Geoscience Reference
In-Depth Information
Consider next theNavier-Stokes
equation (1.26)
, whichwe shall write in the form
∂
2
∂
u
i
∂t
+
˜
∂
u
i
˜
u
j
∂x
j
˜
1
ρ
p
∂x
i
+
∂
˜
u
i
∂x
j
∂x
j
.
˜
=−
ν
(3.38)
Applying the spatial filter and using its commutativity with differentiation yields
u
i
u
j
)
r
∂x
j
p
r
∂x
i
+
∂
2
u
i
∂
˜
u
i
˜
˜
˜
∂(
1
ρ
∂
˜
∂t
+
=−
ν
∂x
j
∂x
j
.
(3.39)
As in
Eq. (1.39)
, we write
(
u
j
τ
ij
ρ
u
j
)
r
u
i
˜
u
j
+
u
j
)
r
u
i
˜
u
i
˜
u
j
−
(
u
i
˜
˜
=˜
u
i
˜
˜
−˜
=˜
,
(3.40)
with
τ
ij
a Reynolds stress due to spatial filtering. From its definition in
Eq. (3.40)
we can write this generalized Reynolds stress as
≡
ρ
u
i
u
j
−
( u
i
u
j
)
r
=
ρ
( u
i
u
j
)
s
−
( u
i
u
j
+
u
i
u
j
+
u
i
u
j
)
r
.
τ
ij
(3.41)
τ
ij
is neither a resolved nor a subfilter-scale quantity, for it has both a subfilter-scale
part,
ρ(
u
j
)
r
. But it does vanish in
the high-resolution limit, so it is called the subfilter-scale (sfs) Reynolds stress.
Substituting
Eq. (3.40)
into
Eq. (3.39)
and reintroducing the viscous stress yields
u
i
˜
u
j
)
s
, and a resolved part,
u
i
˜
u
j
+˜
u
i
˜
u
j
+˜
u
i
˜
˜
−
ρ(
˜
u
i
u
j
−
ρ
−
ν
∂
u
j
∂x
i
u
i
u
i
∂
˜
p
r
∂x
i
.
∂
˜
˜
∂
∂x
j
τ
ij
1
ρ
∂
˜
∂t
+
∂x
j
+
=−
(3.42)
Equation (3.42)
and the resolved continuity
equation (3.36)
comprise four equa-
tions, but because of the structure of the sfs Reynolds stress the unknowns exceed
four in number. Thus one needs a model for
τ
ij
in order to solve
Eqs. (3.42)
and
(3.36)
numerically.
As discussed in
Chapter 16
, with discrete spatial filtering techniques
τ
ij
can be
measured (
Tong
et al
.
,
1999
), and it can also be computed (at relatively small
R
t
values) from numerical-simulation fields. Thus there is also some observational
and computational guidance that can be brought to such
subgrid
or
subfilter-scale
modeling, as it is called.
We shall see in
Chapter 6
that even if
τ
ij
/ρ
is much smaller than
u
j
,which
is the case with very high spatial resolution (small filter width), we still need to
include it in the filtered
equation (3.42)
. It is responsible for the extraction of kinetic
energy from the resolved scales, the manifestation here of the energy cascade that
is a key property of turbulence.
u
i
˜
˜
