Implementation of Direct Numerical Simulation Framework in the Finite Volume Methods-Based Open-Source Code Athena++ for Astrophysical Flows

Abstract

This thesis investigates the performance of the open-source code Athena++ for the simulation of viscous incompressible flows across laminar and turbulent regimes, with particular emphasis on adaptive mesh refinement (AMR). Benchmark test cases, including steady-state channel flow, the lid-driven cavity flow, and the Taylor–Green vortex (TGV) in both two and three dimensions, were employed to assess the solver’s accuracy, stability, and computational efficiency. Spatial and temporal discretization schemes were systematically evaluated through grid convergence studies, supported by Richardson extrapolation and Grid Convergence Index (GCI) analysis, to verify the numerical implementation and establish the expected order of accuracy. An AMR strategy based on a local grid Reynolds number (Regrid) was implemented and tested in both laminar and turbulent TGV flows. For two-dimensional cases, computational savings of up to 44% were achieved, while in three dimensions with moderate Reynolds numbers, the savings were up to 35%. The AMR solutions successfully reproduced global flow characteristics such as kinetic energy decay and dissipation rate, but also highlighted the need for higher-order reconstruction and time integration schemes, as well as the limitations of a single refinement criterion. Finally, under-resolved direct numerical simulations (DNS) of the three-dimensional TGV at Re=1600 demonstrated that Athena++, equipped with high-order reconstruction (PPM) and time integration (RK4), can capture turbulence dynamics in close agreement with benchmark solutions when using grid resolutions consistent with Kolmogorov theory. However, AMR revealed an unphysical increase in the spectral energy density at dissipative scales, possibly due to numerically generated instabilities introduced as a result of mesh-size variations. The findings confirm that Athena++ provides accurate and robust solutions for viscous flows and highlight the potential of AMR to reduce computational cost in under-resolved DNS, provided that the mesh refinement and de-refinement criteria are guided by the physical scales of turbulence.