Inverse Euler equations

Abstract

Following previous work by Keller [2], that is extended to compressible flow, the general time-independent Euler equations for inviscid fluid flow are first written in a perfectly antisymmetric form, using a pair of stream functions as the dependent variables. In a second step the equations are written in an inverse form, using the two stream functions and the natural coordinate as independent variables. As a special case the Bragg-Hawthorne equation for axisymmetric flow is first extended to compressible flow and also transformed to its inverse form. The main advantage of using these inverse equations is associated with the possibilities of using static pressure distributions, Mach number distributions, geometric constraints, etc., or any combination of geometric constraints and specifications of physical quantities to define the boundary conditions. In contrast to conventional inverse methods, that are based on iterative approximations to a desired pressure distribution along the surface of a flow device, for example, the use of inverse Euler equations offers the possibility to arrive at the solution for any kind of boundary conditions in a single step. Furthermore, there is no need for complicated grid generation procedures, because the domain of definition in inverse space is typically a cube with Cartesian coordinates. In the original space, the surfaces on which the natural coordinate is constant are orthogonal to the streamlines. As a consequence, the computation time can be kept small and the accuracy is remarkably high. This semi-orthogonal curvilinear grid is generated automatically together with the solution. The density of grid lines is automatically getting large in domains where gradients are large. Possible difficulties with using inverse Euler equations are mainly related to the topology of the flow field. The transform to inverse coordinates must correspond to a one-to-one mapping. Hence, if the domain of definition is not simply connected it must be cut suitably to obtain piecewise domains for which one-to-one mappings exist.