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about 20.3% lower than the computed value for a flow at the same conditions past the reference airfoil NACA0012 (CD= 1.22×10-3), due to a considerable reduction of shock- wave strength for the flow past the optimal shape with respect to the reference flow past the NACA0012. It also significantly improves previous results returned by the GA (about 1.1×10-3 in average). The overall CPU time for hybrid optimization runs required about 1300 fitness function evaluations, 800 required by the GA and 500 by the BFGS. This is due to the relatively high number of design variables used to parameterize the airfoil shape. In the above, it was shown that for flows characterized by very low wave drag (typically for dense gas flows, but also for perfect gas flows at Mach numbers slightly above the critical one) the solution of the drag minimization problem is very slow and sensitive to numerical parameters used in the flow solver, such as numerical dissipation introduced by the scheme and the computational mesh used to evaluate the fitness function. It was also shown that the use of a more accurate scheme and of finer meshes allows reducing GA- hardness of a given problem. Unfortunately, the use of very fine meshes considerably increases computational costs. In order to alleviate this problem, a simple and effective method has been proposed in 3.3.2. The above strategy is now applied to one of the more difficult test cases of the previous section. Precisely, we consider again the drag minimization problem for a BZT van der Waals gas flow over a symmetric airfoil, with freestream conditions M∞=0.95, α=0°, p∞ / pc = 1.0696, ρ∞ / ρc = 0.735 . Firstly, a reference solution is computed by using a very fine mesh composed by 200×64 cells. CPU time required for a single fitness evaluation on this grid is about 40 minutes on a Pentium V processor. The whole optimization process requires about 800 fitness evaluations. Then, the same optimization run is performed by applying the multigrid strategy described above: for each individual, the fitness function is first computed using a coarse grid of 50×16 cells, then using a finer grid of 100×32 cells, finally a third value is extrapolated via Richardson’s formula. The total CPU time required to obtain the final estimate of the fitness function is about 5 minutes! This noticeable reduction in computational time is due, on the one hand to the lower number of operation required to compute solutions on the coarser grids and, on the other hand to higher convergence rates of the flow solver. An important role is also plaid by initialization of the medium grid computation with the coarse grid solution. Let us recall that the computed FDC parameter of the GA for this 113PDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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