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V
U
σ V inj
jet cotan(α′ )
V
U
σ V suc
jet cotan(β ′ )
σ V inj
jet
0 0
S W
INJECTION SIDE
S W
σ Vjet
suc
0 0
S W
SUCTION SIDE
S W
Figure 7. Representation of the uniform model described by Eq. 10 to 13: UM2.
negligible between the intermediate model and the UM2 model.
C. Implementation of the LES models
The models proposed in section III.B, UM1 and UM2, are implemented in the AVBP code
(section IIA).
To make the coupling easier, one assumes that the surface meshes on the injection and the
suction sides coincide (see Fig.8). To determine the operating conditions at a liner point, only
the values at this node and at the corresponding node on the other side of the plate (same
streamwise and spanwise coordinates on the other side) are used. At each iteration, the mass
flow rate per surface unit through the plate, ϕ, is computed from the pressure drop across
the liner, assessed as the difference between the nodal pressures P inj and P suc (see Fig.8).
For doing so, ϕ is related to the micro-jets velocity V j , viz. ϕ = ρV j sin(α) σ. Introducing
1
the discharge coefficient C D to express V j as a function of ∆P = P suc − P inj , viz. ρV 2 =
√
2
C 2 D ∆P, the mass flow rate per unit wall surface is then ϕ = sin(α) σ 2ρ suc C 2 D ∆P. Note
that in this latter relation, the density is assessed at the suction side, viz. ρ = ρ suc .
Once ϕ is known, the following quantities are imposed:
• at the suction side: the only variable imposed is the normal velocity, computed as
V suc
W
= ϕ ρ suc
. As only one quantity can be imposed for an outlet boundary condition,
both uniform models are implemented with the same boundary conditions for the
suction side, corresponding to the UM2 model (Eq. 12 and 13).
• at the injection side: ρ is determined from the temperature T s uc (assumed to be also
the fluid temperature at the hole outlet) and the pressure P inj . Then the normal
velocity is V inj
W
UM1 and U inj
W
= ϕ ρ inj
and the streamwise velocity is set to U inj
W
= V inj
W
cotan(α′ ) in UM2.
= V inj
W
cotan(α) in
The number of imposed quantities corresponds to what is usually done for classical inlets/outlets.
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