Geoscience Reference
In-Depth Information
The scalar variance budget,
Eq. (5.7)
, can be very useful in applications. For
example, knowing only the mean concentration
C
of a diffusing hazardous species
can be insuffici
ent
, for
C
gives no information on locally and instantaneously large
values of
c
.In
c
2
we also have a measure of such fluctuations. One of the most
difficult challenges in mod
el
ing
Eq. (5.7)
is formulating closures that maintain the
positive-definite nature of
c
2
.
˜
5.3.2 Scaling guidelines
The following are guidelines for scaling terms in the evolution equations for covari-
ances of energy-containing-range variables. We'll discuss the scaling of terms in
covariance equations for dissipative-range quantities in
Chapter 14
.
(u
i
u
i
)
1
/
2
; conserved scalar fluctuations scale with
1. Veloc
ity
fluctuations scale with
u
=
ρu
2
.
2. The correlation coefficients of fluctuating velocity components and conserved scalar
fluctuations are
O(
1
).
†
3. Spatial variations in mean quantities scale with
.
4. Mean scalar gradients and mean velocity gradients scale with
s/
and
u/
, respectively.
5. Constants can be ignored.
6. Mean-advection and local-time-change terms can involve externally imposed scales
L
and
τ
e
not directly related to the turbulence scales
and
u
.
7. Avoid attempting to scale a “mixed-scale” covariance, one involving a small-scale
(dissipative-range) property and a large-scale (variance-containing range) property, for
its correlation coefficient is not
O(
1
)
.
(c
2
)
1
/
2
; pressure fluctuations scale with
p
s
=
=
An example
‡
of a “mixed-scale” covariance is
∂c
2
/∂x
2
c∂c/∂x.
The left side
is the spatial derivative of a mean quantity, so its order is
s
2
/.
The right side is a
mixed-scale covariance;
c
=
s
is a large-scale property, while we shall see shortly
that
∂c/∂x
is a small-scale property. In this case we can rewrite it to determine its
order
(Problem 5.17)
, but it is not always this simple.
The scales
u
and
s
in Guideline 1 are the simplest, most direct measures of the
fluctuation level of velocity and a conserved scalar in turbulent flow. Similarly, 3 is
the simplest scaling of spatial variations in that it associates the same length scale
with the energy-containing eddies and with variations in mean quantities. Guideline
4 incorporates the mixing-length notion that fluctuations are due to eddy motions
in the presence of mean gradients.
∼
†
O(
1
)
is a mathematical term that in turbulence we can interpret as meaning “approaches a constant as
R
t
→∞
.”
In general
O(
1
)
implies nothing about the magnitude of this constant. Schwartz's inequality (
Part III
) limits
correlation coefficients to 1 in magnitude.
‡
This is adapted from
Tennekes and Lumley
(
1972
).
