these simulation numerique et modelisation de l'ecoulement autour ...

Suction side:
U = U inj
jet = V inj
jet cotan(α) over S h and U = 0 over S s . (2)
V = V suc
jet =
q
S h ρ suc
over S h and V = 0 over S s , (3)
U = U suc
jet = U c2 over S h and U = 0 over S s . (4)
V
V inj
jet
U
V inj
jet cotan(α)
V
V suc
jet
U
U c2
0 0 0 0
S s
S h
S s
S s
S h
S s
S s S h
INJECTION SIDE
SUCTION SIDE
S s
S s
S h
S s
Figure 5. Representation of the intermediate model described by Eq. 1 to 4.
ρ inj and ρ suc are the surface-averaged mass density at the hole outlet/inlet. Eq. 4 can be
rewritten as U = U suc
jet = V suc
jet cotan(β), where (β) is the angle formed by e x and the vector
(U c2 ; V suc
jet ).
Indeed, from Eq. 1 to 4, the inviscid flux of streamwise momentum can be assessed and
compared to the numerical results obtained in the reference simulations. The inviscid flux of
2
V j Π d 2
inj
streamwise momentum at the injection side is ρ inj (recall that Vjet = V j /2),
4 tan(α) 4 sin(α)
Π cos(α)
which gives an adimensional value of = 0.68. This value of 0.68 must be compared
16 sin 2 (α)
to the value of 0.89 (111.5% of 0.8, see table 2) in the reference simulation.
The inviscid flux of streamwise momentum at the suction side is −ρ suc Vjet suc U Π d 2
c2
4 sin(α) ,
which gives an adimensional value of −
Π U c2
= −0.305. This must be compared to the
8 V j sin(α)
value of −0.28 (90.4% of 0.31, see table 2) in the reference simulation.
From the analysis of the reference numerical database, 30 it appears that the errors of this
crude model have two main sources:
• the flat profile assumption, viz. the approximation of the flow in the hole by constant
values of velocity. Implicitly, it has been considered that:
∫
1
ρUV ds = ( 1 ∫
ρ ds)( 1 ∫
U ds)( 1 ∫
V ds). (5)
S S S S S S S S
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