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depends on the scalar variance cascade rate, which is equal to the molecular destruc-
tion rate
χ
c
; wavenumber
κ
; and the energy cascade rate, which is equal to the
viscous dissipation rate
. On dimensional grounds it follows that
βχ
c
−
1
/
3
κ
−
5
/
3
,
E
c
=
(7.9)
with
β
a constant. Equivalently, for eddy scales
r
in the inertial range the scalar
intensity
c(r)
is hypothesized to depend only on
χ
c
,
,and
r
,sothat
χ
1
/
2
−
1
/
6
r
1
/
3
,
c(r)
∼
(7.10)
c
which is the counterpart of
Eq. (2.66)
for
u(r)
.
Turbulent advection and diffusion effects on a conserved scalar eddy of size
r
are in the ratio
†
turbulent advection
molecular diffusion
=
c(r)u(r)/r
γ c(r)/r
2
u(r)r
γ
=
≡
Co(r).
(7.11)
When
γ
is the thermal diffusivity the dimensionless group
u(r)r/γ
is an eddy
Péclet number. When
γ
is the diffusivity of any conserved scalar we'll define it as
the eddy
Corrsin
number.
‡
It is for scalar eddies what the eddy Reynolds number
Re(r)
is for velocity eddies. In analogy with
Re
t
, we'll also define
Co
t
=
u/γ
as
the turbulence Corrsin number.
As we did for the eddy Reynolds number in
Eq. (7.7)
, we can write the eddy
Corrsin number,
Eq. (7.11)
,as
r
η
oc
4
/
3
u(r)r
γ
Co(r)
∼
∼
,
(7.12)
(γ
3
/)
1
/
4
a microscale made from the scalar diffusivity
γ
. Introduced
independently by
Obukhov
(
1949
)and
Corrsin
(
1951
), it is called the
Obukhov-
Corrsin
scale.
with
η
oc
=
7.1.5 The scalar spectrum beyond the inertial subrange
When
γ
is a mass diffusivity the ratio
ν/γ
is called the Schmidt number
Sc
;when
it is the thermal diffusivity it is called the Prandtl number
Pr
. This ratio can range
from very small values to very large.
Yu e n g
et al
.
(
2002
) indicate that in diverse
applications
Sc
can range from 10
−
3
to thousands; likewise,
Pr
ranges from very
†
This is a scaling approach used by
Corrsin
(
1951
).
‡
Stanley Corrsin (1920-1986), an engineering professor at Johns Hopkins University, was a well-known
turbulence researcher (
Lumley and Davis
,
2003
).
