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V
U
σ V inj
jet
0 0
S W
INJECTION SIDE
σ V inj
jet cotan(α)
S W
V
σ V jet
suc
σ Vjet suc cotan(β)
0 0
S W
U
SUCTION SIDE
S W
Figure 6. Representation of the uniform model described by Eq. 6 to 9: UM1.
In this model, the velocity averaged over the total surface are the same as in the intermediate
model (Eq. 1 to 4). The above expressions lead to the following approximations
Π cos(α)
for the inviscid fluxes of streamwise momentum: σ for the injection side and
16 sin 2 (α)
−
Π U c2
σ for the suction side, viz. σ times the flux corresponding to the flat profiles
8 V j sin(α)
discussed in the previous sub-section. Thus, modeling the discrete effusion with a uniform
injection imposing the proper mass flow rate and injection/suction angles is not appropriate
to reproduce the fluxes at the perforated wall. As we will see in section IV, this model not
allow to reproduce the correct effect of effusion cooling in the main flow.
In order to retrieve the same streamwise momentum flux as the inhomogeneous model of
section IIIA, another uniform condition (UM2) with modified injection (resp. suction) angle
α ′ (resp. β ′ ) is proposed (see Fig. 7):
Injection side:
Suction side:
V = V inj
W
= σ V inj
jet =
q
S h ρ inj
σ over S W , (10)
U = U inj
W = V cotan(α′ ) over S W . (11)
V =
q σ
S h ρ suc
over S W , (12)
U = V cotan(β ′ ) over S W . (13)
These angles are directly related to (α) and (β) through: tan(α ′ ) = tan(α)σ and
tan(β ′ ) = tan(β)σ. This uniform injection/suction model UM2 injects the same mass flow
rate as the model of Eq. 1 to 4 but the angle of injection is modified to ensure proper streamwise
momentum flux through the plate. Note also that UM2 does not allow reproducing the
vertical momentum flux corresponding to Eq. 1 to 4. However, as the vertical momentum
flux is dominated by a pressure term that does not change in both models, the difference is
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