Geoscience Reference
In-Depth Information
In each of
(6.91)
and
(6.92)
the first pair of terms has the form required for
interscale transfer, a contraction of stress and strain-rate tensors. They are equal
in magnitude but of opposite signs, also as required. We designate these as the
interscale-transfer terms and label them as
−
I
and
I
, respectively:
u
i
u
j
s
ij
−
u
i
u
j
s
ij
.
I
=
(6.93)
As we shall discuss in
Chapter 7
, interscale transfer of energy exists in both two-
and three-dim
ensiona
l t
urbulen
ce
, but their phy
sic
al mech
anisms differ.
Noting that
u
i
u
i
u
j
+
u
i
u
i
u
j
u
i
u
i
(u
j
+
u
j
)
u
i
u
i
u
j
, the second pairs sum to
=
=
turbulent transport, as required:
1
1
2
(u
i
u
i
u
j
)
,j
−
(u
i
u
i
u
j
)
,j
−
2
(u
i
u
i
u
j
)
,j
−
(u
i
u
i
u
j
)
,j
−
2
u
i
u
i
u
j
+
u
i
u
i
u
j
,j
2
u
i
u
i
u
j
,j
.
1
1
2
u
i
u
i
u
j
+
=−
=−
(6.94)
u
i
(u
i
u
j
)
,j
,
Eqs. (6.91)
and
(6.92)
, are each the sum
of an interscale-transfer term and a turbulent-transport term. In the fully homoge-
neous problemof
Section 6.3
the turbulent-transfer terms vanish. In the horizontally
homogeneous boundary layer problem of
Section 6.4
there is turbulent transport in
the inhomogeneous vertical direction.
The analogous procedure with the conserved scalar conservation equation yields
the expression for the rate of interscale transfer of scalar variance:
u
i
(u
i
u
j
)
,j
Thus,
−
and
−
2
c
r
u
j
c
,j
−
2
c
s
u
j
c
,j
.
I
c
=
(6.95)
Here the interscale transfer mechanisms are the same in two- and three-dimensional
turbulence (
Chapter 7
).
Questions on key concepts
6.1
Explain why the space average, unlike the ensemble average, does not
eliminate randomness.
6.2
Explain why a spatial filter cannot be sharp in both physical and wavenumber
space, as is reflected in
Figures 6.1
and
6.2
.
6.3
The running-mean and wave-cutoff filters,
Figures 6.1
and
6.2
,
are positive
definite in one space but not in the other.What filter is positive in both spaces?
6.4
Explain physically why wave-cutoff filtering makes the r and s parts of a
filtered variable uncorrelated.
6.5
Explain the relationship between
Eqs. (6.71)
and
(5.39)
,
(5.41)
which
concern two forms of the Reynolds stress tensor.
