A comparative computational study of different formulations of the compressible Euler equations for mesoscale atmospheric flows in a finite volume framework

We consider three conservative forms of the weakly compressible Euler equations, called CE1, CE2 and CE3, with the goal of understanding which leads to the most accurate and robust pressure-based solver in a finite volume environment. Forms CE1 and CE2 are both written in density, momentum, and specific enthalpy, but employ two different treatments of the buoyancy and pressure gradient terms: for CE1 it is the standard pressure splitting implemented in open-source finite volume solvers (e.g., OpenFOAM®), while for CE2 it is the typical pressure splitting found in computational atmospheric studies. Form CE3 is written in density, momentum, and potential temperature, with the buoyancy and pressure terms addressed as in CE2. For each formulation, we adopt a computationally efficient splitting approach. The three formulations are thoroughly assessed and compared through six benchmark tests involving dry air flow over a flat terrain or orography. We found that all three models are able to provide accurate results for the tests with a flat terrain, although the solvers based on the CE2 and CE3 forms are more robust. As for the mountain tests, CE1 solutions become unstable, while the CE2 and CE3 models provide results in very good agreement with data in the literature, the CE3 model being the most accurate. Hence, even when using a pressure-based approach and space discretization by a finite volume method, the CE3 model is the most accurate, reliable, and robust for the simulation of mesoscale atmospheric flows.