Civil Engineering Reference
In-Depth Information
scale
22
. The velocity scale characteristic of
b
ij
A
∝
K
3/2
ε
int
K
and
d
ij
is
K
. Consequently [ANT 94]
(
)
−
1/ 2
λ
K
−
1/ 2
⎛
⎞
⎛
⎞
A
K
2
K
T
−
1/ 2
int
=
Re
=
=
⎜
⎟
⎜ ⎟
(
)
⎜
⎟
t
ν
νε
A
K
⎝ ⎠
⎝
⎠
K
int
where
Re
t
is the (local) turbulent Reynolds number. Antonia
et al.
[ANT 94] demonstrated a linear correspondence
d
ii
∝
Re
t
−1/2
b
ii
, only in the outer layer where
b
ii
and
Re
t
−1/ 2
simultaneously tend toward zero. A more general relation
linking
b
ij
to
d
ij
is obtained by Hallbäck
et al.
[HAL 90],
suggesting
⎛
⎞
⎧
⎨
⎫
⎛
⎝
⎞
⎛
⎝
⎞
⎠
2
cRe
t
,
∂
U
∂
1
3
2
3
I
2
b
δ
ij
d
ij
=
b
ij
−
⎩
2
I
2
b
+
b
ij
+
b
ik
b
kj
+
⎬
⎜
⎟
⎜
⎟
⎜
⎟
⎠
y
⎭
⎝
⎠
In this relation, the constant
c
depends on the turbulent
Reynolds number and on the mean shear, and
I
2
b
is the
second invariant of the tensor
b
ij
. Liu and Pletcher
[LIU 08] give an empirical relation
()
=
cy
+*
, ∂
U
+
(
)
( )
b
ij
y
+
y
+
d
ij
y
+*
∂
where
y
+*
is a function of
y
+
which coincides relatively well
with the results of DNS up until a value of
Re
τ
=
2, 000
.
22 In order to help readers remember the different scales of turbulence, let
us recap that the integral length scale depends on
(
)
A A
, while
the Kolmogorov scale depends only on the dissipation and viscosity
(
=
K
,
ε
int
int
K
)
. The Taylor scale is intermediary and depends simultaneously
on the dissipation, the kinetic energy and the viscosity
η
=
ηνε
,
(
)
.
λλν ε
=
,,
K
T
T
K
The integral velocity scale and Taylor scale is
K
, whereas the
1/ 4
(
)
Kolmogorov velocity scale is obviously
. Any linked
characteristic is determined by simple dimensional analysis.
u
=
νε
Ko
K
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