Geoscience Reference
In-Depth Information
can be designed to remove eddies smaller than a
cutoff scale
, say, chosen such
that
η
.
The reduction in computational requirements that results from averaging comes
at a price. As we saw in
Chapter 2
, the averaged Navier-Stokes equation has a
Reynolds stress
that must be approximated, or
modeled
, before the equation can be
solved. The resulting flow models fall into two broad classes.
Reynolds-averaged
Navier-Stokes
(RANS), also called
second-order closure
, discussed in
Chapter 5
,is
based on ensemble averaging;
large-eddy simulation
(LES), discussed in
Chapter 6
,
uses space averaging.
3.2 Ensemble-averaged equations
We begin with the continuity equation, which for a constant-density fluid reduces
to a zero-divergence requirement for the full velocity field
u
i
:
˜
∂
u
i
˜
∂(U
i
+
u
i
)
∂x
i
=
=
0
.
(3.1)
∂x
i
Ensemble averaging this equation, using the rules in
Chapter 2
, yields
∂
u
i
˜
∂
u
i
˜
∂U
i
∂x
i
=
∂x
i
=
∂x
i
=
0
,
(3.2)
so the mean field has zero divergence. Subtracting this from
Eq. (3.1)
then yields
∂u
i
∂x
i
=
0
,
(3.3)
so the fluctuating field also has zero divergence.
We consider next the Navier-Stokes
equation (1.26)
, which by incompressibility
we can write as
∂
2
˜
∂
u
i
˜
u
j
∂x
j
˜
˜
˜
∂
u
i
1
ρ
p
∂x
i
+
∂
u
i
∂t
+
=−
ν
∂x
j
∂x
j
.
(3.4)
Ensemble averaging yields
∂
2
∂
u
i
∂t
+
˜
∂
u
i
˜
u
j
∂x
j
˜
1
ρ
p
∂x
i
+
∂
˜
u
i
∂x
j
∂x
j
.
˜
=−
ν
(3.5)
Using
u
i
=
U
i
,
p
=
P
and applying the ensemble-averaging rules to
u
i
u
j
yields
∂x
j
U
i
U
j
+
u
i
u
j
=−
∂
2
U
i
∂x
j
∂x
j
.
∂U
i
∂t
+
∂
1
ρ
∂P
∂x
i
+
ν
(3.6)
