Proceedings of COBEM 2005 18th International Congress of Mechanical Engineering Copyright © 2005 by ABCM November 6-11, 2005, Ouro Preto, MG VISCOUS-INVISCID INTERATION FOR THE SOLUTION OF THE FLOW ABOUT AIRFOILS Paulo Vatavuk Faculdade de Engenharia Civil, Arquitetura e Urbanismo, UNICAMP, Cx. Postal 6021, CEP 13083-852, Campinas, SP, Brasil. pvatavuk@fec.unicamp.br Márcio Teixeira Mendonça Instituto de Aeronáutica e Espaço, CTA, Pç Mal. Eduardo Gomes, 50, CEP 12228-904, São José dos Campos, SP, Brasil Marcio_tm@yahoo.com João Luiz F. Azevedo Instituto de Aeronáutica e Espaço, CTA, Pç Mal. Eduardo Gomes, 50, CEP 12228-904, São José dos Campos, SP, Brasil azevedo@iae.cta.br Abstract. Viscous-inviscid iteration is a technique for the analysis of flows that although not being as precise as a complete Navier-Stokes solution allows very small computational times and that makes it an ideal tool for preliminary or optimization studies. This work reports the coupling of an Euler solver with a differential boundary layer code, using the Cebeci-Smith turbulence model. Results are presented for the RAE 2822 airfoil in transonic and in subsonic flow with a high angle of attack. The results show that the coupling methodology is adequate and that convergence was obtained in critical conditions like the presence of shocks and separation. In the transonic flow, there is some disagreement with the experimental results for the pressure coeficient, previously published work indicate that this is probably due to the use of a coarse mesh. Keywords: Aerodynamics, Viscous-inviscid iteration, Boundary-layer, Euler flow. 1. Introduction The importance of computational fluid dynamics (CFD) as a tool for the study of aerodynamic shapes has grown steadily in the last decades. At present the main focus of the applications is the solution of the Reynolds averaged Navier-Stokes (RANS) equations, with a suitable turbulence model. Some time ago, when the computational power was more limited, one of the most important tools was the solution of the inviscid flow using an Euler or a full-potential formulation. This solution requires much less computational resources because the mesh doesn’t need to be so refined near the walls as in the complete Navier-Stokes formulation, and because the viscous terms in the equations tend to increase the number of interations for convergence. The disadvantage of using an inviscid flow formulation alone is that the results become unrealistic when the viscosity has an important effect in the flow, like for instance in the calculation of shear. To improve the results, the inviscid flow may be solved coupled to a boundary layer solver, a technique called viscous-inviscid interaction (VII). Normally the solution of the boundary layer increases only a small percentage in the computational time so VII remains much more fast than a RANS solution. Lock and Williams (1987) cite a reduction of up to 40 times in CPU time using VII compared to RANS. So this technique is very useful in preliminary and optimization studies where a large number of shapes must evaluated and a small error can be accepted. The viscous-inviscid interaction involves coupling of an inviscid solver that uses either an Euler or a full-potential formulation to a boundary layer code that may be based in either the differential or the integral formulation of the boundary layer. The coupling of the inviscid and viscous flows is not simple because the inviscid flow will supply the external flow boundary condition for the solution of the boundary layer. On the other hand, the boundary layer will influence the inviscid flow, since the presence of the boundary layer will tend to deviate the streamlines away from the surface, like if the thickness of the airfoil was increased by the presence of the boundary layer. This influence can be taken into account in two ways: a) Changing the geometry of the airfoil so as to include the displacement thickness of the boundary layer in the thickness of the airfoil. b) Including, in the boundary conditions of the inviscid flow, a transpiration velocity, normal to the surface of the airfoil, which will produce the same deviation of the streamlines as the boundary layer. In this case, the transpiration velocity, v n , will be given by: ( ) s u v e e e n ∂ ∂ = * 1 δ ρ ρ (1) In the above equation, u e is the speed outside the boundary layer, ρ e is the specific mass outside the boundary layer, s is the longitudinal arc length coordinate, and δ* is the displacement thickness that is defined as: