Geoscience Reference
In-Depth Information
transport, and pressure transport; on the right are tilting production, gradient pro-
duction, and pressure destruction. The gradient production term is interesting in
that it produces a scalar flux that need not be in the direction of the gradient; we
can call this a “tensor diffusivity” term
(Problem 6.29)
.
6.6.3 Modeling subfilter-scale fluxes
As with any turbulence moment equations,
Eqs. (6.71)
-
(6.73)
involve further
unknowns and cannot be solved directly. But they can provide useful insight into
modeling the subfilter-scale fluxes.
Lilly's (
1967
) “first-order theory” for
Eq. (6.71)
in effect assumes a quasi-steady,
locally homogeneous state in which the rates of gradient production and pressure
destruction of
τ
ij
are in balance. If we write this as
†
τ
ij
2
e
3
˜
s
ij
=
T
s
,
(6.74)
with
T
s
a time scale, this yields the eddy-viscosity model
2
e
3
T
s
˜
τ
ij
s
ij
s
ij
.
=
=
K
˜
(6.75)
Lilly suggested that the
Smagorinsky
(
1963
) model for eddy viscosity
K
be used,
(k)
2
D
√
2
,D
2
s
ij
˜
s
ij
,
K
=
=˜
(6.76)
with
the grid-mesh spacing (interpreted today as the filter cutoff scale) and
k
a constant. By requiring that the spectrum of the resolved motion near the cutoff
follow the Kolmogorov inertial subrange form (
Chapter 7
), Lilly showed that
k
is
related to the inertial subrange velocity spectral constant (now generally known as
the Kolmogorov constant
α
1
.
5) by
0
.
23
α
−
3
/
4
k
0
.
17
.
(6.77)
Lilly
(
1967
) also sketched out a “second-order theory” for
Eq. (6.71)
. It includes
time-change and advection but again not the tilting-production terms, so it continues
to make the subgrid deviatoric stress
τ
ij
proportional to the resolved strain-rate
s
ij
. Whether his neglect of tilting production here or in the first-order theory
that gave
Eq. (6.75)
was intentional is not clear from his paper.
tensor
˜
†
The “WET” model (
Chapter 5
) used later in second-order closure has this spirit.
