INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2000; 32: 51–78 A new finite element formulation for both compressible and nearly incompressible fluid dynamics P.A.B. De Sampaio* and M.L. Moreira Instituto de Engenharia Nuclear, Comissa ˜o Nacional de Energia Nuclear, Cidade Uniersitaria, Ilha do Fundao, CEP 21945 -970, CP 68550, Rio de Janeiro -RJ, Brazil SUMMARY A new finite element formulation designed for both compressible and nearly incompressible viscous flows is presented. The formulation combines conservative and non-conservative dependent variables, namely, the mass – velocity (density  velocity), internal energy and pressure. The central feature of the method is the derivation of a discretized equation for pressure, where pressure contributions arising from the mass, momentum and energy balances are taken implicitly in the time discretization. The method is applied to the analysis of laminar flows governed by the Navier – Stokes equations in both compressible and nearly incompressible regimes. Numerical examples, covering a wide range of Mach number, demonstrate the robustness and versatility of the new method. Copyright © 2000 John Wiley & Sons, Ltd. KEY WORDS: finite element formulation; compressible flow; nearly incompressible flow; mass – velocity 1. INTRODUCTION Most numerical methods for the analysis of compressible fluid dynamics present difficulties when applied to low-speed (nearly incompressible) flow problems. The difficulties originate from the fast propagation of pressure waves, as flow conditions approach the incompressible limit. In such cases, the accurate representation of pressure transients requires the use of extremely small time steps, which are unaffordable in practical applications. Conversely, the use of time steps larger than the typical pressure time scale results in errors that may lead to numerical instability. In particular, if an explicit time approximation of pressure is adopted — a usual choice in algorithms for high-speed compressible flows — such errors grow during the computation and stability is rapidly lost. Nearly incompressible flows, i.e. flows characterized by very small Mach number, are usually approximated as fully incompressible. Thus, compressibility effects are eliminated from the start, at the modelling level, prior to considering any particular discretization method. This * Correspondence to: Instituto de Engenharia Nuclear, Comissa ˜o Nacional de Energia Nuclear, Cidade Universitaria, Ilha do Fundao, CEP 21945-970, CP 68550, Rio de Janeiro-RJ, Brazil. CCC 0271–2091/2000/010051 – 28$17.50 Copyright © 2000 John Wiley & Sons, Ltd. Receied March 1998 Reised July 1998