Numerical solutions of the triple-deck equations for laminar trailing-edge stall

Abstract

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...

Interacting Laminar and Turbulent Boundary Layers

“…This has been confirmed recently by Korolev (1989) who was able to extend the numerical solutions of the triple deck problem into the separated flow regime. According to these results the wall shear on the upper plate surface at x = 0-reaches a minimum value Tw(O-) = O for a = 0.47 in agreement with the estimate for incipient separation given by Chow and Melnik (1976). For larger values of a the boundary layer on the upper plate surfaces separates but reattaches upstream of the trailing edge so that Tw(O-) now increases with increasing angle of attack.…”

“…The most simple example is provided by the flow of an incompressible fluid past a flat plate at incidence. If the angle of attack a* is small, the flow ne ar the trailing edge can stiH be investigated by means of triple deck theory, Brown and Stewartson (1970), Chow and Melnik (1976). In contrast to the case of aligned flow, however, two triple decks, one on the suction si de and one on the pressure side of the plate, have to be considered separately.…”

“…function al(a) has been determined numerically by Chow and Melnik (1976). If a is small the flow remains attached as in the case of aligned flow.…”

“…Incipient separation was estimated to occur for a ~ 0.47 but numerical difficulties prevented the calculation of solutions up to this value. It has been conjectured (Brown and Stewartson (1970), Chow and Melnik (1976)) that the rapid increase of al near the point of the incipient separation which is associated with a rapid decrease of eL heralds the onset of catastrophic stall. This point of view was contested by Smith (1983) who provided strong analytical and numerical arguments that in the case of one-sided separat ing flow boundary layer attachment will occur immediately upstream of the trailing edge.…”

Marginal Separation

“…This is the flow past a thin airfoil at an angle a* of incidence, the airfoil being suitably rounded at the leading edge to prevent separation there and having a sharp trailing edge. Chow and Melnik [3] have shown that as a* increases a solution for the trailing-edge region is only possible if a< 0.47, where a* = >...9/ 8 R-…”