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this means the algorithm has converged to a population of almost identical geometries. The optimized airfoil shape for each freestream Mach number is plotted in Figure 20 along with the NACA0012 airfoil, the low-drag airfoil and the sonic arc taken from [25]. It is noteworthy to observe the optimal shape for M ∞ = 0.94 (indicated in the following as OA094) turns out to be almost superimposed with Rusak and Wang’s low-drag airfoil; consequently, this latter shape will be retained in the following analysis as the one producing minimal drag at M ∞ = 0.94 . Reducing the drag for higher values of the free- stream Mach number leads to airfoil shapes that are getting closer to the limiting case of the sonic arc, as far as allowed by the airfoil shape representation retained in this study. Since, ideally, a consistent drag reduction is expected for the whole range of freestream Mach numbers, the wave drag produced by these modified shapes, which are optimized for a specific value of M ∞ , are next checked for different values of M ∞ , taken in the interval [0.9,0.995]; this a posteriori analysis is performed using both the SGS and the UGS for cross-validation purpose. The computed evolutions of the drag coefficient with increasing values of the freestream Mach number are plotted in Figure 21: the optimal shape at M ∞ = 0.95 (OA095) allows to obtain lower drag levels than the OA094 airfoil for the whole range of freestream Mach number, both shapes being far superior to the original NACA0012 airfoil; on the contrary, the optimal shape at M ∞ = 0.98 (OA098) displays a very poor performance for lower values of the freestream Mach number. All of the drag curves fall within the envelope corresponding to the sonic arc performance. The MOGA strategy offers the possibility of looking for an airfoil that would simultaneously minimize the drag coefficient at M ∞ = 0.95 and M ∞ = 0.98 . Such an airfoil should hopefully provide a good drag performance over a larger range of free-stream Mach numbers. To obtain such an airfoil, a bi-objective optimization run is computed, using the same GA parameters as in the previous mono-objective runs; among the set of optimal shapes provided by the MOGA, a typical trade-off solution has been retained (OAB), which is plotted in Figure 20. The associated drag evolution is also displayed in Figure 21: the OAB airfoil turns to yield lower drag levels than the previous single operation point optima for the whole range of freestream Mach number, because its geometry is actually closer to that of the sonic arc. The wall pressure and Mach number distributions at M ∞ = 0.95 and M∞ =0.98 are plotted in Figure 22 for the various geometries under study. At 70PDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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