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

34 S. Mendez and F. Nicoud
geometrical angle. Over the first half of the row, the orientation of the flow progressively
changes and is nearly aligned with the hole direction between y = −1.2 d and y = −0.4 d
(29 ◦ ). Near the hole outlet, the angle continues to decrease to reach 28 ◦ at the hole outlet.
Note that α is different from the classical definition of the jet angle, which is assessed
at the location of maximum velocity magnitude, as the jet trajectory (see § 4.5). The
jet angle in the aperture is also reported in figure 22. The jet angle is close to α at the
hole outlet (27.5 ◦ ) but it is more than 90 ◦ at the hole inlet. This is consistent with the
pressure gradient at the hole inlet, which is almost vertical near the sharp edge of the
hole, where the velocity magnitude is maximum.
It follows from figure 22 that the geometrical angle is not relevant to the flow behaviour
near the hole inlet. On the contrary, the flow is almost aligned with the hole at
the outlet: the streamwise velocity at the hole outlet is strongly related to the vertical
velocity, through the hole angle, the angle mismatch being only 2 ◦ . Note however that
this difference is not without consequence: the averaged streamwise velocity at the outlet
plane U ts would most likely be of interest in any model aiming at reproducing the momentum
transfer through the plate. Assessing this quantity from the averaged vertical
velocity at the outlet plane V ts (easily assessed from the global injection mass flow rate)
and the hole angle as U ts = V ts /tanα g would lead to an error of approximately 10%
since tan30/ tan28 ≈ 1.1.
Putting the previous discussion into a modelling perspective, the following conclusions
can be drawn:
(a) momentum fluxes over the suction plane and the injection plane are dominated by
inviscid contributions. Wall friction is not the first-order effect for permeable plates, at
least when the blowing ratio is not very small. Note however that the viscous contribution
through wall friction is small but not negligible,
(b) vertical and spanwise momentum fluxes can easily be estimated and do not demand
any modelling effort, at least as long as the main flows are aligned with the aperture midplane,
(c) regarding the injection side of the plate, the non-viscous streamwise momentum
flux can be approximated by:
∫
−ρ UV t 2
m˙
h
n 2 ds ≈
(5.5)
S h
ρ tan α g
where m˙
h is the mass flow rate per unit area in the hole, viz. m˙
h = (1/S h ) ∫ S h
ρV t ds.
This approximation leads to an error of 18 %. The difference is mainly due to the two
assumptions that α can be assessed by α g and UV ts by U ts V ts at the hole outlet. In
order to improve the assessment of the inviscid contribution to the streamwise momentum
flux on the injection side, one should improve these two assumptions,
(d) at the suction side of the plate, the flow direction is not controlled by the hole
orientation. Instead, it is directly related to the streamwise velocity in the cold crossflow.
Approximating the non-viscous streamwise momentum flux at the suction side by:
∫
−ρ UV t n 2 ds ≈ U 2 m˙
h
S h
(5.6)