number is that based on free stream velocity
Re0 = U0δ
ν
(3.21)
(3.22)