these simulation numerique et modelisation de l'ecoulement autour ...
these simulation numerique et modelisation de l'ecoulement autour ... these simulation numerique et modelisation de l'ecoulement autour ...
LES of a bi-periodic turbulent flow with effusion 31 Figure 20. Time-averaged skin friction coefficient C f on both sides of the liner. The hole outlet/inlet is represented with a hatched ellipse. (a): View of the liner from the injection side. (b): View of the liner from the suction side. (σ ≈ 4%). Note that the velocity term in the hole does not modify this repartition: it is small compared to the pressure term. This is true as long as the pressure drop is small compared to the operating pressure, which is the case in gas turbine applications, (c) due to the symmetry of the problem, the spanwise momentum flux should be zero. The computation is almost symmetrical, the spanwise momentum flux being 1000 times smaller than the streamwise momentum flux. Even if the porosity of the plate is small (σ ≈ 4%), table 5 shows that wall friction is not the main effect at the wall. However, the skin friction at the wall can be locally high, as shown in figure 20, where the distributions of the skin friction coefficient over the suction and injection walls are presented. The skin friction coefficient is C f = 2 τ w / ( ρ j V j 2 ) , where τ w is the total wall shear stress. As expected from the velocity field analysis, the friction at the wall is strongly inhomogeneous. On the injection side (figure 20a), the field of wall friction shows a structure corresponding to the characteristics of the flow described in §4.5: upstream of the jet, the flow is lifted off and low values of skin friction coefficient are obtained. The primary counter-rotating vortex pair is observed thanks to the high values of wall shear stress. The CVP accelerates the flow near the wall, resulting in higher wall shear stress downstream of the aperture edges. Downstream of the jet separation, the velocity near the wall is small and low values of wall shear stress are observed. Just downstream of the hole, two lobes exhibit very low value of wall shear stress. This feature has to be related with downstream spiral separation node vortices (Peterson & Plesniak 2004a,b). The structure of the wall shear stress is very similar to the one observed experimentally by Peterson & Plesniak (2004b). Note however that due to the reattachment induced by the entrainment of the main flow towards the wall, the region of skin friction deficit stops 3.5 diameters downstream of the hole while it extends more than 10 diameters in the normal hole case (Peterson & Plesniak 2004b). Downstream of the deficit region, traces of the secondary pair labelled (6) in figure 15 are detected in figure figure 20a. On the suction side (figure 20b), the presence of lowvelocity zones is also observed, with small values of wall shear stress. Maximum values are observed all around the hole inlet, where the acceleration of fluid induced by the aspiration is the strongest. Just downstream of the hole inlet centre, the fluid reattaches, showing high values of wall shear stress. These features are consistent with the flow structure described in §4.2.
32 S. Mendez and F. Nicoud Expression −ρ UV ts n 2 −ρ U ts V ts n 2 −ρ (U t) s ( t) s s V n 2 −ρ (U) t (V ) tts n 2 Injection 5.24 × 10 −1 89.2 10.8 < 1% Suction −1.56 × 10 −1 109.6 −9.6 < 1% Table 8. Decomposition of the non-viscous contribution of the streamwise momentum flux per unit surface from Run C: First column: values of the total contribution (in ρ jV j 2 ) on both sides of the plate. Columns 2–4: relative contributions (in %) of the terms detailed in equation 5.4. 5.2. Assessment of the non-viscous fluxes at the perforated plate A key step for the modelling of effusion cooling is to estimate mass/momentum/energy fluxes at both sides of the perforated wall. The amount of air passing through the holes is related to the pressure drop across the plate, through a discharge coefficient. The object of this section is not to propose an assessment of the discharge coefficient but to determine the momentum fluxes at the wall at a given mass flux through the hole. In views of the results presented in tables 5 to 7, assessing the vertical and the spanwise momentum fluxes is straightforward: the vertical momentum flux at the wall is directly related to the value of the pressure at the plate, a quantity that can easily be estimated in the RANS context from the pressure value at the first off-wall mesh point. As the perforation does not have any spanwise orientation, the spanwise momentum flux at both sides of the plate is null. On the contrary, the streamwise momentum flux cannot be determined easily. In the remainder of this section, focus is made on the possibility to propose a rough estimation of the main contribution to this flux, viz. its inviscid part through the hole inlet/outlet. Note first of all that for any quantity φ(x,t) that depends on both space and time, the following decompositions can be considered: φ(x,t) = φ t (x) + (φ) t (x,t) (5.2) φ(x,t) = φ s (t) + (φ) s (x,t) (5.3) where t is the time-averaging