The closely coupled canard-wing interaction, iucluding dihedral effects, has been modeled using the transonic small disturbance theory with cross derivative terms retained. A numerical mapping procedure has been applied to treat the leading and trailing edges of the canard-wing, and their mean chord planes as constant coordinates in the computational domain. This is a local mapping concept, in contrast to the global mapping procedures such as the wing shearing transformation, which require separate computational domains for the canard and the wing, based on their planform shapes. Local mapping concepts, widely used in the Euler and full potential studies, and applied to the small disturbance theory in this study, offer flexibility in treating complex geometries. Transonic results are presented for two different configurations, one with zero dihedral, and the other with dihedral. Introduction A number of highly maneuverable fighter configurations, such as the experimental HiMAT and the forward swept wing X-29A technology demonstrator have been proposed with closely coupled canard systems which can lead to several advantages such as higher trimmed-lift capability, improved pitching moment characteristics, and reduced trim drag. The associated interaction due to such surfaces with the wing as well as those from conventional tail planes involve important nonlinear phenomena in the supercritical speed regime. These effects can significantly change spanwise load distributions as well as the effective incidence field. Corresponding modifications of aerodynamic performance and stability characteristics are therefore to be anticipated not only for fighter configurations, but with tails interacting with large-aspect-ratio wings typical of transFort arrangements as well. Computational modeling of the transonic canard-wing problem, due to its geometric complexity, has been restricted to the use of modified transonic small disturbance theory. More exact models like the full potential theory or Euler theory for canard-wing configurations require compli* Manager, Computational Fluid Dynamics, Associate t Member Technical Staff Cop>iighi ? 4 m e r i ~ m Inriitute of heroniuiirrnnd .~s~ronaulics. Inc.. 1085. AIi riplilr reserved. Fellow AIAA 1 cated body-fitted coordinate mapping routines, and therefore, are not attempted at this point. Even the simplistic modified small disturbance theory approach requires coordinate transformations to map the arbitrarily shaped leading and trailing edges of the wing and canard, and their mean chord planes into constant coordinates. For a typical canard-wing geometry the usual procedure is to use a separate shearing transf~riiiation'-~ (global mapping concept) for the canard and wing that maps the arbitrary planform into a rectangle. In this approach, the computational domains for the canard and the wing are separated due to the use of different transformation for each lifting surface. The information transfer from the canard to the wing and vice versa takes place indirectly through the use of a global crude grid that embeds both the transformed canard and wing doninins. This global mapping concept, though widely used in small disturbance theory studies, gets very complicated in its computational implementation as the complexity of the geometry is increased. To alleviate some of the computational difficulties in treating closely coupled canard-wing geometries, an alternate approach based on the local mapping concept, is employed in this paper. Local mapping procedures, hitherto have been applied to the full potential and Eder e q u a t i o ~ s ~ ~ . The extension of the local mapping concept to the small disturbance framework of this present study is, however, new. In what follows, the paper describes the small disturbance theory equation in Cartesian coordinates, appropriate for treating canard-wing geometries with dihedral and presents the numerical transformation procedure employed to perform the local mapping. The method for applying the boundary conditions within a modified form of the small disturbance theory with cross terms is described. Results are presented for two different canard-wing geometries. One is a canard-wing research model with zero dihedral and the other is the HiMAT configuration with dihedral. Comparisons of the present computational results are made with experimental data. OPGENOMEN IN uEAUTOMAT;I:C.CERDE Formulation In a Cartesian system, the modified transonic small disturbance equation in conservation form can he written as E, +Fu + G, = O (1) F = FCSD + Fczv G = GCSD + ccz; where the subscript C S D stands for 'classical small disturbance" and Czy and Czz denote cross terms in (x-y) and (x-z) planes, respectively. The various terms in Eq. (1) can be described as The (x-y) cross terms of Q. (2) were first introduced hy Lomax' and later used hy Ballhaus8. In the present study, the cross terms in (x-z), similar to the ones in (x-y), are introduced to effectively treat canard-wing geometries with dihedral, as well as pylon or winglet usually aligned with the (x-z) plane. In the present analysis, x is downstream, y is spanwise, and z is vertical. In order to handle geometries with dihedral, Figure 1, a general mapping is introduced to align the leading and trailing edges of the canard and the wing, and their mean chord planes into constant coordinates, Figure 2. The mapping can be expressed in the form F = F ( z , y , z ) rl = rl(? Y , 4 . (3) f = f (z ,Y ,4 It can be shown that the transformed form of Eq. (1) can he written in conservation form as Ee + Fv + G , = o (4) --