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The 2-D plane case (2)
Use geometric flux of relative velocity: • Assume structure is fixed and
fluid is at rest at nodes I-1 and
N 2
I+1, while it has velocity v of
slope β at node I.
• The fluid flux “entering” side
N 1
L 1
is proportional to:
π
Φ
1
= vN i
1
= vL1cos( + α
1− β ) =
2
= vL sin( α −β)
• Fluid mass is conserved when: Φ
1+Φ 2
= 0
L1sinα1+
L2sinα2
tan β =
L cosα
+ L cosα
1 1 2 2
1 1
• The fluid flux “entering” side
L 2
is proportional to:
π
Φ
2
= vN i
2
= vL2cos( + α
2
− β ) =
2
= vL sin( α −β)
2 2
• The angle β is the slope of the line connecting nodes I-1 and I+1
sinα1+
sinα
α
2
1+
α2
• When L 1
= L 2
= L this reduces to: tan β =
, i.e.: β =
cosα
+ cosα
2
1 2
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Exercise 1 – the 2-D axisymmetric case
• Find analytical expression of
the normal direction in 2-D
axisymmetric geometry.
n 1
n
n 2
L 2
• Does the geometrical property
(connecting line) hold also in
this case?
z
r
L 1
• Show that the obtained expression
tends to the one for plane geometry
as the radius tends to infinity.
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