Dimensional Analysis - Transport in the Atmosphere-Vegetation-Soil Continuum - page 358

For the preceding equality to be true, the exponents of each fundamental dimen-
sion, in each dimensionless group should add up to zero. In principle this requires
the solution of a system of equations, but often the solution is easily found. For this
example:

• b 1 equals -1/2 to cancel the time dimension of the velocity gradient; as a consequence, a 1
needs to be +1 to cancel the length dimension of the stress. Thus Π 1

u
z

z
u ′′ /

= ∂
∂ ( )

.

12

• b 2 equals -3/2 to cancel the time dimension of the buoyancy term; as a consequence, a 2
should be equal to one. Thus Π 2

g z

uw

w ′′ .

=

v

( ) ′′ /

32

θ

v

C.3 Do an Experiment
To determine the relationship between the Π 1 and Π 2 , one or more experiments are
needed in which all variables occurring in Π 1 and Π 2 are measured simultaneously.
For the example at hand this was done for the irst time in 1968 in Kansas (Businger
et al., 1971 ).

C.4 Find the Relationship between Dimensionless Groups
Dimensional analysis does not give a prediction of the relation between the dimen-
sionless groups. If there is one dimensionless group it will be constant, and the pur-
pose of the experiment will only be to ind the value of that constant. If there are two
or more dimensionless groups (as in our example), the experiment also serves to ind
the functional relationship.
Figure 3.14 (in Chapter 3 ) shows the results of the Kansas 1968 experiment for
dimensionless groups in our example.