Control of Sonic Surfaces and Shock Waves

In a time long before the arrival of the digital computer, the model equations for compressible flow were derived. Since then, we know the Reynolds-averaged Navier-Stokes for the full prob­lem, the Euler equations for their inviscid simplification and the Potential equation for a further simplification to isoenergctic flows. The latter extended the classical knowledge base of hydro­dynamics into the compressible flow regime. A necessity to find solutions to these equations then led to several attempts to transform them, for instance to reduce the formidable difficulties stem­ming from the nonlinearity of the potential equation. In 2D flow, the hodograph transformation leads to an inversion of the problem, trading linear equations for nonlinear boundary conditions. Several mathematical methods were developed to create the first transonic airfoils. Elegant prob­lem formulations could not hide the fact that solving mapped counterparts of real world problems never became very popular with the aerospace design engineer. Nevertheless, in a time when usually only numerical discretization of complex problems is seen as the way to get deeper in­sight into flow problems, some of these mapped model equations still have some value. One form of the "near sonic" model equations was found particularly useful, because it not only gave a number of flow models for transonic phenomena in closed analytical form hut also led the way toward design principles for practical airfoils and wings.

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