A Comparative Study of the Nonuniqueness Problem of the Potential Equation

where a is the speed of sound, p is the density, _ is the potential, and the subscripts denote partial deriva-The nonuniqueness problem occurring at transonic tives. In the absence of shock waves, both forms of the speeds with the conservative potential equation is in-equation are equivalent. When shock waves are present, vestigated numerically. Evidence is given supporting use of the conservative form in conjunction with an "en-tile thesis that the nonuniqueness problem is inherent tropy" condition guarantees (see ref. 1) that the proper _ in the potential differential equation rather than a con-shock jump condition is satisfied. On the other hand, sequence of the finite-difference approximation. Results use of the nonconservative form leads to an ambigu-are presented from an extensive comparative study be-ous jump across the shock waves. With few exceptions, tween potential and Euler calculations for flow past present-day state-of-the-art working transonic potential two-dimensional airfoil profiles. This study indicates equation codes use the conservative form. that the nonuniqueness problem is not an inviscid phe-For subcritical flows, Bers (ref. 2) has shown that nomenon, but results from the approximate treatment the solution for flow past an airfoil section satisfying of shock waves inherent in the conservative potential the Kutta condition is unique. In the transonic regime, model. A more restrictive bound on the limit of valid-little has been proven rigorously in terms of uniqueness. ity of the conservative potential model is suggested. However, in a recent paper (ref. 3), Steinhoff and Jame