An Embedded Cartesian Grid Euler Solver with Radial Basis Function for Boundary Condition Implementation

Abstract

A Cartesian grid approach for the solution of the Euler equations within the framework of a patched, embedded Cartesian field mesh is described. As Cartesian grids are not necessarily body-aligned, an accurate representation for the surface boundary is important. In this paper a gridless boundary treatment using a cloud of nodes in the vicinity of the body combined with the multiquadric radial basis function (RBF) for the conserved flux variables for boundary implementation is proposed. In the present work, the RBF is applied only at the boundary interface, while a standard structured Cartesian grid approach is used everywhere else. Flow variables for solid cell centers for boundary condition implementation are determined via the use of reflected node involving a local RBF fit for a cloud of grid points. RBF is well suited to approximate multidimensional scattered data without any mesh accurately. Compared to the least-square method, RBF offers greater flexibility in regions where point selection may be very limited since the resulting matrix will be non-singular regardless of the sampling point’s location. This is particularly important in the context of computations involving complex geometries where eligible points selected may be very close to one another. It is also shown that it provides similar accuracy with less cloud of points. The use of a Cartesian field mesh for the non boundary regions allows for effective implementation of multigrid methods, and issues associated with global conservation are greatly mitigated. Several two and three-dimensional problems are presented to show the efficiency and robustness of the method.