A gridless boundary condition method for the solution of the Euler equations on embedded Cartesian meshes with multigrid

The solution of the Euler equations using a “gridless” boundary condition treatment on a patched, embedded Cartesian field mesh is described. The gridless boundary treatment is implemented by means of a least squares fitting of the conserved flux variables using a cloud of nodes in the vicinity of the body. The method allows for accurate treatment of the surface boundary conditions without the need for excessive refinement of the Cartesian mesh. Additionally, the method does not suffer from problems associated with thin body geometry or extremely fine cut cells near the body. Unlike some methods that consider a gridless (or “meshless”) treatment throughout the entire domain, the use of a Cartesian field mesh allows for effective implementation of multigrid acceleration, and issues associated with global conservation are alleviated. Results are presented for transonic flow about single and dual NACA 0012 airfoil configurations including a convergence study indicating the effectiveness of multigrid acceleration within the construct of the gridless boundary treatment. Where applicable, comparisons to the FLO52 body-fitted Euler code are presented to gauge the accuracy of the method.