The atmospheric surface layer - Turbulence in the Atmosphere - page 218

η , ,and ν ;wetakeittobe η 1 / 4 /ν 3 / 4 . Since there is only one quantity, it is a
function of nothing - i.e., it is a constant:

ν 3 1 / 4

η 1 / 4
ν 3 / 4

= constant , η = constant × . (10.7)

With the constant chosen as 1 this is Kolmogorov's result, Eq. (1.34) .
This is as simple a dimensional analysis problem as you'll meet, so simple that
it doesn't require a formalism to work out. You can write down the answer: If the
length scale η depends only on and ν , then the result (10.7) follows because it is
the only length scale that can be made from and ν .
The pipe-friction problem in Chapter 1 demonstrates better the power of the
Buckingham Pi Theorem. On what parameters does the mean wall stress in tur-
bulent flow in a circular pipe depend? The o b vious ones are pipe diameter D ,
mean velocity averaged over the cross section, u ave , fluid density ρ , and kinematic
viscosity ν . Another is the characteristic height h r of the roughness elements on
the pipe wall, which protrude into the diffusive sublayer and add drag. Therefore
(Problem 10.19)

τ wall = τ wall (D, u ave ,ρ,ν,h r ). (10.8)

Here we have m − 1 = 5 governing parameters and n = 3 dimensions, so there are

m 3 independent dimensionless quantities that are functionally related. It is

conventional to choose these as

− n =

2 τ wall
ρu ave

u ave D
ν

h r

f = ,Re = ,

D .

(10.9)

Then we can write the friction factor f as

f = f(Re,h r /D). (10.10)

It is conventional to plot f against Re with h r /D as a parameter, as in Figure 1.2 .

10.2.2 M-O governing parameters
What we now call the Monin-Obukhov (M-O) similarity hypothesis ,orsimply
M-O similarity, rests on the Obukhov ( 1946 ) paper and a later one by Monin and
Obukhov ( 1954 ). Foken ( 2006 ) has discussed its history and evolution in detail. In
the M-O hypothesis five parameters govern the quasi-steady turbulence structure
immediately above a flat, horizontally homogeneous land surface: the length scale