A solution of the euler equations on an unstructured grid system

A numerical algorithm is presented for the solution of the one and two-dimensional Euler Equations based on a conservative formulation. The objective of the study is to combine the theory of characteristics with a conservative finite volume algorithm resulting in a numerical scheme with increased accuracy through the use of enhanced shock capturing capability, and reduced levels of numerical dissipation. Characteristic boundary conditions are developed for subsonic and supersonic flow regimes, and for solid wall boundaries. The numerical scheme is further broadened by the use of an unstructured triangular grid system with generalized placement of internal boundary conditions. The approach is to cast the Euler Equations in an uncoupled or approximately uncoupled form based on the classical eigenvalue study. Using the similarity transformation matrices from this study, a matrix operator is derived which splits a conservatively calculated residual into components which are propagated along specified paths. The matrix operator is derived for the quasi one-dimensional Euler Equations as well as the two-dimensional Euler Equations. Boundary conditions are derived based on a first order approximation of the compatibility equations for both the one and two-dimensional forms and shown to be consistent with the interior scheme. The numerical algorithm is validated by various test cases with comparisons with theory and other numerical flow solvers. Tlte algorithm is shown to give near exact solutions for the one-dimensional steady state ducts with area variation. Salient features of the two-dimensional algorithm include excellent shock capturing capability, use of general unstructured grids for complex geometries, as well as the ability to accurately model low and high speed flows with little or no artificial dissipation required for stability.