The term v ′x v ′y is the time-average of the product of v ′x and v ′y at a point in the flow. It is
called the Reynolds shear stress.
Prandtl (1925) suggested that v ′x and v ′y are related to the velocity gradient dv/dy shown in
Figure 2.9b. He proposed to characterize the turbulence with a dimension called the "mixing
length" l, which is assumed to be the same in both the x and y directions. Accordingly,
dv
v ′x ~ l
(2.70)
dy
dv
v ′y ~ l
(2.71)
dy
and
2
dv
τ ~ ρ l2
dy
(2.72)
If it is assumed that the mixing length can be represented by the product of a constant κ and y
(i.e., l = κy), then for steady uniform turbulent flow,
2
dv
τ = ρκ y
dy
2
2
(2.73)
Using different reasoning von Karman (1930) derived the same equation. Equation 2.73 can be
rearranged to the form:
τo / ρ
dv
=
(2.74)
κy
dy