Geoscience Reference
In-Depth Information
16.3.1 A simple approach for the velocity field
We represent the undistorted flow (that existing in the absence of the body) as
˜
u
i
(t)
u
i
(t)
, the usual sum of ensemble-mean and fluctuating parts, with
the
x
1
-axis chosen in the mean-flow direction. We denote the distorted flow near
the body (but not in its wake) with a superscript d:
=
U
1
δ
i
1
+
u
i
(t)
U
i
(
x
)
u
i
(
x
,t)
.The
˜
=
+
mean distorted flow need not be in the
x
1
-direction.
Now we make a key assumption: that
/U
1
, the time scale of the fluctuating
velocity signal at a point near the body, is much larger than
a/U
1
, the time scale of
the response of the distorted flow to those velocity fluctuations:
/U
1
a/U
1
,
(16.83)
which is equivalent to
a
. This ensures that the distorted flow near the body
sees the turbulence in the approach flow as varying so slowly that it “tracks” its
time variations perfectly. Thus we can treat the distorted flow near the body as
quasi-steady.
We now write the distorted velocity at the measurement position
x
near the body
in a Taylor series about a base state of steady, nonturbulent approach flowof velocity
(U
1
,
0
,
0
)
:
u
i
(
x
,t)
=
u
i
(
x
)
0
+
0
u
i
(
x
)
∂U
j
∂
˜
u
j
(t)
+···
.
(16.84)
The subscript 0 means evaluated in the base state. In typical applications we need
keep only the linear terms in the expansion, so we truncate
Eq. (16.84)
and write
it as
u
i
(
x
)
0
+
u
i
(
x
,t)
˜
˜
a
ij
(
x
)u
j
(t).
(16.85)
The
flow-distortion coefficients a
ij
are defined as
0
u
i
(
x
)
∂U
j
∂
˜
a
ij
(
x
)
=
.
(16.86)
For geometrically simple bodies the
a
ij
can be determined through the analytical
solution for potential flow; in this way
Wyngaard
(
1981
) found them for the flow
ahead of a circular cylinder. For somewhat more complex shapes (e.g., bodies of
revolution) one can calculate the
a
ij
numerically (
Wyngaard
et al
.
,
1985
). For the
muchmore complicated geometries of typical
in-situ
probes the
a
ij
can bemeasured
directly (
Hogstrom
,
1982
).
u
i
(
x
)
0
=
In the absence of flow distortion
˜
U
1
δ
i
1
,a
ij
(
x
)
=
δ
ij
and
Eq. (16.85)
reduces to
u
i
(
x
,t)
˜
=
U
1
δ
i
1
+
δ
ij
u
j
(t)
=
U
1
δ
i
1
+
u
i
(t)
=˜
u
i
(t),
(16.87)
as required.
