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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 .
 
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