Transonic Euler computation in streamfunction co-ordinates

A new approach has been developed to calculate two-dimensional steady transonic flows past aerofoils using the Euler equations in streamfunction co-ordinates. Most existing transonic computation codes require the use of a grid generator to determine a suitable distribution of grid points. Although simple in concept, the grid generation may take a considerable proportion of the CPU time and storage requirements. However, this grid generation step can be avoided by introducing the von Mises transformation, which produces a formulation in streamwise and natural body-fitting co-ordinates. In this work a set of Euler equivalent equations in streamfunction co-ordinates is formulated, consisting of three equations with three unknowns; one is a geometric variable, the streamline ordinate y, and the other two are physical quantities, the density ρ and the vorticity ω. To solve these equations, typedependent differencing, development of a shock point operator, marching from a non-characteristic boundary and successive line overrelaxation are applied. Particular attention has been paid to the supercritical case where a careful treatment of the shock is essential. It is shown that the shock point operator is crucial to accurately capture shock waves. The computed results show excellent agreement with existing experimental data and other computations.