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

30 S. Mendez and F. Nicoud
Region total plate hole solid wall
∫
∫
∫
Expression
∫S (< −ρUV + τ12 >) n2 ds − < ρUV > n 2 ds < τ 12 > n 2 ds < τ 12 > n 2 ds
S h S h Ss
Injection 7.21 × 10 −1 114.1 −0.1 −14.0
Suction −2.83 × 10 −1 86.8 0.0 13.2
Table 5. Time averaged wall fluxes for streamwise momentum from Run C: First column:
expression and values of the total flux (in ρ jV j 2 d 2 ) on both sides of the plate (total surface S).
Columns 2–4: relative contributions (in %) of the terms involved in the wall fluxes.
Region total plate hole solid wall
Expression
∫
S < −P − ρV 2 + τ 22 >) n 2 ds
∫
S h
− < P + ρV 2 > n 2 ds
∫
− < P > n 2 ds
Ss
Injection 3.42 × 10 3 4 96
Suction −3.46 × 10 3 4 96
Table 6. Time averaged wall fluxes for vertical momentum from Run C: First column: expression
and values of the total flux (in ρ jV j 2 d 2 ) on both sides of the plate (total surface S).
Columns 2–3: relative contributions (in %) of the non-viscous terms involved in the wall fluxes.
Region total plate hole solid wall
∫
∫
∫
Expression
∫S (< −ρV W + τ32 >) n2 ds − < ρV W > n 2 ds < τ 32 > n 2 ds < τ 32 > n 2 ds
S h S h Ss
Injection −1.21 × 10 −4 38.1 −3.6 65.5
Suction 9.95 × 10 −5 116.9 −4 −12.9
Table 7. Time averaged wall fluxes for spanwise momentum from Run C: First column:
expression and values of the total flux (in ρ jV j 2 d 2 ) on both sides of the plate (total surface S).
Columns 2–4: relative contributions (in %) of the terms involved in the wall fluxes.
viscous stress tensor:
τ ij = µ( ∂V i
∂x j
+ ∂V j
∂x i
) − 2 3 µ∂V k
∂x k
δ ij (5.1)
where µ is the dynamic viscosity (µ = 1.788 × 10 −5 kg m −1 s −1 ), V i (i = 1, 2, 3) the
velocity components, x i (i = 1, 2, 3) the coordinates and δ ij the Kronecker symbol. Recall
that denotes time averaged quantities.
Several statements can be made from tables 5 to 7:
(a) streamwise momentum < ρU >: the non-viscous streamwise momentum flux (table
5, second column) is the main term for both the suction and the injection sides of the
perforated plate. The viscous term over the hole surface is very small. The wall friction
over the solid wall is approximately 8–10 times smaller than the non-viscous aperture
term for the operating point considered. This means that one can only focus on the inviscid
part of the flux when developing a first order model for effusion. In other words,
assuming that the turbulent transfers scale as the wall friction, turbulence is not a firstorder
issue when dealing with discrete effusion, which is of course significantly different
from the classical case of an attached boundary layer over a solid plate,
(b) vertical momentum < ρV >: the flux of normal momentum involves a pressure
term that is clearly dominant. As pressure is almost constant, the repartition of fluxes
between hole surface and solid wall surface corresponds to the porosity of the plate