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We enforce this condition on the reconstructed field at
discrete locations on the boundary of the annulus.
Let
v
j
=
(u
j
,
v
j
)
,
j
=1,
...
,
N
, denote the normal-
ized PIV measurements of the horizontal velocity field
at any horizontal level
of the cylindrical tank and let
x
j
=
(x
j
,
y
j
)
denote the corresponding normalized loca-
tions of the measured field. Here we have normalized so
that the outer radius of the tank is unity. Since the PIV
data do not include measurements on the boundary, we
must define these points. We choose the boundary points
to be equally spaced on the inner and outer circles of the
annulus with a density that is comparable to that of the
interior points. We denote these boundary points by
ξ
k
,
k
=1,
...
,
M
, and we let
n
k
denote the corresponding unit
outward normal vector at
ξ
k
. The matrix-valued kernel
approximation of the field then takes the form
fitting the measurements. All evaluations of
v
are done
with
ν
n
=
ν
f
to filter out the noise. The resulting filtered
approximation then satisfies the boundary conditions.
Presently, there is no theory for selecting
ν
n
and
ν
f
in
an “optimal” manner. Instead, the choice is somewhat by
trial and error. In the experiments that follow, we found
that
ν
n
= 3.5 and
ν
f
= 5.5 gave good results for several dif-
ferent flow parameter regimes and vertical measurement
levels. Half-integer choices for the smoothness parame-
ter also lead to significant simplifications in computing
(17.15) [
Fasshauer
, 2007].
Because of the properties of
div
ν
and
curl
ν
, the expan-
v
curl
in (17.18) are divergence and curl free,
respectively. Thus, the
v
div
and
sions
v
mimics the Helmholtz-Hodge
decomposition theorem. Furthermore, an approximation
to the divergence-free or curl-free parts of the field can be
obtained from these respective expansions.
In Figure 17.13, we show the reconstruction and decom-
position of the velocity field for two sets of parameters
measured with PIV close to the surface at
z
= 120 mm.
For these data we set the shape parameter to
α
= 20.91,
which corresponds to the inverse of the minimum of
the pairwise distances between the normalized sample
locations.
Figures 17.13a,c show contour plots of the stream-
function for the divergence-free part
N
M
div
(
div
ν
f
v
(
x
;
ν
n
,
ν
f
)
=
ν
n
(
x
,
x
j
)
a
j
+
(
x
,
ξ
k
)
n
k
)d
j
j
=1
k
=1
v
div
(
x
;
ν
n
,
ν
f
)
N
M
curl
ν
n
(
curl
ν
f
+
(
x
,
x
j
)
a
j
+
(
x
,
ξ
k
)
n
k
)c
j
,
j
=1
k
=1
v
div
of the 120 mm
fields, while Figures 17.13b,d show contours of the veloc-
ity potentials for the curl-free part of the fields
v
curl
(
x
;
ν
n
,
ν
f
)
(17.18)
v
curl
.
As can be seen, the main pattern of the flow is quasi-
geostrophic dynamics which is divergence free. The curl-
free patterns shown in 17.13b,d can be interpreted as
a deviation from pure quasi-geostrophic flow. We see
that these deviations are strongest at the inner and outer
boundaries of the annulus. There prominent axial flows
can be expected due to the heating and cooling of the
boundaries. The axial gradients of this flow component
induce a horizontal divergence. While the divergence-free
part shown in Figures 17.13a,c is rather robust, the curl-
free part is more delicate and already small effects can per-
turb the symmetry of the patterns. Still, the curl-free part
of both experiments is rather smooth and no small-scale
wavelike features can be seen. The reason for this might be
that the spatial resolution of the PIV observations is not
high enough to resolve the transient, nongeostrophically
balanced part of the flow.
We conclude this section by noting that we can also
use (17.18) to compute the divergence of the recon-
structed velocity field at any location in the 2D slice of
the cylindrical tank. These approximations can be com-
bined with the incompressibility assumption of the full
3D fluid in the rotating annulus to reconstruct the full
velocity field of the fluid (see
Harlander et al.
[2012b] and
the extended abstract on http://ltces.dem.ist.utl.pt/lxlaser/
lxlaser2012/upload/92_paper_ecvgbw.pdf for details).
T
,
c
k
,and
d
k
are determined by the
following constraints:
where
a
j
=
[
a
j
b
j
]
v
(
x
i
;
ν
n
,
ν
f
)
=
v
i
,
i
=1,
...
,
N
,
v
div
(
ξ
i
;
ν
f
,
ν
f
)
·
n
i
=0,
i
=1,
...
,
M
,
(17.19)
v
curl
(
ξ
i
;
ν
f
,
ν
f
)
·
n
i
=0,
i
=1,
...
,
M
.
These constraints can be arranged into a
(
2
N
+2
M)
×
(
2
N
+2
M)
symmetric linear system of equations for
determining the unknown coefficients.
We have introduced two smoothness parameters
ν
n
and
ν
f
in (17.18) to provide a mechanism for filtering
the reconstructed field. The method we use for filtering
is adapted from a technique first proposed by
Beatson
and Bui
[2007] for scalar-valued RBF approximations.
It involves fitting the noisy data with one smoothness
parameter
ν
n
and then evaluating the resulting approx-
imation with a larger smoothness parameter
ν
f
. This
means the data is fit with one kernel but evaluated with
a smoother yet similar kernel. As discussed by
Beat-
son and Bui
[2007], this kernel replacement technique
corresponds to applying a low-pass filter to the approxi-
mations. Since the measurementsare noisy and the bound-
ary conditions are not, we only use
ν
n
in (17.19) when
