these simulation numerique et modelisation de l'ecoulement autour ...
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This equality is almost verified at the suction si<strong>de</strong> ( 1.5% error) but not at the injection<br />
si<strong>de</strong>. At the injection si<strong>de</strong>, using the hypothesis of Eq. 5 leads to an error of 15% on<br />
the estimation of the inviscid streamwise momentum flux through the hole outl<strong>et</strong>.<br />
• the estimation of the tangential velocity (here the streamwise velocity): the assumption<br />
that the geom<strong>et</strong>rical angle (α) is also relevant to the velocity vector at the injection<br />
si<strong>de</strong> is not perfectly true. Assuming that U inj<br />
j<strong>et</strong> = V inj<br />
j<strong>et</strong> cotan(α) (Eq. 2) introduces an<br />
error of 10% on the estimation of the streamwise velocity at the j<strong>et</strong> outl<strong>et</strong>: the j<strong>et</strong> angle<br />
in the reference <strong>simulation</strong> is 62 ◦ instead of 60 ◦ for the hole angle. On the suction si<strong>de</strong>,<br />
assuming that U suc<br />
j<strong>et</strong> = U c2 leads to an error of approximately 10%<br />
Eventually, the mo<strong>de</strong>l <strong>de</strong>scribed by Eq. 1 to 4 assesses the inviscid streamwise momentum<br />
flux with an error of 9% on the suction si<strong>de</strong> and 24% on the injection si<strong>de</strong>. This last error<br />
is not small but is consi<strong>de</strong>red to be acceptable, given the simplicity of the mo<strong>de</strong>l.<br />
The aim is now to <strong>de</strong>sign a uniform condition that applies over the entire surface of the<br />
plate and that has the same characteristics in terms of resulting momentum fluxes.<br />
B. Construction of the uniform mo<strong>de</strong>l for full-scale <strong>simulation</strong>s<br />
A mo<strong>de</strong>l for effusion cooling has to be provi<strong>de</strong>d in or<strong>de</strong>r to perform <strong>simulation</strong>s involving a<br />
perforated plate without resolving the flow insi<strong>de</strong> and near the plate. The mo<strong>de</strong>l for FCFC<br />
has thus to be homogeneous to be adapted to any grid in the near-wall region: the holes<br />
and the solid wall are not distinguished. The mass flow rate is injected through the entire<br />
plate: the injection surface is 1/σ larger in the uniform mo<strong>de</strong>l. As a consequence the normal<br />
injection velocity is multiplied by σ to ensure that the proper mass flow rate crosses the<br />
equivalent boundary. The obvious uniform mo<strong>de</strong>l (named UM1) proposed is (see Fig. 6):<br />
Injection si<strong>de</strong>:<br />
Suction si<strong>de</strong>:<br />
V = V inj<br />
W<br />
U = U inj<br />
W<br />
= σ V inj<br />
j<strong>et</strong> =<br />
q<br />
S h ρ inj<br />
σ over S W , (6)<br />
= V inj<br />
W cotan(α) over S W. (7)<br />
V = V suc<br />
W<br />
U = U suc<br />
W<br />
= σ V suc<br />
j<strong>et</strong> =<br />
q<br />
S h ρ suc<br />
σ over S W , (8)<br />
= V suc<br />
W cotan(β) over S W. (9)<br />
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