Geoscience Reference
In-Depth Information
with
C
,j
the gradient of
C
. The scalar gradient follows the equation
(Problem16.13)
DC
,i
Dt
∂C
,i
∂t
+
∂U
j
∂x
i
C
,j
,
=
U
j
C
,ij
=−
(16.93)
which shows that the scalar gradient is not conserved in flow distortion.
DC
,i
U
i
Dt
C
,i
∂U
i
=
.
(16.94)
∂t
Thus if the flow is steady the quantity
C
,i
U
i
is conserved during flow distortion,
any change in velocity along a trajectory being accompanied by a compensating
change in the scalar gradient.
If we denote a distorted field with a superscript d, then by integrating
Eq. (16.94)
in time along a trajectory from
x
0
in the free stream to a point
x
m
in a region of
flow distortion we can write
C
,i
U
i
(
x
m
,t
+
t)
=
C
,i
U
i
(
x
0
,t),
(16.95)
where
t
is the travel time between the two points. Using
Eq. (16.92)
in the free
stream and in the flow-distortion region gives
∂C
d
∂t
+
∂C
∂t
+
U
i
C
,i
,
U
i
C
,i
=
(16.96)
which with
Eq. (16.95)
gives
∂C
d
∂t
∂C
∂t
(
x
0
,t).
(
x
m
,t
+
t)
=
(16.97)
This means that in a steady velocity field the time series of conserved scalar fluc-
tuations in a flow-distortion region has a constant time delay relative to the time
series in the free stream. Therefore their frequency spectra are identical.
16.3.2.2 Quasi-steady turbulent flow, low frequencies
In turbulent flow
Eq. (16.94)
is
D
c
,i
˜
˜
u
i
c
,i
∂
u
i
∂t
=
˜
=˜
0
,
(16.98)
Dt
c
,i
˜
˜
so
u
i
is not conserved under flow distortion.
Wyngaard
(
1988
) discussed the turbulent case when the velocity and scalar sig-
nals are low-pass filtered at the “probe frequency”
f
p
=
U/
2
πa
associated with
