Geoscience Reference
In-Depth Information
•
The shear stress,
u
′′
(for two reasons: a) the shear stress vertically exchanges momen-
tum, and hence inluences the gradient of
u
; b) the shear stress is part of the shear pro-
duction term in the TKE equation (
3.10
)
The buoyancy
g
w
θ
′′
•
term in the TKE equation:
θ
v
C.2 Make Dimensionless Groups
The construction of dimensionless groups follows a straightforward recipe. This is
called Buckingham's pi-theorem.
1. Determine the number of dimensionless groups
n
. This depends on the number of
selected quantities
m
(in our case four) and the number of fundamental dimensions
r
in the following way:
n = m - r
. The number of fundamental dimensions requires
some explanation. For the present subject, the units of all variables can be expressed
as a combination of the fundamental dimensions of mass (
M
), length (
L
), time (
T
)
and temperature (
Θ
). For example, the SI units of force is a Newton, which is equal
to kg m s
-2
, or in general terms [
M L T
-2
]. If there is only one dimensionless group, we
know beforehand that it should be constant. The value of the constant still needs to be
determined experimentally.
2. For each dimensionless group one so-called key quantity needs to be selected, such that
all key quantities together contain all fundamental dimensions that are present in the
physical quantities selected in step 1.
3. Each dimensionless group is the product of a key quantity and the remaining quantities,
each raised to some power. This power should be chosen such that the entire dimension-
less group is indeed dimensionless.
Returning to our example, the number of selected quantities is four. The number of
fundamental dimensions of the four quantities is only two
(
L
and
T
; the
Θ
contained
in the virtual heat lux is cancelled by the division by
θ
V
). Hence the number of
groups is 4 - 2 = 2.
In the selection of the key quantities there is some arbitrariness, but we will choose
u
g
∂
∂
z
and
w
′′. This yields the following expressions for the dimensionless groups
θ
θ
V
(which are identiied by a capital pi, Π):
u
z
=
∂
∂
a
b
Π
[][
z
]
uw
1
′′
1
1
g
=
Π
w
′ ′
θ
[] [
z
a
]
b
uw
′′
2
2
v
2
θ
v
To ind out what the values for
a
1
, a
2
, b
1
and
b
2
should be, we need to analyse the fun-
damental dimensions:
−
1
a
2
−
2
b
[] [
−=
−=
TLLT
LT
][][
]
1
1
23
−
a
22
−
b
[] [
][][
LLT
]
2
2
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