Geoscience Reference
In-Depth Information
The first application of Lilly's ideas appears to be Deardorff's (
1970a
) numeri-
cal study of turbulent channel flow. He used 6720 grid points
(
24
20
)
and
the eddy-viscosity closure of
Eq. (6.75)
, with the eddy viscosity
K
specified as in
Eq. (6.76)
. He found it necessary to reduce the constant
k
in
Eq. (6.76)
from its
predicted value of 0.17 to 0.10, due in part to the relatively coarse resolution. He
next turned to the convective boundary layer (
Deardorff
,
1970b
), using more grid
points (16000) and the same subgrid model but with some refinements. Here he
used
k
×
14
×
0
.
21, double his original value and closer to the prediction of
Eq. (6.77)
.In
subsequent papers
Deardorff
(
1974a
,
1974b
) usedmodeled versions of the SFS con-
servation
equations (6.71)
and
(6.73)
rather than eddy-diffusivity approximations,
but at the expense of a 2.5-fold increase in computation time.
More recent applications of LES have almost exclusively used eddy-diffusivity
closures,
Eq. (6.75)
for the deviatoric SFS stress and its counterpart for SFS
scalar flux,
=
K
c
∂c
r
f
i
=−
∂x
i
.
(6.78)
The eddy diffusivities
K
and
K
c
are typically taken as
K
=
C
u
e
1
/
2
,
K
c
=
C
c
e
1
/
2
,
(6.79)
with
C
u
and
C
c
constants. The subfilter-scale TKE,
e
, is typically obtained from
a modeled version of its conservation
equation (6.72)
.
Lilly
(
1967
) showed that
C
u
can be related to the constant for the inertial subrange of the velocity spectrum
(
Chapter 7
).
Schumann
et al
.
(
1980
)and
Moeng and Wyngaard
(
1988
) showed that
when the filter scale lies in the inertial range the constants are related by
C
u
C
c
=
β
α
,
(6.80)
with
β
and
α
the inertial range three-dimensional spectral constants for the scalar
and for velocity, respectively.
6.6.4 Measuring subfilter-scale fluxes
Subfilter-scale fluxes can be measured through the “array technique” (
Tong
et al
.
,
1998
). As discussed in more detail in
Chapter 16
, here the signals from a horizon-
tal array of anemometers are filtered in the lateral direction and in time (through
Taylor's hypothesis the latter is a surrogate for streamwise filtering) to obtain
resolved and subfilter-scale variables. The first such experiment, called HATS
(Horizontal Array Turbulence Study), was carried out in 2000 near Kettleman City,
