Finite element analysis of incompressible, compressible, and chemically reacting flows, with an emphasis on the pressure approximation.

Abstract

A finite element numerical method is developed for the modelling of compressible flows with locally incompressible regions, which avoids the pressure oscillations frequently exhibited in these areas. Unification of the modelling of pressure in the finite element approximation of compressible and incompressible flows is investigated through the appropriate combinations of approximation spaces and integration schemes. The penalty method for incompressible flows is re-examined in the context of a slightly compressible fluid, yielding a formulation that is consistent for both the Navier-Stokes and Stokes equations, and providing an accurate method for calculation of pressure that is faster than the solution of a pressure Poisson equation. Extension of concepts from the reformulated penalty method to the compressible Navier-Stokes equations leads to an algorithm using piecewise constant density and pressure, with bilinear velocity and temperature. Further investigation shows that bilinear density with selective reduced integration also avoids pressure oscillations, while providing improved shock capture. Selective reduced integration of all terms related to compressibility is shown to be a key element in the avoidance of pressure oscillations. The identification of the isotropic component of the stress tensor as a compressibility term, not to be combined with viscous terms, is emphasized. Reduced integration of divergence terms is shown to yield conservation of mass on the element level as well as on the assembled level, whereas full integration conserves mass only on the assembled level. The compressible flow algorithm is coupled with a chemistry solver, for the study of chemically reacting flows, where an oscillation free flow solution is essential since unphysical oscillations may cause premature ignition. Computational efficiency is obtained by iterating a fluid flow step with frozen chemistry, and a chemical reaction step with frozen flow. The algorithm is applied to the study of the ram accelerator concept, a technique for accelerating a projectile in a tube to extremely high velocities by using a shock wave to initiate combustion. The viability of the ram accelerator is demonstrated through calculations at various velocities, pressures, and gas mixtures.