Analysis of aerodynamic problems of unspecified geometry based on a Lagrangian computational method

In the aeronautical applications, many problems involving boundaries of unspecified geometry are of interest, such as the indirect problems of determining the shape of an airfoil to generate a specified pressure distribution, supersonic nozzle design based on reflection-suppression condition, flexible-membrane airfoils or wings, jet-flapped airfoils, and others. In these cases, the utilization of the Eulerian formulations of the Euler equations of motion, may lead to long iterative computations for successive shapes of the boundaries, until the final geometry is reached. For this type of problems, the Lagrangian formulations using the streamline coordinates are more suitable, since the geometrically-unspecified boundaries are always represented by streamlines.Computational methods based on Lagrangian formulations have been recently developed for supersonic flows for the solution of the system of Euler equations. These Lagrangian formulations use the stream-function and the Lagrangian-time or -distance coordinates. In our present study, the numerical method applied to solve aerodynamic problems of unspecified geometry, is based on a finite volume discretization in the Lagrangian coordinates (stream-function and Lagrangian-distance), in which the flux values are determined by using the Riemann problem solution. Improvements leading to a better resolution of the shock waves are included.The method has been first validated for nozzles and airfoils of specified geometry, by comparison with analytical results and previous solutions obtained using Eulerian formulations. Then, the method has been applied for the solution of the above mentioned problems of unspecified geometry. In the cases of the flexible-membrane airfoils and the jet-flapped airfoils, an analytical solution has been developed in addition to the numerical solution