Geoscience Reference
In-Depth Information
it must bemodeled. Because turbulencemodels are not perfect, we expect that (other
things being equal) an ensemble-averaged model is inherently less reliable than a
spatially filtered model. Thus the two approaches have two important differences:
• A spatially filtered model requires three-dimensional, time-dependent calculations. The
ensemble-averaged model requires only as many spatial dimensions as there are inho-
mogeneous directions, and can be time independent if the flow is steady. Thus the
computational requirements for a space-averaged model can be much larger.
• Since a spatially filtered model resolves part of the turbulent flux, and since the reliability
of turbulent-fluxmodels is uneven, other things being equal it tends to be themore reliable.
3.5 Summary
The equations for the ensemble-averaged fields of velocity and a conserved scalar
in an incompressible turbulent flow are
∂U
i
∂x
i
=
0
,
(3.2)
U
i
U
j
+
ν
∂U
i
∂U
i
∂t
∂
∂x
j
∂U
j
∂x
i
1
ρ
∂P
∂x
i
,
+
u
i
u
j
−
∂x
j
+
=−
(3.7)
U
i
C
∂C
∂t
∂
∂x
i
γ
∂C
∂x
i
+
+
−
=
u
i
c
0
.
(3.13)
Without the new terms involving
u
i
u
j
and
u
i
c
,
Eqs. (3.7)
and
(3.13)
are the Navier-
Stokes and scalar conservation
equations (1.26)
and
(1.30)
, which have turbulent
solutions at large Reynolds number. But here the turbulent-flux terms, being of
leading order, ensure smooth, nonturbulent solutions that reveal little, if anything,
about the instantaneous structure of a turbulent flow.
The spatially filtered equations are
u
i
∂
˜
∂x
i
=
0
,
(3.36)
ρ
−
ν
∂
u
j
∂x
i
u
i
u
i
∂
˜
p
r
∂x
i
,
∂
˜
˜
∂
∂x
j
τ
ij
1
ρ
∂
˜
u
i
u
j
−
∂t
+
∂x
j
+
=−
(3.42)
c
r
∂t
+
c
r
∂x
i
∂
˜
∂
∂x
i
γ
∂
˜
u
i
˜
c
r
˜
+
f
i
−
=
0
.
(3.43)
Here the spatial filtering has produced the new terms
τ
ij
/ρ
=
u
i
u
j
−
( u
i
u
j
)
r
and
f
i
=
( u
i
c)
r
−
u
i
c
r
involving the subfilter-scale turbulence.
