Accuracy and stability of a finite element pseudo-compressibility CFD algorithm for incompressible flows

The term "pseudo-compressibility" refers to a technique used in numerically solving incompressible flows. In this approach, the pressure field solution is obtained by introducing a pressure time derivative term into the continuity equation to form a set of hyperbolic-type equations. The objective of this thesis is to implement the method through modification of an existing 2-D compressible flow Navier-Stokes finite element code and to evaluate the algorithm's accuracy and stability. The non-dimensional form of the pseudo-compressibility equation set is presented and some theoretical issues concerning the pseudo-compressibility parameter are discussed. A companion conservation law system is derived via identification of truncation error terms in a Taylor series expansion. A weak statement is then formed, and is subsequently semi-discretized in space by the finite element method. Finally, the algebraic construction of the algorithm is discussed, including an implicit time integration scheme and boundary condition enforcement. Three benchmark test cases are considered for code and algorithm verification. The results obtained compare well with those documented in the literature. Stability and convergence problems were encountered, however, pertaining to oscillations in the pseudo-pressure field, especially near boundary corner geometrical singularities. This and other computational issues axe evaluated and discussed.