Civil Engineering Reference
In-Depth Information
where it must be remembered that
is the rotation of the
vortex filament situated at a distance
y
ζ
a
from the wall.
The velocity near to the stagnation point
=
is
x
=
3
a
expressed by
u
to the first order, and the stress rate in
∞
=
γ
x
inner variables is
+
∂
u
x
Re
⎛
⎞
γ
+
=
=
3
4
T
[4.17]
⎜
⎝⎠
+
2
∂
a
x
=
3
a
Note that we have introduced the Reynolds number of the
vortex
Re
T
. The Reynolds number of the QSVs in the
inner layer is typically
= ζ ν
and
a
+
=
20
Re
=
Re
=
22
T
QSV
γ
+
=
[ROB 91a], which gives us
0.02
. It is interesting to note
that the timescale linked to the process is
t
+
=
γ
+
=
50
,
which is nothing but the duration between the ejections (the
inverse of the frequency) in the inner layer. The stagn
ation
flow induces a local boundary layer of thickness
11
1
γ
+
.
The wall friction near to the stagnation point is, in this
case
12
,
δ
+
∝
1
+
Re
u
+
+
+
+
T
[4.18]
τ
∝∝
∞
u
γ
∝
u
∞
∞
+
+
δ
a
The analysis performed here, which recaps the arguments
advanced in [ORL 94], gives a view which is undeniably
qualitative, but nevertheless interesting, of the repercussion
11 The exact relation depends on the nature of the stagnation flow. For a
Hiemenz flow, for instance,
+
+
[SCH 76]. However, the analysis
of order of magnitude presented here is generally applicable, including for
a 3D stagnation flow [DRA 06].
12 The exact relation for the local friction in a plane stagnation Hiemenz
δ
=
2.4
γ
+
+
+
. This equation must be multiplied by
3
at an
flow is
τ
=
1.233
u
γ
∞
equivalent distance
x
for an axisymmetrical stagnation flow.
Search WWH ::
Custom Search