Geoscience Reference
In-Depth Information
a tilde,
a(
x
,t)
, and representing its mean and fluctuating parts with upper- and
lower-case symbols:
a(
x
,t)
˜
=
A(
x
,t)
+
a(
x
,t).
(2.2)
Several types of averages have been used to define mean values in turbulence.
Reynolds
(
1895
) used a volume average. Somewhat later (in the 1930s, accord-
ing to
Monin and Yaglom
(
1971
)), Kolmogorov and his school, and Kampé de
Fériet, brought the ensemble average of statistical physics to turbulence; it is
conceptually the most elegant.
Tennekes and Lumley
(
1972
) used a time aver-
age in steady conditions. A time average is almost always used with quasi-steady
observations, and space averages in homogeneous directions are convenient with
numerical-simulation results.
2.2.1 The ensemble average
Turning on the blower that drives a laboratory turbulent flow generates a
realization
of that flow. A flow property
˜
a(
x
,t)
,where
t
is time measured from the instant the
blower is turned on, say, is random - i.e., different in every realization. We indicate
this randomness bywriting the flowproperty as
a(
x
,t
;
α),α
denoting the realization
number.
The ensemble average (also called the
expected value
)of
a
is defined as the limit
˜
of the average of a large number of samples of
a
:
˜
N
1
N
˜
≡
≡
1
˜
;
a(
x
,t)
A(
x
,t)
lim
N
→∞
a(
x
,t
α).
(2.3)
α
=
As indicated in
Eq. (2.3)
, the ensemble average can depend on both position and
time. Being linear, it commutes with other linear operations such as differentiation
and integration,
b
b
˜
;
∂
a(
x
,t
α)
∂
∂t
˜
=
a(
x
,t),
a(
x
,t
˜
;
α) dt
=
a(
x
,t)dt,
˜
(2.4)
∂t
a
a
andsoforth
(Problem 2.6)
.
In the literature the ensemble parameter is often not explicitly indicated, any
unaveraged quantity being taken as an arbitrary member of the ensemble. In
later chapters we shall follow this convention, suppressing the ensemble index.
Unless stated otherwise, by
average
or
mean
we generally intend the ensemble
average.
