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1. Introduction
During the last few years there has been a tremendous increase in the development and application of finite element technology for the solution of fluid flow problems, particularly on adaptively defined unstructured meshes. Some of the current finite element codes can handle large-scale threedimensional simulations of compressible or incompressible flows with very complicated geometry, general boundary conditions, and hundred of thousands to millions of degrees of freedom[1,2].
A successful general analysis code that attempts to handle realistic problems must be equipped with four main properties: modelling flexibility, numerical stability, accuracy and efficiency. Flexibility in modelling general geometries, material properties, boundary conditions etc. is an important capability which requires special non-trivial treatment, especially in three dimensions. Stability and accuracy are essential ingredients in establishing the reliability of the numerical results. High computing speed is a critical necessity for practical reasons of obtaining results within acceptable times. Various tools have been devised in recent years in the context of finite element analysis to address each of these issues. The scheme described in this paper incorporates a number of tools which cover all four aspects, and thus leads to a powerful solution technique.
Traditionally compressible and incompressible viscous flow problems are handled by two separate codes, since these two types of problems typically involve different concerns and numerical difficulties. However, recently Hauke and Hughes[3] showed how the compressible and incompressible cases can be combined, by approaching the incompressible limit while using either primitive (pressure-velocity-temperature) variables or entropy variables (but not conservation variables). In this work, we have adopted this approach and have developed a finite element scheme capable of solving both the compressible and incompressible Navier-Stokes equations. While a direct approach to incompressible problems may be more efficient in some cases[1], the combined compressible-incompressible approach provides a unified and convenient analysis framework, and is especially attractive when buoyancy is introduced into the incompressible Navier-Stokes equations via the Boussinesq approximation.
In the present paper, we propose a new finite element framework for the large-scale analysis of compressible and incompressible viscous flows. The scheme is based on a combined compressible-incompressible Galerkin leastsquares (GLS) space-time variational formulation. Unstructured threedimensional spatial meshes of tetrahedral elements are employed, with linear spatial interpolation in all the variables. This setup enables straightforward generalization...