Computational challenge - Euler solution for ellipses

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Abstract
T HIS paper presents a difficult flow problem for the nu- merical solution of the Euler equations: the inviscid flow past an ellipse. The basic result obtained here is a lifting solu- tion for any combination of grid and/or angle of attack which is nonsymmetric. The purpose of this paper is to present this flow as a challenge to the computational fluid dynamics com- munity, where the hope is that someone can explain this unu- sual behavior. I. Introduction The purpose of this paper is to introduce a flow problem for the Euler equations that should be relatively simple to solve numerically, but turns out to be very difficult. The problem is inviscid subcritical flow past an elliptical two-dimensional sur- face at angle of attack. In particular, a 6:1 ellipse at Mx = 0.2, a: = 5 deg is considered, where conventional finite-difference or finite-volume schemes are employed. Assuming initial and boundary conditions which are irrotational and contain no cir- culation, one would expect that the inviscid flow at any angle of attack would remain irrotational, and not generate any cir- culation. In practice, every numerical solution obtained to date by this author and many others has produced results with nonzero lift. The results obtained are very sensitive to algo- rithm parameters, mesh definition, and algorithm type. Even on the same mesh, various codes produce a wide scatter of lift values all of which are nonzero. It can be shown, however, that the solutions obtained are equivalent to potential flow past an ellipse with added circulation defined by the lifting nu- merical solution. This suggests that the numerical solutions obtained are reasonable solutions to the Euler equations with some mechanism which sets the lift. The mechanism responsi- ble for the circulation generation is not understood at this time, and this paper stands as a challenge for other researchers to define the mechanism or to produce numerical results which are nonlifting for this class of problems.

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