Geoscience Reference
In-Depth Information
−
(U
i
+
u
j
)
r
u
i
)
r
(U
j
+
u
j
)
r
u
i
u
j
−
(u
i
u
j
)
r
,
(U
i
+
u
i
)(U
j
+
=
(6.65)
=
(C
u
j
)
r
c)
r
(U
j
+
u
j
)
r
(cu
j
)
r
c
r
u
j
.
+
c)(U
j
+
−
+
=
−
f
j
(C
u
i
+
u
i
,c
c
r
c
s
With the further decomposition
u
i
=
=
+
these become
u
i
u
j
s
u
i
u
j
+
u
i
u
j
r
τ
ij
ρ
=
u
i
u
j
+
−
,
(6.66)
c
r
u
j
s
c
r
u
j
+
c
s
u
j
r
c
s
u
j
+
f
j
=−
+
.
Equation (6.66)
shows that
τ
ij
/ρ
and
f
j
depend only on the filtered turbulence
fields, but beyond that the expressions are not easy to interpret. But two aspects
are evident:
1. Because the wavenumbers of the Fourier components of a product involve the sum of
the wavenumbers of the Fourier components of each term in the product, multiplication
of
u
i
by
u
j
, and
c
r
by
u
j
, generates spectral content at wavenumbers of magnitude up
to 2
κ
c
, with
κ
c
the filter-cutoff wavenumber. Thus the first term of each of
(6.66)
has
Fourier components of wavenumber magnitudes from
κ
c
to 2
κ
c
.
2. The second term of each of
(6.66)
involves filtered products of the r and s fields; each of
these can be nonzero. Interactions of this type are sketched in the second and third panels
of
Figure 6.3
.
The subfilter-scale variables in the last term in this group can involve large,
computationally unresolvable wavenumbers.
An alternative form of the filtered equation set
(6.64)
is
u
j
r
u
j
r
τ
ij,j
ρ
τ
ij
1
ρ
˜
u
i,t
+
u
i
˜
p
,i
,
u
i
˜
u
j
)
r
,
˜
˜
,j
−
=−
ρ
=
˜
−
(
u
i
˜
˜
(6.67)
u
j
r
u
j
r
c
,t
+
c
r
f
j,j
=
f
j
u
j
)
r
c
r
˜
˜
˜
,j
+
0
,
=
(
c
˜
˜
−
˜
˜
.
In this case the expressions for the Reynolds fluxes have only the resolved-scale
term in
Eq. (6.66)
:
u
i
u
j
+
u
i
u
j
r
τ
ij
ρ
u
i
u
j
+
=−
,
(6.68)
c
r
u
j
+
c
s
u
j
r
f
j
c
s
u
j
+
=
.
Both
τ
ij
,τ
ij
and
f
j
,f
j
are called
subfilter-scale
fluxes because they involve
subfilter-scale fields. From their definition
(6.68)
τ
ij
and
f
j
are resolved quantities,
however. This illustrates the difficulty of naming such variables.
