Geography Reference
In-Depth Information
5.1.2
Reynolds Averaging
In a turbulent fluid, a field variable such as velocity measured at a point generally
fluctuates rapidly in time as eddies of various scales pass the point. In order that
measurements be truly representative of the large-scale flow, it is necessary to
average the flow over an interval of time long enough to average out small-scale
eddy fluctuations, but still short enough to preserve the trends in the large-scale
flow field. To do this we assume that the field variables can be separated into
slowly varying mean fields and rapidly varying turbulent components. Following
the scheme introduced by Reynolds, we then assume that for any field variables,
w and θ , for example, the corresponding mea
ns
are indicated
b
y overbars and the
fluctuating components by primes. Thus, w
θ
. By definition,
the means of the fluctuating components vanish; the product of a deviation with a
mean also vanishes when the time average is applied. Thus,
w
and θ
=
w
+
=
θ
+
w
θ
w
θ
=
=
0
where we have used the fact that a mean variable is constant over the period of
averaging. The average of the product of deviation components (called the
covari-
ance
) does not generally vanish. Thus, for example, if on average the turbulent
vertical velocity is upward where the potential
temp
erature deviation is positive
and downward where it is negative, the product w
θ
is positive and the variables
are said to be correlated positively. These averaging rules imply that the average
of the product of two variables will be the product of the average of the means plus
the product of the average of the deviations:
w
)(θ
θ
)
w
θ
wθ
=
(w
+
+
=
wθ
+
Before applying the Reynolds decomposition to (5.1)-(5.4), it is convenient to
rewrite the total derivatives in each equation in flux form. For example, the term
on the left in (5.1) may be manipulated with the aid of the continuity equation (5.5)
and the chain rule of differentiation to yield
u
∂u
Du
Dt
=
∂u
∂t
+
u
∂u
v
∂u
w
∂u
∂v
∂y
+
∂w
∂z
∂x
+
∂y
+
∂z
+
∂x
+
∂u
2
∂x
+
∂u
∂t
+
∂uv
∂y
+
∂uw
∂z
=
(5.6)
Separating each dependent variable into mean and fluctuating parts, substituting
into (5.6), and averaging then yields
u
u
u
v
u
w
Du
Dt
=
∂
u
∂t
+
¯
∂
∂x
∂
∂y
∂
∂z
u
¯
u
¯
+
+
u
¯
v
¯
+
+
u
¯
w
¯
+
(5.7)
