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M ∞ = 0.95 , all of the airfoils except the sonic arc are beyond their respective critical Mach number; in fact, the NACA0012, Rusak-Wang and OA098 airfoils are also beyond the drag divergence Mach number, whereas the OA095 and OAB airfoils still yield isentropic flows. Note the OA098 airfoil displays a shock in the leading edge region down to M ∞ = 0.90 . At M ∞ = 0.98 , all of the airfoils, including the sonic arc, are subjected to the transonic drag rise: the OA095 and OAB wall distributions are similar to the ones associated with the sonic arc up to 70% of chord, where the shapes of the three airfoils start to strongly differ (see Figure 20). A similar optimization process has been carried out for a flow of perfect gas with γ = 1.4 as well as a dense gas flow with p∞ / pc = 1.0696 , ρ∞ / ρc = 0.73502 modeled using the more realistic MAH equation of state. While the VDW equation of state was retained up to now in order to allow a comparison with the results obtained in [25], the analysis performed with MAH and PFG indicates the improvement in terms of drag reduction that can be expected (for inviscid flows) when switching from air to PP10 with airfoil shapes optimized according to the nature of the fluid. Various optimization strategies were applied to the PFG case: single-point drag minimization for different free-stream Mach numbers in the transonic drag rise region, simultaneous minimization of drag for two values of M∞ above the drag divergence Mach number of the baseline NACA0012 airfoil. The overall best performance, for the shape representation described at the start of this section, is plotted in Figure 23 : the optimal airfoil, denoted OA083, is specifically targeted to achieve minimal (zero) drag at M ∞ = 0.83 so that it suffers from a slight increase in wave drag at M∞ =0.82 due to the appearance of a weak shockwave. For the MAH case, the best performance was obtained with an airfoil (OAB) designed to minimize drag at M ∞ = 0.925 and M ∞ = 0.95 . As shown in Figure 23, the optimal airfoil increases the drag divergence number from M ∞ = 0.91 for the NACA0012 airfoil up to almost M ∞ = 0.93 . Figure 24 displays the strong discontinuities produced by the baseline NACA0012 airfoil in the drag divergence region associated with each gas model (PFG, VDW and MAH) and presents the pressure contours of the corresponding shock-free airfoils (just before the drag divergence Mach number is attained). 71PDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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