Conformal Cartesian Grids for Symmetric Bodies: A Novel Boundary Fitted Grid Method

Abstract

A novel Cartesian grid discretization method is developed to simulate two and three dimensional problems, governed by partial differential equations. In the present approach, the grid points lie exactly onto the surface of an immersed object or of the domain’s boundaries, allowing for accurate imposition of the surface boundary conditions. This is well demonstrated for symmetric objects, but it can be also extended for non-symmetric shapes. The method intrinsically possesses higher accuracy than the conventional body fitted or Immersed Boundary Methods, since the implemented grid is universally orthogonal and the boundary conditions are imposed precisely onto the surface, without any interpolation or tuning of the governing equations. Conformal Cartesian Grids bridge the topology of Cartesian grid methods with the treatment of the surface boundary conditions, which is adopted in conventional bodyfitted grid approaches. Emphasis is given in a two dimensional fluid dynamics problem to demonstrate this approach. A finite difference code has been developed, which encompasses the present methodology. Space discretization is performed via the second order accurate central difference scheme and time discretization by the fourth order accurate Runge-Kutta method. The flow past a cylinder at low Reynolds number is resolved to validate the accuracy and performance of the method. Two different flow regimes are thoroughly investigated at Re numbers varying from 10 up to 100 based on the cylinder’s diameter. Computed results agree well with the available measurements and numerical computations in literature. Three dimensional results are also briefly presented mainly for revealing the applicability of the method.