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In this paper, we consider the problem of determining the effect of laminar boundary layers on the lift of thin wings in subsonic flow at high Reynolds numbers°The viscous flow about an airfoil at high Reynolds numbers is largely controlled by a strong local interaction between the boundary layer, wake, and external potential flow near the trailing edge. The standard (weak) interaction theory for laminar flows develops singularities in the second-order inviscid solution and is not uniformly valid at trailing edges.The singularities are such that the pressures induced by the boundary layer are unbounded at trailing edges, and, consequently, a Kutta condition cannot be enforced in the second-order solution and the lift correction cannot be determined.Brown and Stewartson considered the problem in Ref. i, where they developed a formal asymptotic theory for strong viscous interactions at airfoil trailing edges in the ~imit e-+ O, where e is related to the Reynolds number by e = R'~. The theory led to a three-layer or "triple deck" structure of the flow near the trailing edge over a streamwise extent of e 3. The triple-deck structure is now known to be a common feature of many laminar interaction problems (e.g., see Ref. 2). The theory of Brown and Stewartson represents an extension to the lifting case of the earlier works of Stewartson (Ref. 3) and Messiter (Ref. 4) for a flat plate at zero incidence.This theory results in a completely consistent formulation for determining the viscous modification to the Kutta condition.A nonlinear boundary-value problem was formulated that must be solved to determine the lift correction and to predict laminar trailing-edge stall. The boundary-value problem depends on a single parameter, a, defining the relative incidence of the plate where ~ = e-~ % ~ ~ , % is the Blasius constant (% = 0.33206), and ~* is the incidence angle. Coordinate expansions were developed in Ref. I to describe the analytic behavior of the solution in the far field and near an important singularity at the trailing edge. Although a very approximate linear solution was obtained in Ref. i, no attempt was made to construct accurate numerical solutions of the nonlinear boundary-value problem.Numeri.cal solutions for the triple-deck equations for the symmetric problem (~ = O) were obtained in three independent investigations: by Jobe and Burggraf (Ref. 5), Veldman and van de Vooren'(Ref. 6), and by the present authors (Ref. 7). Finite-difference methods were employed in all three studies. Although the methods used in the studies differed in many details, very good agreement was obtained between the three sets of results. The method developed by the present authors was also 136 applicable to the general problem for a plate at incidence, and a single solution for ~ = 0.10 was presented in Ref. 7o Additional numerical experiments performed after Ref~ 7 was written indicated the need to modify the program to improve the accuracy.These modifications have been made, and additional numerical computations have been carri...

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Summarywhere a is the speed of sound, p is the density, _ is the potential, and the subscripts denote partial derivaThe nonuniqueness problem occurring at transonic tives. In the absence of shock waves, both forms of the speeds with the conservative potential equation is inequation are equivalent. When shock waves are present, vestigated numerically.Evidence is given supporting use of the conservative form in conjunction with an "entile thesis that the nonuniqueness problem is inherent tropy" condition guarantees (see ref. 1) that the proper _ in the potential differential equation rather than a conshock jump condition is satisfied. On the other hand, sequence of the finite-difference approximation.Results use of the nonconservative form leads to an ambiguare presented from an extensive comparative study beous 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 pheFor 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 validlittle 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-