Vassberg, John C.
Boeing Technical Fellow, The Boeing Company, AIAA Fellow
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Published in 2011
Non-unique solutions of the Euler equations were originally discussed by Jameson in 1991 for several highly cambered airfoils which were the result of aggressive shape optimization. In 1999 Hafez and Guo found non-unique solutions for a symmetric parallel sided airfoil, and subsequently Kuzmin and Ivanova have discovered some fully convex symmetric airfoils that provide non-unique solutions. In this article four new symmetric airfoils, all of which exhibit non-unique solutions in a narrow band of transonic Mach numbers, were studied. The first, NU4 was the result of shape optimization. The second, JF1 is an extremely simple parallel sided airfoil. The third JB1, is also parallel sided but has continuous curvature over the entire profile. The fourth, JC6, is convex and C1 continuous. CL − plots of these airfoils exhibit three branches of zero angle of attack, the P, Z and N-branches with positive, zero and negative lift respectively. At some Mach numbers no stable Z-branch could be found. When the P-branch is continued to negative in some cases there is a transition to the Z-branch, while in other cases there is a direct transition from the P to N-branch.Download PDF