Environmental Engineering Reference
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which leads to
ε
= 2
σ
x T
(3.219)
x
v
x
= 2
L
ε
σ
T
(3.220)
ij
y
y
v
ε
=
σ σ
v
(3.226)
y
ij
V
V
i
j
ε
= 2
σ
z T
(3.221)
z
v
z
This relationship is particularly useful in estimating
turbulent diffusion coefficients from measurable
Eulerian velocity characteristics. Comparing Equations
(3.217) and (3.226), it is clear that the Eulerian velocity
correlation length scale, L ij , is related to the Lagrangian
velocity correlation time scale, T ij , by the approximate
relation
In a similar manner to which correlation time scales are
used to measure time lags over which velocity fluctua-
tions remain correlated in a Lagrangian reference frame,
correlation length scales are used to measure the dis-
tance lags over which velocity fluctuations remain cor-
related in an Eulerian reference frame. Correlation
length scales, L ij , in Eulerian reference frames are
defined by the relation
L
ij
T
ij
v
0
( )
L
=
R s ds
(3.222)
ij
ij
where R ij ( s ) is the Eulerian velocity correlation function
given by
EXAMPLE 3.23
V
( )
x x
σ σ
V
′ +
(
s
)
i
j
R s
( ) =
(3.223)
The Eulerian covariance function of the velocity fluc-
tuations in an ocean is given in Table 3.3, where the
spatial lags are measured in the flow direction. If the
mean flow velocity is 25 cm/s, then estimate the correla-
tion length scale, the correlation time scale, and the
diffusion coefficient in the flow direction.
ij
V
V
i
j
V j are the deviations of the i and j com-
ponents of the Eulerian velocity from the respective
means, V i and V j ; s is the spatial lag between velocities
measured in the mean flow direction, and σ V i and σ V j
are the standard deviations of V i and V j , respectively. If
the velocity field is stationary and homogeneous, then
the variances of the Eulerian and Lagrangian velocities
are equal, and the frozen turbulence assumption leads
to the following relationship between the Lagrangian
velocity covariance function, ρ ij ( τ ), and the Eulerian
velocity covariance function, R ij ( s ),
where
V i and
Solution
Since the velocity variance is equal to the covariance at
zero lag, then the velocity variance in the direction of
the flow velocity, σ v 2 , is equal to 225 cm 2 /s 2 . The Eule-
rian correlation function, R 11 ( s ), is related to the Eule-
rian covariance function, C 11 ( s ) by the relation
ρ τ
( )
R
(
V
τ
)
(3.224)
= C
σ
( )
s
ij
ij
11
R s
( )
11
2
1
Combining Equation (3.224) with Equation (3.208)
leads to the following approximate relation between the
turbulent diffusion coefficient, ε ij , and the parameters of
the Eulerian velocity field
v
and therefore the correlation function, R 11 ( s ), is given
by Table 3.4.
Numerical integration of R 11 ( s ) yields the following
estimate of the correlation length scale, L 11 in the flow
direction
1
( )
(3.225)
ε
σ σ
R s ds
ij
V
V
ij
i
j
V
0
TABLE 3.3
Covariance, C 11 (cm 2 /s 2 )
225
110
53
19
9
5
1
0.4
−0.01
0.01
0.01
lag, s (m)
0
1
2
3
4
5
6
7
8
9
10
TABLE 3.4
R 11 ( s )
1
0.49
0.24
0.084
0.04
0.022
0.0044
0.0018
−0.00004
0.00004
0.00004
lag, s (m)
0
1
2
3
4
5
6
7
8
9
10
 
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