Geoscience Reference
In-Depth Information
acts to drive it toward isotropy. Similarly, in an isotropic turbulence field the com-
ponent velocity varian
ces
are
eq
ual. In
Eqs. (5.59)
and
(5.60)
pressure transfer is
the p
rin
cipal source of
w
2
and
v
2
, respectively, acting to drive them toward equality
with
u
2
.
In constant-density flows the rms value
σ
p
of pressure fluctuations is of the order
of
ρu
2
. In the atmospheric boundary layer with
ρ
1kgm
−
3
and
u
=1ms
−
1
this
1newtonm
−
2
,or10
−
5
atmospheres (10 microbars). These fluctuations
are far smaller than the hydrostatic pressure changes that drive atmosphericmotions.
Bradshaw
(
1994
) has described turbulent pressure fluctuations as one of the
“Great Unmeasurables,” because in disturbing the flow near it a pressure probe
generates spurious pressure fluctuations that can be as large as those being mea-
sured. It does appear, however, that in lower-atmospheric applications turbulent
pressure fluctuations can now be reliably measured with a special “quad disc pres-
sure port” (
Nishiyama and Bedard
,
1991
;
Wyngaard
et al
.
,
1994
). We'll discuss this
more in
Part II
.
gives
σ
p
5.6 From the covariance equations to turbulence models
5.6.1 Background
The covariance
equations (5.7)
,
(5.39)
,and
(5.41)
describe exactly the spatial and
temporal evolution of turbulent fluxes and other second moments. However, they
cannot be directly solved because their turbulent-transport, pressure-covariance,
and molecular-destruction terms are unknown. By the same technique one could
derive equations for these unknowns, but those equations would involve yet further
unknowns. This is a manifestation of the
closure problem
in turbulence.
Nonetheless these covariance equations provide important insights into turbulent
fluxes. For example, the presence of the mean-gradient production and tilting-
production terms in the scalar flux-conservation
equation (5.39)
suggests that the
turbulent scalar flux need not be aligned with the mean scalar gradient, contrary to
a common assumption.
Experience with these covariance equations has also shown (
Launder
,
1996
)
that the sign of a covariance is usually the sign of its principal production
term. Thus if we write the conservation equation for a covariance
C
, say, in its
lowest-order form
∂C
∂t
principal production term
−
principal destruction term
,
(5.62)
and approximate the principal destruction term as
−
C/T
, with
T
a time scale, we
have
∂C
∂t
C
T
,
P
−
(5.63)
