Civil Engineering Reference
In-Depth Information
This quantity can be reduced to the von Kárman constant,
κ
, which should be independent of the Reynolds number
in a
universal
logarithmic sublayer.
5
Figure 1.7 shows the
distributions of
ℑ=
−
1
()
+
found by direct numerical simulation
by Bauer [BAU 14]. Note the emergence of a zone beginning
at approximately
ℑ
y
y
+
−
1
()
+
varies little but
does not remain perfectly constant. The size of this zone
increases with increasing
Re
=
50
in which
ℑ
y
, as might be expected.
τ
Let us attempt to clarify the situation by analyzing
Figure 1.8, which shows the distribution of
()
on the
basis of the wall-normal distance in inner units. The figure is
adapted from [OST 00], and includes the experimental data
on the boundary layer in a wide range of Reynolds number
values, based on the momentum thickness covering
2, 500
ℑ
y
+
does indeed tend
toward a universal value independent of the Reynolds
number, but quite far from the wall at
Re
≤≤
27, 000
. We note that
ℑ=
κ
y
+
≥
(the data are
averages over different profiles with different
Re
values).
Although they are slight, the viscous effects extend from the
end of the buffer sublayer
200
y
+
y
+
, depending on
the Reynolds number. A
universal logarithmic
sublayer of
significant extent would, therefore, exist only beyond
200
=
30
to
=
200
, and then only if the Reynolds number is sufficiently
high - typically
y
+
=
.
6
The logarithmic zone would not
be entirely free of viscous effects at
Re
θ
≥
6, 000
, and the von
Kárman constant would depend on the Reynolds number, as
shown in [OST 00] and Figure 1.8.
5 Readers can consult [TAR 11a] and [TAR 11b], and the references cited
therein, for a detailed discussion of the state of the art on the structure of
the mean velocity distribution in different sublayers. In this chapter, we
content ourselves with a very restricted overview.
6 Using the semi-empirical relation
Re
θ
≤
6, 000
δ
+
==
Re
0.30
Re
from [GEO 97], we
τ
θ
obtain the limit
.
Re
τ
>
1,800
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