Development of a Class of Higher Order Compact (HOC) Finite Difference Schemes for complex Fluid flow Problems

Abstract: Higher Order Compact (HOC) finite difference schemes are gradually gaining popularity because of their high accuracy and advantages associated with compact difference stencils. These schemes were basically developed for tackling convection–diffusion equations. As the Navier–Stokes (N–S) equations that govern the incompressible viscous flows, easily fits into convection–diffusion type, they were well equipped to simulate such flows with ease. A high-order compact finite difference scheme is one with accuracy greater than two and whose stencil consists of grid points located only directly adjacent to the node about which the differences are taken. Because of their higher-order accuracy coupled with the compact difference stencils, they yield highly accurate numerical solutions on relatively coarser grids with greater computational efficiency. Recently, a transformation-free HOC scheme for steady-state convection diffusion problems on non-uniform polar grids was developed by Ray and Kalita (IJNMF, 2010). The scheme is then extended for incompressible viscous flows governed by the unsteady N–S equations on non-uniform polar grids (Kalita and Ray, JCP, 2009). This scheme for unsteady N–S equations is temporally second-order and spatially at least third-order accurate. Both the schemes very efficiently and accurately reproduce the complex flow phenomena.

 In this presentation, the development of HOC scheme for steady state convection diffusion equation on non-uniform polar grids will be shown first. Then the scheme will be extended for solving unsteady flow problems. The validation of the scheme will be shown by solving flow past an impulsively started circular cylinder problem. At last, an extension of the HOC scheme for solving Immersed Interfaced problems (governed by differential equations with various discontinuities) will be discussed. In this process, present computed results will be validated and compared with existing experimental and standard numerical results in the literature.