Geoscience Reference
In-Depth Information
but since we saw in
Chapter 15
that in an isotropic field
F
1
=
F
2
=
F
3
=
F
, say,
we can simply write this as
∞
(c)
2
=
2 [1
−
cos
(κr)
]
F(κ)dκ.
(16.63)
−∞
In the inertial subrange
F
is
3
10
βχ
c
−
1
/
3
κ
−
5
/
3
,
F(κ)
=
(
15
.
43
)
and with this form
Eq. (16.63)
can be integrated to give
(Problem 16.7)
2
.
4
βχ
c
−
1
/
3
r
2
/
3
C
c
2
r
2
/
3
.
(c)
2
=
=
(16.64)
C
c
2
is known as the structure-function parameter for the scalar
c
.The
κ
−
5
/
3
behavior
of the scalar spectrum in the inertial range of wavenumbers corresponds to the
r
2
/
3
behavior of the difference variance in the inertial range of spatial separations.
Temperature is often used as the scalar in turbulent flows. It typically is measured
with a fine-wire sensor of temperature-dependent resistance, operated in a bridge
circuit. If the wire has a time constant
τ
w
, the measured temperature fluctuation
θ
m
follows the equation
τ
w
∂θ
m
θ
m
(t)
+
=
θ(t),
(16.65)
∂t
with
θ(t)
the fluid temperature at the sensor. A difference filter passes primarily
temperature eddies whose scale in the separation direction is smaller than the sep-
aration. Since smaller temperature eddies appear in the signal as higher frequency
components, the time constant
τ
w
influences the difference filter output. We'd like
to know how it affects its variance, for example.
One way to proceed is to use Taylor's hypothesis
(Chapter 2)
in the form
∂θ/∂t
=
−
U
1
∂θ/∂x
1
, with
U
1
the mean velocity in the streamwise direction
x
1
,toconvert
Eq. (16.65)
into an equation in
x
1
:
τ
w
U
1
∂θ
m
(
x
)
∂x
1
θ
m
(
x
)
−
+
=
θ(
x
).
(16.66)
The Fourier-Stieltjes components
dZ
m
and
dZ
of the measured and true signals
are then related by
−
iκ
1
τ
w
U
1
)
dZ
m
(
κ
)
=
dZ(
κ
).
(
1
(16.67)
