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solutions, this implies that the GA evolves during a given number of generations, and finally stagnates, generating random individuals in the neighborhood of the optimum without never reaching it. The two optimal solutions obtained in the two runs are represented in Figure 47(a) along with the classical NACA0012 airfoil shape, which is taken as a reference. The corresponding Mach number distributions at the wall are displayed in Figure 47(b). The drag coefficients obtained in the two cases are equal to 7.02×10-3 and 6.94×10-3 (it equals 1.64×10-2 for the reference NACA0012 airfoil). In order to evaluate the fitness distance correlation of the present minimization problem, an estimation of the solution global optimum is required. This is obtained by refining the GA solution by means of a gradient-based method (see 3.2). The subset of designs used to compute the FDC is the global GA population generated during 30 generation for both optimization runs (about 1800 individuals, some repeated). The resulting FDC value, equal to about 0.7, is relatively far from the ideal value FDC=1: as a consequence, the problem can be considered to be GA-hard. GA convergence difficulties can be related to the fact that very weak shocks characterize the best individuals, which provide an almost shock- free flow. Because of numerical errors, a numerical “viscous” drag adds to the small physically relevant wave drag. For sufficiently similar individuals, it is reasonable to believe that this numerical drag, related to the chosen grid density and numerical scheme, is essentially independent from the particular airfoil shape, contrarily to wave drag that is more sensitive to shape changes. In the neighborhood of the global optimum, the effect of numerical drag is to “smooth out” the fitness function minimum, i.e. to reduce fitness function concavity. In these conditions, a GA will continue to generate randomly individuals characterized by more or less the same small drag coefficient without being able to detect the optimal one. In order to check the influence of numerical errors on GA convergence, optimization runs are repeated using a finer grid of 150×50 cells, mean height of the closest cell to the wall 10-3 chords. Grid refinement reduces numerical errors and provides sharper shock waves. Optimal shapes obtained using the finest grid are much closer each other (1.9% differences) and to the optimal value determined by refining the solution with the gradient-based method. The FDC for this case is 0.852, a closer value to 1. This result supports the conclusion that, for transonic flows of perfect gas characterized by weak shock waves, numerical viscosity behaves as a low-pass filter that levels a variety of low-drag profiles close to the optimum to the same degree of fitness. 108PDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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