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4.3. Shape Optimization of an isolated Airfoil 4.3.1. Inviscid Case 4.3.1.1. Drag Minimization for a symmetric airfoil The problem is to find the minimal value of the drag coefficient CD (Σ)where Σ denotes an airfoil symmetric shape. In the present case, this shape is constrained to satisfy the following conditions: i) the coordinates of the leading edge and trailing edge normalized by the airfoil chord are respectively (0, 0) and (1, 0), ii) the thickness-to-chord ratio is 12%. The upper airfoil surface is represented by a Bezier curve which is determined in the present study by the coordinates of 8 Bezier points: the fixed leading edge P0 and trailing edge P , 5 control points P (x , y ), regularly spaced along the chord, with the x 7 k=2,6 k k k coordinates varying in distinct sub-intervals of ]0,1[ and a control point P (0, y ), which 11 ensures the upper surface of the profile is tangent to the y-axis at leading edge. A fixed 12% thickness-to-chord ratio is obtained by ad-hoc normalization of the airfoil maximum thickness. The family of airfoil shapes considered in the present study is therefore entirely described by 11 parameters, which vary continuously between prescribed limiting values; the shape representation is such that it allows recovering the NACA0012 airfoil as well as the low-drag airfoil proposed in [25]. The minimization of CD (Σ) is first performed for three different values of the freestream Mach number, successively taken equal to M ∞ = 0.94, 0.95, 0.98 . In each case, the flow over a population of evolving (half)-airfoil shapes is computed using the UGS on the grid made of 6400 quadrilateral cells with a grid size of 5×10−3c at the wall, where c denotes the airfoil chord, 120 points along the airfoil surface and an outer boundary about 10 chords away from the airfoil. VDW equation of state is used with γ =1.02 and the freestream thermodynamic conditions normalized by their critical values are taken equal to p ∞ / p c = 1.0696 , ρ ∞ / ρ c = 0. 73502 i n or d e r t o al l o w a c o mpa r i s o n w i t h t he r e s u l t s obtained in [25]. For each run, the GA-driven population appears to have reached a stabilized optimal (minimal) value of the drag coefficient after the prescribed number of generations, as can be deduced from Figure 19: since the maximum and mean value of the drag coefficient in the final population of airfoil shapes are almost the same in each case, 69PDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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