Geometric Conservation Law and Its Application to Flow Computations on Moving Grids

TLDR

Boundary‑conforming coordinate transformations map flow domains onto computational grids, but time‑dependent mappings from moving boundaries complicate global conservation and local volume element calculations. The study introduces a differential geometric conservation law (GCL) to govern the spatial volume element under arbitrary mappings. The GCL is solved numerically alongside flow conservation laws using conservative difference operators. Numerical experiments demonstrate the method for implicit unsteady Navier–Stokes and explicit steady supersonic flow solutions.

Abstract

Boundary-conforming coordinate transformations are used widely to map a flow region onto a computational space in which a finite-difference solution to the differential flow conservation laws is carried out. This method entails difficulties with maintenance of global conservation and with computation of the local volume element under time-dependent mappings that result from boundary motion. To improve the method, a differential ''geometric conservation law (GCL) is formulated that governs the spatial volume element under an arbitrary mapping. The GCL is solved numerically along with the flow conservation laws using conservative difference operators. Numerical results are presented for implicit solutions of the unsteady Navier-Stokes equations and for explicit solutions of the steady supersonic flow equations.