Geoscience Reference
In-Depth Information
The symmetric tensor
u
i
u
j
, which is shorthand for
u
i
(
x
,t)u
j
(
x
,t)
, involves only the
fluctuating part of the velocity field. As di
scus
sed in
Chapter 1
, it can be interpreted
in terms of stress or momentum flux.
ρu
i
u
j
is the mean force (in the
i
-direction)
per unit a
rea (
whose normal is in the
j
-direction) due to the fluctuating turbulent
motion.
ρu
i
u
j
can be interpreted as themean
i
-direction turbulent flux of
j
-direction
turbulent momentum.
G. I. Taylor called
−
ρu
i
u
j
a
virtual mean stress
because it exists only as an
ensemblemean. Today it ismore commonly called the
Reynolds stress
after Osborne
Reynolds, who first identified it in 1895 in the space-averaged equations.
We can also write the ensemble-averaged Navier-Stokes
equation (3.6)
in
flux
form
:
−
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
U
j
∂U
j
/∂x
i
)
is the mean kinematic momentum flux
tensor. In the simplest situation, where the length and velocity scales of both the
mean flow and the turbulence are
and
u
, the magnitudes of its turbulence and
viscous contributions are in the ratio
u
i
u
j
ν(∂U
i
/∂x
j
u
2
νu/
∼
turbulence contribution
viscous contribution
u
ν
=
∼
R
t
1
.
(3.8)
This says that away from solid boundaries (where velocity fluctuations vanish) the
viscous stress in the ensemble-averaged Navier-Stokes equation can be neglected
in comparison to the Reynolds stress.
In turbulent flow the Reynolds-stress term in the averaged Navier-Stokes equa-
tion is generally as important as any other term. As a result, the fidelity of any
numerical solution of
(3.6)
can depend strongly on the fidelity of the model used
for the Reynolds stress.
3.2.1 Example: Steady turbulent flow in a channel
Figure 3.1
shows turbulent flow in a channel, the first flow studied through LES
(
Deardorff
,
1970
). The channel length in the streamwise (
x
1
or
x
)
†
direction is much
greater than its depth 2
D
, so the flow is homogeneous in
x
and
∂U/∂x
0. The
channel width in the
y
-direction is also much greater than 2
D
, so that away from
the lateral walls the flow is homogeneous in
y
as well. That implies
∂V/∂y
=
=
0
,
and from the symmetry of the mean flow about the
y
=
0 plane that implies
V
=
0. The mean continuity
equation (3.2)
then reduces to
∂W/∂z
=
0
.
From the
symmetry of the mean flow about the
z
=
0 plane we conclude that
W
=
0
†
As is common in the literature, we'll tend to write velocity and position components as
u, v, w
and
x,y,z
.