operator and s denotes the spatial-averaging operator over the hole inlet or outlet surfaces (S h ). Consistently, () t and () s denote the fluctuations from the time and spatial averages. Note eventually that t is nothing but the < > operator used in the previous sections. Combining these two decompositions, the inviscid flux per unit hole surface can be written as (the mass density ρ is supposed to be constant in space and time over the hole inlet/outlet surface): ( −ρ UV ts n 2 = −ρ U ts V ts + (U t) s ( V t) s s + (U) t (V ) tts) n 2 (5.4) The first term is the product of time and spatial averaged quantities, the second one estimates the non-uniformity and correlation of time averaged velocity components over the hole surface. The third term is the spatial average of the classical Reynolds stress based on time averaging. The values of each of these terms are reported in table 8, for the hole outlet (injection) and inlet (suction). A reasonable estimation of the non-viscous streamwise momentum flux per unit surface of aperture is obtained from the product of time and spatial-averaged quantities. Assessing UV ts by U ts V ts leads to an error of approximately 10%. This difference is mainly due to the non-uniformity of the timeaveraged velocity field over the hole inlet and outlet surfaces. The term of fluctuations in time does not contribute in the mean although the turbulence activity is substantial,
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LES of a bi-periodic turbulent flow with effusion 31<br />
Figure 20. Time-averaged skin friction coefficient C f on both si<strong>de</strong>s of the liner. The hole<br />
outl<strong>et</strong>/inl<strong>et</strong> is represented with a hatched ellipse. (a): View of the liner from the injection si<strong>de</strong>.<br />
(b): View of the liner from the suction si<strong>de</strong>.<br />
(σ ≈ 4%). Note that the velocity term in the hole does not modify this repartition: it is<br />
small compared to the pressure term. This is true as long as the pressure drop is small<br />
compared to the operating pressure, which is the case in gas turbine applications,<br />
(c) due to the symm<strong>et</strong>ry of the problem, the spanwise momentum flux should be zero.<br />
The computation is almost symm<strong>et</strong>rical, the spanwise momentum flux being 1000 times<br />
smaller than the streamwise momentum flux.<br />
Even if the porosity of the plate is small (σ ≈ 4%), table 5 shows that wall friction is not<br />
the main effect at the wall. However, the skin friction at the wall can be locally high, as<br />
shown in figure 20, where the distributions of the skin friction coefficient over the suction<br />
and injection walls are presented. The skin friction coefficient is C f = 2 τ w / ( ρ j V j<br />
2 ) ,<br />
where τ w is the total wall shear stress. As expected from the velocity field analysis, the<br />
friction at the wall is strongly inhomogeneous. On the injection si<strong>de</strong> (figure 20a), the<br />
field of wall friction shows a structure corresponding to the characteristics of the flow<br />
<strong>de</strong>scribed in §4.5: upstream of the j<strong>et</strong>, the flow is lifted off and low values of skin friction<br />
coefficient are obtained. The primary counter-rotating vortex pair is observed thanks to<br />
the high values of wall shear stress. The CVP accelerates the flow near the wall, resulting<br />
in higher wall shear stress downstream of the aperture edges. Downstream of the j<strong>et</strong><br />
separation, the velocity near the wall is small and low values of wall shear stress are<br />
observed. Just downstream of the hole, two lobes exhibit very low value of wall shear<br />
stress. This feature has to be related with downstream spiral separation no<strong>de</strong> vortices<br />
(P<strong>et</strong>erson & Plesniak 2004a,b). The structure of the wall shear stress is very similar to<br />
the one observed experimentally by P<strong>et</strong>erson & Plesniak (2004b). Note however that<br />
due to the reattachment induced by the entrainment of the main flow towards the wall,<br />
the region of skin friction <strong>de</strong>ficit stops 3.5 diam<strong>et</strong>ers downstream of the hole while it<br />
extends more than 10 diam<strong>et</strong>ers in the normal hole case (P<strong>et</strong>erson & Plesniak 2004b).<br />
Downstream of the <strong>de</strong>ficit region, traces of the secondary pair labelled (6) in figure 15<br />
are d<strong>et</strong>ected in figure figure 20a. On the suction si<strong>de</strong> (figure 20b), the presence of lowvelocity<br />
zones is also observed, with small values of wall shear stress. Maximum values<br />
are observed all around the hole inl<strong>et</strong>, where the acceleration of fluid induced by the<br />
aspiration is the strongest. Just downstream of the hole inl<strong>et</strong> centre, the fluid reattaches,<br />
showing high values of wall shear stress. These features are consistent with the flow<br />
structure <strong>de</strong>scribed in §4.2.
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