Civil Engineering Reference
In-Depth Information
Townsend [TOW 76], which we have analyzed in Chapter 4.
The predicted turbulent intensity in internal variables is
+
2
*
+
2
2
+
2
*
+
2
+
2
*
+
2
+
*
+
+
uu u uu uu uu
=+
α
+
β
+
2
β
+
2
α
+
p
OM
OM
OM
OM
[6.6]
*
++
2
+2
αβ
uu
OM
obtained simply on the basis of equation [6.4]. Given the
dissimilarity between the scales of the large structures and
those responsible for
u
+
and
u
are two independent random variables, whi
ch is
also
confirmed by the measurements [MAT 09]. Hence,
u
, it is logical to assume that
OM
u
++
*
=
0.
OM
Locally, the overall mean of equation [6.4] gives us
+
*
+
*
+
+
+
*
+
+
uu uu u u u
=+
β
+
α
=+
α
≡
0
p
OM
OM
OM
by constructio
n
13
,
and
the mean of the individual
components is
u
+
are
independent, the random variables defined by the arbitrary
functions
*
+
+
=≡
. However, if
u
+
and
uu
0
*
OM
OM
(
)
(
)
f u
+
*
and
gu
+
are also independent and
OM
decorrelated.
14
Thus:
uu uu uu uu uu uu
*2
++
2
=
*2
+ +
2
;
*2
++
=
*2
+ +
=
0;
*
++
=
*
++
=
0
OM
OM
OM
OM
OM
OM
*
++
2
*
+ +
2
uu uu
=
=
0
OM
OM
By substituting this back into equation [6.6], we obtain:
(
)
()
()
(
)
⎡
⎤
+
2
+
*
+
2
+
+
2
2
*
+
2
+
⎦
[6.7]
uy euyu e uy
τ
,
=
+
αβ
+
p
OM
τ
⎣
13 Note that the concepts discussed in this section differ from the
approach taken by Panton
[PAN 07], which we have discussed in Chapter
4.
14 If two variables are independent, then they are decorrelated. The same
reasoning cannot generally be applied
vice versa
.
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