A unified method for the numerical analysis of compressible and incompressible viscous flows

Abstract

A new unified numerical method is presented for the analysis of both compressible and incompressible viscous flows. The proposed method has two key features. One is the energy equation expressed in terms of pressure. The other is the description of the governing equations in non-conservative forms. The both features contribute greatly to the construction of the computational method. The temporal discretization of the governing equations are based on the finite difference method. The procedure for advancing flow field variables in a time step consists of two phases, namely an advection phase and a non-advection phase, and accordingly the governing equations are split into the advection and non-advection equations. First, the non-advection phase is calculated. The non-advection equations are discretized in space by using the Galerkin FEM. Those discrete equations are solved in an implicit manner to yield the intermediate values of the variables. These intermediate values are corrected by solving the advection equations. The advection equations derived from the momentum equations are discretized in space by the SU/PG-FEM, while the advection equations from the continuity and the energy equations are discretized by the finite volume method with a first-order upwinding scheme. The proposed method is demonstrated on four numerical examples of compressible and incompressible flows; a shock-tube problem, a supersonic flow over a forward-facing step, incompressible flows over a backward-facing step and in a lid-driven cavity. The accuracy of the proposed method has been assessed by comparing our numerical results with other numerical results, analytical solutions and available experimental data. Stable and accurate computations have been attained.