Steady-State Flow Solutions for Delta Wing Configurations at High Angle of Attack Using Implicit Schemes

Finding fully converged, steady-state solutions of the compressible Reynolds Averaged Navier-Stokes (RANS) equations for aerodynamic configurations on the border of the flight envelope often poses serious challenges to solution algorithms that have proven robust and successful for configurations at cruise conditions. Examples of such cases are agile configurations at high angles of attack. When trying to compute solutions in these scenarios, one often observes that the solution process breaks down after few iterations or that a steady-state RANS solution, although it may exist, cannot be reached with the employed solution algorithm. While, in general, no clear reason for this behavior can be identified, the complexity of these flows seems to be significantly greater compared to flows around transport aircraft in cruise flight. The flow fields are dominated by the interaction of shock waves with a system of vortices emanating from the leading edges on the upper surface of the wing, leading to massive flow separation. These flow features tend to be inherently unsteady and can be assumed to cause problems in computing a converged solution using an algorithm designed to find steady-state solutions of the RANS equations. To avoid these problems, it is not uncommon to calculate such configurations in an unsteady mode, which often comes at a rather high computational cost. This article demonstrates the necessity for implicit smoothers to approximate fully converged solutions of these challenging simulations. A numerical example is given to confirm that convergence is only possible using an exact derivative together with a suited preconditioner.