Further Studies of Airfoils Supporting Non-Unique Solutions in Transonic Flow

Non-unique solutions of the Euler equations were originally discussed by Jameson in 1991 ("Airfoil Admitting Non-Unique Solutions to the Euler Equations," AIAA Paper 1991-1625, June 1991) for several highly cambered airfoils which were the result of aggressive shape optimization. In 1999 Hafez and Guo ("Nonuniqueness of Transonic Flows," Acta Mechanica, Vol. 138, Nos. 3-4, 1999, pp. 177-184) found non-unique solutions for a symmetric parallel sided airfoil, and subsequently Kuz'min and Ivanova ("The Structural Instability of Transonic Flow Associated with Amalgamation/Splitting of Supersonic Regions," Theoretical and Computational Fluid Dynamics, Vol. 18, No. 5, 2004, pp. 335-344) 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, are 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 C-infinity continuous. C-L - alpha plots of these airfoils exhibit three branches at 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 a in some cases there is a transition to the Z-branch, while in other cases there is a direct transition from the P- to the N-branch.