# Transonic computation using Euler equations in stream function coordinate system.

1992

Doctoral Thesis

Ph.D.

## Department

Mathematics and Statistics

## Keywords

Physics, Fluid and Plasma.

## Rights

info:eu-repo/semantics/openAccess

In this dissertation, a new approach has been developed to calculate the two dimensional steady transonic flow past airfoils using the Euler equations in a stream function coordinate system. Due to the importance of the transonic flow phenomenon in aeronautical practice, transonic flow computation has been an upsurging topic of the past two decades. 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 takes a significant proportion of the CPU time and storage requirements. However, this time-consuming step can be completely avoided by introducing the von Mises transformation and the corresponding stream function coordinate system because this particular transformation combines the flow physics and flow geometry and produces a formulation in streamwise and body-fitting coordinates, without performing any conventional grid generation. In the present work, a set of the Euler equivalent equations in stream function coordinates is formulated. It consists of three equations with three unknowns. One unknown is a geometric variable, the streamline ordinate y, and the other two are physical quantities, density $\rho$ and vorticity $\omega$. For irrotational fluid flow, the Euler formulation is simplified to the full potential formulation in which only two unknowns are solved--y and R, the generalized density. To solve these equations, several numerical techniques are applied: type-dependent differencing, shock point operator, marching from a non-characteristic boundary and successive line overrelaxation, etc. 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 for both analysis and design problems show good agreement with existing experimental data. The limitations of the approach and further investigations have been discussed.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1992 .A555. Source: Dissertation Abstracts International, Volume: 54-05, Section: B, page: 2560. Thesis (Ph.D.)--University of Windsor (Canada), 1992.