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

Region total plate hole solid wall
Expression
∫ ∫
∫
S W
(−P − ρV 2 + τ 22 )n 2 dxdz
S h
−(P + ρV 2 )n 2 dxdz
S s
−P n 2 dxdz
Injection 3.64 × 10 3 4 96
Suction −3.69 × 10 3 4 96
Table 3. Wall fluxes for the vertical momentum: First column: expression and values of the
total flux (in ρ j V j 2 d 2 ) on both sides of the plate (total surface S W ). Columns 2–5: relative
contributions (in %) of the terms involved in the wall fluxes.
repartition between hole and solid surface fluxes is completely related to the porosity of the
plate σ = 0.04: pressure is almost constant over the whole wall.
Note that owing to the symmetry of the problem, the spanwise momentum flux should
be zero. In the computation, it is not perfectly so but for injection the spanwise momentum
flux is approximately 50 times smaller than the streamwise momentum flux. This ratio is
even smaller for the suction side.
An appropriate model has thus to reproduce the two main effects of the flow around a
perforated plate: the non-viscous streamwise momentum flux due to injection and the nonviscous
vertical momentum flux that can be reduced to a pressure effect. All the other terms
are negligible, at least in a first modeling effort. As it is usual in wall bounded flows, we will
consider that the outer pressure is a good measure of the pressure in the vicinity of the wall.
This is verified in the small-scale LES results. Thus, the pressure term can be easily related
to the pressure values obtained in the calculation. Hence, the main modeling effort consists
in obtaining a good estimation of the non-viscous streamwise momentum flux in the hole.
We are going to present the models in terms of equivalent boundary condition, answering
the following question: what is the equivalent injection/suction over the whole plate surface
that better represents the real injection/suction of fluid through cooling holes.
Before constructing homogeneous models over the whole surface, an intermediate model
that takes two different constant velocity values over the hole zone and the solid wall zone is
built. The wall-normal vertical velocity to impose is directly related to the mass flow rate.
The ratio between the vertical velocity and the streamwise velocity over the hole surface
depends on the side of the plate: indeed, the velocity at the hole outlet is strongly orientated
by the hole angle; on the contrary, at the hole entrance, the streamwise velocity highly
depends on the crossflow velocity. Let U c2 be the average streamwise velocity on the suction
channel. We assume the streamwise The intermediate model we propose reads:
Injection side:
V = V inj
jet =
q
S h ρ inj
over S h and V = 0 over S s , (1)
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