Geoscience Reference
In-Depth Information
3.2.2 Conserved scalars
Substituting the mean-turbulent decomposition into
Eq. (1.29)
for a scalar con-
stituent
c
in a turbulent flow and ensemble averaging yields
˜
U
i
C
∂C
∂t
∂
∂x
i
γ
∂C
∂x
i
+
+
−
=
u
i
c
0
.
(3.13)
U
i
C
γ∂C/
∂x
i
is the total mean flux of
c
-stuff, a vector. It has a contribution
due to turbulence,
u
i
c
,the
turbulent flux of c
. In a flow with velocity and length
scales
u
and
, the turbulent and molecular fluxes are in the ratio
u/γ
.Ifthis
ratio is large, then we can again neglect the molecular flux except very near solid
boundaries.
In
Chapter 2
we discussed the turbulent diffusion of a conserved dye that was
initially concentrated in the lower half of a tank of water heated from below. This
problem is horizontally homogeneous with
U
+
u
i
c
−
=
V
=
W
=
0, so from
(3.13)
the
ensemble-mean concentration of dye follows
wc
∂C
∂t
+
∂
∂z
γ
∂C
∂z
−
=
0
.
(3.14)
We assume the dye does not penetrate the tank surface, so the molecular flux
vanishes there. Thus, if
u/γ
is large the molecular flux is everywhere negligible
and
Eq. (3.14)
simplifies to
∂C
∂t
+
∂wc
∂z
=
0
.
(3.15)
This says that the mean concentration evolves solely due to the divergence of the
turbulent flux of
c
.
Figure 3.2
sketches the evolution of this turbulent mass flux
profile. We commonly call this process
turbulent diffusion
.
3.3 Interpreting the ensemble-averaged equations
The physical processes underlying the conservation equations of
Chapter 1
are
straightfoward. But the ensemble-averaged equations describe what Taylor might
have called “virtual physics,” which can be more difficult to understand.
3.3.1 Example: A conserved scalar diffusing from a point source
A conserved scalar released from a point source in a turbulent flow creates a
highly irregular, filament-like instantaneous plume downstream.
Figure 3.3
shows
an artist's sketch of such an instantaneous plume.
